Risk Management
Second Revised and Enlarged Edition
Michael Frenkel ´ Ulrich Hommel
Markus Rudolf (Editors)
Risk Management
Challenge and Opportunity
Second Revised and Enlarged Edition
With 100 Figures
and 125 Tables
12
Professor Dr. Michael Frenkel
Professor Dr. Markus Rudolf
WHU
Otto Beisheim Graduate School of Management
Burgplatz 2
56179 Vallendar
mfrenkel@whu.edu
mrudolf@whu.edu
Professor Dr. Ulrich Hommel
EUROPEAN BUSINESS SCHOOL International University
Stiftungslehrstuhl
Unternehmensfinanzierung und Kapitalmårkte
Schloss Reichartshausen
65375 Oestrich-Winkel
ulrich.hommel@ebs.de
Cataloging-in-Publication Data
Library of Congress Control Number: 2004114544
ISBN 3-540-22682-6 Springer Berlin Heidelberg New York
ISBN 3-540-67134-X 1st edition Springer Berlin Heidelberg New York
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In my "Word of Greeting" of the first edition of this book which was dedicated to
Günter Dufey, I pointed out that I appreciate Günter Dufey as someone who builds
bridges between Germany and the United States. Meanwhile, almost 5 years have
gone by. Günter Dufey's significance as an academic intermediary between the
continents has even increased since then. Due to his efforts, the cooperation between high ranked U.S. business schools and the WHU - Otto Beisheim
Hochschule in Germany have been intensified. The joint summer MBA program
on the WHU campus is attended by 45 U.S. students every year. This number is
still growing. Moreover, since the issue of the first edition, Günter Dufey has
enlarged his activity spectrum also to Asia. In 2002 until 2003 Günter Dufey
joined the Singapore Office of the firm as Senior Advisor, supporting the Corporate Governance Practice of the firm in the Region. Since then he was appointed
as Professor of Banking and Finance at the Nanyang Business School - Nanyang
Technological University and as Principal of the Pacific International Business
Associates. Last but not least, he is now an ordinary member of the Singapore Institute of Directors.
It is impressive to see the energy and the enthusiasm with which Günter Dufey
travels restlessly around the world, once eastwards, another time westwards.
Without any doubt, it is quite unusual that a Festschrift sells out. The first edition
of this book has been written by a global selection of financial experts. They
wanted to act as a sign of friendship by honoring Günter Dufey's 60th birthday. I
am very happy, that the first edition was so successful because this indicates also
the importance of the book' content.
Much has been changed in the field of risk management since then. Probably
most significantly, there has been an intensive discussion between financial institutions dealing with modified rules of determining the adequate amount of equity
capital for risks. The committee of banking supervision located at the Bank of International Settlement in Basel, Switzerland, has initiated several proposals known
under the short form "Basel II". One of the core questions in the context of these
capital adequacy rules is the capital requirement for credit risks. According to
Basel II, in the future this will be closer related to the rating of transaction counterparties enhancing the role of the rating process and the rating industry. Another
important risk category which is addressed by Basel II for the first time is operative risk. The terror attack on the World Trade Center on 11 September 2001
showed drastically how significant external and unpredictable events can be on the
operations of any company, particularly of banks. Moreover, risks form internal
processes, people, or systems contribute to the success or to the failure of the
business. All these risks are summarized as new risk type in Basel II, namely as
VI
A Word of Greeting
operative risk. Operative risk is much more difficult to measure than credit and
market risks. But they nevertheless affect the performance of financial institutions.
The deadline for implementing Basel II in national laws is year-end 2006. It is
obvious that this development is intensively covered in the second edition of the
book. I hope that this book will help to understand the complex and new aspects of
risk management better. And I am happy that such an instructional content is associated to the name of my former student in Würzburg, Günter Dufey.
Otmar Issing
2TGHCEG
Michael Frenkel, Ulrich Hommel, Markus Rudolf
The success of the first edition of this book encouraged us to update and extend
this volume in order to provide an up-to-date and comprehensive survey of the
major areas of risk management issues. Since the first edition of this book, a number of changes in the area of risk management took place. Some of them are reflected in the discussions on the “Basel II” rules. The new edition takes these new
developments into account. Given the wider scope of the new edition, we decided
to structure the book according to the type of risk management the various aspects
are most narrowly related to. More specifically, we distinguish four broader topics. Part 1 focuses on bank risk management, part 2 on insurance risk management, part 3 on corporate risk management, and part 4 on systemic issues of risk
management. In the following, a very brief outline of the papers is presented.
Part 1 begins with the analysis by Thomas Hartmann-Wendels, Peter Grundke
and Wolfgang Spörk of the Basel II rules and their consequences on bank lending.
Then, Ingo Walter looks at conflicts of interest involving financial services firms.
He shows the conditions that can cause or limit exploitation of conflicts of interest
and argues that external regulation and market discipline can be both complements
and substitutes. A normative theory or risk management in banks is the subject of
the contribution of Gerhard Schröck and Manfred Steiner. Then, Claudia Holtorf,
Matthias Muck and Markus Rudolf present a case study that analyses the new
Basel capital accord requirements by applying the RiskMetricsTM. Value at Risk
is the core of several papers in part 1. Alois Paul Knobloch surveys applications of
this concept for risk measurement purposes; John Bilson uses to concept to review
fixed income portfolios, Robert Härtl and Lutz Johanning examine risk budgeting,
and Jack Wahl and Udo Broll examine the implications of Value at Risk for the
optimum equity level of a bank. Wolfgang Drobetz and Daniel Hoechle compare
alternative estimates of conditional return expectations. Subsequently, Ludger
Overbeck surveys modelling of credit risk portfolios. A critical evaluation of
credit risk models is the topic of the paper by Hergen Frerichs and Mark Wahrenburg. Related to this type of risk is the analysis of Stefan Huschens, Konstantin
Vogl, and Robet Wania, who look at possibilities to estimate default probabilities
and default correlations. The subsequent two papers examine operational risk in
the context of Basel II. While Carol Alexander gives an overview of the different
dimensions of this risk type, Wilhelm Kross addresses practical issues for management dealing with such risk. In the last paper of this part, Christoph Kaserer,
Niklas Wagner and Ann-Kristin Achleitner investigate possibilities to measure
private equity returns under conditions of illiquidity.
Part 2 focuses on insurance risk management. Martin Nell and Andreas Richter
discuss three issues related to the management of catastrophic risk which stem
from the terror attacks of September 11, 2001. Subsequently, Christopher Culp
surveys products and solutions that represent the convergence or integration of
VIII
Preface
capital markets and traditional insurance. Such products are referred to as alternative risk transfer. Ulrich Hommel and Mischa Ritter address a similar area of risk
management. They analyze the main forces behind the securitization of catastrophic insurance risk and derive conclusions as to how other forms of insurance can
be transferred to financial markets. In recent years, demographic changes in a
number of advanced economies have been discussed intensely. The paper of Petra
Riemer-Hommel and Thomas Trauth addresses this issue by analyzing possibilities of managing longevity risk associated with pension, annuity and long-term
care products. Particularly in the German market, another problem of life insurance companies generate from unknown capital market developments and the simultaneously issued interest rate guarantees of traditional life insurance products.
Peter Albrecht and Carsten Weber investigate the implications of this constellation
on the asset allocation decision.
Part 3 includes papers that discuss a variety of issues of corporate risk management. In the first paper, Fred Kaen addresses the relationship between risk management and corporate governance and makes the point that risk management not
only helps a firm to survive but also serves broader policy objectives. In the next
paper, Christian Laux examines how corporate risk management can be integrated
into the objective of maximizing firm value. Subsequently, Ulrich Hommel investigates the more fundamental question why the management of corporate risk
should be managed at all and why it should be managed on the firm rather than the
investor level. Focusing on a German regulation requiring firms to implement risk
management systems, Jürgen Weber and Arnim Liekweg discuss critical implementation issues for non-financial firms. How risk analysis and risk aggregation
enters value-based corporate risk management is the topic of the paper by Werner
Gleißner. A more macroeconomic focus of risk is presented by Lars Oxelheim and
Clas Wihlborg who emphasize the importance of exchange rates, interest rates and
inflation rates in estimating corporate risk. This issue is taken one step further in
the paper of Matthias Much and Markus Rudolf as they include international issues of corporate risk management. They use the case study of three airlines to
emphasize commodity and exchange rate risk and show the effects on corporate
risk by applying the “Earnings at Risk” concept. A consequence of corporate risk
is the use of financial derivatives in risk management. In this context, real options
represent alternatives to financial hedging. The paper of Alexander Triantis uses a
specific example to discuss the implications of these alternatives. Operational and
managerial flexibility in international supply chains also contribute to real options.
Arnd Huchzermeier uses a case study to illustrate the value of such flexibility for
risk management. In the international context, exchange rate exposure represents a
major risk, when cross-border acquisitions are considered. Stefan Hloch, Ulrich
Hommel, and Karoline Jung-Senssfelder show that this risk stems from considerable time lags between the acquisition decision and its implementation due to, e.g.,
the process of regulatory clearance by the antitrust authorities. In the following
paper, Christian Geyer and Werner Seifert describe electricity derivatives as new
risk classes to organized exchanges and explain why the German Stock Exchange
(Deutsche Börse) intends to establish an exchange for energy derivatives. Foreign
exchange risk is more closely examined in two contributions. While Martin Glaum
Preface
IX
presents an empirical study on the measuring and management of foreign exchange risk in large German non-financial corporations, Kathryn Dewenter,
Robert Higgins and Timothy Simin show that, contrary to many studies presented
earlier in the literature, there is a negative influence of the value of the dollar and
stock returns of U.S. multinational firms. The subsequent paper by Wolfgang
Breuer and Olaf Stotz addresses the problem of securing the real value rather than
the nominal value of assets in risk management. The last paper of part 3 focuses
on capacity options. Stefan Spinler and Arnd Huchzermeier explain how options
on capacity can be used in capital intensive industries for risk management.
Part 4 focuses on more systemic risk aspects with which firms have to deal in
the national and the international environment. Adrian Tschoegl argues in his contribution that financial debacles in the mid-1990s are the result of management
failures and suggests that risk management has to take into account that such errors are the result of human nature. While this emphasizes a microeconomic element of risk management, Michael Frenkel and Paul McCracken show that a currency union as represented by the European Monetary Union exerts several
additional risks which firms operating in this area have to recognize. Whether risk
management itself makes financial markets riskier is discussed in the paper by Ian
Harper, Joachim Keller and Christian Pfeil. The authors argue that both on theoretical and empirical grounds there are indications that this is indeed possible. In
the same direction, Torben Lütje and Lukas Menkhoff analyze risk management
of institutional investors may lead to the behaviour of rational herding. A final
look at systemic risk aspects is presented by Mitsuru Misawa. He looks at the
Japanese experience in the 1990s when financial markets suffered significant
damage due to the burst of the asset price bubble and evaluates Japan’s big bang
financial reform.
Although this book covers a variety of diverse aspects of risk management, no
book on this broad and complex issue can cover all aspects. Therefore, we were
forced to be selective in certain areas. In addition, new topics may come up in the
future, as further risk categories may continue to evolve and both risk management and policies will also further develop.
Such a volume cannot be completed without the help of many individuals. We
thank all authors and those that have given us suggestions for the new edition. We
are very grateful to Kerstin Frank who showed enormous commitment and patience in preparing the manuscript. We are also thankful to Gudrun Fehler for
proofreading a number of papers of this volume and to Martina Bihn representing
the publisher for her support and patience in making this new edition possible.
$TKGH6CDNGQH%QPVGPVU
A Word of Greeting
Preface
V
VII
Part 1: Bank Risk Management
Basel II and the Effects on the Banking Sector
Thomas Hartmann-Wendels, Peter Grundke and Wolfgang Spörk
3
Conflicts of Interest and Market Discipline in Financial Services Firms
Ingo Walter
25
Risk Management and Value Creation in Banks
Gerhard Schröck and Manfred Steiner
53
The New Basel Capital Accord
Claudia Holtorf, Matthias Muck, and Markus Rudolf
79
Value at Risk:
Regulatory and Other Applications, Methods, and Criticism
Alois Paul Knobloch
99
Parsimonious Value at Risk for Fixed Income Portfolios
John F. O. Bilson
125
Risk Budgeting with Value at Risk Limits
Robert Härtl and Lutz Johanning
143
Value at Risk, Bank Equity and Credit Risk
Jack E. Wahl and Udo Broll
159
Parametric and Nonparametric Estimation of Conditional
Return Expectations
Wolfgang Drobetz and Daniel Hoechle
169
Credit Risk Portfolio Modeling: An Overview
Ludger Overbeck
197
Evaluating Credit Risk Models
Hergen Frerichs and Mark Wahrenburg
219
Estimation of Default Probabilities and Default Correlations
Stefan Huschens, Konstantin Vogl, and Robert Wania
239
Managing Investment Risks of Institutional Private
Equity Investors – The Challenge of Illiquidity
Christoph Kaserer, Niklas Wagner and Ann-Kristin Achleitner
259
XII
Brief Table of Contents
Assessment of Operational Risk Capital
Carol Alexander
279
Operational Risk: The Management Perspective
Wilhelm Kross
303
Part 2: Insurance Risk Management
Catastrophic Events as Threats to Society:
Private and Public Risk Management Strategies
Martin Nell and Andreas Richter
321
New Approaches to Managing Catastrophic Insurance Risk
Ulrich Hommel and Mischa Ritter
341
Alternative Risk Transfer
Christopher L. Culp
369
The Challenge of Managing Longevity Risk
Petra Riemer-Hommel and Thomas Trauth
391
Asset/Liability Management of German Life Insurance Companies:
A Value-at-Risk Approach in the Presence of Interest Rate Guarantees
Peter Albrecht and Carsten Weber
407
Part 3: Corporate Risk Management
Risk Management, Corporate Governance and the Public Corporation
Fred R. Kaen
423
Integrating Corporate Risk Management
Christian Laux
437
Value-Based Motives for Corporate Risk Management
Ulrich Hommel
455
Value-based Corporate Risk Management
Werner Gleißner
479
Statutory Regulation of the Risk Management Function in Germany:
Implementation Issues for the Non-Financial Sector
Jürgen Weber and Arnim Liekweg
495
A Comprehensive Approach to the Measurement of
Macroeconomic Exposure
Lars Oxelheim and Clas Wihlborg
513
Foreign-Exchange-Risk Management in German
Non-Financial Corporations: An Empirical Analysis
Martin Glaum
537
Brief Table of Contents
XIII
Estimating the Exchange Rate Exposure of US Multinational Firms:
Evidence from an Event Study Methodology
Kathryn L. Dewenter, Robert C. Higgins and Timothy T. Simin
557
International Corporate Risk Management:
A Comparison of Three Major Airlines
Matthias Muck and Markus Rudolf
571
Corporate Risk Management: Real Options and Financial Hedging
Alexander J. Triantis
591
The Real Option Value of Operational and Managerial Flexibility
in Global Supply Chain Networks
Arnd Huchzermeier
609
Managing Acquisition-Related Currency Risk Exposures:
The E.ON-Powergen Case
Stefan Hloch, Ulrich Hommel, and Karoline Jung-Senssfelder
631
Introducing New Risk Classes to Organized Exchanges:
The Case of Electricity Derivatives
Christian Geyer and Werner G. Seifert
651
Was Enron’s Business Model Fundamentally Flawed?
Ehud I. Ronn
671
“Real” Risk Management:
Opportunities and Limits of Consumption-based Strategies
Wolfgang Breuer and Olaf Stotz
679
Capacity Options: Convergence of Supply Chain Management
and Financial Asset Management
Stefan Spinler and Arnd Huchzermeier
699
Part 4: Systemic Issues of Risk Management
The Key to Risk Management: Management
Adrian E. Tschoegl
721
Economic Risks of EMU
Michael Frenkel and Paul McCracken
741
Does Risk Management Make Financial Markets Riskier?
Ian R. Harper, Joachim G. Keller, and Christian M. Pfeil
765
Risk Management, Rational Herding and Institutional Investors:
A Macro View
Torben Lütje and Lukas Menkhoff
Revitalization of Japanese Banks – Japan’s Big Bang Reform
Mitsuru Misawa
785
801
6CDNGQH%QPVGPVU
A Word of Greeting
Preface
V
VII
Part 1: Bank Risk Management
Basel II and the Effects on the Banking Sector
Thomas Hartmann-Wendels, Peter Grundke and Wolfgang Spörk
1. Overview on the New Basel Capital Accord
1.1 Why Do We Need a More Sophisticated Banking Supervision?
2. The Standardized Approach
3. The Internal Ratings-Based Approach
3.1 The IRB Approach for the Corporate Asset Class
3.1.1 Basic Structure of the IRB Approach for the Corporate Asset
Class
3.1.2 The Risk Components
3.1.3 The Risk Weight Function
3.2 The IRB Approach for the Retail Asset Class
4. Consequences of Basel II
4.1 Consequences on the Lending Margins
4.2 Consequences for the Banking Industry
Conflicts of Interest and Market Discipline in Financial Services Firms
Ingo Walter
1. A Conflict of Interest Taxonomy
1.1 Conflicts of Interest in Wholesale Financial Markets
1.2 Conflicts of Interest in Retail Financial Services
1.3 Wholesale-Retail Conflicts
2. Conflicts of Interest and Strategic Profiles of Financial Firms
2.1 Potential Conflicts of Interest in Multifunctional Client
Relationships
3. Constraining Exploitation of Conflicts of Interest
3.1 Regulation-Based Constraints
3.2 Market-Discipline Constraints
3.3 Intersection of Regulation and Market-Based Constraints
4. Conclusion
Risk Management and Value Creation in Banks
Gerhard Schröck and Manfred Steiner
1. Introduction
2. Necessity for a Framework on Risk Management in Banks
at the Corporate Level
3. RAROC as Capital Budgeting Rule in Banks
3.1 Evolution of Capital Budgeting Rules in Banks
3.2 Definition of RAROC
3.3 Assumptions and Deficiencies of RAROC
4. Overview of New Approaches
3
3
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6
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9
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XVI
Table of Contents
5. Implications of the New Approaches on Risk Management
and Value Creation in Banks
5.1 Implications for Risk Management Decisions
5.2 Implications on Capital Budgeting Decisions
5.3 Implications on Capital Structure Decisions
6. Foundations for a Normative Theory for Risk Management in Banks
7. Conclusion
The New Basel Capital Accord
Claudia Holtorf, Matthias Muck, and Markus Rudolf
67
68
71
71
72
74
79
1. Introduction
2. VaR Calculation
3. Regulatory Reporting, VaR, and Capital Requirement
4. Internal vs. Standard Model
5. Credit Risk
6. Operational Risk
7. Summary and Outlook
79
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89
91
94
97
97
Value at Risk:
Regulatory and Other Applications, Methods, and Criticism
Alois Paul Knobloch
99
1. The Concept of Value at Risk and its Role in
Contemporary Risk Management
1.1 Value at Risk: Definition and Risks of Concern
1.2 Applications and Regulatory Background
2. Calculating Value at Risk: Methods and Inherent
Sources of Inaccuracy
2.1 Delta-normal and Delta-gamma Approach
2.2 Simulation Methods: Historical and Monte Carlo Simulation
3. Risk Reduction and Capital Allocation Within a Value at Risk
Framework
3.1 Minimizing Value at Risk
3.2 Allocating VaR to Business Units
4. Shortcomings of Value at Risk as a Measure of Risk
5. Conclusion
Parsimonious Value at Risk for Fixed Income Portfolios
John F. O. Bilson
1. Introduction
1.1 A Simple Example
1.2 The Key Rate Duration Model
1.3 The Level, Slope, and Curvature (LSC) Model
1.4 LSC Risk Analysis
1.5 Conclusion
Risk Budgeting with Value at Risk Limits
Robert Härtl and Lutz Johanning
1. Introduction
2. Definition of Value at Risk Limits
3. The Structure of the Simulation Models
4. Adjusting Risk Limits for Time Horizons and
Profits and Losses
99
100
101
103
104
107
109
110
112
114
121
125
125
126
130
133
137
140
143
144
145
147
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Table of Contents
5. Incorporating Asset Correlations Into Risk Budgets
6. Conclusion and Practical Implications
Value at Risk, Bank Equity and Credit Risk
Jack E. Wahl and Udo Broll
1. Introduction
2. A Banking Firm
2.1 The Economic Setting
2.2 The Stochastic Setting
2.3 Value at Risk and the Bank’s Profit
3. Optimal Capital Requirement
4. Value Maximization and Bank Equity
5. Conclusion
Parametric and Nonparametric Estimation of Conditional
Return Expectations
Wolfgang Drobetz and Daniel Hoechle
1. Introduction
2. Parametric versus Nonparametric Regression – A Simple Derivation
2.1 Conditional Mean, Econometric Loss, and Weighted Least Squares
2.2 The Parametric Approach: An Unusual Representation of OLS
2.3 Nonparametric Regression Analysis
2.4 The Multivariate Case
2.5 Bandwidth Selection for Nonparametric Regression Estimators
3. Data Description
4. Empirical Results
4.1 In-sample Results
4.2 Out-of-sample Results
5. Conclusion
6. Acknowledgement
Credit Risk Portfolio Modeling: An Overview
Ludger Overbeck
1. Purpose of Credit Risk Modeling
1.1 Enterprise Risk Management
1.1.1 Economic Capital
1.1.2 Capital Allocation
1.2 Integration of Risk Types
1.3 Loss Distribution
1.4 Risk Measure
1.5 Portfolio Transactions
2. Basic Components of Credit Risk Modeling
2.1 Inputs
2.1.1 Exposure at Default
2.1.2 Loss Given Default
2.1.3 Default Probability
2.1.4 Dependency Concept
2.1.5 Event Versus Time Series Correlation
2.2 Output
2.2.1 Economic Capital
XVII
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Table of Contents
2.2.2 Value-at-Risk
2.2.3 Expected Shortfall
2.2.4 Coherent Risk Measures
2.2.5 Capital Allocation
2.2.6 Contribution to Volatility and Contribution to VaR,
Capital Multiplier
2.2.7 Contribution to Expected Shortfall
3. Portfolio Models
3.1 Actuarial Approach
3.1.1 Specification of Severity and Frequency Distributions
3.1.2 Dependence
3.1.3 Extensions
3.2 Structural Approach
3.2.1 Default Event
3.2.2 Dependencies
3.2.3 Loss Distribution
3.2.4 Extensions
4. Summary
Evaluating Credit Risk Models
Hergen Frerichs and Mark Wahrenburg
1. Introduction
2. Backtests Based on the Frequency of Tail Losses
3. Backtests Based on Loss Density Forecasts
4. Forecast Evaluation Approaches to Backtesting
5. Conclusion
204
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206
206
207
207
208
208
209
209
210
210
211
213
215
216
219
219
221
225
231
236
Estimation of Default Probabilities and Default Correlations
Stefan Huschens, Konstantin Vogl, and Robert Wania
239
1. Introduction
2. Estimation of Default Probabilities
2.1 Single-Period Case
2.2 Multi-Period Case
2.3 Multi-Group Case
3. Estimation of Default Correlation
3.1 Concepts of Dependent Defaults
3.2 Estimation in a General Bernoulli Mixture Model
3.3 Estimation in a Single-Factor Model
4. Simultaneous Estimation
4.1 General Bernoulli Mixture Model
4.2 Single-Factor Model
5. Conclusion
239
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249
249
250
253
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257
Managing Investment Risks of Institutional Private
Equity Investors – The Challenge of Illiquidity
Christoph Kaserer, Niklas Wagner and Ann-Kristin Achleitner
1. Introduction
2. Measuring Private Equity Returns and Risk
2.1 Asset Value Based Returns
2.2 Smoothed Proxy Observations
2.3 Noisy Smoothed Proxy Observations
259
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261
263
264
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Table of Contents
2.4 Cash Flow Based Returns
3. Risk Management and Asset Allocation
3.1 Specific Issues in Risk Management
3.2 Specific Issues in Asset Allocation
4. Conclusion
Assessment of Operational Risk Capital
Carol Alexander
1. The Operational Risk Capital Model
1.1 Frequency, Severity and the Loss Distribution
1.2 Operational Risk Capital Calculation
2. Dealing with Operational Risk Data
2.1 Choosing the Functional Form of the Loss Model
2.2 Data Filtering and Scaling
2.3 Risk Self-Assessment
2.4 Data-Oriented AMA
3. Aggregation of Operational Risks
3.1 Identification of Dependencies
3.2 The Effect of Dependencies on the Aggregate ORC
3.3 Aggregating Operational Risks with Other Risks
4. Summary and Conclusions
Operational Risk: The Management Perspective
Wilhelm Kross
1. Introduction
1.1 Commonly Practiced Approaches to OpRisk
1.2 Pitfalls on the Road to AMA Compliance
1.3 Inefficiencies in AMA Compliance Management
1.4 Desirable Side-Effects in OpRisk Management
1.5 Priorities and Maximized Value in OpRisk Management
1.6 Generic Roadmap towards Effective OpRisk Management
1.7 Conclusions and Recommendations
XIX
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Part 2: Insurance Risk Management
Catastrophic Events as Threats to Society:
Private and Public Risk Management Strategies
Martin Nell and Andreas Richter
1. Introduction
2. Insurance-linked Securities
3. State Guarantees for Catastrophic Risk?
4. Problems with Catastrophe Insurance Demand
5. Conclusion
New Approaches to Managing Catastrophic Insurance Risk
Ulrich Hommel and Mischa Ritter
1. Introduction
2. CAT-Linked Securities – A New Asset Class
3. Traditional and ART-Based CAT Reinsurance
4. Optimizing the Issuer’s Risk Portfolio
5. Risk Management Strategies Using CAT-Linked Securities
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XX
Table of Contents
5.1 Ex-Post Capital Provision and Funding Cost Reduction with
CAT-linked Bonds
6. Valuation Issues
7. Concluding Remarks
Alternative Risk Transfer
Christopher L. Culp
1. Introduction
2. Self-Insurance, Captives, and the Emergence of ART
2.1 Single-Parent Captives
2.2 Other Captive-Like Structures
2.2.1 Mutualized Structures
2.2.2 Rent-A-Captives and Protected Cell Companies
3. Finite Risk
3.1 Typical Finite Risk Structures
3.2 Potential Benefits to Corporates
3.3 The AIG/Brightpoint SEC Settlement
4. Multi-Line Programs and Risk Bundling
4.1 Overcoming Silo-by-Silo Inefficiency
4.2 A Mixed Record
5. Multi-Trigger Programs
6. Structured Finance Solutions
6.1 Asset Securitization
6.2 Risk Securitization
6.3 Future Flow Securitization
6.4 Structured Liabilities
7. Contingent Capital
8. Conclusion
The Challenge of Managing Longevity Risk
Petra Riemer-Hommel and Thomas Trauth
1. Introduction
2. Establishing the Relevance of Longevity Risk to the Insurance Industry
3. Economic Reasons for the (Re)Insurance Gap
3.1 Difficulties in Forecasting Longevity Trends
3.2 Adverse Selection
3.3 Moral Hazard
3.4 Absence of Diversification and Hedging Opportunities
4. Possible Solutions for Longevity Risk (Re)Insurance
4.1 Pricing to Risk
4.2 Finite Reinsurance Solutions
4.3 Capital Market Solutions
5. Conclusion
Asset/Liability Management of German Life Insurance Companies:
A Value-at-Risk Approach in the Presence of Interest Rate Guarantees
Peter Albrecht and Carsten Weber
1. Introduction
2. The Model and its Calibration
3. The Case of German Life Insurance Companies
4. Pure Market Values of Assets
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Table of Contents
5. Book Values of Assets
6. The Riskless Asset
7. Summary
8. Appendix A: Probable Minimum Return
9. Appendix B: Worst Case-Average Return
10. Appendix C: Conversion of Market Values into Book Values
XXI
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Part 3: Corporate Risk Management
Risk Management, Corporate Governance and the Public Corporation
Fred R. Kaen
423
1. Introduction
2. “Scientific” Theoretical Perspective on Risk Management
3. From Theory to Practice: Why Firms Should Manage Risk
3.1 Using Risk Management to Lower Taxes
3.2 Reducing Financial Distress and Bankruptcy Costs
3.3 Using Risk Management to Encourage and Protect
Firm Specific Investments
3.4 Using Risk Management to Monitor and Control Managers
3.5 Using Risk Management to Improve Decision Making
and Capital Budgeting
3.6 Risk Management and Dividends
4. Back to Berle and Means
5. Summary and Conclusions
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424
426
426
427
Integrating Corporate Risk Management
Christian Laux
1. Introduction
2. How Does Risk Management Add Value?
3. Measuring the Value of Risk Management
4. Identifying a Firm’s Collective Risks
5. Interactions Between Risk Management, Financial Structure,
and Operating Decisions
6. Integrated Products
7. Risk Management and Managerial Incentive Problems
Value-Based Motives for Corporate Risk Management
Ulrich Hommel
1. Introduction
2. The Irrelevance Theorem of Modigliani-Miller (MM)
3. Value Based Motives for Corporate Risk Management
3.1 Raising the Efficiency of Financial Contracting
3.1.1 Shareholders vs. Management
3.1.2 Creditors vs. Shareholders
3.2 Reducing the Corporate Tax Burden
3.3 Reducing Transaction Costs
3.3.1 Transaction Cost of Financial Distress
3.3.2 Transaction Cost of Hedging
3.4 Selecting the Optimal Risk Portfolio
3.5 Coordinating Financial and Investment Policies
4. Conclusion
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Table of Contents
Value-based Corporate Risk Management
Werner Gleißner
1. Introduction
2. Tasks and Elements of Corporate Risk Management – Overview
2.1 From Risk Management to Value-Based Management and
Strategic Management
2.2 Analyzing Risks
2.3 Aggregating Risks: Definition of Total Risk Volume
2.4 Coping with Risks
2.5 Designing Risk Management Systems and Monitoring
3. Risk, Cost of Capital and Shareholder Value
3.1 Introducing Considerations, the Shareholder Value
3.2 Enterprise Value and Capital Costs in Efficient Markets
3.3 Model Criticism
3.4 Deriving Realistic Cost of Capital Rates
3.5 Further Consequences of Inefficient Capital Markets
4. Conclusion
Statutory Regulation of the Risk Management Function in Germany:
Implementation Issues for the Non-Financial Sector
Jürgen Weber and Arnim Liekweg
1. Introduction: Statutory Regulations as Cause of a
New German Discussion on Risk–Management
2. Entrepreneurial Risk and Risk Management: A Holistic Approach
2.1 Chance, Risk and their Definitions
2.2 Chance, Risk and their Dimensions
2.3 The Process of Entrepreneurial Chance and Risk Management
2.3.1 The Chance/Risk–Strategy
2.3.2 Chance/Risk–Identification
2.3.3 The Chance/Risk–Analysis
2.3.4 The Chance/Risk–Reporting
2.3.5 Chance/Risk–Management
2.3.6 Chance/Risk–Monitoring
2.4 The Process-External Monitoring and Revision Function
3. Summary: The Critical Factors for the Implementation of the
Risk Management Function
A Comprehensive Approach to the Measurement of
Macroeconomic Exposure
Lars Oxelheim and Clas Wihlborg
1. Introduction
2. Exposure Coefficients
3. The Choice of Dependent Variable
4. The Choice of Independent Variables and Time Horizon
5. Volvo Cars
6. Results, Interpretations and the Use of Coefficients
6.1 Explanatory Factors
6.2 Exposure to Macroeconomic Shocks
6.3 Exposure Under Pegged Versus Flexible Exchange Rates
6.4 What Has Financial Exposure Management Achieved?
6.5 Financial Structure as a Hedge Against Macroeconomic Exposure
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XXIII
7. Using Estimated Coefficients for Future Periods
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8. Concluding Remarks and the Use of MUST Analysis in External Reporting 533
Foreign-Exchange-Risk Management in German
Non-Financial Corporations: An Empirical Analysis
Martin Glaum
1. Introduction
2. Theoretical Framework: Measurement and Management of ForeignExchange Risk
3. Methodology of the Empirical Study
4. Results of the Empirical Study
4.1 Exposure Concepts
4.2 Exchange-Risk-Management Strategies
4.3 The Use of Foreign-Exchange-Rate Forecasts
4.4 Organization of Exchange-Rate Management
4.5 Further Arguments and Hypotheses on Exchange-Risk Management
5. Conclusion
Estimating the Exchange Rate Exposure of US Multinational Firms:
Evidence from an Event Study Methodology
Kathryn L. Dewenter, Robert C. Higgins and Timothy T. Simin
1. Introduction
2. Sample Selection and Event Study Methodology
3. Event Study Measures of Exchange Rate Exposure
4. Determinants of Exchange Rate Exposure
5. Conclusion
International Corporate Risk Management:
A Comparison of Three Major Airlines
Matthias Muck and Markus Rudolf
1. Introduction
2. The Current Situation of the Airlines
2.1 Lufthansa AG Background Information
2.2 United Airlines Background Information
2.3 Qantas Background Information
3. CorporateMetricsTM – Explaining the Model
4. Income Statements
5. Corporate Risk Drivers
6. Hedging Strategies
7. Simulation Results
8.Conclusion
Corporate Risk Management: Real Options and Financial Hedging
Alexander J. Triantis
1. Identification and Classification of Risks
2. Rationales for Managing Risk
3. Using Derivatives and Other Contracts to Manage Risk
4. Using Real Options to Hedge and Exploit Risk
5. Using Real versus Financial Options for Hedging
6. Creating an Integrated Risk Management Strategy
7. Conquering Risk
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XXIV
Table of Contents
The Real Option Value of Operational and Managerial Flexibility
in Global Supply Chain Networks
Arnd Huchzermeier
1. Introduction
2. The Benefit of Operational Flexibility
2.1 The Original Problem
2.2 Supply Chain Network Cost Optimization
2.2.1 The Two-stage Supply Chain Network Model Formulation
2.2.2 The International Two-stage Supply Chain Network Model
2.3 Profit Maximization
2.4 Shareholder Value Maximization
2.5 Transfer Pricing
2.6 Knowledge Management
2.7 Real Exchange Rate Risk
3. The Option Value of Managerial Flexibility
3.1 Demand Risk
3.1.1 Stochastic or Scenario Programming Formulation with Recourse
3.1.2 The Option Value of Managerial Flexibility under Demand Risk
3.1.3 Monte-Carlo Simulation Study
3.2 Exchange Rate Uncertainty
3.2.1 Local Pricing
3.2.2 World Pricing
3.2.3 Home-Country or US$-Pricing
3.2.4 The Option Value of Managerial Flexibility under Demand Risk
and Price/Exchange Rate Uncertainty
3.2.5 Monte-Carlo Simulation Study
4. Summary
Managing Acquisition-Related Currency Risk Exposures:
The E.ON-Powergen Case
Stefan Hloch, Ulrich Hommel, and Karoline Jung-Senssfelder
1. E.ON’s Acquisition of Powergen
2. Currency Risk Exposures in Cross-Border Acquisitions
2.1 Currency Exposure Defined
2.2 Contingent Exposure
2.3 Translation Exposure
3. Introducing an Acquisition-Related Approach to Managing
Currency Risk Exposures
3.1 Exposure Identification
3.2 Policy Formulation
3.3 Exposure Measurement
3.4 Exposure Monitoring and Reporting
3.5 Exposure Control
3.5.1 Foreign Debt
3.5.2 Currency Options
3.5.3 Currency Forwards, Futures and Cross-Currency Swaps
3.5.4 “Acquisition Companies”
4. Concluding Remarks
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Table of Contents
Introducing New Risk Classes to Organized Exchanges:
The Case of Electricity Derivatives
Christian Geyer and Werner G. Seifert
1. Introduction
2. Building on New Paradigms
2.1 The Integration of the Markets is Accelerating
2.2 Consolidation of European Market Infrastructures
2.3 A New Understanding of Roles, New Technologies, and
New Abilities Need a Different Form of Capitalization
3. New Risk Classes in Electricity
3.1 Challenges and Opportunities in the Emerging Power Market
3.2 Competition in the Electricity Industry
3.3 Opportunities Offered by an Electricity Exchange
3.4 Why Power is Different
3.5 Determinants of Power Prices and Related Risks
3.6 Limitations of Black/Scholes With Respect to Electricity
4. Price Discovery: Reshaping the Power Industry
4.1 The Role of the Forward Curve
4.2 Price Discovery in Bilateral and Exchange Markets
4.3 Reshaping of the Energy Industry has Begun
4.4 The Creation of the European Energy Exchange
5. Transfer to Other Risk Classes
5.1 The Future of Deutsche Börse: Developer and Operator of Markets for
Tradable Products
Was Enron’s Business Model Fundamentally Flawed?
Ehud I. Ronn
1. Overview
2. Causes for Market-Value Losses Known Prior to Oct. 16, 2001
3. Corporate Governance and the Slide towards Bankruptcy:
Business Practices Brought to Light Subsequent to Oct. 16, 2001
4. The Aftermath of Enron for Merchant Energy
5. The Economic Role of Markets: Price Discovery, Risk Management
and Price-Signaling
6. Was Enron’s Business Model Fundamentally Flawed?
“Real” Risk Management:
Opportunities and Limits of Consumption-Based Strategies
Wolfgang Breuer and Olaf Stotz
1. Onassis and the Numéraire Problem
2. Consumption-Oriented Utility Functions
3. Onassis’ Decision Problem Reconsidered
4. Consumption Oriented Utility and International Invitations for Tenders
4.1 The General Setting
4.2 Capital Market Data
4.3 Entrepreneurial Data
4.4 Risk Management Situations
4.4.1 Active Risk Management Only at t = 1
4.4.2 Active Risk Management Only at t = 0
5. Conclusion
XXV
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XXVI
Table of Contents
Capacity Options: Convergence of Supply Chain Management
and Financial Asset Management
Stefan Spinler and Arnd Huchzermeier
1. Introduction
2. Supply Contracting: Emergence of Forward Buying,
Contractual Flexibility and Risk Hedging
2.1 Pricing Issues
2.2 Long-Term Investment vs. Short-Term Flexibility
2.3 Contractual Flexibility
2.4 Management of Demand Uncertainty
3. Capacity Options and Risk Management
3.1 A Model for Capacity Options
3.2 Risk Hedging via Flexibility Contracts
3.3 Trading Opportunities for Flexibility Contracts
3.4 Contract Portfolios
4. Summary
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Part 4: Systemic Issues of Risk Management
The Key to Risk Management: Management
Adrian E. Tschoegl
1. Introduction
2. Some Examples of Financial Debacles
2.1 Barings Brothers
2.2 Daiwa Bank
2.3 Sumitomo Corporation
3. Conceptualizing Debacles and their Prevention
4. Conclusion
Postscript: Allied Irish Bank
Economic Risks of EMU
Michael Frenkel and Paul McCracken
1. Introduction
2. Risks Stemming from Excessive Government Borrowing
3. Risks of High Adjustment Costs Stemming from European Labor Markets
4. Risks Associated with EMU Enlargement
5. Risks in EMU Financial Markets
6. Conclusion
Does Risk Management Make Financial Markets Riskier?
Ian R. Harper, Joachim G. Keller, and Christian M. Pfeil
1. Introduction
1.1 Increased Risk through Risk Management?
2. Market Risk as a Regulatory Concern
3. The Measurement of Market Risk
3.1 Some Comments on Different Approaches to VaR
3.2 VaR as an Amplifier of Volatility?
4. Some Empirical Results on Volatility in Major Stock Markets
4.1 Model Set-up, Data and Hypotheses
4.2 Estimation Results
5. Conclusion
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Table of Contents
Risk Management, Rational Herding and Institutional Investors:
A Macro View
Torben Lütje and Lukas Menkhoff
1. Introduction
2. Incentives towards Rational Herding of Institutional Investors
3. Evidence on Herding of Fund Managers
4. Survey Findings on Herd Behavior
4.1 Evidence of Herding Among Institutional Investors
4.2 Relation between the Perception of Herding and the
Institutional Investors' Characteristics
4.3 Perception of Herding and the Sources of Information
5. Consequences for the Management of Macro Risks
Revitalization of Japanese Banks – Japan’s Big Bang Reform
Mitsuru Misawa
1. Current Status
2. Demise of the High Growth Period and Birth of the Bubble Economy
3. The Japanese Big Bang (Financial Overhaul)
4. Reforming the Financial System
4.1 Shift toward the “Business-Category Subsidiary” System
4.2 Legalization of Financial Holding Companies
5. Revitalization through Coordination and Consolidation
6. Risk Management by Deferred Tax Accounting
7. A Case of Major Bank’s Default – Risk Avoiding by Nationalization
8. Future of Japan’s Big Bang Financial Reform
Authors
XXVII
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PART 1
Bank Risk Management
$CUGN++CPFVJG'HHGEVUQPVJG$CPMKPI5GEVQT
Thomas Hartmann-Wendels, Peter Grundke and Wolfgang Spörk1
1
University of Cologne, Department of Banking, Albertus-Magnus-Platz,
50923 Cologne, Germany
Abstract: Basel II will dramatically change the allocation of regulatory equity
capital to credit risk positions. Instead of an uniform 8 % capital charge regulatory equity capital will depend on the size of the credit risk, measured either by external or by internal rating systems. This will lead to a dramatic change in the
bank-debtor relation. Credit spreads will widen and for high risk borrowers it
may become difficult to get new loans. The major Basel II rules are surveyed and
their consequences for bank lending are discussed.*
JEL classification: G18, G21, K23
Keywords: Basel II, Capital Requirements, Credit Risk, Ratings
1XGTXKGYQPVJG0GY$CUGN%CRKVCN#EEQTF
In January 2001 the Basel Committee on Banking Supervision has issued a
consultative paper on the New Basel Capital Accord, that, once finalized, will replace the current Basel Accord from 1988 (Basel Committee 2001a). After an intensive consultation period with the banking industry and several modifications,
the Basel Committee has outlined the future regulation of credit risks in the socalled third consultative paper that will be the basis for the final Basel II Accord.
The proposed regulatory framework is based on three – mutually reinforcing – pillars (see Fig. 1.1. ).
*
The survey on the Basel II rules is based on the information released by the Basel Committee on Banking Supervision until the submission deadline of this contribution in October 2003.
4
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
The New Basel Capital Accord
PILLAR I
PILLAR II
PILLAR III
minimum capital
requirements
supervisory
review
market
discipline
• credit risks
• market risks
(unchanged)
• operational risks
• review of the
institution´s
capital adequacy
• enhancing
transparency
through rigorous
disclosure rules
• review of the
institution´s
internal assessment
process
Fig. 1.1. Overview on the New Basel Capital Accord
In the first pillar the rules for quantifying the necessary amount of capital to cover
the risk exposure of the various risk types are specified. Besides the already (explicitly) regulated credit and market risk positions, operational risk is included as a
new risk type. Operational risk is defined as “the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events”
(Basel Committee 2003b, p. 2). As the methodologies to capture market risk remain nearly unchanged compared to the 1996 update on the Basel Accord (Basel
Committee 1996), the most far-reaching changes in this pillar stem from credit
risk. The consequences of the refined measurement framework and risk coverage
for the banking business are in the focus of the following discussion.
Although pillars II and III have not been to the same extend controversially discussed in the public, they have enormous consequences for the banking industry,
especially in the credit area. The proposals of pillar II display a strong qualitative
element of supervision in opposite to the current regulation, in which a nearly sole
quantitative regulation approach can be observed. For example, banks are encouraged to develop – based on a supervisory catalogue of qualitative criteria – own
methods to measure credit and operational risk. Besides for an internal use (for
risk management), these can also be used for regulatory purposes, when the supervisor testifies the compliance with the criteria catalogue. This fundamentally new
way of supervision will lead to a merger of internal and regulatory risk management systems. The widened disclosure requirements, especially concerning the
used risk management systems that are proposed in the third pillar, will enhance
the transparency and thereby allow a deeper insight into an institute’s risk profile.
Basel II and the Effects on the Banking Sector
5
Due to the increased transparency market participants may be able to evaluate the
individual risk profile and the corresponding capital coverage of an institute more
rigorously and sanction it adequately. This market discipline effect will lead to an
additional banking supervision through the market participants and will transfer
some of the supervisory duties to the markets.
At early stages of negotiating an internationally accepted update of the 1988
Capital Accord, it was planned to implement Basel II in 2004. But the ambitious
goal to integrate different cultures, varying structural models and the complexity
of public policy and existing regulation led to the need for an intensive discussion
process. This need for aligning the original proposals was also raised by various
Quantitative Impact Studies (Basel Committee 2001b, 2002, 2003c,d), which
showed that the aim of a lower capital requirement for more ambitious approaches
in measuring the various risks was not met. Moreover, the public pressure and the
lobbying work of organizations representing small and medium sized enterprises
(SME) resulted in an integration of political ideas in an originally economical
concept. Currently it is planned that the Basel Committee will publish the final
version of the New Basel Capital Accord by no later than midyear 2004. There is
hope that Basel II will be implemented by various national bank supervisors by
the end of 2006.
9J[&Q9G0GGFC/QTG5QRJKUVKECVGF$CPMKPI5WRGTXKUKQP!
By integrating operational risks in the new regulatory framework and increasing
the transparency, some of the systematic weaknesses of the 1988 Capital Accord
can be reduced. But the most obvious changes of the revised Capital Accord are
related to the area of credit risks or, to be more precise, to the credit risks of the
banking book. The by far largest positions in the banking book are traded and nontraded claims on sovereigns, banks and other customers. As the potential loss from
these positions is determined by the amount outstanding and the creditworthiness
of the borrower, the capital charge under the current regulation results from the
product of both parameters multiplied with the solvability coefficient of 8%. The
creditworthiness of each borrower is quantified by a system of standardized risk
weights. For sovereigns the risk factor is 0% or 20%, for banks 20% and for all
other borrowers – independent from their individual creditworthiness – 100%.
It is quite obvious that a flat risk weight of 100% for all non-banks (and nonsovereigns) can not catch adequately the individual credit quality. One reason for
such an undifferentiated regulation approach may be found in the trade-off between the complexity of implementation of a regulation and the degree of accuracy in assessing the individual creditworthiness. Another argument in favor of
such a simple risk measurement system stems from the hope that the flat risk
weights may not reflect a proper measurement for an individual borrower, but on
average (over all claims in the banking book) the total credit risk is captured at
least approximately correct. From the point of view of a bank supervisor it is satisfying that the total amount of credit risk of the whole banking book is covered.
6
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
Unfortunately, many banks (ab)used the undifferentiated risk measurement system for regulatory arbitrage. For example they sold their “good” claims of the
banking book (usually via asset backed transactions; ABS). This resulted in a
situation in which more “bad” claims than on average remained in the banking
book for which an average risk weight is obviously not high enough. Moreover, an
increasing divergence of economic and regulatory capital was observable.
Another consequence of the undifferentiated way of assessing the credit quality
of borrowers can be seen in the lending margins. As the banks have to apply the
same risk weight for all non-banks, independent of their real credit quality, they
have to cover different risk exposures with the same amount of regulatory capital.
This divergence of economic and regulatory capital leads to a subsidization of
“bad” borrowers by “good” borrowers, i.e. good borrowers pay too much for their
loans.
To overcome the weaknesses of the current regulation, banks can use in the future one of the three following approaches to measure the credit risk of their banking book. In the standardized approach the risk weights are derived from credit
assessments from qualified rating agencies (see Basel Committee 2003a, pp. 14-15
for an catalogue of eligibility criteria), whereas in the internal ratings based approaches (IRB) the risk weights are estimated by the institute itself. Banks can
choose between a foundation and an advanced IRB approach. These three approaches will be introduced in the next chapters.
6JG5VCPFCTFK\GF#RRTQCEJ
The standardized approach is merely a modification of the current regulatory
framework. Analogue to the current methodology, the required amount of capital
to cover a claim on the banking book is calculated by:
exposure at default (EAD)
•
risk weight (RW)
•
8%
risk weighted asset
Although the structure of the formula is equivalent to the approach in the current
version of the Capital Accord and the solvability coefficient is unchanged, there
are some differences in the definitions of the components which enter the calculation of the risk weighted assets. This is due to the fact, that implicitly all “other
risks” have been covered with the amount of capital for credit risks under the current regulation. As most of the risks formerly known as “other risks” are captured
now explicitly as operational risks, but the total capital requirement to cover all
risks should remain unchanged, the definition of the credit exposure is changed by
recognizing credit risk mitigations as a mean to lower the exposure.
Quite similar to the current regulation, the claims in the banking book are categorized in claims on sovereigns and on banks. The former group of “non-banks” is
Basel II and the Effects on the Banking Sector
7
split into a category of corporate claims and a “regulatory retail portfolio”. All
claims on private customers and on SMEs with a total exposure of less than € 1
million belong to this new group. In addition to the current categories, ABS are
treated as a separate category.
In opposite to the 1988 Capital Accord, different risk weights are applied to the
exposures in each of the categories. The individual risk weight is determined by
the credit assessment of an external rating agency. To all not-rated borrowers a
standardized risk weight is applied. This risk weight is in most cases higher in
comparison to the current regulation (see table 1.1 for details).
Beside the categories that are based solely on the status of the borrower, mortgages on residential property are treated with a risk weight of 35%. For mortgages
on non-residential property a flat risk weight of 100% has to be applied. The assignment of the risk weights to a claim outlined in Tab. 1.1 can be changed in a
more favorable way, if the claim’s credit risk is either transferred to a third party
or collateralized by qualified assets.
Table 1.1. Risk weights in the standardized approach
external
rating
AAA to
AAA+ to ABBB+ to
BBBBB+ to
BBB+ to B-
sovereigns
option
1 (a)
0
20
20
50
risk weights (in %)
banks
option 2 (b)
corporates
M<3 m. M>3 m.
20
20
retail
portfolio
20
20
50
50
50
50
100
100
100
100
50
100
below B-
150
150
150
150
unrated
100
100
20
50
ABS
75
350
150
1,250
100
(a) option 1: banks will be assigned a risk weight one category less favorable than that
assigned to claims of that country
(b) option 2: the risk weights are solely based on the banks external rating; a preferential
risk weight, that is one category more favorable, may be applied to claims with an original
maturity (M) of 3 month (m) or less, subject to a floor of 20%
The standardized approach allows for a wider recognition of credit risk mitigants
(CRM) for regulatory purposes than under the current regulation. The three ways
to reduce the amount of risk weighted assets through CRM are presented in Fig.
2.1. (based on Hartmann-Wendels 2003, p. 39):
8
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
financial collaterals
netting agreements
EAD is reduced to the
net amount owed to
the other party
comprehensive
approach:
EAD is reduced by
the adjusted value of
the collaterals
exposure at default
(EAD)
credit derivatives/guarantees
simple approach:
RW of the claim is
substituted by the risk
weight of the
collateral
RW of the claim is
substituted by the
RW of the third party
that buys or
guarantees the risk
risk weight (RW)
risk weighted assets
Fig. 2.1. The influence of CRM-techniques on the risk weighted assets
CRM techniques are recognized only under a restrictive set of requirements (see
Basel Committee 2003a, pp. 17ff. for details) concerning the legal certainty in the
case of a credit event. Whereas netting agreements can only reduce the EAD and
credit derivatives and guarantees lead to a situation in which a more favorable risk
weight can be applied to the credit exposure, financial collaterals can influence
each of the parameter of the risk weighted assets, depending on the used approach.
Financial collaterals are defined as highly liquid assets with a low volatility concerning their values (see Basel Committee 2003a, pp. 22 ff. for the list of eligible
instruments).
Using the simple approach, banks may apply the more favorable risk weight of
the collateral for the collateralized credit exposure. If a bank decides for the comprehensive approach, more financial assets are eligible, e.g. equities that do not
belong to a main index. The market value of these collaterals has to be reduced by
so-called “haircuts”, which should capture possible changes in value. Only the adjusted value of the collateral reduces the EAD of the collateralized credit exposure. The collateralized exposure is calculated as follows:
E* = max {0, [E ⋅ (1+He) – C ⋅ (1 – Hc – Hfx)]}
where: E*=
E=
C=
Hx =
(1)
the exposure value after risk mitigation
the exposure value before risk mitigation
the current value of the collateral received
haircut appropriate to the exposure (x = e); for currency mismatch between the collateral and exposure (x = fx); to the collateral (x = c)
Banks have the choice between standard supervisory haircuts (see Basel Committee 2003a, p. 24 for details) and own estimations of potential changes in value of
the various categories of eligible financial instruments. When a bank decides for
an own model to estimate the haircuts, the use of this model is subject to the permission of the supervisory authorities.
Basel II and the Effects on the Banking Sector
9
Summarizing, one can state that even with the standardized approach a more
differentiated concept to measure the credit risks was designed, which overcomes
some of the weaknesses of the current regulation. Especially the revised definition
of the credit exposure (by recognizing CRM-techniques for regulatory purposes)
leads to a more realistic risk measurement. Moreover, the introduction of a new
asset-category for ABS with very high risk weights for tranches rated BB+ and below, will give no more incentives for regulatory arbitrage. Nevertheless, the standardized approach is – at least for Germany and most of the European countries –
no suitable regulatory framework, because only a very low percentage of all companies (and by definition no private obligor) has a rating. As a consequence, the
number of applicable risk weights is similar to the current regulation, as for nearly
all claims on the banking book the risk weights from the “non-rated” category
have to be applied. Therefore, this approach will be used only by a very small
number of (especially smaller) institutes. The bulk of banks will use an internal
ratings-based approach (IRB), which will be introduced in the next chapter.
6JG+PVGTPCN4CVKPIU$CUGF#RRTQCEJ
Beside the standardized approach, banks are also allowed to use an approach for
determining the capital requirement for a given exposure that is based on their
own internal assessment of the credit quality of an obligor. After publication of the
first consultative paper in 1999, this was a central demand of the European and
especially the German supervisory authorities.
If a bank uses the internal ratings-based (IRB) approach to credit risk, it has to
categorize its banking book exposures into five broad asset classes. These asset
classes are corporate, sovereign, bank, retail, and equity. The corporate asset class
contains five sub-classes of specialized lending, and the retail asset class is completely divided into three sub-classes. Based on the information given in the third
consultative paper (Basel Committee 2003a), we restrict ourselves in the following to the explanation and discussion of the proposals for the corporate asset class
(without specialized lending), which are very similar to those one for the sovereign and bank asset class, and the retail asset class.
Once a bank uses the IRB approach partially, the bank will be expected to extend the use to all their asset classes and business units. Hence, a cherry picking
between the standardized and the IRB approach is not possible.
6JG+4$#RRTQCEJHQTVJG%QTRQTCVG#UUGV%NCUU
$CUKE5VTWEVWTGQHVJG+4$#RRTQCEJHQTVJG%QTRQTCVG#UUGV
%NCUU
The IRB approach for the corporate asset class consists of a foundation and an advanced approach. These two approaches mainly differ with regard to the number
of risk components for which a bank can use its own internal estimate as opposed
10
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
to a supervisory value and the treatment of the effective maturity of an exposure.
The minimum requirements that must be met by a bank in order to be allowed to
use the IRB approaches are the highest for the advanced IRB approach. For the retail asset class, there is no distinction between a foundation and an advanced approach (see section 3.2).
As in the standardized approach, the risk weighted assets for corporate exposures under the IRB approach equal the product of the risk weight and the exposure at default (EAD). An important difference to the standardized approach is
that the risk weight does not only depend on the obligor’s rating, but on several
risk components. In the foundation approach, the risk weight is a continuous function of the one-year probability of default (PD) of the internal rating grade an obligor belongs to, the loss given default (LGD), and, for exposures to small and
medium enterprises (SME), the firm’s total annual sales (S). In the advanced approach the risk weight additionally depends on the effective maturity of the exposure, whereas in the foundation approach an average maturity of 2.5 years is assumed. Hence, characteristics of an exposure, which are relevant for its credit risk,
are recognized in much more detail under the IRB approach than under the standardized approach, which results in more differentiated capital requirements under
the IRB approach. Banks applying the foundation approach can only use their own
internal estimate of the PD, but are required to use supervisory values for LGD
and EAD, whereas the advanced approach completely relies on bank internal estimates of all risk components.
6JG4KUM%QORQPGPVU
Probability of Default
For both IRB approaches, a bank has to be able to estimate the one-year default
probabilities of their internal rating grades, which are based on a supervisory reference definition of default. According to this reference definition, an obligor’s
default has occurred when the obligor is past due more than 90 days on any material credit obligation or even when the bank considers that the obligor is unlikely
to pay its credit obligations in full. An indication of unlikeliness to pay is for example an account-specific provision, which the bank has made for the credit obligation due to a decline in the obligor’s credit quality.
Banks are allowed to use three different techniques to estimate the average PD
for each of their internal rating grades. First, a bank may use internal default data.
Second, a bank can map their internal rating grades to the scale used by an external credit assessment institution and then employ the corresponding default rates
observed for the external institution’s grades for their own rating grades. Of
course, a bank using this method has to ensure that the rating criteria and the default definition applied by the external institution are compatible with their own
practices. Third, a bank can use statistical default prediction models, with which it
estimates the individual default probability of each obligor. In this case, the PD of
a rating grade equals the average of the individual default probabilities. Irrespective of the applied method for the PD estimation, the underlying historical observation period must cover at least five years (during a transition period of three
Basel II and the Effects on the Banking Sector
11
years starting on the date of the implementation of the New Accord shorter time
periods are sufficient). In order to account for the uncertainty in the estimation
process of the PD, a floor of 0.03% has been proposed. The PD of a defaulted obligor is set equal to 100%.
If there is a recognized credit risk mitigation in the form of a guarantee or a
credit derivative, a bank applying the foundation IRB approach has to split an exposure into a covered and into an uncovered portion. For the covered portion the
bank has to take the PD appropriate to the guarantor’s internal rating grade and the
risk weight function appropriate to him, whereas the uncovered portion of the
exposure gets the risk weight associated with the underlying obligor. A bank using
the advanced IRB approach can take into account guarantees or a credit derivatives either through adjusted PD values or through adjusted LGD estimates. Under
either approach, the effect of double default must not be recognized. Despite the
fact that (beside in the case of a perfect positive correlation) the joint probability
of a default of the protection provider and the underlying obligor is smaller than
each of the individual default probabilities, the adjusted risk weight of the covered
portion of an exposure must not be less than that of a comparable direct exposure
to the protection provider. Under the foundation IRB approach the range of eligible guarantors is the same as under the standardized approach, whereas under the
advanced IRB approach there are no restrictions to the range of eligible guarantors, but minimum requirements with regard to the type of guarantee have to be
satisfied.
Loss Given Default
The loss given default equals the expected economic loss per unit exposure of default a bank has to bear if a default occurs. The economic loss includes discount
effects and costs associated with collecting on the exposure. As opposed to the
PD, the LGD can vary for different exposures to the same obligor, for example if
the exposures exhibit a different seniority.
Under the foundation approach, banks have to use for all senior corporate exposures without recognized collateral a standardized supervisory LGD value of 45%,
whereas all subordinated claims on corporates not secured by a recognized collateral are assigned a supervisory LGD value of 75%. If there are recognized collaterals, these supervisory LGD values can be reduced to collateral specific minimum LGD values. The range of eligible collaterals consists of those financial
collaterals that are also recognized in the standardized approach and, additionally,
IRB specific collaterals, such as receivables or specified residential and commercial real estates. Other physical collaterals may be recognized, too, but two basic
requirements must be fulfilled in any case: Existence of a liquid market and existence of publicly available market prices for the collateral.
Under the advanced IRB approach, a bank can use its own estimate of the LGD
for each facility, but these must be based on a data observation period of at least
seven years. The range of eligible collaterals is not limited, but the collaterals have
to meet some qualitative requirements.
12
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
Time to Maturity
Under the foundation IRB approach, an average time to maturity of 2.5 years is
assumed, whereas under the advanced IRB approach, the risk weight explicitly
depends on the facility’s effective time to maturity. But there are two possible exceptions: First, national supervisors can choose to require all banks to adjust risk
weights for the effective time to maturity even under the foundation IRB approach. Second, national supervisors can decide to exclude facilities to small domestic obligors from the explicit maturity adjustment of the risk weights under the
advanced IRB approach. The prerequisite is that the obligor’s total annual sales as
well as the total assets of the consolidated group of which the firm is a part are
less than €500 million. The effective maturity of a facility with a predetermined
cash flow schedule is defined as M=Σn⋅CFn/ΣCFn where CFn denotes the cash
flow (principal, interest, and fees) due at time n. Options (e.g. call privileges),
which can cause a reduction of the remaining time to maturity, are not recognized
in this definition. A cap of five years has been fixed for the time to maturity entering the risk weight formula under the advanced IRB approach. The floor is one
year, but there are exceptions for certain short-term exposures, which are defined
by each supervisor on a national basis.
Total Annual Sales
For exposures to SMEs with total annual sales of less than €50 million, there is a
reduction of the risk weight: The lower the total annual sales, the lower the risk
weight and, hence, the capital requirement. The maximal reduction is reached for
firms with total annual sales of €5 million. If the total annual sales are no meaningful indicator of firm size and if the national supervisor agrees, the total annual
sales can be substituted by the total assets as an indicator of the firm’s size.
Exposure at Default
In contrast to the standardized approach, the exposure at default is the amount legally owed to the bank, i.e. gross of specific provisions or partial write-offs. Onbalance sheet netting of loans and deposits of an obligor are recognized subject to
the same conditions as under the standardized approach. The EAD of traditional
off-balance sheet positions, such as commitments, is the committed but undrawn
line multiplied by a product-specific credit conversion factor (CCF). Under the
foundation IRB approach, only the use of standardized CCFs is allowed, whereas
under the advanced approach, banks can use their own internal CCF estimates
provided the exposure has not a supervisory CCF of 100% in the foundation approach. The internal CCF estimates must be based on a time period no shorter than
seven years. The EAD of innovative off-balance sheet positions, such as interest
rate or equity derivatives, is calculated as under the current Basel Accord, i.e. as
the sum of replacement costs and potential future exposure add-ons, where the latter depend on the product type and the maturity.
Basel II and the Effects on the Banking Sector
13
6JG4KUM9GKIJV(WPEVKQP
The continuous function, which combines the risk components PD, LGD, S, and
M to a risk weight, is one of the key elements of the IRB approach proposed by
the Basel Committee. Irrespective of the asset class or its sub classes and irrespective of the chosen approach (foundation versus advanced), the risk weight function
always has the same basic structure (Hartmann-Wendels 2002):
RW =
1
⋅ LGD ⋅ VaR ⋅ MF
0.08
(2)
The term VaR in the above general risk weight function (2) can be interpreted as a
Value-at-Risk because it equals the loss per unit EAD, which in an infinitely large
portfolio, where all positions exhibit a time to maturity of one year and an assumed LGD of 100%, is not exceeded within one year with a probability of
99.9%. Hence, applying the risk weight formula (2) guarantees that the probability
of the sum of expected and unexpected losses per year being larger than the bank’s
regulatory capital is smaller than 0.1%. In order to interpret the term VaR actually
as a Value-at-Risk, a simplified version of the credit portfolio model CreditMetrics™ has to be applied (Gordy 2001, Bluhm et al. 2003, pp. 83-94). The default
of an obligor is modeled as insufficient asset value return, which is below some
critical level at the risk horizon. It is assumed that the asset return of each obligor
can be represented as the sum of one systematic and one firm-specific risk factor,
which are both normally distributed. Conditional on a realization of the systematic
credit risk factor, the asset returns of all obligors and, hence, the default events are
assumed to be stochastically independent. Together with the assumed infinity of
the portfolio (and on additional technical assumption making sure that in the limit
the portfolio exhibits no dominating single exposure) this latter assumption ensures that the (strong) law of large numbers can be applied. Using the law of large
numbers, it can be shown that the random variable which represents the percentage portfolio loss equals almost surely the conditional (on a realization of the systematic risk factor) default probability. Important for the interpretation of the term
VaR as a Value-at-Risk is finally the assumption that there is only one single systematic risk factor driving the asset returns of all obligors and the monotony of the
conditional default probability as a function of this single systematic risk factor.
Within the assumed credit portfolio model, the term VaR also corresponds to the
obligors’ default probability conditional on an especially bad realization of the
systematic risk factor.
The proposal of the Basel Committee that capital for unexpected as well as for
expected losses has to be hold for has caused much criticism shortly after the publication of the second consultative paper, especially from the German supervisory
authorities and banks. It has been argued that expected losses are usually covered
by provisions and risk premiums paid by the obligor. Partly, these arguments have
been considered in the third consultative paper, e.g. provisions made by a bank
can reduce the capital charge for expected losses. Meanwhile, it seems as if the US
American supervisory authorities themselves, who originally favored capital re-
14
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
quirements for expected and unexpected losses, re-open the discussion whether
capital requirements for the expected part of the credit losses are really necessary.
As the term VaR corresponds to the Value-at-Risk of a portfolio of positions
with a time to maturity of one year, but the foundation IRB approach assumes an
average time to maturity of 2.5 years and the advanced IRB approach requires an
explicit adjustment of the risk weight for the remaining time to maturity, the above
general risk weight formula (2) additionally contains a maturity adjustment factor
MF, which is intended to control for the effect the exposures’ time to maturity has
on the Value-at-Risk.
Calculation of the Risk Weight for Corporate Exposures under the Foundation IRB Approach
Table 3.1 shows the specifications of the factors VaR and MF in the general risk
weight formula (2) for corporate exposures under the foundation IRB approach.
N(⋅) denotes the cumulative distribution function of the standard normal distribution and N-1(⋅) the inverse of this function. S stands for the firm’s total annual
sales in million €. The values for PD (and also for LGD in (2)) are entering the
formulas as decimals rather than whole numbers (e.g. 0.01 instead of 1%).
Table 3.1. Factors VaR and MF in the general risk weight formula (2) for corporate exposures under the foundation IRB approach
VaR
 N −1 (PD)

ρ(PD,S)
⋅ N −1 (0.999)  where
N
+


1 − ρ(PD,S)
 1 − ρ(PD,S)

correlation ρ(PD,S) of the asset returns
= 0.12 ⋅
MF
 1 − e −50⋅PD
1 − e −50⋅PD
+ 0.24 ⋅  1 −
−50
1− e
1 − e −50

MF(PD)found =

 max{5;S} − 5 
;0 
 − 0.04 ⋅ max 1 −
45



1
1 − 1.5 ⋅ ( 0.08451 − 0.05898 ⋅ ln(PD) )
2
As table 3.1 shows, the asset return correlation, modeled by the joint dependency
on the systematic risk factor, of obligors with total annual sales over €50 million is
assumed to be a monotonously decreasing function of the PD, where the minimal
correlation value is 12% and the maximal value 24%. This dampens the increase
of the risk weight function for increasing PD values. Exposures to corporates
where the reported total annual sales for the consolidated group of which the firm
is a part are less than €50 million are classified as exposures to SMEs and receive
a size dependent reduction of their risk weight. This is achieved by reducing the
asset return correlation parameter with decreasing sales S. The maximal asset
correlation reduction of 4% is reached for firms with total annual sales of €5
million; reported sales below €5 million are treated as if they were equal to €5
million. For small PD and S values the reduction of the risk weight for an
Basel II and the Effects on the Banking Sector
15
small PD and S values the reduction of the risk weight for an exposure to a SME
obligor can come to over 20% of the risk weight for a non-SME obligor.
The empirical findings concerning the firm size- and PD-dependency of the asset return correlation are partially contradictory and the reasons for these contradictions are still not clear. Overall, it seems as if the proposed decrease of the asset
return correlation with decreasing firm size can be empirically confirmed (Düllman and Scheule 2003 and partially Dietsch and Petey 2003), but the results concerning the PD-dependency are ambiguous. For example, Düllmann and Scheule
2003 rather find that the asset correlation is increasing with rising PD, especially
for medium and large firms, whereas Lopez 2002 confirms the relationship assumed by the Basel Committee. The specification of the corporate exposure risk
weight function shown in table 3.1 is the result of several modifications (in comparison to the second consultative paper), which were judged to be necessary after
Quantitative Impact Studies (Basel Committee 2001b, 2002, 2003c,d) had shown
that the capital requirements would overall increase and that there would be no incentive to apply the more sophisticated IRB approaches. The absolute level of the
current risk weight function is lower and the function less steep than the originally
proposed version (beside for small PD values) so that less regulatory capital per
unit EAD is necessary and the increase of the capital requirements for more risky
obligors is reduced. The following Fig. 3.1 shows the risk weights as a function of
the PD for various total annual sales S.
S≤5
S=27.5
S≥50
Fig. 3.1. Risk weights under the foundation IRB approach as a function of the PD for various total annual sales S (LGD=0.45)
Calculation of the Risk Weight for Corporate Exposures Under the Advanced
IRB Approach
Under the advanced IRB approach, the risk weight explicitly depends on the remaining time to maturity of an exposure. Table 3.2 shows the specifications of the
16
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
factors VaR and MF in the general risk weight formula (2) under the advanced
IRB approach.
Table 3.2. Factors VaR and MF in the general risk weight formula (2) for corporate exposures under the advanced IRB approach
VaR
MF
identical with the foundation IRB approach


2
MF(PD)adv = MF(PD)found ⋅  1+ ( 0.08451 − 0.05898 ⋅ ln(PD) ) ⋅ (M − 2.5) 
 144444244444

3
=∆


For times to maturity of less than 2.5 years the risk weight under the advanced
IRB approach is smaller than the corresponding one under the foundation approach, and for times to maturity of more than 2.5 years the ranking is reversed.
The risk weight under the advanced IRB approach is assumed to increase linearly
in the maturity (see Fig. 3.2). The proposed positive sensitivity ∆ of the risk
weight to the time to maturity depends on the credit quality of the obligor: The
higher the obligor’s PD, the lower is the sensitivity of the risk weight to the time
to maturity. Hence, a variation of the time to maturity causes an up or down scaling of the risk weight of the foundation IRB approach, which is smaller the lower
the obligor’s credit quality.
advanced
IRB approach
foundation
IRB approach
Fig. 3.2. Risk weights under the foundation and the advanced IRB approach as a function
of the time to maturity M (PD=0.01, LGD=0.45, S=5)
A sensitivity of the Value-at-Risk to the time to maturity of the positions in the
portfolio can only be observed in so-called Mark-to-Market (MtM) models, but
not in pure Default Mode (DM) models. In a MtM model the value of a bond or a
loan at the risk horizon depends on the future credit quality of the obligor. For example in CreditMetrics™, the simulated asset return indicates in which rating
Basel II and the Effects on the Banking Sector
17
class an obligor is at the risk horizon, and then the corresponding risk-adjusted
forward rates, observed today, are used for discounting the future cash flows of
the bond or the loan which are due beyond the risk horizon. In contrast, DM models, such as CreditRisk+™, only differentiate whether an obligor has defaulted until the risk horizon or not. In the former case, the position’s value at the risk horizon equals a fraction of its face value, and in the latter case, the value is identical
to the face value. Hence, for both credit quality states considered in DM models
the remaining time to maturity beyond the risk horizon has no influence on the future position’s value. If additionally the time to maturity is irrelevant for the obligor’s default probability until the risk horizon, the maturity has no influence at all
on the Value-at-Risk. In a MtM model the sensitivity of the Value-at-Risk to the
time to maturity decreases with worsening credit quality because the probability
rises that the obligor defaults until the risk horizon and that the loan or the bond is
set equal to a value, a fraction of its face value, which is independent from the remaining time to maturity.
The definition of retail exposures under the IRB approach is similar to that one
under the standardized approach. Eligible for retail treatment are loans to individuals or small firms, where the exposure must be one of a large pool of loans,
which are managed by a bank on a pooled basis. This requirement is intended to
guarantee a sufficient granularity of the retail portfolio. Under the IRB approach,
there is no explicitly stated upper percentage of the total exposures of the pool,
which the single exposure must not exceed. For loans to individuals, an upper absolute limit of the single exposure size is also not given, whereas loans to small
firms can only qualify for the retail asset class if the total exposure to the firm is
less than €1 million, which under the standardized approach is also the upper
absolute limit for loans to individuals. Furthermore, the bank has to treat the loan
to a small firm in its internal risk management system in the same manner as other
retail exposures. In contrast to the standardized approach, residential mortgage
loans belong to the retail asset class regardless of the exposure size as long as the
loan is given to an individual who is owner and occupier of the property.
Under the IRB approach, the retail asset class is divided up into three subclasses:
1. exposures secured by residential properties,
2. qualifying revolving exposures, and
3. all other retail exposures.
In order to qualify for a retail treatment, a revolving exposure must be an unsecured and uncommitted exposure to an individual with a volume of less than
€100,000. Furthermore, the future margin income must be high enough to cover
the sum of expected losses and two standard deviations of the annualized loss rate
of the sub-class.
For calculating the risk weight of a retail exposure, the proposals of the Basel
Committee do not differ between a foundation and an advanced IRB approach.
18
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
Banks are expected to provide internal estimates of the PD (as for corporate exposures with a supervisory floor of 0.03%) and the LGD for each of the identified
pools to which the single exposures are assigned. The minimum data observation
period for PD and LGD estimates is five years (during the transition period shorter
time periods are sufficient).
For each of the three sub-classes of the retail asset class a separate risk weight
function has been proposed. Basically, these three risk weight functions have the
same structure as the general risk weight formula (2), but without an explicit maturity adjustment (MF=1). The VaR terms of the three retail risk weight functions
are shown in table 3.3.
Table 3.3. Factor VaR in the general risk weight formula (2) for retail exposures under the
IRB approach
sub-class of the retail asset
class
residential mortgage exposures
VaR
 N -1 (PD)

0.15 −1
⋅ N (0.999) 
N 
+
0.85
 0.85

qualifying revolving expo−
1
 N (PD)

ρ(PD)
sures
⋅ N −1 (0.999)  − 0.75 ⋅ PD
N
+
 1 − ρ(PD)

1 − ρ(PD)


where ρ(PD) =0.02 ⋅
other retail exposures
 1 − e −50⋅PD 
1 − e −50⋅PD
+ 0.11 ⋅ 1 −

−50
1− e
1 − e −50 

 N −1 (PD)

ρ(PD)
⋅ N −1 (0.999) 
N
+
 1 − ρ(PD)

1 − ρ(PD)


where ρ(PD) =0.02 ⋅
 1 − e −35⋅PD 
1 − e −35⋅PD
+ 0.17 ⋅ 1 −

−35
1− e
1 − e −35 

Similar to the IRB approach for corporate exposures, the asset return correlation
proposed for calculating the risk weight for qualifying revolving and other retail
exposures are decreasing with worsening credit quality. Only for residential mortgage exposures a constant correlation of 15% is assumed. But on average, the asset return correlation values for retail exposures are much lower than those of corporate exposures. An additional difference to the IRB approach for the corporate
asset class exists for qualifying revolving exposures: For exposures in this subclass, 75% of the expected losses have not to be covered by regulatory capital. Instead, by definition of this sub-class, it is assumed that expected losses are mainly
covered by future margin income. As a consequence of the missing explicit maturity adjustment, the lower asset return correlation values and the special treatment
of the expected losses in the sub-class revolving exposures, the IRB risk weights
for retail exposures are significantly lower than those for corporate exposures (see
Fig. 3.3). Through this, the higher diversification of the retail asset class compared
to the corporate asset class is accounted for. In the case of residential mortgage
Basel II and the Effects on the Banking Sector
19
exposures this is true only for PD values up to 2%-5.5% (depending on the firm
size S). For larger PD values the risk weight for residential mortgage exposures is
higher than that for corporate exposures because the risk weight decreasing effect
of the missing maturity adjustment is overcompensated by the higher constant asset correlation of 15% compared to the with the PD value decreasing asset correlation for corporate exposures. But practically, PD values larger than 2%-5.5% can
only be observed for less creditworthy speculative grade obligors.
foundation IRB approach
(S=50)
residential mortgage exposures
foundation IRB approach
(S=5)
other retail exposures
qualifying revolving exposures
Fig. 3.3. Risk weight functions under the IRB approach for retail exposures and under the
foundation IRB approach for corporate exposures (LGD=0.45)
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As shown in the previous chapters, the New Basel Capital Accord will have farreaching effects on the methods to quantify credit risks and the resulting minimum
capital requirements. The consequences for the lending margins, the lending behavior and the risk management of banks will be discussed in the next sections.
The minimum interest rates that have to be earned with a loan can be divided in
four components (see Fig. 4.1). Besides the refinancing costs, a risk premium, that
covers the expected loss from the loan, the operating costs for handling the loan,
the minimum interest rate also has to include the costs for equity to cover the unexpected losses.
20
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
minimum
equity costs
return on equity, that covers the
unexpected losses
risk premium
risk premium, that covers the
expected losses
operating costs
costs for handling and
supervising the loan
interest
rate
refinancing costs
Fig. 4.1. Components of the minimum interest rate
Operating Costs
Due to the increased complexity of the revised regulation, the banks face very
high costs associated with developing, maintaining and continuously upgrading a
new system to quantify credits risks. Moreover, the costs for collecting data and
maintaining the data base for the internal rating system will lead to an additional
increase of the fixed costs. Making the staff (and also many of especially smaller
corporate clients) familiar with the new system and the contents of Basel II will
also cause costs. Nevertheless, although the fixed costs will increase due to the
new regulatory framework, there is a chance that the variable costs for handing out
and supervising loans may be reduced, because these processes can be standardized and streamlined. Summarizing, one can say that the operating costs will – especially during the first years after Basel II is in force – slightly increase.
Costs for Equity
The most significant changes in the various cost components are expected for the
equity costs. Although the Basel Committee seeks a solution, in which the average
capital charge for credit risks and operational risks will be constant in comparison
to the current capital charge for credit risks, one will see tremendously varying
capital charges for individual loans, depending on the individual credit quality.
Very creditworthy borrowers will benefit from (slightly) decreasing loan conditions, whereas lower rated customers will be confronted with sharply increasing
interest rates. As the leverage ratio is an important parameter influencing a rating,
young and small companies will be affected mostly by this increase. Although this
effect is dampened for SMEs, due to an eased regulation for this group of claims,
Basel II and the Effects on the Banking Sector
21
the capital charge will be more risk sensitive, resulting in a widened spreading of
the individual coverage ratios.
Risk Premium
The risk premium is under the current regulation the only cost component, that influences the lending margins of obligors with different credit qualities. It can be
observed that banks calculate so-called “standard risk costs” for obligors belonging to different credit quality classes. These standard risk costs, that reflect the average expected loss ratio per class, are used as an orientation for negotiating the
lending margins with the potential customer. Due to the more sophisticated measurement framework of Basel II with a higher number of risk classes, a more risk
sensitive and precise quantification of credit risk will be possible. This will lead to
a decrease of the average risk premia, because the uncertainty premium for potentially inadequate risk measurement techniques will be abandoned or at least reduced. Nevertheless, also in the area of risk premia, Basel II will lead to a huge
widening of the range of add-ons for expected losses. Especially obligors with a
high probability of default will have to accept high surcharges, that will be similar
to those asked for in the bond market, but can currently not be realized in the German loan market.
cost components
Costs for Refinancing
Basel II will not cause a short term effect on the refinancing costs, but may lead to
(slightly) decreasing refinancing costs. This can be explained by the improved
transparency concerning an institutes risk exposure and the enhanced (permanent)
supervision from the authorities and the markets, which can create an even higher
degree of investor trust.
risk premia
equity costs
operating costs
obligor´s credit quality
Fig. 4.2. Add-ons to the refinancing costs depending on the obligor’s creditworthiness
(current regulation)
cost components
22
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
risk premia
equity costs
operating costs
obligor´s credit quality
Fig. 4.3. Add-ons to the refinancing costs depending on the obligor´s creditworthiness
(Basel II)
Summarizing the consequences of the New Capital Accord on the lending margins, one can state that on average – caused by the increased operating costs – the
interest rates for loans will slightly increase. This effect will be compensated subsequently by shrinking risk premia and refinancing costs. Analyzing the situation
of individual obligors we will clearly see a huge widening in the range of individual loan interests. This is due to the fact that the costs for equity and the risk premia will mainly depend on the individual borrower´s creditworthiness. The lending margins resulting from the changed regulatory framework are compared in
Fig. 4.2 and 4.3.
!
The New Capital Accord will – despite the still existing weaknesses – improve the
stability and soundness of the national and international financial markets. Especially the IRB approaches are sophisticated tools, with which credit risks can be
quantified adequately. This, combined with more transparency, will ease the trading of credit risks. Moreover, input data for credit portfolio models can be collected, which allow for an improved active credit risk management (e.g. diversification by reducing clump risks). This will speed up the process of transforming
the role of banks from risk takers into risk traders.
The widening of the range of possible lending margins will lead to a situation
in which refinancing by loans becomes more attractive for corporates with a very
high credit quality, because their required minimum interest rates will shrink to a
Basel II and the Effects on the Banking Sector
23
risk adequate level, after the cross subsidization of bad obligors has ended. High
risk customers, who are currently virtually excluded from the bank loan market,
might get an access to bank loans, when they are willing and able to accept the
very high but risk adequate lending margins. Hereby a high risk/high return segment – similar to the junk bond markets outside Germany – that is quite interesting with respect to diversification, can be created. Losers of the revised regulation
are especially small and medium sized companies with high leverage ratios and/or
without a long lasting lender relationship or those companies without an existing
reporting system that can easily produce key indicators for evaluating the companies with regard to their credit quality. On the one hand, financial institutions may
loose parts of their loan business to this group of companies, but on the other
hand, they may gain market shares in the area of loan alternatives (e.g. leasing,
factoring and private equity) and by providing consulting for these ways of financing.
Finally, it has to be said that the ongoing statements in the public discussion that
Basel II will lead to higher minimum capital requirements and therefore higher
lending margins, which will cause a credit squeeze – especially for SMEs – cannot
be supported. It is the pronounced goal of the Basel Committee that the capital requirements remain on average unchanged in comparison to the current regulation.
But even if this goal is not met, this will not necessarily lead to higher loan interests. In this case, institutes would need more (expensive) equity, but simultaneously the risk exposure per unit equity would decrease, implying a reduced required return on equity. In a perfect capital market world, both effects compensate
each other perfectly (Modigliani and Miller 1958), i.e. although the capital charge
of a given risk exposure is raised, the total equity costs for this risk exposure remain constant. Even taking into account that banks do not work in such a perfect
capital market world, may not lead to the opposite statement that there is an independency between the required return on equity and the risk exposure per unit equity.
4GHGTGPEGU
Basel Committee on Banking Supervision (1996) Amendment to the Capital Accord to
incorporate market risks, Basel January 1996
Basel Committee on Banking Supervision (2001a) The New Basel Capital Accord, Basel
January 2001
Basel Committee on Banking Supervision (2001b) Results of the Second Quantitative Impact Study, Basel November 2001
Basel Committee on Banking Supervision (2002) Results of Quantitative Impact Study 2.5,
Basel June 2002
Basel Committee on Banking Supervision (2003a) The New Basel Capital Accord, Basel
April 2003
24
Thomas Hartmann-Wendels, Peter Grundke, and Wolfgang Spörk
Basel Committee on Banking Supervision (2003b) Sound Practices for the Supervision of
Operational Risk, Basel February 2003
Basel Committee on Banking Supervision (2003c) Quantitative Impact Study 3 – Overview
of Global Results, Basel May 2003
Basel Committee on Banking Supervision (2003d) Supplementary Information on QIS 3,
Basel May 2003
Bluhm C, Overbeck L, Wagner C (2003) An Introduction to Credit Risk Modeling. Chapman and Hall, New York
Dietsch M, Petey J (2003) Should SME exposures be treated as retail or corporate exposures? A comparative analysis of probabilities of default and assets correlations in
French and German SMEs. Working Paper, University Robert Schuman of Strasbourg
and University of Lille
Düllmann K, Scheule H (2003) Determinants of the Asset Correlations of German Corporations and Implications for Regulatory Capital. Working Paper, Deutsche Bundesbank
and Department of Statistics, University of Regensburg
Gordy MB (2001) A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules.
Working Paper, Board of Governors of the Federal Reserve System
Hartmann-Wendels T (2002) Basel II – Implications for the Banking Industry (in German).
WISU 4/02: 526-536
Hartmann-Wendels T (2003) Basel II – The New Basel Accord (in German). Economica,
Heidelberg
Lopez JA (2002) The Empirical Relationship between Average Asset Correlation, Firm
Probability of Default and Asset Size. Working Paper, Federal Reserve Bank of San
Francisco
Modigliani F, Miller MH (1958) The cost of capital, corporation finance, and the theory of
investment, American Economic Review 48: 261-297
Ingo Walter 1
1
New York University, USA*
Abstract: There has been substantial public and regulatory attention of late to apparent exploitation of conflicts of interest involving financial services firms based
on financial market imperfections and asymmetric information. This paper proposes a workable taxonomy of conflicts of interest in financial services firms, and
links it to the nature and scope of activities conducted by such firms, including
possible compounding of interest-conflicts in multifunctional client relationships.
It lays out the conditions that either encourage or constrain exploitation of conflicts of interest, focusing in particular on the role of information asymmetries and
market discipline, including the shareholder-impact of litigation and regulatory
initiatives. External regulation and market discipline are viewed as both complements and substitutes – market discipline can leverage the impact of external
regulatory sanctions, while improving its granularity though detailed management
initiatives applied under threat of market discipline. At the same time, market discipline may help obviate the need for some types of external control of conflict of
interest exploitation
JEL G21, G24, G28, L14
Keywords: Conflicts of Interest, Financial Regulation, Financial Services, Banking
Potential conflicts of interest are a fact of life in financial intermediation. Under
perfect competition and in the absence of asymmetric information, exploitation of
conflicts of interest cannot rationally take place. Consequently, the necessary and
sufficient condition for agency costs associated with conflict of interest exploita*
Paper originally presented at a Federal Reserve of Chicago - Bank for International Settlements conference on “Market Discipline: Evidence Across Countries and Industries,”
October 30 - November 1, 2003. Yakov Amihud, Edward Kane, Anthony Saunders, Roy
Smith, William Silber, Lawrence White, Clas Wihlborg, provided valuable comments on
earlier drafts of this paper.
26
Ingo Walter
tion center on market and information imperfections. Arguably, the bigger and
broader the financial intermediaries, the greater the agency problems associated
with conflict-of-interest exploitation. It follows that efforts to address the issue
through improved transparency and market discipline are central to creating viable
solutions to a problem that repeatedly seems to shake public confidence in financial markets.
In recent years, the role of banks, securities firms, insurance companies and asset managers in alleged conflict-of interest-exploitation – involving a broad array
of abusive retail market practices, in acting simultaneously as principals and intermediaries, in facilitating various corporate abuses, and in misusing private information – suggests that the underlying market imperfections are systemic even
in highly developed financial systems. Certainly the prominence of conflict-of- interest problems so soon after the passage of the US Gramm-Leach-Bliley Act of
1999, which removed some of the key structural barriers to conflict exploitation
built into the US regulatory system for some 66 years, seems to have surprised
many observers.
Moreover, recent evidence suggests that the collective decision process in the
management of major financial firms impairs pinpointing responsible individuals,
and that criminal indictment of entire firms runs the risk of adverse systemic effects. Monetary penalties and negotiated agreements neither admitting nor denying
guilt seem to have emerged as the principal external mechanisms to address conflict of interest exploitation. Market discipline operating through the share price
may, under appropriate corporate governance, represent an important additional
line of defense.
Part 1 of this paper proposes a taxonomy of conflicts between the interests of
the financial firm’s owners and managers and those of its clients, including situations where the firm is confronted by conflicts of interest between individual clients or types of clients. Some of these conflicts have been discussed extensively in
the literature,1 while others seem to have surfaced more recently. Mapped onto
this taxonomy is the distinction between conflicts of interest that arise in wholesale and retail domains, characterized by very different degrees of information
asymmetry and fiduciary responsibility, and conflicts that arise on the interface
between the two domains. Part 2 of the paper relates this conflict of interest taxonomy to the strategic profile of financial services firms, linking potential conflicts of interest exploitation to the size and breadth of financial firms and illustrating how those conflicts can be compounded in large multi-line financial
institutions. Part 3 reviews regulatory and market discipline-based constraints on
conflict of interest exploitation, including issues of granularity and immediacy,
and considers linkages between the two types of constraints. Part 4 presents the
conclusions and some implications for public policy.
1
See for example Edwards [1979], Saunders [1985], and Rajan [1996]. A general survey
of the literature on corporate conflicts of interest is presented by Demski [2003].
Conflicts of Interest and Market Discipline in Financial Services Firms
27
There are essentially two types of conflicts of interest confronting firms in the financial services industry under market imperfections.
Type 1 - Conflicts between a firm’s own economic interests and the interests of its clients, usually reflected in the extraction of rents or mispriced transfer of risk.
Type 2 - Conflicts of interest between a firm’s clients, or between types
of clients, which place the firm in a position of favoring one at the expense of another.2
They may arise either in interprofessional activities carried out in wholesale financial markets or in activities involving retail clients. The distinction between these
two market “domains” is important because of the key role of information and
transactions costs, which differ dramatically between the two broad types of market participants. Their vulnerability to conflict-exploitation differs accordingly,
and measures designed to remedy the problem in one domain may be inappropriate in the other. In addition there are what we shall term “transition” conflicts of
interest, which run between the two domains – and whose impact can be particularly troublesome. In the following sections, we enumerate the principal conflicts
of interest encountered in financial services firms arranged by type and by domain
(see Figure 1).
Wholesale Domain
Retail Domain
Type-1 - Firm-client conflicts.
Principal transactions.
Tying.
Misuse of fiduciary role.
Board memberships.
Spinning.
Investor loans.
Self-dealing.
Front-running.
Type-1 - Firm-client conflicts.
Biased client advice.
Involuntary cross-selling.
Churning.
Laddering.
Inappropriate margin lending.
Failure to execute.
Misleading disclosure and reporting.
Privacy-related conflicts.
Type-2 - Inter-client conflicts.
Misuse of private information.
Client interest incompatibility.
Domain-Transition Conflicts
Type-1 - Firm-client conflicts.
Suitability.
Stuffing.
Conflicted research.
Laddering.
Bankruptcy-risk shifting
Fig. 1. A Conflict of Interest Taxonomy
2
Firm behavior that systematically favors corporate clients over retail investors in the
presence of asymmetric information is a prominent example of this type of conflict.
28
Ingo Walter
In wholesale financial markets involving professional transaction counterparties,
corporates and sophisticated institutional investors, the asymmetric information
and competitive conditions necessary for conflicts of interest to be exploited are
arguably of relatively limited importance. Caveat emptor and limited fiduciary obligations rule in a game that all parties fully understand. Nevertheless, several
types of conflicts of interest seem to arise.
Principal transactions. A financial intermediary may be involved as a principal
with a stake in a transaction in which it is also serving as adviser, lender or underwriter, creating an incentive to put its own interest ahead of those of its clients
or trading counterparties. Or the firm may engage in misrepresentation beyond the
ability of even highly capable clients to uncover.3
Tying. A financial intermediary may use its lending power to influence a client
to use its securities or advisory services as well – or the reverse, denying credit to
clients that refuse to use other (more profitable) services.4 Costs are imposed on
the client in the form of higher-priced or lower-quality services in an exercise of
market power. This differs from cross-subsidization, in which a bank (possibly
pressured by clients) engages in lending on concessionary terms in order to be
considered for securities or advisory services. There may be good economic reasons for such cross-selling initiatives, whose costs are borne by the bank’s own
shareholders. The line between tying and cross-selling is often blurred,5 and its effectiveness is debatable. In 2003 the Federal Reserve clarified the concept of tying
and imposed a fine of $3 million on WestLB for violating anti-tying regulations. 6
3
4
5
6
The classic case involves complex Bankers Trust’s derivative transactions with Procter &
Gamble Inc. and Gibson Greetings Inc. in 1995, which triggered major damage to the
Bank’s franchise, key executive changes, and arguably led to the Bank’s takeover by
Deutsche Bank AG in 1999.
A 2002 survey of corporations with more than $1 billion in annual sales found that 56%
of firms that refused to buy fee-based bank services had their credit restricted or lending
terms altered adversely, and 83% of the surveyed CFOs expected adverse consequences
should they refuse to buy non-credit services. [Association for Financial Professionals,
2003].
In the United States the first type of linkage is prohibited under the anti-tying provisions
of the Bank Holding Company Act Amendments of 1970 and by the Federal Deposit Insurance Act, while reducing the price of credit to benefit an investment banking affiliate
violates Section 23B of the Federal Reserve Act. However, the courts have generally
upheld allegations of tying only where abuse of market power has been demonstrated.
Since anti-trust cases in wholesale banking are difficult to make in light of the industry’s
competitive structure, very few allegations of tying have been found to violate the law.
Tying can also have some perverse competitive consequences. [Stefanadis , 2003] There
are no prohibitions on tying bank lending to trust services, deposit balances, etc. and investment banks are in any case exempt from anti-tying constraints and have actively used
senior debt to obtain fee-based business. For a review, see Litan [2003].
Banks may not use their lending power “in a coercive manner” to sell non-lending services, although they may link lending and non-lending services when clients seek such
Conflicts of Interest and Market Discipline in Financial Services Firms
29
Misuse of fiduciary role. Mutual fund managers who are also competing for
pension fund mandates from corporations may be hesitant to vote fiduciary shares
against the management of those companies, to the possible detriment of their own
shareholders. Or the asset management unit of a financial institution may be pressured by a corporate banking client into voting shares in that company for management’s position in a contested corporate action such as a proxy battle.7 The potential gain (or avoidance of loss) in banking business comes at the potential cost
of inferior investment performance for its fiduciary clients, and violates its duty of
loyalty.8
Board interlocks. The presence of bankers on boards of directors of nonfinancial companies may cause various bank functions such as underwriting or equity
research to differ from arms-length practice.9 This displacement may impose costs
7
8
9
“bundling.” Even so, they cannot tie a given loan to a given non-lending product without
allowing the client “meaningful choice.” In the WestLB case, the bank required
participation in debt underwriting as a condition of lending in a series of structured
finance transactions. See “Fed Fines WestLB $3m for ‘Tying’ Loans to Products,”
Financial Times, August 28, 2008.
Example: The 2001-02 Hewlett-Packard Co. effort to acquire Compaq Computer Corp.
was bitterly opposed by the son of one of the co-founders, William R. Hewlett. Hewlett
assembled sufficient backing to force the contest down to the wire. H-P began to lobby
one of the large institutional shareholders – the investment arm of Deutsche Bank AG,
which had opposed the merger – to change its vote. Other Deutsche Bank units, notably
the corporate finance division, favored H-P in the merger. But the Chinese wall between
the dealmakers and the asset managers apparently held firm. Shortly before the proxy
vote, H-P CEO Carly Fiorina was quoted as saying “...we need a definite answer from the
[Deutsche Bank] Vice Chairman, and if it’s the wrong one, we need to swing into action.... See what we can get, but we may have to do something extraordinary to bring
them over the line here.” [Burrows, 2003] Deutsche then set up meetings with both H-P
and Walter Hewlett and, after some heated internal debate, changed its votes in favor of
H-P. The vote-switch, along with a similar story at Northern Trust Co., was investigated
by the SEC and the US Attorney’s Office for the Southern District of New York. The
SEC fined Deutsche Bank $570,000 in July 2003 for not disclosing its conflict of interest
in the matter. See Deborah Solomon and Pui-Wing Tam, “Deutsche Bank Unit is Fined
Over H-P,” Wall Street Journal, August 19, 2003.
In a very different example, prior to WorldCom’s 2002 bankruptcy filing the investment
banking unit of Citigroup was the lead advisor and banker to the firm. Citigroup also
served as the exclusive administrator of the WorldCom executive stock option plan. Executive stock options were generally exercised early in January, and the firm’s fund administrators allegedly passed information on their size and timing to Citigroup’s equity
trading desk, allowing traders to front-run the WorldCom executives’ transactions.
[Morgenson, 2001]
A high-profile case emerged in 2002, when a member of the ATT Board, Citigroup
Chairman and CEO Sanford Weil, allegedly urged the firm’s telecom analyst, Jack
Grubman, to rethink his negative views on the company’s stock – ATT CEO Michael
Armstrong also served on the Citigroup Board. ATT shares were subsequently up-rated
by Grubman, and Citigroup coincidentally was mandated to co-manage a massive issue
of ATT Mobile tracking stock. Grubman down-rated ATT again not long thereafter, and
30
Ingo Walter
on the bank’s shareholders10 or on clients. Although constrained by legal liability
issues, director interlocks can compound other potential sources of conflict, such
as simultaneous lending, advisory and fiduciary relationships.11
Spinning. Securities firms involved in initial public offerings may allocate
shares to officers or directors of client firms on the understanding of obtaining future business, creating a transfer of wealth to those individuals at the expense of
other investors.12
Investor loans. In order to ensure that an underwriting goes well, a bank may
make below-market loans to third-party investors on condition that the proceeds
are used to purchase securities underwritten by its securities unit.
Self-dealing. A multifunctional financial firm may act as trading counterparty
for its own fiduciary clients, as when the firm’s asset management unit sells or
buys securities for a fiduciary client while its affiliated broker-dealer is on the
other side of the trade.13
Front-running. Financial firms may exploit institutional, corporate or other
wholesale clients by executing proprietary trades in advance of client trades that
may move the market.14
Weill himself narrowly averted being named in subsequent regulatory investigations of
the case. See Schiesel & Morgenson [2002].
10
For shareholders these costs in the United States could come through the legal doctrines
of “equitable subordination” and “lender liability” in case of financial distress, which
must be offset against the relationship-related and private-information benefits that board
membership may contribute. This is given as a reason why bankers tend to be present
mainly on the boards of large, stable corporations with low bankruptcy risk. [Krozner and
Strahan, 1999]
11
In 1979 there were 182 separate director interlocks between the five largest banks and the
five largest US nonfinancial corporations. [Saunders, 1985]. Of the ten largest US nonfinancial corporations (by market capitalization) in 2002, 7 had senior bankers or former
bank CEOs on their boards in 2002. [Corporate Library, 2003].
12 In the literature, see Ritter & Welch [2002] and Loughran & Ritter [2002].
13
The 1974 Employee Retirement Income Security Act (ERISA) bars transactions between
asset management units of financial firms that are fiduciaries for defined-benefit pension
plans and affiliated broker-dealers, despite possible costs in terms of best-execution.
[Saunders et al., 2001] Trades between US mutual funds and affiliated securities units of
the same firm must be fully disclosed.
14
Example: In April 2003 investigations by the SEC and the NYSE were aimed at floor
specialists allegedly violating their “negative obligation” or “affirmative obligation” in
assuring firm and orderly markets in listed securities, and instead “trading ahead” of customer orders –long-standing rumors of suspicious specialist behavior. Criminal frontrunning charges had been filed as far back as 1998 against NYSE floor brokers. Included
in the 2003 investigation were specialist affiliates of major financial firms including
FleetBoston Financial Group, Goldman Sachs Group and Bear Stearns Cos. [Kelly and
Craig, 2003] Spear, Leeds & Kellogg had been fined $950,000 in 1998 (prior to its acquisition by Goldman Sachs) by the NASD for intentionally delayed trade reporting and
was again fined $435,000 in 2003 by the American Stock Exchange for trading misconduct during the years 1999-2002.
Conflicts of Interest and Market Discipline in Financial Services Firms
31
All of the foregoing represent exploitation of Type 1 conflicts, which set the
firm’s own interest against those of its clients in wholesale, interprofessional
transactions. Type 2 conflicts dealing with differences in the interests of multiple
wholesale clients center predominantly on two issues:
Misuse of private information. As a lender, a bank may obtain certain private
information about a client. Such proprietary information may be used in ways that
harm the interests of the client. For instance, it may be used by the bank’s investment banking unit in pricing and distributing securities for another client, or in advising another client in a contested acquisition.15
Client interest incompatibility. A financial firm may have a relationship with
two or more clients who are themselves in conflict. For example, a firm may be
asked to represent the bondholders of a distressed company and subsequently be
offered a mandate to represent a prospective acquirer of that corporation. Or two
rival corporate clients may seek to use their leverage to impede each other’s competitive strategies. Or firms may underprice IPOs to the detriment of a corporate
client in order to create gains for institutional investor clients from whom they
hope to obtain future trading business.16
Asymmetric information is intuitively a much more important driver of conflictof-interest exploitation in retail financial services than in interprofessional wholesale financial markets. Retail issues appear to involve Type 1 conflicts, setting the
interests of the financial firm against those of its clients.
15
Examples: In 2003 Dana Corp. sued to prevent UBS from advising on a hostile bid by
ArvinMeritor Corp. on grounds of “breach of duty” and “breach of contract” due to the
bank’s relationship with Dana. UBS argued that its ArvinMeritor relationship predated its
relationship with Dana, which in any case was non-exclusive. See “UBS Sued Over Role
in Bitter Battle,” Financial Times, August 6, 2003. In 1988 Sterling Drug Company was
the object of a hostile takeover bid by F. Hoffmann La Roche of Switzerland, advised at
the time by J.P. Morgan, which also had a banking relationship with Sterling. During the
three-week battle, Sterling blasted Morgan for providing investment banking services to
Roche. CEO John M. Pietruski sent a letter to Morgan Chairman Lewis T. Preston indicating that he was shocked and dismayed by what he considered to be Morgan's unethical
conduct in aiding and abetting a surprise raid on one of its longtime clients. Morgan, he
suggested, was "privy to our most confidential financial information," including shareholder lists, and asked "How many relationships of trust and confidence do you have to
have with a client before you consider not embarking on a course of action that could be
detrimental to [its] best interest?" The Sterling chairman said his company was reviewing
"all our dealings" with Morgan, and intended to "bring the matter to the attention" of
other Morgan clients. See “A Picture Perfect Rescue,” Time, February 1, 1988.
16
In 2003 revelations, some investor clients kicked back a significant part of their IPO
gains to the underwriting firms in the form of excessive commissions on unrelated secondary market trades. [Attorney General of the State of New York, 2003].
32
Ingo Walter
Biased client advice. When financial firms have the power to sell affiliates’
products, managers may fail to dispense "dispassionate" advice to clients based on
a financial stake in promoting high-margin “house” products. Sales incentives
may also encourage promotion of high-margin third-party products, to the ultimate
disadvantage of the customer. The incentive structures that underlie such practices
are rarely transparent to the retail client.17 Even when the firm purports to follow a
so-called “open architecture” approach to best-in-class product selection, such arrangements normally will be confined to suppliers of financial services with
whom it has distribution agreements.
Involuntary cross-selling. Retail clients may be pressured to acquire additional
financial services on unfavorable terms in order to access a particular product,
such as the purchase of credit insurance tied to consumer or mortgage loans. Or
financial firms with discretionary authority over client accounts may substitute
more profitable services such as low-interest deposit accounts for less profitable
services such as higher-interest money market accounts, without explicit instructions from the client.
Churning. A financial firm that is managing assets for retail or private clients
may exploit its agency relationship by engaging in excessive trading, which creates higher costs and may lead to portfolio suboptimization. Commission-based
compensation is the usual cause of churning, which can also arise in institutional
portfolios – average US equity mutual fund turnover rose from 17% annually in
the 1950s to almost 110% in the early 2000s.18
Inappropriate margin lending. Clients may be encouraged to leverage their investment positions through margin loans from the firm, exposing them to potentially unsuitable levels of market risk and high credit costs. Broker incentives tied
to stock margining usually underlie exploitation of this conflict of interest.
Failure to execute. Financial firms may fail to follow client instructions on
market transactions if doing so benefits the firm. Or payments may be delayed to
increase the float.19
Misleading disclosure and reporting. Financial firms may be reluctant to report
unfavorable investment performance to clients if doing so threatens to induce outflows of assets under management. Whereas a certain degree of puffery in asset
17
Following SEC, NASD and Massachusetts securities regulators investigations of its mutual fund sales practices civil charges were settled in September 2003 by Morgan Stanley
in connection with the use of sales contests to sell in-house back-end loaded funds -- in
direct violation of 1999 rules barring such practices. The firm was fined $2 million in the
matter. See “Morgan Stanley to Face Charges Over Context, Wall Street Journal, August 11, 2003; and “Morgan Stanley Fined Over Mutual Funds,” Financial Times, September 17, 2003.
18 John C. Bogle, “Mutual Fund Directors: The Dog That Didn’t Bark,” Vanguard, January
28, 2001.
19
The brokerage firm of E.F. Hutton was criminally indicted for check kiting in 1985, and
subsequently was absorbed by Shearson Lehman Bros. Regulatory enforcement in the
brokerage industry tightly circumscribes failure to execute.
Conflicts of Interest and Market Discipline in Financial Services Firms
33
management performance reviews is normal and expected, there is undoubtedly a
“break-point” where it becomes exploitive if not fraudulent.
Violation of privacy. The complete and efficient use of internal information is
central to the operation of financial services firms, including such functions as
cross-selling and risk assessment. This may impinge on client privacy concerns or
regulatory constraints on misuse of personal information, and raises potentially serious conflict-of-interest issues, which tend to be increasingly serious as the activity-lines of a particular firm become broader.20
Conflicts of interest between the wholesale and retail domains – characterized by
very different information asymmetries – can be either Type 1 or Type 2, and
sometimes both at the same time.
Suitability. A classic domain-transition conflict of interest exists between a
firm’s “promotional role” in raising capital for clients in the financial markets and
its obligation to provide suitable investments for retail clients. Since the bulk of
compensation usually comes from capital-raising side, and given the information
asymmetries that exist, exploiting such conflicts can have adverse consequences
for retail investors.
Stuffing. A financial firm that is acting as an underwriter and is unable to place
the securities in a public offering may seek to ameliorate its exposure to loss by allocating unwanted securities to accounts over which it has discretionary authority.
[Schotland, 1980] This conflict of interest is unlikely to be exploited in the case of
closely-monitored institutional portfolios in the wholesale domain. But in the absence of effective legal and regulatory safeguards, it could be a problem in the
case of discretionary trust accounts in the retail domain.
Conflicted research. Analysts working for multifunctional financial firms wear
several hats and are subject to multiple conflicts of interest. In such firms, the researcher may be required to: (1) Provide unbiased information and interpretation
to investors, both directly and through retail brokers and institutional sales forces;
(2) Assist in raising capital for clients in the securities origination and distribution
process; (3) Help in soliciting and supporting financial and strategic advisory activities centered in corporate finance departments; and (4) Support various management and proprietary functions of the firm. These diverse roles are fundamentally incompatible, and raise intractable agency problems at the level of the indivi20
The 1999 Gramm-Leach-Bliley Act eliminating functional barriers for US financial services firms contains privacy safeguards with respect to sharing personal information with
outside firms, but not intra-firm among banking, brokerage, asset management and insurance affiliates. The Fair Credit Reporting Act of 1970 (as amended in 1996) allows sharing of certain data within multifunctional financial firms. This issue is complicated in the
US by state blue-sky laws versus federal authority, “opt-in” versus “opt-out” alternatives
with respect to client actions, the need to track credit histories, and efforts to combat
identity theft.
34
Ingo Walter
dual analyst, the research function, the business unit, and the financial firm as a
whole.
The extent of this incompatibility has been reflected, for example, in the postIPO performance of recommended stocks [Michaely & Womack, 1999], contradictory internal communications released in connection with regulatory investigations, evidence on researcher compensation, and the underlying economics of the
equity research function in securities firms.21 Other evidence seems to suggest that
efforts to exploit this conflict of interest are generally unsuccessful in terms of investment banking market share and profitability. [Ljungqvist et al., 2003] It is argues that equity research conflicts are among the most intractable. Researchers
cannot serve the interests of buyers and sellers at the same time. No matter how
strong the firewalls, as long as research is not profitable purely on the basis of the
buy-side (e.g., by subscription or pay-per-view), the conflict can only be constrained but never eliminated as long as sell-side functions are carried out by the
same organization. And even if research is purchased from independent organizations, those organizations face the same inherent conflicts if they expect to develop further business commissioned by their financial intermediary clients.
Market-timing and late-trading. Important clients tend to receive better service
than others, in the financial services sector as in most others. When such discrimination materially damages one client segment to benefit another, however, a conflict of interest threshold may be breached and the financial firm’s actions may be
considered unethical or possibly illegal, with potentially serious consequences for
the value of its franchise. Such cases came to light in 2003, involving both criminal fraud charges and civil settlements regarding “late trading” and “market timing” by hedge funds in the shares of mutual funds, shifting returns from ordinary
investors to the hedge funds in exchange for other business solicited by the mutual
fund managers involved.22
21
Firms argue, for instance, that expensive research functions cannot be paid for by attracting investor deal-flow and brokerage commissions, so that corporate finance and other
functions must cover much of the cost. Moreover, researcher compensation levels that
have been far in excess of anything that could possibly be explained by incremental buyside revenues at prevailing, highly competitive commission rates provide inferential evidence for the agency problems involved.
22 One hedge fund reached a $40 million settlement with the New York State Attorney
General, basically a disgorgement of illicit profits from “late trading” or “market timing”
in shares of mutual funds managed by Bank of America, Strong Capital Management,
BancOne, Janus Capital Group, Prudential Securities and Alliance Capital Management
(AXA Financial) – altogether representing 287 mutual funds with $227 billion in assets
under management. Late trading allowed the hedge fund to execute trades at daily closing 4 pm net asset values (NAV) as late as 9 pm, enabling the hedge fund to profit from
news released during the interval. Other fund investors were obliged to trade at the opening NAV on the following day. The practice transferred wealth from ordinary shareholders to the hedge fund in question. The investigation also uncovered “market timing” in
mutual fund shares -- a practice usually prohibited in prospectuses -- involving rapid-fire
trading by hedge funds in shares of international mutual funds across time-zones, for example, a practice that increases mutual fund expenses which have to be borne by all in-
Conflicts of Interest and Market Discipline in Financial Services Firms
35
Laddering. Banks involved in initial public offerings may allocate shares to institutional investors who agree to purchase additional shares in the secondary
market, thereby promoting artificial prices intended to attract additional (usually
retail) buyers who are unaware of these private commitments.23 A related conflict
involves providing bank loans to support the price of a security in the aftermarket.
[Saunders, 1995]
Shifting bankruptcy risk. A bank with credit exposure to a client whose bankruptcy risk has increased, to the private knowledge of the banker, may have an incentive to assist the corporation in issuing bonds or equities to the general public,
with the proceeds used to pay-down the bank debt.24 Such behavior can also serve
to redistribute wealth between different classes of bondholders and equity investors, and represents one of the “classic” conflicts of interest targeted by the 1933
separation of commercial and investment banking in the United States.
vestors, not just the market-timers. In some of the revelations the mutual fund management companies facilitated market-timing trades by revealing to the hedge funds the portfolio weights (allowing them to take short positions) as well as providing direct-access
terminals. Various responsible executives were fired. An employee of Bank of America
was indicted on criminal securities fraud charges while a hedge fund manager pleaded
guilty of criminal violations. See “Ex-Broker Charges in Criminal Fraud Case,” New
York Times, 28 September 2003; and “Fund Probe Reaches Prudential,” Wall Street
Journal, October 2, 2003.
23 In October 2003, for example, JP Morgan settled SEC laddering charges with a $25 million fine in one of several laddering allegations against IPO underwriters, supported by email evidence suggesting a quid pro quo linking IPO allocations to aftermarket purchases
at specific stock price targets. In fact, the case was brought under relatively obscure SEC
Regulation M, Rule 101, a technical violation of securities underwriting procedures, as
opposed to Securities Exchange Act Rule 10b-5, indicating securities fraud.
24
For example, in 1995 Den Danske Bank underwrote a secondary equity issue of the Hafnia Insurance Group, stock which was heavily distributed to retail investors, with the
proceeds being used to pay-down the bank’s loans to Hafnia even as the insurer slid into
bankruptcy. The case came before the Danish courts in a successful investor litigation
supported by the government -- for a discussion, see Smith and Walter [1997]. Historically, there appears to be little evidence that this potential conflict of interest has in fact
been exploited, at least in the United States. During the 1927-29 period investors actually
paid higher prices for bonds underwritten by commercial banks subject to this potential
conflict of interest than from independent securities firms, and such bonds also had lower
default rates. [Puri, 1994] The same finding appeared in the 1990s, when commercial
bank affiliates were permitted to underwrite corporate bonds under Section 20 of the
Glass-Steagall Act prior to its repeal in 1999. [Gande et al., 1997]. The reason may be
that information emanating from the credit relationship allows more accurate pricing, less
costly underwriting and reinforced investor confidence. [Puri 1996, Gande et al., 1999].
36
Ingo Walter
We posit that the broader the activity-range of financial firms in the presence of
imperfect information, (1) the greater the likelihood that the firm will encounter
conflicts of interest, (2) the higher will be the potential agency costs facing clients,
and (3) the more difficult and costly will be the internal and external safeguards
necessary to prevent conflict exploitation. If true, competitive consequences associated with conflict-exploitation can offset the realization of economies of scope
in financial services firms. Scope economies are intended to generate benefits on
the demand side through cross-selling (revenue synergies) and on the supply side
through more efficient use of the firm’s business infrastructure (cost synergies).
Commercial lender
Commercial lender
Loan arranger
Loan arranger
Wholesale
Debt underwriter
Debt underwriter
Equity underwriter
M&A advisor
Equity underwriter
A
M&A advisor
Strategic financial advisor
Strategic financial advisor
Equity analyst
Equity analyst
Debt analyst
Debt analyst
Board member
Board member
Institutional asset manager
Institutional asset manager
Insurer
Insurer
Reinsurer
Reinsurer
Clearance & settlement provider
Clearance & settlement provider
Custodian
Custodian
Transactions processor
Transactions processor
Deposit taker
Deposit taker
Stockbroker
Stockbroker
D
Retail
Life insurer
Private banker
Life insurer
P&C insurer
P&C insurer
B
Private banker
Retail lender
Retail lender
Credit card issuer
Credit card issuer
Mutual fund distributor
Financial adviser
Principal Investor / Trader
C
E
Mutual fund distr.
Financial adviser
PI / T
Fig. 2. Indicative Financial Services Conflict Matrix
As a result of conflict exploitation the firm may initially enjoy revenue and profitability gains at the expense of clients. Subsequent adverse legal, regulatory and
reputational consequences – along with the managerial and operational cost of
complexity – can be considered diseconomies of scope.
The potential for conflict-of-interest exploitation in financial firms can be depicted
in a matrix such as Figure 2. The matrix lists on each axis the main types of retail
and wholesale financial services, as well as infrastructure services such as clear-
Conflicts of Interest and Market Discipline in Financial Services Firms
37
ance, settlement and custody. Cells in the matrix represent potential conflicts of
interest. Some of these conflicts are basically intractable, and remediation may require changes in organizational structure. Others can be managed by appropriate
changes in incentives, functional separation of business lines, or internal compliance initiatives. Still others may not be sufficiently serious to worry about. And in
some cases it is difficult to imagine conflicts of interest arising at all.
For example, in Figure 2 cell D is unlikely to encompass activities that pose serious conflicts of interest. Others cells, such as C, have traditionally been ringfenced using internal compliance systems. Still others such as B and E can be
handled by assuring adequate transparency. But there are some, such as A, which
have created major difficulties in particular circumstances (such as advising on a
hostile takeover when the target is a banking client), and for which easy answers
seem elusive.
The foregoing discussion suggests that conflicts of interest are essentially twodimensional – either between the interests of the firm and those of its client (Type
1), or between clients in conflict with one another (Type 2). They can also be multidimensional, however, spanning a number of different stakeholders and conflicts
at the same time. Figures 3 and 4 provide two examples from the rich array of
corporate scandals that emerged during 2001-2003.
In the Merrill Lynch - Enron case (Figure 3), a broker-dealer was actively involved in structuring and financing an off-balance-sheet special-purpose entity
(LJM2), which conducted energy trades with Enron and whose CEO was simultaneously Enron’s CFO. Merrill was both a lender to and an investor in LJM2 – as
were a number of senior Merrill executives and unaffiliated private and institutional investors advised by the firm. Merrill also structured a repurchase transaction for Enron involving a number of barges in Nigeria. Allegedly, the sole purpose of the highly profitable LJM2 and Nigerian barge transactions was to
misrepresent Enron’s financials to the market.25
At the same time, Merrill performed a range of advisory and underwriting services for Enron, provided equity analyst coverage, and was one of Enron’s principal derivatives trading counterparties. Conflicts of interest in this case involved
Merrill and Enron shareholders, investors in Enron and LJM2 debt, Merrill executives, as well as unaffiliated institutional and private shareholders in the LJM2
limited partnership.
25
See Healy & Palepu [2003]. In a similar case, Enron’s Mahonia Ltd. Special-purpose entity, JP Morgan Chase in 2003 agreed to pay a fine of $25 million to avoid prosecution
on criminal charges in a settlement with the New York District Attorney under the 1921
Martin Act. A criminal indictment would have terminated a broad array of fiduciary relationships and triggered large-scale client defections, possibly endangering the continued
viability of the bank.
38
Ingo Walter
Equity
stake
Nigeria Barge
Repo Contract
Merrill Lynch Executives
Personal
Investments
($16.7 MM)
Principal
Investment
($5 MM)
Lender
($10mm)
LJM2 SPE
Trades
Corporate finance
advisory assignments
$QDO\VW
Private
placements
Private
Investors
Securities underwriter
Energy derivatives counterparty
Fees 1999-2001: Underwriting $20 million; Advisory $18 million; Fund raising ($265 million out of a total of
$387 million for LJM2,
Fig. 3. Multilateral Conflicts of Interest: Merrill Lynch – Enron
Such structures were instrumental in Enron’s 2001 Chapter 11 bankruptcy filing,
with pre-petition on- and off-balance sheet liabilities exceeding $60 billion. [Batson, 2003b] As a consequence, the financial firms that helped design and execute
them (and in some cases actively marketed them to other clients) have been in the
regulatory spotlight -- in July 2003 JP Morgan Chase and Citigroup agreed to pay
$192.5 million and $126.5 million, respectively, in fines and penalties (without
admitting or denying guilt) to settle SEC and Manhattan District Attorney charges
of financial fraud, which in turn encouraged civil suits and risked some of the
banks’ Enron loans with “equitable subordination” in the bankruptcy proceedings.26
26
According to the report of Enron bankruptcy examiner Neal Batson [2003a], Citigroup
alone was involved in over $3.83 billion in Enron financing, including “prepays” and
other questionable transactions. The final report [Batson 2003b] concluded that both
Citigroup and JP Morgan (1) “…had actual knowledge of the wrongful conduct of these
transactions;” (2) Helped structure, promote, fund and implement transactions designed
solely to materially misrepresent Enron’s financials; and (3) Caused significant harm to
other creditors of Enron.
Conflicts of Interest and Market Discipline in Financial Services Firms
g
Proprietary trader
39
p
Exclusive
pension
fund
adviser
Financial and
strategic adviser
Lender
$QDO\VW
Securities underwriter
Fig. 4. Multilateral Conflicts of Interest: Citigroup – Worldcom
In the Citigroup - WorldCom case (Figure 4), a global financial conglomerate was
serving simultaneously as equity analyst supplying assessments of WorldCom to
institutional and (through the firm’s brokers) retail clients while advising WorldCom management on strategic and financial matters – at times participating in
board meetings. As a major telecommunications-sector commercial and investment banking client, WorldCom maintained an active credit relationship with
Citigroup and provided substantial securities underwriting business. As already
noted, Citigroup also served as the exclusive pension fund adviser to WorldCom
and executed significant stock option trades for WorldCom executives as the options vested, while at the same time conducting proprietary trading in WorldCom
stock. Simultaneous conflict of interest vectors in this instance relate to retail investors, institutional fund managers, WorldCom executives, and WorldCom shareholders as well as Citigroup’s own positions in WorldCom lending exposure and
stock trades prior to its $103 billion bankruptcy in 2002.
Such examples suggest that the broader the range of services that a financial
firm provides to an individual client in the market, the greater the possibility that
conflicts of interest will be compounded in any given case, and (arguably) the
more likely they are to damage the market value of the financial firm’s business
franchise once they come to light.
40
Ingo Walter
From a public policy perspective, efforts to address exploitation of conflicts of interest in the financial services sector should logically focus on improving market
efficiency and transparency. Compelling arguments have been made that regulation can materially improve the efficiency of financial systems. The greater the information asymmetries and transaction-cost inefficiencies that exist (inefficiencies
that are at the core of the conflict of interest issue), the greater is the potential gain
from regulation that addresses these inefficiencies. [Kane, 1987] In the United
States, periodic efforts in this direction go back almost a century, often in response
to perceived market abuses. A recent example is Regulation FD (“fair disclosure”)
of 1999, governing the flow of corporate information to the financial markets,
with a clear potential for ameliorating conflicts of interest.
Nonetheless, the history of US and other relatively well-developed financial
markets chronicles conflict of interest exploitation involving all of the majorbracket US securities firms, four of the top-six UK merchant banks (prior to their
acquisition by larger financial firms), all of the major Japanese securities houses,
as well as various commercial banks, asset managers, insurance companies and financial conglomerates.27 So what is left of market imperfections and information
asymmetries, under intense competition and regulatory oversight, appears to allow
plenty of scope for continued conflict exploitation on the part of financial intermediaries – suggesting a continuing role for external control through firm-specific
regulation and market discipline and internal control through improved corporate
governance, incentive structures, and compliance initiatives.
!
The regulatory overlay of the financial services sector can be conveniently depicted in terms such as Figure 5. The right-hand side of the diagram identifies the
classic policy tradeoffs that confront the design and implementation of a properly
structured financial system. On the one hand, financial regulation must strive to
achieve maximum static and dynamic efficiency. This implies low levels regulation consistent with a competitive market structure, creating persistent pressure on
financial intermediaries to achieve operating cost and revenue efficiencies and to
innovate. On the other hand, regulation must safeguard the stability of, and confidence in, the financial system and its institutions. Safety-net design is beset with
difficulties such as moral hazard and adverse selection, and can become especially
problematic when different types of financial services shade into each other, when
on- and off-balance sheet activities are involved, when some of the regulated firms
are multifunctional financial conglomerates, and when business is conducted
across national and functional regulatory domains and may exploit “fault lines”
between them.
27
For a chronology, see Smith & Walter [1997].
Conflicts of Interest and Market Discipline in Financial Services Firms
41
Static and
Dynamic
Efficiency
Objectives
Financial
Services
Firm
Supervisory
Applications
Regulatory
Techniques
Systemic
Stability and
Market Conduct
Objectives
Fig. 5. Regulatory Tradeoffs, Techniques and Control
Regulators continuously face the possibility that “inadequate” regulation will result in costly failures, on the one hand, and on the other hand the possibility that
“overregulation” will create opportunity costs in the form of financial efficiencies
not achieved, which by definition cannot be measured. Since any improvements in
financial stability can only be calibrated in terms of damage that did not occur and
external costs that were successfully avoided, the argumentation surrounding financial regulation is invariably based on “what if” hypotheticals. In effect, regulators face the daunting task of balancing the unmeasurable against the unknowable.
The principal tools depicted in Figure 5 that regulators have at their disposal include (1) “Fitness and properness” criteria, under which a financial institution are
chartered and allowed to operate, (2) Frequency and speed of financial reporting,
(3) Line-of-business regulation as to what types activities financial institutions
may engage in, (4) Adequacy of capital and liquidity, (5) limits on various types
of exposures, and (6) Rules governing valuation of assets and liabilities. But regulatory initiatives may create financial market distortions of their own, which can
become problematic when financial products and processes evolve rapidly and the
regulator can easily get one or two steps behind.
A third issue depicted in Figure 5 involves the regulatory machinery itself, including self-regulatory organizations (SROs) and public oversight by regulators
with civil and criminal enforcement powers. The proper role of SROs is often debated, especially when there are problems in financial markets.28 “Regulatory cap28
Examples: (1) In 1994 the UK Investment Management Regulatory Organisation
(IMRO), which regulates pension funds, failed to catch the disappearance of pension assets from Robert Maxwell’s Mirror Group Newspapers PLC. The UK Personal Investment Authority (PIA) for years failed to act against deceptive insurance sales practices.
(2) In 1996 NASDAQ, one of the key US markets regulated by the National Association
of Security Dealers (NASD), and some of its member firms were assessed heavy monetary penalties in connection with rigging OTC equity markets, eventually leading to im-
42
Ingo Walter
ture” is a clear problem with SROs, suggesting greater reliance on publicoversight for financial regulation. But this too is subject to regulatory capture,
since virtually any regulatory initiative is likely to confront powerful vested interests attempting to bend the rules in their favor [Kane, 1987; White, 1991].
Further tradeoffs are encountered between regulatory and supervisory alternatives. Some regulatory techniques (for example, capital adequacy rules) are fairly
easy to supervise but full of distortive potential given to their broad-gauge nature.
Others (for example, fitness and properness criteria) may be cost-effective but difficult to supervise. Some supervisory techniques involve higher compliance costs
than others. Regulators must deal with these tradeoffs under conditions of ongoing
market and industry change, blurred institutional and activity demarcations, and
functional as well as international regulatory fault-lines.
Within this setting, regulatory control of conflicts of interest tends to be applied
through both SROs and public agencies, and are generally anchored in banking,
insurance, securities, and consumer protection legislation that is supposed to govern market practices. Its failure to prevent serious exploitation of conflicts of interest came into particularly sharp relief in the US during the early 2000s with serial revelations of misconduct by financial intermediaries. Most of the regulatory
initiatives in these cases were taken not by the responsible SROs or by the national
regulators, but by the New York State Attorney General under the Martin Act, a
1921 state law that was aimed at securities fraud and that survived all subsequent
banking and securities legislation and was bolstered in 1955 with criminal penalties.29 The de facto ceding of enforcement actions by the SROs and the SEC to a
state prosecutor (later joined by several others) focused attention on gaps in external regulation and led to a burst of activity by the SEC, the NYSE, the NASD, and
the Congress, including the 2002 Sarbanes-Oxley Act and the 2003 “Global Settlement” with 12 major banks and securities firms.
Both the Martin Act prosecutions and the Sarbanes-Oxley legislation appear
flawed. The “Global Settlement” allowed financial firms to settle without determination of guilt or innocence, thereby creating no new legal ground.30 The Sar-
portant changes in regulatory and market practices. (3) In 2001 Moody’s (which, along
with other rating agencies, is increasingly a part of the regulatory infrastructure) pleaded
guilty to criminal charges of obstruction of justice in connection with an SEC investigation of the firm’s unsolicited ratings practices. (4) And in 2003 the New York Stock Exchange faced a series of governance issues including the composition of its board, remuneration of its CEO, and alleged conflict of interest exploitation by specialists central to
its trading system.
29 The Act contains extremely broad “fraud” provisions and conveys unusually wide discovery and subpoena power, but had been largely dormant until the 2001-02 revelations
of the excesses in market practices and corporate governance.
30 The SEC, supported by lobbyists for financial intermediaries, was quick to promote
legislation to strip state securities regulators and prosecutors of the authority to pursue future malfeasance or impose rules on the capital markets, specifically including conflict of
interest requirements -- the Securities Fraud Deterrence and Investor Restitution Act of
2003. The SEC clearly felt the need to regain the initiative in regulation of national fi-
Conflicts of Interest and Market Discipline in Financial Services Firms
43
banes-Oxley Act was drafted in haste and quickly triggered unintended consequences, including international regulatory repercussions and high compliance
costs imposed on financial intermediaries and their clients.
"
If external regulatory constraints on conflict of interest exploitation are often politicized and difficult to devise, calibrate and bring to bear on specific problems
without collateral damage, what is the alternative? As a general matter, it can be
argued that regulatory constraints and litigation are relatively blunt instruments in
dealing with exploitation of conflicts of interest in financial firms, conflicts that
are often extremely granular and sometimes involve conduct that is “inappropriate” or “unethical” rather than “illegal.” So the impact of conflict exploitation on
the franchise value of a financial firm may provide a more consistent and durable
basis for firm-specific, internal defenses against exploitation of conflicts of interest than those mandated by the regulators or implemented through the firm’s compliance infrastructure by legal staff reporting to senior management. Here we shall
argue that constraints on conflicts of interest that are rooted in market discipline
can be substantially more cost-effective and surgical than constraints based on external regulation. Given the persistence of market inefficiencies and information
asymmetries they can, in combination, have powerful deterrent effects on conflict
of interest exploitation.
First, exploitation of conflicts of interest, whether or not they violate legal and
regulatory constraints, can have a powerful reputation effect, leading to revenue
erosion as clients defect to competitors. In the case of Bankers Trust’s 1995 exploitation of conflicts of interest in derivatives trading with Procter & Gamble Inc.
and Gibson Greetings Inc., revenue losses from client defections dwarfed the $300
million in customer restitution the firm was forced to pay. It left the firm mortally
wounded, subsequently acquired by Deutsche Bank AG in 1999. In the case of
conflict-of-interest exploitation at Arthur Andersen in 2002, reputation losses and
client defections virtually assured the liquidation of the firm well before its indictment and conviction on criminal charges
Second, on the cost side, increased regulatory pressure or market-impacts of
conflict exploitation force the reinforcement of an acceptable compliance
infrastructure and other managerial safeguards that may reduce operating
efficiency, including organizational changes and separation of functions that may
impair realization of revenue economies of scope. Compliance itself is an
expensive business in terms of direct outlays as well as separation of business
units by “Chinese walls” or into distinct legal entities, which can raise costs. Also
on the cost side is the impact of regulatory penalties in civil and criminal litigation
nancial markets, and followed with a series of draft proposals that would simplify conflict of interest rules.
44
Ingo Walter
the impact of regulatory penalties in civil and criminal litigation and class action
settlements.31
J.P. Morgan
Securities, Inc.
& Subsidiaries
Morgan Guaranty
Trust Company
of New York
1. Investor and
General
Partner
2. Fund Manager
Corsair Fund, L.P.
4. Corp. Finance
Advisory Assignments
3. Securities
Underwriter
($500 MM)
5. Equity
Shareholding
($162 MM = 7.9%)
Banesto
Private
Investors
6. Credit
Relationship
7. Board
Representation
Financial holdings
Nonfinancial holdings
Fig. 6. Measuring the Stock Price-Effects of Conflicts of Interest
Third, the likelihood of exploitation of conflicts of interest and its consequences
clearly has to be incorporated by the market in the valuation of financial firms in
the marketplace. A high degree of sensitivity to conflict exploitation and its revenue and cost impacts should be associated with greater earnings volatility and reduced share price valuation, all else equal.
How these factors may come together to damage a firm’s market value can be
illustrated by a 1993 case, depicted in Figure 6, in which J.P. Morgan simultaneously acted as commercial banker, investment banker, and adviser to Banco
Español de Crédito (Banesto) in Spain, as well as serving as an equity investor and
fund manager for co-investors in a limited partnership (the Corsair Fund, L.P.)
holding shares in the firm. Additionally, Morgan’s Vice Chairman served on
Banesto’s Supervisory Board. The potential conflicts of interest imbedded in the
31
Probably the leading example is the aforementioned $1.4 billion “global settlement” between the regulators and major banks and securities firms involving various allegations
of conflicts of interest, as well as smaller amounts of $100 million each that had previously been assessed against Merrill Lynch and CSFB. In turn, financial firms provisioned
well over $5 billion to cover hundreds of civil cases filed against them alleging conflicts
of interest in financial market practices and aiding and abetting financial fraud.
Conflicts of Interest and Market Discipline in Financial Services Firms
45
complex relationship may have affected the Morgan share price immediately after
the Bank of Spain, concerned about a possible Banesto collapse, announced a
takeover of the bank on December 28, 1993.32 Abnormal returns attributable to the
Banesto event for JP Morgan shareholders represented a cumulative loss of about
10% of the market value of equity at the time, a drop in JP Morgan market capitalization of approximately $1.5 billion as against a maximum direct after-tax loss
of about $10 million. [DeLong & Walter 1993]. This is consistent with the findings of an earlier event study by Smith [1992] of the Salomon Brothers Treasury
bond auction scandal in 1991, which was associated with a one-third share price
drop and contributed to Salomon’s ultimate absorption by Travelers, Inc.33
! !
One can argue that regulation-based and market-based external controls, through
the corporate governance process, create the basis for internal controls which can
be either prohibitive (as reflected in Chinese walls and compliance systems, for
example) or affirmative, involving the behavioral “tone” and incentives set by senior management together with reliance on the loyalty and professional conduct of
employees. The more complex the financial services organization – perhaps most
dramatically in the case of massive, global financial services conglomerates where
comprehensive regulatory insight is implausible – the greater the challenge of sensible conflict-of-interest regulation, suggesting greater reliance on the role of market discipline. The logic runs as follows: First, market discipline can leverage the
effectiveness of regulatory actions.34 When they are announced -- and especially
when they are amplified by aggressive investigative reporting in independent media -- regulatory actions can have a serious adverse effect on a financial firm’s
share price as well as its debt rating. In turn, this affects its cost of capital, its ability to make strategic acquisitions, its vulnerability to takeover, and management
compensation. Such effects simply reflect the market’s response to the prospective
32
Banesto’s CEO, Mario Condé was later convicted on charges of financial fraud and imprisoned.
33 More recent examples that are less amenable to event study methodology are precipitous
declines during 2002 in Merrill Lynch and Citigroup share prices relative to cohorts immediately following release of new information regarding exploitation of analyst conflicts of interest.
34 For example, following the 2003 Global Settlement and its widespread coverage in the
media, the proportion of “sell” recommendations rose abruptly, in the US from less than
1% in mid-2000 to about 11% in mid-2003. In Europe the percentage of “sell” recommendations rose from 12% to 24% in Germany, from 13% to 21% in France, and from
6% to 16% in the UK over the same period. See “Your Stock Stinks, But We Want the
Dean,” Wall Street Journal, July 24, 2003. On the other hand, there was evidence that
several of these same firms continued to engage in prohibited sales practices involving
analysts outside the United States. See “Wall Street Accord Isn’t Global,” Wall Street
Journal, June 6, 2003.
46
Ingo Walter
impact of regulatory actions on revenues, costs (including derivative civil litigation) and exposure to risk.35
Assuming appropriate corporate governance, boards and managements should
be sensitive both to regulatory constraints and prospective market-reactions with
regard to exploitation of conflicts of interest. That is, they should be aware that
violations of regulatory constraints designed to limit conflict-of-interest exploitation may be greatly amplified by market reactions – in the extreme including absorption by other firms, breakup, or bankruptcy.36 This awareness ought to be reflected in compensation arrangements as well as organizational design.
Second, even in the absence of explicit regulatory constraints, actions that are
widely considered to be “unfair” or “unethical” or otherwise violate accepted behavioral norms will tend to trigger market discipline. In a competitive context, this
will affect firm valuation through revenue and risk dimensions in particular.
Avoiding conflict of interest exploitation is likely to reinforce the value of the firm
as a going concern and, with properly structured incentives, management’s own
compensation. In a firm well known for tying managers’ remuneration closely to
the share price, Citigroup CEO Sanford Weill noted in a message to employees
“There are industry practices that we should all be concerned about, and although
we have found nothing illegal, looking back, we can see that certain of our activi-
35
Civil litigation can be an important component of market discipline and its reinforcement
of regulatory sanctions. This was evident in the link between the release of the 2003
Global Settlement “findings of fact,” the prospects of massive civil claims against the financial intermediaries and their corporate clients, and a $1 billion restitution offer negotiated with some 300 companies issuing IPOs in the late 1990s – possibly to be recouped
from subsequent civil settlements with the underwriters. Indeed, some of the entrepreneurial characteristics of US tort litigation can be regarded as an important aspect of market
discipline relating to conflicts of interest. See “$1 Billion Offered to Settle Suit on IPOs,”
The New York Times, June 27, 2003. However, by no means all civil suits are justified, as
seen in a 2003 stinging rebuke to plaintiffs in a class action filed against Merrill Lynch
by Judge Milton Pollack. Reuters, 2 July 2003.
36 A prominent example of weak internal controls in a firm removed from market discipline
is the former Prudential Insurance Company of America – since demutualized and renamed Prudential Financial. The firm’s securities affiliate, Prudential Securities, was
fined $371 million (including $330 million in restitution) in 1993 for mis-selling limited
partnerships. In 1996 Prudential was fined $65 million by state regulators for mis-selling
life insurance policies, followed in 1997 by a $2.6 billion class action settlement on behalf of 640,000 clients. The firm was fined $20 million in 1999 by NASD for mis-selling
variable life insurance, and censured (and fined $10,000) in 2001 by NASD for failing to
enforce written policies regarding the sale of annuities. New probes on variable annuity
sales practices were launched in 2003 and notified to NASD and state insurance commissioners. It can be argued that persistently misaligned internal incentives would have been
a less serious problem if Prudential had been subject to market discipline all along. See
Smith and Walter [2000] and “NASD Investigates Prudential,”Wall Street Journal, May
30, 2003.
Conflicts of Interest and Market Discipline in Financial Services Firms
47
ties do not reflect the way we believe business should be done. That should never
be the case, and I’m sorry for that.”37
Third, since they tend to be more granular and provide constant reinforcement in
metrics that managers can understand (market share, profitability, the stock price)
market discipline constraints can reach the more opaque areas of conflict-ofinterest exploitation, and deal with those issues as they occur in real time, which
external regulation normally cannot do.
Fourth, since external regulation bearing on conflicts of interest tends to be
linked to information asymmetries and transaction costs, it should logically differentiate between the wholesale and retail domains, discussed earlier. Often this is
not possible, resulting in overregulation in some areas and underregulation in others. Market discipline-based constraints can help alleviate this problem by permitting lower overall levels of regulation and bridging fault-lines between wholesale
and retain financial market segments. Few things are as reputation-sensitive as
hawking the “risk-free” rump-ends of structured asset-backed securities deals -so-called “toxic waste” -- to retirees in trailer homes trying to make ends meet.
Moreover, just as market discipline can reinforce the effectiveness of regulation, it
can also serve as a precursor of sensible regulatory change.
Finally, market structure and competition across strategic groups can help reinforce the effectiveness of market discipline. For example, inside information accessible to a bank as lender to a target firm would almost certainly preclude its affiliated investment banking unit from acting as an adviser to a potential acquirer.
An entrepreneur may not want his or her private banking affairs handled by a bank
that also controls his or her business financing. A broker may be encouraged by a
firm’s compensation arrangements to sell in-house mutual funds or externallymanaged funds with high fees under “revenue-sharing” arrangements, as opposed
to funds that would better suit the client’s needs.38 Market discipline that helps
avoid exploitation of such conflicts may be weak if most of the competition is
coming from a monoculture of similarly-structured firms which face precisely the
same issues. But if the playing field is also populated by a mixed bad of aggressive insurance companies, commercial banks, thrifts, broker-dealers, fund managers, and other “monoline” specialists, market discipline may be much more effective – assuming competitors can break through the fog of asymmetric information.
Based on a taxonomy of potential conflicts of interest in financial services firms,
how these conflicts relate to their strategic positioning, and the conditions that underlie their exploitation, we conclude that market discipline -- though the reputa37
38
As quoted in The New York Times, September 6, 2002.
Such conflicts of interest are particularly problematic in the mutual funds industry due to
limited or non-disclosure of fees, incentives and other compensation arrangements, revenue-sharing agreements, trading costs and soft-dollar commissions to brokers.
48
Ingo Walter
tion-effects on the franchise value of financial intermediaries -- can be a powerful
complement to external regulation.
Firms can benefit from conflict-exploitation in the short term, to the extent that
business volumes and/or margins are increased as a result. Conflict management is
a costly and complicated (non-revenue-generating) business, and various types of
walls between business units and functions promote inefficient use of proprietary
information. On the other hand, reputation losses associated with conflict-exploitation can cause serious damage, as demonstrated by repeated “accidents” and
contribute to weaker market valuations among the most exposed financial services
firms. The fact that such events repeat with some regularity suggests that market
discipline is no panacea. The reasons have to do with lapses in corporate governance among financial services firms.
In the end, management of financial services firms must be convinced that a
good defense is as important as a good offence in determining sustainable competitive performance. This is something that is extraordinarily difficult to put into
practice in a highly competitive environment, and seems to require an unusual degree of senior management leadership and commitment. [Smith & Walter, 1997]
Internally, there have to be mechanisms that reinforce the loyalty and professional
conduct of employees. Externally, there has to be careful and sustained attention
to reputation and competition as disciplinary mechanisms.
Market discipline is an often-overlooked sanction. It greatly reinforces regulatory sanctions, particularly when too-big-to-fail considerations or criminal prosecution are involved. In turn, it relies on a favorable legal framework, including
controversial elements such as the Martin Act and class action litigation. Alongside measures to improve transparency and market efficiency, an important public
policy objective is to make market discipline more effective, notably through better corporate governance and internal reward systems more closely aligned to the
interests of shareholders. Still, “accidents” will continue to happen, sometimes repeatedly and sometimes repeatedly within the same firm. There is no panacea.
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4KUM/CPCIGOGPVCPF8CNWG%TGCVKQPKP$CPMU
Gerhard Schröck1 and Manfred Steiner2
1
Mercer Oliver Wyman, Bleichstrasse 1, D – 60313 Frankfurt am Main, Germany,
e-mail: gschroeck@mow.com.*
2
University of Augsburg, Universitätsstrasse 16, D – 86159 Augsburg, Germany,
e-mail: manfred.steiner@wiwi.uni-augsburg.de
Abstract: Previous academic work has focused on why risk management at the
corporate level is necessary and desirable from a value creation perspective
rather than on how much or what sort of risk management is optimal for a particular firm/bank. Therefore, we develop in this chapter the foundations for a
normative theory of risk management in banks. We first explain the need for a
consistent framework for risk management at the corporate level in banks. We
then move on to defining and examining RAROC (Risk-Adjusted Return on Capital), a capital budgeting rule currently widely used in the banking industry. We
then introduce new approaches to capital budgeting and deduct implications from
applying these new approaches in banks.
Keywords: Banks, Risk Management, Value Creation, Valuation, Capital Budgeting, Capital Structure
*
All views presented in this chapter represent the author’s view and do not necessarily reflect those of Mercer Oliver Wyman. The author would like to thank Mercer Oliver
Wyman’s Finance & Risk Practice and especially Jens Kuttig for challenging discussions. Parts of this chapter are adapted from Schröck, Risk Management and Value Creation in Financial Institutions, ©2002 John Wiley & Sons; this material is used by permission and the author would also like to thank John Wiley & Sons, Inc.
54
Gerhard Schröck and Manfred Steiner
+PVTQFWEVKQP
Banks are – by their very nature – in the risk business and we can observe as an
empirical fact that they do conduct risk management. Both facts constitute a positive theory for risk management in banks, but the central role of risk in the banking business is merely a necessary condition for the management of risks. Only the
fact that risk management can also create value makes it a sufficient condition, assuming that value maximization is the ultimate objective function in banks.1
However, there is very little known from a theoretical point of view on where
and how a bank can create value by managing risks. Also, there is anecdotal but –
due to data limitations – only weak or inconclusive empirical evidence for a link
between risk management and value creation (Schröck 2002). Therefore, the focus
of this chapter is to examine risk management at the corporate level in the light of
the sufficient condition, i.e., as to whether and how risk management can be used
as a device to increase the value of banks. This chapter aims at exploring whether
there is also a normative theory for risk management that offers (more) detailed
instructions of how to achieve value creation, and how this compares to what is already done in practice.
We do so by first explaining the need for a consistent framework for risk management at the corporate level in banks (Section 2). We then move on to defining
and examining RAROC (Risk-Adjusted Return on Capital) in Section 3, a capital
budgeting rule2 currently widely used in the banking industry. Section 4 introduces new approaches to capital budgeting, Section 5 deducts implications from
applying these new approaches and Section 6 investigates that these can form the
foundation for a normative theory of risk management in banks. Section 7 offers
some concluding remarks.
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$CPMUCVVJG%QTRQTCVG.GXGN
When examining whether financial theory can offer rationales for conducting risk
management at the bank level in order to enhance value, we find that the neoclassical finance theory offers no foundation for such an approach. The neoclassical theory with its strict assumptions has laid the foundation for the development of extremely useful theories like the Capital Asset Pricing Model (CAPM
– Sharpe 1964, Lintner 1965, and Mossin 1966) and the Modigliani and Miller
(M&M – Modigliani and Miller 1958) propositions. However, in such a world risk
1
2
Despite other stakeholders’ interests being both deviating and strong (e.g., regulators and
depositors typically want to have ensured a bank’s safety and survival rather than its
value being maximized), the academic literature agrees with this conclusion (Schröck
2002).
Even though RAROC is also often applied as a performance metric, the focus of this
chapter is its application as a capital budgeting tool.
Risk Management and Value Creation in Banks
55
management at the bank level is irrelevant, unnecessary and can even be harmful
with respect to the corporate objective of value creation, because investors could
replicate or reverse all of a bank’s risk management actions at no (extra) cost. Incurring (higher) costs for conducting risk management at the bank level would be
therefore a value-destroying proposition (Damodaran 1997). Additionally, in the
neo-classical world capital budgeting, capital structure, and risk management decisions can be separated and the application of the traditional Discounted Cash
Flow (DCF)-approach is justified as the capital budgeting rule, since only the systematic risk to a broad market portfolio counts.
Under the strict assumptions of the neo-classical theory there would be also no
reason for banks to exist (Mason 1995). Exploring the neo-institutional finance
theory, that relaxes many of the unrealistic assumptions of the neo-classical world,
we can find that various market imperfections can build the rationale for the existence of banks and for conducting risk management. Here, managing risk at the
corporate level can increase the value of a bank, because it can reduce (the present
value of) agency costs of equity3 and debt4 as well as that of transaction costs.
The central component of these transaction costs is the likelihood of default and
the (direct and indirect) costs associated with financial distress situations. Avoiding “lower tail outcomes” that incur these costs via risk management seems to
provide the most profound value gains. This is especially true for banks.5 The central role of (relative) creditworthiness in the provision of financial services (Mason
1995) and the potential loss of their franchise value lead to high default costs and
to high costs for (unexpected) external financing (which is specifically costly for
banks in situations when it is needed most). Both of these costs cause banks to behave as if they were risk-averse6, and – since these costs are higher for banks than
for other firms – to conduct relatively more risk management.
However, neither the neo-classical nor the neo-institutional theory offer a general framework that can be used to guide risk management strategies and that
gives detailed instructions of how to apply these concepts in practice (= normative
theory of risk management). This is due to the fact that previous academic work
has focused on why risk management at the corporate level is necessary and desirable from a value creation perspective, rather than on how much or what sort of
risk management is optimal for a particular firm/bank (Froot et al. 1993).
Rather than simply demonstrating that there is a role for risk management, a
well-designed and coherent risk management strategy – both in terms of the
amount of risk management and the instruments used (see box below)7 – can en-
3
4
5
6
7
This allows banks to increase their leverage without increasing the probability of default.
Risk management is used in this context as equity substitute.
This is implied by both the results of various studies (see e.g. James 1991) and the reductions in market values during e.g. the Russian crisis in 1998 that exceeded the credit exposure of various banks by a multiple.
Note that banks are not risk-averse by themselves.
Even the best risk management programs will incur losses in some trades. However,
more severe is the opportunity cost for using the wrong instrument and taking positions
56
Gerhard Schröck and Manfred Steiner
able a bank to maximize its value by providing specific answers on the logically
prior questions (Froot et al. 1993):
• Which risks should be hedged and which risks should be left unhedged?
• And to what degree (partially or fully)?
• What kind of instruments and trading strategies are appropriate?
There are various ways to conduct risk management in banks. The figure below provides
a general overview and indicates that there are two broad categories that need to be distinguished when discussing the various options: Firstly, the bank needs to determine which
approach or set of actions it wants to apply when managing risks and secondly, the bank
then has a set of instruments available to actually manage these risks.
Ways
Waysto
toConduct
ConductRisk
RiskManagement
Management
Approaches/Actions
Approaches/Actions
Instruments
Instruments
Hedge/Sell
Eliminate/Avoid
Diversify
Transfer
Insure
Set Policy
Absorb/Manage
Hold Capital
Overview of Ways to Conduct Risk Management
We will now discuss the three approaches or set of actions and within them the various
instruments (adapted from Mason 1995 and Allen and Santomero 1996) that are available
to banks and how they can be applied.
1. Eliminate/Avoid:
The bank can decide to eliminate certain risks that are not consistent with its desired financial characteristics or are not essential to a financial asset created.8 Any element of the
(systematic) risk that is not required or desired can be either shed by selling it in the spot
market or hedged by using derivative instruments like futures, forwards, or swaps (including securitizations). Moreover, the bank can use portfolio diversification9 in order to elimi-
8
9
in derivatives that do not fit well with the corporate strategy, meaning that a (coherent)
risk management strategy needs to be integrated with the overall corporate strategy.
Banks do bundle and unbundle risks to create new assets (Merton 1989).
Note that diversification is something shareholders and other stakeholders can do on their
own, but potentially only at a higher cost than the bank can – see below.
Risk Management and Value Creation in Banks
57
nate specific risk.10 Additionally, it can decide to buy insurance in the form of options (Mason (1995) classifies options as insurance) or (actuarial) insurance e.g. for event risks. Furthermore, the bank can choose to avoid certain risk types up-front by setting certain business practices/policies (e.g. underwriting standards, due diligence procedures, process
control) to reduce the chances of certain losses and/or to eliminate certain risks ex ante.
2. Transfer:
Contrary to the bank’s decision to (simply) avoid some risks, the transfer of risks to other
market participants should be decided on the basis of whether or not the bank has a competitive advantage in a specific (risk) segment and whether or not it can achieve the fair
market value for it. The alternative to transferring risks is to keep (absorb) them internally,
which will be discussed in the subsequent point.
The transfer of risk eliminates or (substantially) reduces risk by selling (or buying) financial claims (this includes both selling in the spot market and hedging via derivative instruments as well as buying insurance – as described above). Note that diversification is no
means of transferring risks to other market participants for obvious reasons. As long as the
financial risks of the asset (created) are well understood in the market, they can be sold easily to the open market at the fair market value. If the bank has no comparative advantage in
managing a specific kind of risk, there is no reason to absorb and/or manage such a risk,
because – by definition – for these risks no value can be created. Therefore, the bank should
transfer these risks.
3. Absorb/Manage:
Some risks must or should be absorbed and managed at the bank level, because they have
one or more of the following characteristics (Allen and Santomero 1996):
a) They cannot be traded or hedged easily (i.e. the costs of doing so would exceed the benefits)
b) They have a complex, illiquid or proprietary structure that is difficult, expensive, or impossible to reveal to others (this is due to disclosure or competitive advantage reasons)
c) They are subject to moral hazard11
d) They are a business necessity. Some risks play a central role in the bank’s business purpose and should therefore not be eliminated or transferred.12
In all of the four circumstances (a) – d)) the bank needs to actively manage these risks by
using one of the following three instruments:
(1) Diversification: The bank is supposed to have superior skills (competitive advantages),
because it can provide diversification more efficiently/at a lower cost than individual investors could do on their own. This might be the case in illiquid areas where
shareholders cannot hedge on their own. We know that banks care about the internal diversification of their portfolios and especially the management of their credit
portfolio, because the performance of a credit portfolio is not (only) determined by
exogenous but rather endogenous factors like superior ex ante screening capabilities
and ex post monitoring skills (Winton 2000). Diversification, typically, reduces the
10
Usually, risk elimination is incomplete, because some portion of the systematic risk and
that portion of the specific risk, that is an integral part of the product’s unique business
purpose, remain (Allen and Santomero 1996).
11 For instance, even though insurance is provided for a certain risk type, other stakeholders
may require risk management as a part of standard operating procedures to make sure
that management does not misbehave.
12 For instance, if the bank offers an index fund, it should – by definition of the product –
keep exactly the risks that are contained in the index and should not try to manage e.g.
the systematic part of the constituent stocks (Allen and Santomero 1996).
58
Gerhard Schröck and Manfred Steiner
frequency of both worst-case and best-case outcomes, which generally reduces the
bank’s probability of failure.13
(2) Internal insurance: The bank is supposed to have superior risk pooling skills (Mason
1995) for some risks, i.e. it is cheaper for the bank to hold a pool of risks internally
than to buy external insurance.
(3) Holding capital: For all other risks that cannot be diversified away or insured internally and which the bank decides to absorb, it has to make sure that it holds a sufficient amount of capital14 in order to assure its probability of default is kept at a sufficiently low level (note that equity finance is costly).
The decision to absorb risks internally should always be based on competitive advantages
vis-à-vis the market that reimburse the bank more than the associated costs, i.e. when value
is created.15 A bank should have an appropriate metric to identify uneconomic risk taking
that allows it to decide when risk absorption is not the right choice and to decide when it is
better to transfer risk to the market or when to avoid it altogether (Allen and Santomero
1996). The complete hedging of all risks should almost never be an option, or as Culp and
Miller (1995) put it “most value-maximizing firms do not hedge”.
We have presented here in a general way that there are many more ways to conduct risk
management than just hedging. The decision as to which approach is most appropriate and
which instrument should be chosen is discussed in more detail in Sections 5 and 6 below.
Adapted from Schröck (© 2002 John Wiley & Sons) – This material is used by permission of John Wiley & Sons, Inc.
In the neo-institutional world with costly external finance and where lower tail
outcomes matter (as they are costly due to financial distress costs), total risk (including specific risk) is the relevant risk measure. Hence, not only systematic
risks, but also unmarketable (i.e. non-hedgable) idiosyncratic risks will impose
real costs on the firm. Therefore, firms can increase their value through risk management by decreasing these total risk costs. Capital-budgeting (and risk management) procedures should hence take the cost of a project’s impact on the total risk
of the firm into account (Froot et al. 1993 and Stulz 1999).
However, this makes risk management inseparable from capital budgeting decisions and the capital structure choice (see Figure 2.1).16 As both risk management
and capital structure decisions can influence total risk costs, capital budgeting can
also no longer – as in the neo-classical theory – only be concerned with the systematic drivers of a firm’s cash flow.
13
Winton (2000) shows that “pure” diversification in credit portfolios into areas where the
bank does not have these superior screening and monitoring skills can result in an increase in the bank’s probability of failure.
14 A conservative financial policy is considered to be an alternative to the other instruments
of risk management (Tufano 1996).
15 Hedging/selling in liquid markets is a zero NPV transaction and does not create value in
itself; it just shifts the bank along the Capital Market Line. It seems problematic to earn
systematically a positive return in highly liquid and transparent markets that exceed the
costs of doing so.
16 For instance, negative NPV projects under this paradigm could be turned into positive
NPV projects by reducing their contribution to total risk (Stulz 1999 and Perold 2001).
Risk Management and Value Creation in Banks
Systematic
RISK
59
Specific
Capital
Structure
Value Creation
Capital
Budgeting
Risk
Management
Fig. 2.1. Dependency of Capital Budgeting, Capital Structure, and Risk Management when
Risk Management Can Create Value (Schröck; © 2002 John Wiley & Sons – This material
is used by permission of John Wiley & Sons, Inc.)
Therefore, the CAPM and the traditional DCF methodology might no longer be
universally valid as a capital budgeting tool (Froot et al. 1993) and the traditional
NPV rule might not always be the correct way to decide whether or not to undertake a project and whether value is created. Removing any of the perfect market
assumptions, typically, but not always, destroys the intellectual foundations for the
capital budgeting rules used in the neo-classical world (Stulz 1999).
Returning to Figure 2.1, it is worthwhile to emphasize that a bank’s capital
structure should be determined by the bank’s exposure to total risk (i.e. to both
systematic and specific risk) and driven by its concern regarding its own creditworthiness. Even though some models realize how critical financial policy / structure can be in enabling companies to make valuable investments17, none of these
models includes the role of risk management in deciding on value-enhancing projects. Additionally, while the current practice in risk management seems to aim
mostly at specific risk, risk management should also aim at systematic risk and
hence the totality of risk.
We can therefore deduct the following requirements from the above discussion:
Firstly, there is a need to define an adequate total risk measure for banks, because (especially in a non-Gaussian world) neither systematic nor specific risk
capture the concern with lower tail outcomes well. We can identify “economic
capital” (also often called “risk capital” as e.g. in Merton and Perold 1993) as a
measure that concentrates on the concern with lower tail outcomes, as it is defined
as the maximum loss, given a certain confidence level. Its similarity to the “Value
at Risk”-idea might be the reason why economic capital has developed as the standard approach at best practice institutions in the financial industry. Schröck (2002)
17
According to Myers (1977) and Myers and Majluf (1984) firms face real trade-offs in
how they finance their investments.
60
Gerhard Schröck and Manfred Steiner
(2002) presents various ways how economic capital can be determined, differentiated by the three risk types typically faced by banks (market, credit, and operational risk) and how the contribution of a transaction to the overall risk of the (existing) bank portfolio can be determined.
Secondly, it is important to define a metric / capital budgeting rule that enables
us to answer the question of whether a (risk management) activity creates value
and that reflects the interrelation of capital budgeting, capital structure and risk
management. It should also provide a consistent merger of the neo-classical and
the neo-institutional theories, i.e. encompasses market as well as bank internal
portfolio considerations. We will therefore investigate in the following sections
whether a measure that is currently used in banks in practice and that uses the risk
measure Economic Capital – Risk-Adjusted Return on Capital (RAROC) – can
help to decide if risk borne within a bank is more valuable than risk borne outside.
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Capital budgeting rules in banks developed over time in the following way: 18 Until
the 1970s many banks took a purely accounting driven approach and focused on
measuring their (net) revenues or earnings. This obviously set the incentive to
maximize earnings by increasing the bank’s assets. Since this approach lacks the
link to a reference point, banks subsequently set these (net) revenues in relation to
their assets as determined in their balance sheets (i.e. calculating a return on assets
[ROA] ratio). As off-balance sheet activities grew substantially and the riskiness
of the underlying assets gradually became more important19, banks realized that
the scarce resource in their business is equity. Therefore, they decided to focus on
ROE (= return on equity) ratios and measured net revenues in relation to their
book equity in order to find out which businesses are most profitable and where to
invest.
The introduction of the BIS regulatory capital requirements (after 1988) reinforced the view throughout the banking industry that assets can have very different
risks. Even though regulatory requirements do not offer a sophisticated modeling
of these risks20, they focused the view on the notion that regulatory required capital can be very different from (current) book equity, that these requirements are
binding restrictions on the banks’ activities, and that the amount of equity should
be linked to the overall riskiness of the bank. These facts subsequently lead to the
18
For a more extensive discussion of this evolution, see e.g. Schröck (1997).
Banks moved – due to increasing pressure on their margins – into higher credit risk types
of lending and experienced increased credit losses, especially during the first country risk
crisis during the 1980s.
20 They are basically determined only by the so-called “Risk Weighted Assets”. The proposed Basle Accord (Basel II) takes a much more risk-oriented view.
19
Risk Management and Value Creation in Banks
61
adjustment of the capital ratios in banks (Berger et al. 1995 and Davis and Lee
1997) and the calculation of return on regulatory (equity) capital numbers as the
capital budgeting rule.21
However, increased shareholder pressure forced banks to focus more and more
on value creation. Financial institutions realized that accounting driven ROE
measures do not have the economic focus of a valuation framework. They fail to
take the actual riskiness of the underlying business, the value of future cash flows
(Crouhy et al. 1999), and the opportunity cost of equity capital – that needs to be
included in order to calculate economic “profits” – into account. Additionally,
banks realized that the traditional DCF-framework22 does not address their fundamental problems and that it also does not work from a theoretical point of view,
because total risk matters to them.
None of the approaches to calculate a bank’s profitability presented so far adjusts for (total) risk in a systematic way (Crouhy et al. 1999 and Grübel et al.
1995). However, economic capital – as briefly introduced above – is a measure
that is calculated to reflect exactly the riskiness of the bank’s transactions and also
the bank’s concern with total risk. Hence, as an obvious next step, banks developed the practical heuristic RAROC (Risk-Adjusted Return on Capital) as a capital budgeting rule which tries to determine a bank’s economic profitability by calculating the return on economic capital. Doing so is often summarized under the
abbreviation RAPM (Risk-Adjusted Performance or Profitability Measures).
Many of the leading institutions around the globe calculate such a modified return
on equity measure (Grübel et al. 1995) and take a purely economic perspective by
trying to link it to a market-determined minimum required return (Schröck 1997)
(so-called hurdle rate23) to find out whether a transaction adds value to the bank or
not. Wills et al. (1999) find that out of 55 selected leading banks world-wide, 59%
have established an "economic capital / RAROC process”, 12% plan to do so, and
only 29% do not use such an approach.
In the following section, we will briefly define and discuss RAROC as we understand it is applied in the banking industry today as a current best practice approach to capital budgeting and how it is linked to value creation.
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Risk measurement and the determination of the amount of economic (risk) capital
that is required to cover the risk contribution of each of the transactions of a bank
is (according to Zaik et al. 1996) necessary for two reasons:
• For risk management purposes to determine how much each transaction
contributes to the total risk of the bank and to determine the capital re21
Possible other alternatives not discussed here are to calculate the return on invested equity capital or the return on a market driven evaluation of equity, such as market capitalization, etc.
22 Recall that this neo-classical approach only considers systematic risk.
23 Wilson (2003) provides a discussion of how these hurdle rates can be determined.
62
Gerhard Schröck and Manfred Steiner
quired by the bank as a whole. Note that the internal betas used in such a
calculation recognize (only) the diversification potential within the existing
bank portfolio.
• For performance evaluation purposes to determine the economic profitability of very different transactions on a comparable, risk-adjusted basis
across different sources of risk. The objective is to reveal the contribution
of a transaction to the overall value creation of the bank in order to provide
the basis for capital budgeting and incentive compensation decisions and to
identify transactions where the bank has a competitive advantage (Wilson
1992).
For the second of these two purposes the banking industry’s best practice is to
employ a measure called RAROC (Risk-Adjusted Return on Capital).
Unfortunately, there is considerable confusion on the correct definition of RAROC. Without discussing and contrasting the details of other RAPM-variants24,
we define (as e.g. Zaik et al. 1996, Kimball 1998, Crouhy et al. 1999):
RAROC =
Risk − Adjusted Net Income
Economic Capital
(1)
Hence, RAROC is a modified return on equity measure, namely the return on economic capital, where
Risk-Adjusted Net Income (in absolute dollar terms)25 =
+ Expected Revenues (Gross Interest Income + Other Revenues (e.g.
Fees))
– Cost of Funds
– Non-interest Expenses (Direct and Indirect Expenses + (Allocated)
Overhead)
± Other Transfer Pricing Allocations26
– Expected (Credit) Losses
+ Capital Benefit27
24
These other variants are e.g. RORAC (= Return on Risk-Adjusted Capital) or
RARORAC (Risk-Adjusted Return on Risk-Adjusted Capital). For a discussion of the
differences and similarities, see e.g. Matten (1996), Groß and Knippschild (1995), Punjabi (1998), Anders (2000). However, from our point of view, all measures try to calculate what is defined in Equation (1), but just have different names (Schröck and Windfuhr
1999).
25 Whereas some definitions of RAROC consider taxes in the risk-adjusted net income, we
are only considering a pre-tax version of this RAPM-measure for the following reasons:
(1) Taxes do not provide a strong rationale for conducting risk management at the corporate level in order to create value (Schröck 2002). (2) As RAROC can be calculated at the
transaction level, it is very difficult – if not impossible – to determine the tax treatment at
this level. (3) For internationally operating banks, taxes can provide a considerable skew
in the comparability of the results. Therefore, many of them use pre-tax RAROC numbers to evaluate business units operating under different tax codes.
26 Kimball (1998) describes the challenges of designing allocation and transfer pricing systems in banks at length.
Risk Management and Value Creation in Banks
63
and Economic Capital (also in absolute dollar terms) as the amount of (risk) capital that is required for a transaction on a marginal basis (as required and defined in
Perold 2001). Note that Economic Capital is a risk measure that is completely
bank-specific.
RAROC can be calculated at the bank level as well as at the single transaction
level, assuming that transfer and allocation methods work correctly. As can be
immediately seen, RAROC is a single period measure.28 Since Economic Capital
is typically calculated at a one-year horizon, the risk-adjusted net income is hence
also determined over the same measurement period. Even though we will not discuss each of the components of the Risk-adjusted Net Income in detail, note that
the only risk-adjustment in the numerator is the deduction of Expected (Credit)
Losses or standard risk costs.29
Given that RAROC is a single period measure calculated at the one-year horizon, it is also often re-written in economic profit (Zaik et al. 1996) or residual
earnings form in the spirit of EVA®30 and Shareholder Value concepts:
Economic Profit = Risk-Adjusted Net Income – Cost of Economic Capital
where
(2)
Cost of Economic Capital =
Economic Capital ⋅ Hurdle Rate and
Hurdle Rate
=
Appropriate rate of return for the investment
as determined e.g. by the CAPM and required by the (equity) investors.31
This assumes that the Risk-Adjusted Net Income is a (good) proxy for the Free
Cash Flows to the shareholders at the end of period 132 and that the Economic
27
Capital benefit is often defined as the cost saving for refinancing assets by using (economic) capital instead of debt. Alternatively, one could assume that the asset is 100% refinanced and that the required economic capital is (as an additional asset) invested into a
risk-free asset, generating a return that is equivalent to the capital benefit.
28 RAROC is very similar to the so-called Sharpe ratio, being defined as (see e.g. Sharpe
and Alexander 1990) Si = (Ri – Rf)/σi, where Si = Sharpe ratio for transaction i; Ri = return of transaction i; Rf = risk-free rate of return; σi = standard deviation of the rate of return of transaction i. Assuming Risk-Adjusted Net Income equals Ri, subtracting Rf from
the RAROC numerator and assuming Economic Capital equals σi, it is easy to show that
some banks apply RAROC (without capital benefit) correctly in the sense that they want
to maximize the Sharpe ratio in order to maximize value. Dowd (1998) discusses the
problems and deficiencies of this view at length.
29 For a definition of Expected Losses see Schröck (2002).
30 For a discussion of Economic Value Added (EVA®) concepts in banks see Uyemura et
al. (1996).
31 Note that the CAPM (beta) does not consider the risk and the costs associated with default (Crouhy et al. 1999). Wilson (2003) develops a framework for estimating these betas differentiated by line of business and a bank’s external agency rating (as a proxy for
the default risk of a financial institution).
32 Uyemura et al. (1996) suggest four adjustments to reported bank accounting earnings to
transform them into a proxy for free cash flows: (1) actual charge-offs instead of loan
loss provisions, (2) cash taxes rather than tax provisions, (3) exclusion of securities gains
64
Gerhard Schröck and Manfred Steiner
Capital equals the equity investment in the transaction.33 Note that Economic Profits are neither accounting profits nor cash flows. They rather represent the contribution of a transaction to the value of the firm by considering the opportunity cost
of the capital that finances the transaction. If the economic profit is larger than
zero, this value contribution is positive, otherwise negative (i.e. value is destroyed).
Given this transformation of RAROC into Economic Profits, it is easy to show –
by rearranging the terms – that in order to find out whether a transaction creates or
destroys value, it is sufficient to compare the calculated RAROC with the hurdle
rate (Schröck 1997 and Smithson 2003). As long as the RAROC of a transaction
exceeds the shareholders’ minimum required rate of return (i.e. the cost of equity
or hurdle rate34), then a transaction is judged to create value for the bank (Zaik et
al. 1996). Otherwise, it will destroy value.
#UUWORVKQPUCPF&GHKEKGPEKGUQH4#41%
When we closely examine RAROC, we find that in order to make this riskadjusted performance measure comparable to an equity return, one has to accept a
set of (implicit) assumptions. For instance, even though economic capital is a fictional amount of money, RAROC assumes that it is the same as “cash” equity
capital provided by the shareholders, that the bank holds exactly this amount of
equity in reality and that all cash flows will “flow” to it as well. Even if one accepts these (rigid) assumptions, it can be shown (see e.g. Crouhy et al. 1999 and
Schröck 2002) that the standard RAROC approach is biased and that it may lead
to accepting negative NPV projects (and vice versa).
Moreover, we can identify that there are many more fundamental theoretical
concerns with RAROC when it comes to the determination of value creation.
RAROC (as defined above) compares a risk measure that has its foundations in
the neo-institutional finance theory with a hurdle rate that was derived in the neoclassical world, and hence under very different assumptions. Whereas RAROC
only considers the risk contribution to the total risk of the (existing) bank portfolio, the neo-classical theory is only concerned with the systematic risk to a broad
market portfolio. Obviously, this discrepancy should lead to the development of
new approaches to capital budgeting in banks that are discussed in the next section.
and losses, (4) consideration of non-recurring events as an adjustment to either earnings
or capital.
33 Schröck (2002) discusses this assumption at length.
34 There can be severe consequences of comparing RAROC only to a single (bank-wide)
equity hurdle rate as opposed to an appropriate and differentiated hurdle rate for the
transaction or business unit (Wilson 2003).
Risk Management and Value Creation in Banks
65
1XGTXKGYQH0GY#RRTQCEJGU
Wilson (1992) was one of the first to identify that there is a fundamental problem
with RAROC when it is applied as a single-factor model in combination with a
CAPM-based hurdle rate using only economic capital and ignoring invested (real)
shareholder capital. His solution, however, is trying to fix the problem in a single
factor world. It results in the recognition of real capital while adjusting the confidence level (α) at which economic capital needs to be calculated in order to make
RAROC compatible with the neo-classical world. This adjustment of α in turn
contradicts the bank’s concern with total risk and how (and why) it decided to determine the economically required amount of capital in the first place.
Some other practitioners and academics subsequently realized that economic
capital is costly, but that the CAPM-determined hurdle rate does not reflect these
“total risk costs”, i.e. it does not consider the risk and cost associated with default
(Crouhy et al. 1999).
In this section we will briefly introduce three approaches suggested in the academic literature to date:
(A) Merton and Perold Model:
Merton and Perold (1993) provide the theoretically purest model and combine the
two concerns described above into a two-factor approach. They require the invested cash capital to earn the CAPM-determined rate of return, because this is the
economically correct price for the risk as priced in capital markets. They show that
the cost of risk capital35 is driven by the agency and information costs that make
financial distress costs the main rationale for conducting risk management 36, i.e.
because total risk is expensive and hence external finance is costly. Since they
view risk capital as the provision of (asset) insurance, as long as this (implicit) insurance is bought at the fair market price, there are no economic costs associated
with it.37
However, due to information asymmetries and agency concerns between the
various bank stakeholders, this insurance can only be obtained by paying a spread
over the (actuarial) fair market value.38 These “deadweight losses” are the economic costs of risk capital for the bank (Merton and Perold 1993). However, the
problem with this approach is that, in order to determine these total risk costs, one
would need to apply the theoretically correct (actuarial) model and compare its re35
The full-blown approach to determine risk capital as suggested by Merton and Perold
(1993) can only be applied in practice in a reduced and practical version, i.e. as “economic capital” (Schröck 2002).
36 These transaction and agency-related costs also provide incentives for diversification
within the bank portfolio (Perold 2001).
37
If a bank could buy (asset) insurance at these fair terms, risk capital would not be costly
and hence the model would fall back to a CAPM solution where the firm is indifferent
vis-à-vis risk management.
38 This is mostly due to the fact that banks are opaque institutions (Merton and Perold
1993).
66
Gerhard Schröck and Manfred Steiner
sults against observable market prices to identify these costs. Obviously, this is
impossible to do in practice.
(B) Froot and Stein Model:
Froot and Stein (1998a and 1998b) also present a two-factor model. They argue
along the lines that market frictions make risk management and capital structure
matter. In such a world, bank-specific risk factors should be an important element
of the capital budgeting process (James 1996). They conclude that a transaction’s
contribution to the overall variability of the bank’s own portfolio will affect the
transaction’s hurdle rate or cost of capital in the following way:
Hurdle Ratei = Rf + βi ⋅ (RM – Rf) + λ ⋅ σi,P
(3)
where
Rf = Risk-free rate of return
βi = CAPM beta
RM = Market rate of return
RP − Rf
= Unit cost for volatility of the bank’s portfolio of
λ=
non-hedgable cash flows
σ 2P
σi,P = Covariance of transaction i with the bank portfolio P.
Therefore, the transaction’s hurdle rate reflects the priced (market) risk (like e.g.
in the CAPM) plus the contribution of the project to the overall volatility of the
bank’s cash flows that cannot be hedged in the market. The price for the bank specific risk λ will vary directly with the cost of external financing and depends on
the current capital structure of the bank (James 1996).
Model (B) obviously comes to the somewhat extreme conclusion that a bank
should hedge all tradable risks as long as they can be hedged at little or no cost in
the capital markets (see Proposition 1 in Froot and Stein 1998a). This is because
the bank’s required price for bearing tradable risk will exceed the market price for
risk by the contribution of a hedgable risk to the overall variability of the bank’s
portfolio. Hence, the only risk the bank should bear is illiquid or non-tradable risk
– which contradicts reality. There are the following problems with Model (B):
• It is not immediately obvious that the second pricing factor in the model
necessarily reflects total risk costs in the sense developed in this chapter.
• Froot and Stein (1998a) admit that it could be extremely difficult to estimate these costs since they cannot be observed directly in the market.
• Model (B) is very unspecific about when it falls back to one or the other
single factor solution, i.e. when it prices like the market does in the neoclassical solution and when it uses only the internal portfolio as the relevant universe.
• Also, Model (B) is unspecific about the trade-off between the costs of selling hedgable risk in the market and the cost of total risk of keeping it in the
bank’s portfolio.
Risk Management and Value Creation in Banks
67
(C) Stulz Model:
Stulz (1996, 1999, and 2000) also develops a two-factor model. Like Model (A)
he concludes that invested cash capital should be required to make (at least) the
CAPM-determined hurdle rate. Since economic capital is a total risk measure with
regard to the bank’s own portfolio, it should – in addition to the costs of standard
capital budgeting – reflect the costs of the impact of the project on the bank’s total
risk (Stulz 1999). If economic capital is costly, ignoring these costs will lead to a
mistake in the capital budgeting decision-making process. Hence, the value of a
project for a bank is the “traditional” NPV (as determined in the neo-classical
world) minus the (project’s contribution to the) cost of total risk.
Even though Stulz leaves open how these costs of total risk can be quantified in
reality, he proves that the total risk costs can be approximated for small changes in
the portfolio by a constant incremental cost of economic capital per unit of economic capital (Stulz 2000). Note that these costs of total risk do not disappear, irrespective of whether we deal with risk in liquid or illiquid markets. This puts
holding risk within a bank portfolio always at a disadvantage vis-à-vis the market.
We can summarize and compare Models (A) – (C) as follows: All three models
agree that a “total risk” component in addition to the neo-classical capital budgeting approach is necessary in a world where risk management does matter to banks
in order to create value. Even though none of these approaches shows how one
could exactly quantify these total risk costs in practice, Model (C) appears to be
the most plausible and promising for practical purposes for the following reasons:
• It integrates a total risk measure (economic capital) – that is already widely
used throughout the banking industry – into a new capital budgeting decision rule.
• Despite the fact that the total risk component does not vanish in liquid
markets, as e.g. in Model (B)39, it has – as we will show below – the highest potential to identify transactions where the bank has competitive advantages and can really create value.
• As already mentioned above, neither Model (A) nor Model (B) seem appropriate for practical purposes. Both models are impractical because of
the unavailability of observable market data to determine the costs of the
second pricing factor. Additionally, Model (B) seems inappropriate because of its unrealistic conclusion that the bank will only hold nonhedgable risk.
+ORNKECVKQPUQHVJG0GY#RRTQCEJGUQP4KUM/CPCIG
OGPVCPF8CNWG%TGCVKQPKP$CPMU
As already indicated above, our further discussion is based on Model (C). This
two-factor model defines the required rate of return for capital budgeting decisions
of transaction i as the sum of the CAPM-determined rate of return (RE,i) on the in39
As mentioned above, the exact workings of this effect are unclear.
68
Gerhard Schröck and Manfred Steiner
vested shareholder capital (VE,i) and the contribution to the total risk costs. These,
in turn, can be defined as the product of the required (marginal) economic capital
of the transaction (ECi) and the (proportional) financial distress costs of the bank
(DC)40. Therefore:
Required Returni = RE,i ⋅ VE,i + DC ⋅ ECi
(4)
Clearly, in this model holding risk within a bank portfolio is always costly. The
first component of the required return is the fair market price – which is not costly
in an economic sense – whereas the second component reflects the costs associated with the contribution of the transaction to the total risk costs of the bank’s
portfolio, which is driven by the actual capital structure. Hence, the price for holding risk on one’s own books always exceeds the costs as paid in the market.
Even though this insight might contradict conventional financial theory, it can,
on the one hand, explain the interdependence of risk management, capital budgeting, and capital structure decisions in a bank when total risk matters (as depicted
in Figure 2.1 and in the left-hand part of Figure 6.1 below). On the other hand, this
fact sheds some more light on a normative theory for risk management in banks.
We will first consider the implications of such a model on the risk management
decisions of a bank in Section 5.1. We will then discuss the implications on capital
budgeting decisions and for capital structure decisions in Sections 5.2 and 5.3 respectively.
+ORNKECVKQPUHQT4KUM/CPCIGOGPV&GEKUKQPU
As discussed above – since holding risks on the bank’s own books is costly – risk
management can create value when it can reduce these costs. A bank has the following options to do so: It can either:
a) Reduce the (sources of) risk in its own portfolio, hence the amount of required economic capital (Merton and Perold 1993), and therefore the total
risk costs
or it can
b) Reduce the cost of total (default) risk for a given level of economic capital.
The ultimate consequence of option a) would be to sell the entire bank’s business
and invest the proceedings into risk-free assets. Note that this is something Wilson
(1992) predicts as a consequence of using RAROC as a performance measure.
However, this could include selling risks where the bank has a competitive advantage and where it could really create value – despite the fact that it is costly for the
bank to hold these risks. Therefore option a) appears not to be a choice for banks.
Hence, only option b) is viable. It can be achieved in three ways (Stulz 2000 –
note that we have discussed these in a general way in the Box in Section 2):
40
Again, we do not specify here how these costs are determined, and leave this point to further research. We assume that it is a constant percentage assigned to the required amount
of economic capital and is the deadweight cost of (economic) capital (Perold 2001).
Risk Management and Value Creation in Banks
1.
2.
3.
69
Increasing actual capital: An institution’s default risk is inversely related to
its available real equity capital. However, when increasing equity capital,
the exact impacts on economic capital and its associated costs must be considered. If a bank raises its equity to expand its business (at the same riskiness), this does not lower the costs of total risk. Therefore, equity would
have to be invested in projects that have a negative internal beta to the existing portfolio. Neither an investment in risk-free assets, nor the repayment of debt does change the bank’s required economic capital. But both
actions change the cost of total risk, assuming that the other operations are
left unchanged. However, holding equity capital is associated with opportunity cost41 and therefore holding sufficient equity to make risk management irrelevant is not a choice for banks.
Selecting projects according to their impact on total risk: The selection of
projects in order to improve the (internal) diversification of the bank portfolio to manage risk is also expensive. On the one hand, as a market
benchmark, one can observe that the diversification discount for conglomerates vis-à-vis a portfolio of their specialized competitors is about 14%
(Stulz 2000). On the other hand, expanding into unfamiliar sectors can be
very costly, because this often adds an additional (and costly) management
layer or can lead to unexpected high (credit) losses (Winton 2000). However, in our model, these costs have to be balanced against the cost savings
in total risk costs. Note that – on the contrary – Zaik et al. (1996) describe
that RAROC gives the message that internal diversification pays off in any
case – even beyond familiar business segments.
Using derivatives to hedge and other (financial) risk management instruments to shed risks: Applying risk management instruments in liquid markets is the most cost efficient way to reduce firm-wide risk. Therefore, a
bank should evaluate the total risk contribution of a new transaction only
after carrying out these hedging activities. However, and as we will see in
the implications in the subsequent paragraphs, the costs of these instruments have to be lower than the total risk costs of these transactions.
Given the previous discussion of option a) and b), the application of the model as
defined in Equation (4) leads us to the following practical implications for risk
management in banks:
Implication 1 The two-factor model will identify where a bank has competitive advantages and where it can really create value.
41
Shareholders expect the CAPM return on equity – which is not costly in an economic
sense. However, when using equity to buy back debt, this gives a windfall to the existing
debtholders (making their debt safer) and therefore redistributes some of the benefits of
increasing equity capital to other stakeholders. Moreover, part of the tax shield is lost for
the bank when debt is bought back. Other information asymmetries and agency costs
(e.g. managerial discretion) as well as the transaction costs (of issuing new capital) also
make new equity expensive.
70
Gerhard Schröck and Manfred Steiner
Reasoning
As long as a bank is able to expropriate excess (economic) rents from informational advantages that exceed both cost components of Equation
(4), it can actually increase the bank’s value by holding these risks internally. This will most likely happen in illiquid areas (see right-hand part
in Figure 6.1), because – by definition – markets are only liquid when
market players have homogenous expectations and no informational advantages. Therefore, when the bank decides to hold positions in liquid
market risks (where market inefficiencies are very unlikely to occur), the
price needs to cover (at least) both the market costs and the total risk
costs in order to create value. Otherwise, this will be a value-destroying
proposition.
Implication 2 A bank should sell all risks, where it does not have a competitive advantage, i.e., all hedgable or non-compensated42 risks should be sold as long
as the costs for doing so will not exceed the total risk costs.
When the bank will not have informational or competitive advantage(s)
Reasoning
that will compensate for both cost components of Equation (4) for all of
its transactions, it will destroy value by keeping these risks. Again, one
needs to trade off the costs of shedding these risks against the total risk
costs:
• In liquid markets these risks – most likely – trade at their fair market
prices. In this case the decision is obvious: The costs (i.e. the spreads
above the fair market value) for selling off these risks are lower, the
greater the volume of transactions in a given market, the lower the
volatility of the underlying asset price, and the less private information is relevant for pricing the underlying asset. Hence, the bank
should sell or re-distribute these liquid risks using derivative hedging
instruments, because the costs of doing so are almost certainly lower
than the total risk costs incurred when holding these transactions in
the bank’s own portfolio.
• However, not all risks are traded in liquid markets. But there are other
on- and off-balance sheet risk management instruments available that
can be applied across the whole risk spectrum. Securitizations, (internal) diversification, insurance, loss prevention by process control, etc.
should be applied as long as their costs do not exceed the total risk
costs of the underlying transactions.
Note that the conclusion to sell all liquid risks is similar to the result of
Model (B) as described above. However, our model allows for competitive advantages even in liquid markets, whereas the Model (B) would indicate that all liquid risks should be sold off immediately without further
considerations. However, speculating “on the market” in these liquid
segments will require economic capital and is therefore costly. If the
bank, nonetheless, decides to hold on to risks that it could more cheaply
shed, it will destroy value.
42
Schrand and Unal (1996) define compensated risks as those risks where the bank has
comparative advantages with regard to their management. These risks are therefore the
source of the economic profits of the firm. Hedgable risks are, on the contrary, those
risks where the bank cannot extract economic rents (mostly liquid or traded risks).
Risk Management and Value Creation in Banks
71
Implication 3 Hedging specific risks and diversification of the bank’s portfolio can
create value even if it comes at a cost43.
As long as the costs of e.g. diversifying credit risk and managing operaReasoning
tional risk (which is typically highly specific risk, see right-hand part in
Figure 6.1) are lower than the total risk costs they incur, it will pay off
for the bank to do so. This contradicts the conclusion of the neo-classical
theory that spending time and money to eliminate firm-specific risks will
destroy value in any case.44
+ORNKECVKQPUQP%CRKVCN$WFIGVKPI&GEKUKQPU
We have seen above that, when total risk matters, banks can increase their value
through risk management. However, this fact makes risk management inseparable
from capital budgeting decisions. On the one hand, and as also indicated above,
capital budgeting decisions on transactions should only be taken after all noncompensated risks45 have been shed via risk management actions. On the other
hand, the bank may only be able to buy a risk – where it does have a compensated
informational advantage – as a part of a “risk”-bundled product. However, the
bank may not be able to shed the other, non-compensated risks that are also associated with that “bundled” transaction later, because the costs of doing so would
exceed the total risk costs as indicated by Equation (4) and as described in Implication 2 in Section 5.1. However, these risks impose a real cost on the bank.
Therefore, the bank cannot and should not separate the risk management from the
capital budgeting decision. Applying our two-factor model with all its implications ex ante would prevent the bank from investing in such risks beforehand –
unless the compensation of the informational advantage were to exceed the additional total risk costs imposed by the non-sellable risk components of the package.
+ORNKECVKQPUQP%CRKVCN5VTWEVWTG&GEKUKQPU
As the actual capital structure determines the total risk costs for the bank, neither
risk management nor capital budgeting decisions can be made without considering
the actual capital structure. In our model as defined in Equation (4), both real equity and total risk are costly. Decreasing the leverage of the bank (i.e., increasing
the equity capital ratio) decreases the total risk costs, but increases – on the other
hand – the (overall) equity costs. If increasing equity to decrease the cost of total
risk is costly (due to the increase in transaction and agency costs), then – at the
margin – the cost of total risk has to equal the cost of equity and the capital struc-
43
Note that Perold (2001) comes to the same conclusion.
Note that this issue can also be discussed in the light of operational risk where the benefits of self-insurance (internal risk pooling) need to be balanced with the costs of thirdparty insurance for event risks.
45 Non-compensated risks are those risks that are cheaper to sell off than to keep internally.
44
72
Gerhard Schröck and Manfred Steiner
ture has to be adjusted until equilibrium is reached (Stulz 1999).46 However, this
does not mean that economic capital and actual equity capital also have to be
equal. Knowing the required amount of economic capital therefore does not resolve the problem of the actual capital structure choice. Note that since increasing
total risk has a significant cost that has to be taken into account in everything the
bank does, higher capital ratios in banks might be less expensive than is commonly thought of – given that they can lower total risk costs. An extreme conclusion of this discussion is that if the bank holds infinite (real) capital, it would be
risk-neutral as in the neo-classical world. This is something that is not reflected in
RAROC (as defined in Equation (1)) as the economic capital always has to earn
the CAPM-required return.
(QWPFCVKQPUHQTC0QTOCVKXG6JGQT[HQT4KUM
/CPCIGOGPVKP$CPMU
We can draw the following conclusions from the application of the two-factor
model as suggested in Equation (4):
• Risk management can create value. There is a whole spectrum of instruments (see Box in Section 2) – apart from just derivatives in liquid markets –
which can be used as long as the cost of applying them is lower than the total
risk costs associated with the transaction.
• As shown in the left-hand part of Figure 6.1, capital budgeting, capital structure and risk management decisions are interrelated and need to be determined simultaneously – rather than separately as in the neo-classical world.
• As also shown in Figure 6.1, when total risk matters and is costly to the
bank, the world cannot be reduced to just dealing with systematic and specific risks. It is rather a question of whether risks generate – via the bank’s
competitive advantages – enough revenues to compensate both market and
total risk costs, so that it is worthwhile to hold them internally. Even though
these competitive advantages are likely to exist in illiquid markets, where informational asymmetries prevail, they can be achieved across the whole risk
spectrum (as shown in the right-hand part of Figure 6.1).
• The bank should concentrate on these competitive advantages and should
understand where they come from (Stulz 1996) and why they exist (Braas
and Bralver 1990). Risk management allows the bank to concentrate on
these risks, because the capital budgeting decision rule of the two-factor
model encourages the shedding of all other risks whose revenues do not
cover both cost components.47 Only those risks without competitive advan46
For a further discussion of the optimal capital structure in depository institutions, see e.g.
Cohen (2003).
47 Since our model indicates that it is typically advisable to sell off all marketable/liquid
risks, implicitly the model falls back to the neo-classical solution as the bank’s (quasi)
risk aversion does not enter into its decision-making process in these cases.
Risk Management and Value Creation in Banks
73
tages that have little impact on the overall firm risk, but that are expensive to
eliminate, should be kept within the bank’s portfolio.
These conclusions are not dramatically new – as many of them are already practiced in the financial industry. The difference, however, is that they cannot be explained by the tools and theories that are currently available. As suggested above,
two-factor models can therefore provide the foundations for a normative theory of
risk management. Not only can this new approach explain why risk management
can create value at the bank level, but also it provides much more detailed and differentiated (theoretical) arguments of which approach / action and which instruments (as presented in a general way in the box in Section 2) can be applied in order to achieve the ultimate goal of value maximization.
Illiquid
Systematic
Liquid
Risk
Management
Capital
Structure
Credit Risk
Specific
Value
Creation
Capital
Budgeting
RISK
Market Risk
Operational Risk
Fig. 6.1. Overview of the Components of a Normative Theory for Risk Management
(Schröck; © 2002 John Wiley & Sons – This material is used by permission of John Wiley
& Sons, Inc.)
In general, this contradicts traditional intuition. Hedgable risks eat up a fraction of
the overall available risk capacity that could be used to extract economic rents by
using the bank’s comparative advantages. Therefore, the pure and “naïve” implication of the neo-classical world to reduce risks cannot be the goal. It is rather the
right “co-ordination” of risks that is required.48
The need for identifying and concentrating on competitive advantages as well as
the right co-ordination of risks is most obvious in the recent developments in the
area of traditional bank lending. Loan securitizations and the “unbundling” of the
traditional lending business model (Kuritzkes 1998 and 1999) from a “buy-and48
Schrand and Unal (1996) show that financial institutions (given a certain capital base)
hedge interest rate risks in order to be able to take on more credit risks – which is consistent with what our model suggests.
74
Gerhard Schröck and Manfred Steiner
hold” strategy to a separation of loan origination and an active credit portfolio
management (Reading et al. 1998) (in secondary markets) both require the expansion of RAROC for a market component and a decision tool that properly identifies the informational advantages in the credit process.
%QPENWUKQP
Firms and especially banks try to avoid financial distress situations or try to decrease the likelihood of their occurrence by using risk management. Since these
“lower tail outcomes” can be caused by both systematic and (firm-) specific risks,
banks do worry about total risk and the composition of their (existing) bank portfolio matters when they make capital budgeting decisions. Even though both of
these actions are unexplained in the neo-classical financial theory, they can be observed in reality. In a world (as defined in the neo-institutional financial theory)
where these two concerns matter, risk management could indeed increase the
bank’s value by reducing total risk and the costs associated with it. Whereas this
fact can provide a rationale for conducting risk management at the bank level in
order to create value, it is only a partial solution, because it does not provide detailed instructions which risk management instruments should be used and to what
degree (i.e. a normative theory for risk management) and how value creation
should be measured in such a world.
Total risk costs can be influenced by the actual capital structure. Increasing the
(financial) leverage also increases the probability of incurring the costs of financial distress. Therefore, holding (equity) capital commensurate with the risks held
on the bank’s books is sensible from both an economic as well as a regulatory
point of view and can hence be considered as an alternative form of risk management. Additionally, when risk management can create value, it can also influence
capital budgeting decisions. Therefore, capital structure, capital budgeting, and
risk management decisions cannot be separated in such a world and traditional
decision rules might not be applicable.
In a world where total risk matters, a capital budgeting tool needs to include a
component that compensates for both the market price of risk (i.e. the required
market return) as well as a component that reflects the contribution of a transaction to the total risk of the (existing bank) portfolio. As RAROC mixes these two
components into a single-factor model, it does not appropriately capture both of
these effects and hence is not an adequate capital budgeting tool for banks from a
theoretical point of view. Unfortunately, this contradicts what can be observed in
practice, where RAROC is used as an acceptable proxy to indicate value creation
from a practical point of view.
We have seen that two-factor models are better suited to capture both of these
(pricing) components. We discussed an approach derived from the models already
available in the light of its practicability and its implications for risk management
decisions in banks. We concluded that our model is a better decision rule to identify whether a bank should hold a transaction on its own books and whether it can
Risk Management and Value Creation in Banks
75
really create value by doing so. We found that our model allows for much more
detailed instructions on what banks should do exactly and which (risk management) actions can enhance value, because it considers the cost of total risk. It can
hence form the foundation for a normative theory of risk management in banks
and will help banks to focus on their comparative advantages.
We can therefore conclude that when total risk counts and is costly, banks can
indeed increase their value through risk management. The new decision rules deduced from the two-factor model could replace what banks have been doing intuitively for a long time and what is an observable phenomenon in real life. It, however, requires the jointly and endogenous determination of risk management,
capital budgeting, and capital structure decisions.
The difficulty with the two-factor model suggested is that the DCF-approach
can then no longer be the universally valid capital budgeting decision tool, contradicting the principles that were used in corporate finance over the past 30 years.
This chapter only provides the foundations for applying this new paradigm. Much
more research needs to be done in order to parameterize and to make such a model
operational, communicable, and implementable in practice. Until this is the case,
banks may want to apply RAROC as the currently only practicable solution to
capital budgeting problems in banks, but need to be aware that they apply a biased
tool that may deliver only directionally correct answers in some cases.
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6JG0GY$CUGN%CRKVCN#EEQTF
Claudia Holtorf, Matthias Muck, and Markus Rudolf 1
1
Wissenschaftliche Hochschule für Unternehmensführung WHU - Otto Beisheim
Graduate School of Management; Dresdner Bank Chair of Finance, Burgplatz 2,
56179 Vallendar, Germany, Tel.: +49 – (0)261 6509 421, Fax: +49 – (0)261 6509
409, Email: markus.rudolf@whu.edu, Internet: http://www.whu.edu/banking
Abstract: This paper addresses the capital requirements based on the RiskMetricsTM framework and the BIS standard model. A case study is developed which
shows that the capital requirements can be reduced by applying the more accurate
RiskMetricsTM framework. Furthermore it gives an overview of the capital requirement rules for credit risk and operational risk in the Basel II Accord.
JEL classification: G21
Keywords: Capital Requirements, Value at Risk, BIS Standard Model, Basel II, Credit Risk,
Operational Risk
One of the most extensively discussed topics in financial risk measurement is the
new Basel Capital Accord also known as Basel II. The Accord was initiated by the
Committee for Banking Supervision of the Bank of International Settlement in
Basel (BIS) and will replace the first Basel Accord from 1988 which was implemented in international law by over 100 countries worldwide. In June 2004, the
Committee adopted the new Accord after several years of ongoing discussion and
the release of three proposals for consultation (June 1999, January 2001, and April
2003). Its implementation will start at the end of the year 2006. The Accord’s aim
is to develop improved capital standards for financial intermediaries. It consists of
three pillars including Minimum Capital Requirements, Supervisory Review Process, and Market Discipline.
*
We thank Frank Guse for helpful comments.
80
Claudia Holtorf, Matthias Muck, and Markus Rudolf
The first pillar sets minimum capital requirements for three different kinds of
risk (market risk, credit risk, and operational risk). It distinguishes between the
trading book and the banking book. The trading book “consists of positions in
financial instruments and commodities held either with trading intent or in order to
hedge other elements of the trading book.”1 In the 1996 amendment of the proposal, the Committee for Banking Supervision suggested for the first time that internal models may be used in order to determine the capital requirements of the
trading activities. In principle, banks have the choice between a so-called standard
model and internal models which have to be accepted by the national regulation
authorities. In Germany, this proposal has been codified as national law in October
1998. The Federal Banking Supervisory Office in Germany [BAFin] has to confirm the suitability of the model upon the bank’s requests. At the center of the
capital adequacy rules according to internal models is the Value at Risk [VaR]
concept. VaR represents the expected maximum loss of a portfolio over a prespecified holding period and a given confidence level. It aggregates the bank’s total exposure in a single number. The rules for market risk remained more or less
unchanged in the new Basel proposal. The major new aspects address credit risks
in the banking book as well as operational risks.
The second pillar is the Supervisory Review Process. It endows bank supervisors with extensive information rights and manifold possibilities for intervention.
Furthermore, the second pillar also defines interest rate risk as a significant risk
factor within the trading book. However, in contrast to the trading book banks are
not required to hold capital to protect against losses from it. Finally, the third pillar
(Market Discipline) deals with enhanced disclosure.
This paper gives an overview over the new Basel Accord. It focuses on market
risk management and compares the BIS standard model to the more sophisticated
RiskMetricsTM VaR model. Banks may sometimes prefer to apply so-called internal [VaR-] models if they reduce the capital requirements and therefore the costs
of capital. On the other hand, internal models are typically more complex and
therefore difficult to develop, fine-tune and apply. In this paper, a case study is
developed which allows the comparison of the capital requirements in the standard
model and the VaR approach. In Germany, there are (only) 15 banks which use internal market risk models for their capital requirements 2, mainly because the assessment procedure of the suitability of internal risk models is costly and timeconsuming. The paper is structured as follows: Section 2 presents the case study
and illustrates the calculation of the VaR due to market risk. Based on the results
section 3 explains how to compute the capital requirement for market risk. Section
4 shows how the corresponding capital requirement is determined in the standard
approach. Section 5 gives an overview over the procedures put in place in order to
calculate the capital requirement due to credit risk. Section 6 briefly deals with
operational risk. Section 7 concludes.
1
2
See paragraph 685 of the new Basel Accord (Basel 2004).
See BAFin, annual report 2003, p. 102, http://www.bafin.de.
The New Basel Capital Accord
81
VaR represents the maximum potential change in the value of a portfolio within a
certain holding period given a statistical confidence level. There exists a number
of different approaches for determining VaR. The RiskMetricsTM approach is
based on standard deviations, correlations, and durations of financial instruments
and assumes jointly normally distributed returns for the instruments. It goes back
to an initiative by JP Morgan and has evolved to a standard for internal risk management models. RiskMetricsTM is based on a database which is updated on a
daily basis and is distributed via the internet3. The core idea of the RiskMetricsTM
approach is to consider financial instruments as cash-flows which are driven by a
limited number of risk factors. Stock options are for instance driven by the same
systematic risk factors as the underlying share. In the case of DAX options, option
delta and option gamma relate the price changes of the option to the development
of the DAX. Fixed income instruments such as government bonds or interest rate
swaps are driven by interest rates and can be analyzed with duration as the standard sensitivity measure. This proceeding enables us to characterize the risk position of any instrument in terms of the underlying risk factors, no matter how complex and exotic a specific product is.
The RiskMetricsTM framework considers four basic risk categories: (1) Stocks,
(2) interest rates, (3) currencies and (4) commodities. A limited number of risk
factors are relevant within these categories. These risk factors are called the building blocks of the total portfolio risk. For stocks, 31 indices of national stock markets (Argentina, Australia, Canada, Mexico, USA, Switzerland, Denmark, Norway, Sweden, UK, EMU countries, South Africa, Australia, Hong Kong, Indonesia, Japan, South Korea, Malaysia, New Zealand, Philippines, Singapore, Thailand, Taiwan) are available. Any individual share is characterized by the risk of
the respective stock market. This implies that only systematic risks are considered
and unsystematic risks due to insufficient diversification are neglected. The currency risks are expressed with respect to the foreign exchange rates of 21 countries. Furthermore, data for contracts on 11 commodities (aluminum, copper, natural gas, gold, heating oil, nickel, platinum, silver, unleaded gas, light sweet crude,
zinc) with 7 different times to maturity (spot, 1M, 3M, 6M, 12M, 15M, 27M) 4 is
supplied. Finally, 14 interest rates of different maturities (1M, 3M, 6M, 12M, 2Y,
3Y, 4Y, 5Y, 7Y, 9Y, 10Y, 15Y, 20Y, 30Y) are available for each country. These
refinements of the risk categories over-satisfy the Basel Accord requirements but
are based on a limited number of risk factors which can be handled easily. All instruments are treated as portfolios of cash-flows which are assigned to these risk
factors. If a cash-flow cannot be directly assigned, it will be mapped to two adjacent risk factors. If for instance a fixed income cash-flow occurs in 6 years then it
will be partly mapped to the 5Y as well as the 7Y interest rate vertex. The subsequent case study will illustrate in detail how the mapping is to be carried out in
practice.
3
4
See http://www.riskmetrics.com/rmcovv.html.
Months (M), years (Y).
82
Claudia Holtorf, Matthias Muck, and Markus Rudolf
The VaR calculation consists of three steps: First, cashflows are evaluated on a
mark-to-market base. Second, they are mapped onto the standard grid of vertices,
i.e. the risk factors explained above. Finally, the VaR is derived as the portfolio
value for which the cumulative distribution function equals 5% for a 1-day holding period. The case study which is going to be developed in this chapter will involve the following steps:
1. Characterization of the trading book: Summarizes the market values of all long
and short positions in the portfolio.
2. Mapping: All positions are assigned to risk factors and mapped to the RiskMetricsTM vertices. The result is that the portfolio of instruments is transformed into a portfolio of cash-flows which are exposed to different risk factors.
3. Aggregation: The cashflows of all positions are aggregated.
4. Determination of portfolio risk on a €-basis: Taking into account the diversification effects due to non-perfect correlations between the different risk factors,
the market risk of the trading book is determined for a specific confidence
level.
5. Calculation of the VaR of the trading book (diversified and undiversified).
Table 1. Trading book positions as of December 11th, 2003
Price
Bundesanleihe
Land NRW
Zerobond
short (AA)
T-Bill
Face value /
Number of
shares
99.88 €
1’000’000
Time to Mod. Du- Market value
in €
maturity ration /
Beta /
Delta
Bonds
3.50
5Y
4.53
998’760
101.55 €
87.04 €
3’000’000
-300’000
4.00
0.0
4Y
4Y
3.67
3.86
3’046’500
-261’116
98.78 $
3’000’000
0.0
6M
0.49
2’426’630
1.2
346’500
1.1
2’826’073
-0.52
86’750
9’470’097
1.2212
1’068
3’821
256
Coupon
Stocks
DaimlerChrysler
Microsoft
34.65 €
10’000
21.57 $
160’000
Options
DAX put
173.50 €
Sum in €
$/€:
S&P 500:
DAX:
Trading days per year:
500
Source: Datastream, RiskMetricsTM Dataset.
The New Basel Capital Accord
83
Table 1 summarizes the portfolio which will serve as the basis for the case study.
It consists of four fixed income, two share and one stock index option position and
of two currencies (€ and $). It further includes the S&P 500 and the DAX. The
data is based on market information for December 11th, 2003 and has been obtained from Datastream and the RiskMetricsTM dataset.
Among the fixed income positions, there are three with AAA rating: a Bundesanleihe (German government bond), a bond issued by the state of North RhineWestphalia (NRW), and a US T-Bill. The fourth fixed income position is a short
position in a zerobond revealing a double A rating. The yield curves for Euro as
well as US-$ and the interest rate spreads for AA issuers are given in table 2
(source: Bloomberg). Table 1 contains seven financial instruments whose cashflows will be assigned to eight risk factors and which are summarized in table 3.
In order to illustrate the mapping approach, it is assumed, that only the 1M, 1Y,
3Y, and 5Y fixed income risk factors are available. Although the RiskMetrics TM
dataset provides more vertices, the bank regulation authorities do not require the
consideration of the 6M, the 2Y and the 4Y vertex. In addition to five fixed income vertices, table 3 contains the risk figures for the DAX, the S&P 500 and the
€/$ exchange rate. The measurement of VaR is based on the principles of portfolio
theory which have been derived by Markowitz 1952, one of the central issues being that risk is determined by volatilities and coefficients of correlation.
Table 2. Term structure of interest rates in € and $ as of December 11th, 2003
1M
3M
6M
1Y
2Y
3Y
4Y
5Y
€
$
2.14
2.15
2.20
2.41
2.79
3.16
3.49
3.74
1.17
1.17
1.23
1.48
1.95
2.51
3.04
3.37
Spread AA issuer in bp.
for €
0
0
2
2
3
3
4
5
Spread AA issuer in bp.
for $
0
0
8
17
19
21
24
26
Source: Bloomberg.
Table 4 characterizes the cashflows from the first position of the trading book
which is the German government bond (Bundesanleihe). The coupon payments
are 3.5% of the face value of 1’000’000 €. This is 35’000 € per year. The appropriate discount rates are the AAA interest rates in table 2. From this follow the
present values of each of the cashflows. The cashflow after 4 years has to be discounted by 1.0354 which yields 30’511 €. The sum of all discounted cashflows is
991’085 € which slightly differs from the market value (998’760 €) given in table
1. Anyone holding the Bundesanleihe will receive a cashflow after two years. Unfortunately, there is no risk factor given in table 3 this cashflow could be assigned
to. Therefore, the payment in the second year has to be mapped to the adjacent
vertices i.e. to the 1Y and the 3Y Euro vertex. In order to make sure that the risk
84
Claudia Holtorf, Matthias Muck, and Markus Rudolf
characteristic of the second year cashflow remains the same, the fraction x which
is mapped onto the 1Y vertex has to be chosen in such a way that the volatility of
the cashflow remains unchanged. After the mapping, the 2Y cashflow is a portfolio consisting of a 1Y and a 3Y cashflow. According to table 3, the volatility of a
1Y Euro cashflow (multiplied by 1.65 which represents the confidence level of
95%) is 0.0485% per day and 1.65 times the volatility of a 3Y Euro cashflow is
0.2375% per day. The coefficient of correlation between both cashflows is 0.31.
The volatility σ2Y of the second year cashflow then is5
σ22Y = x 2 ⋅ σ12Y + (1 − x )2 ⋅ σ32Y + 2 ⋅ x ⋅ (1 − x ) ⋅ σ1Y ⋅ σ3Y ⋅ ρ1Y,3Y
(
)
(
0 = x 2 ⋅ σ12Y + σ32Y − 2 ⋅ σ1Y ⋅ σ3Y ⋅ ρ1Y ,3Y + 2 ⋅ x ⋅ − σ32Y + σ1Y ⋅ σ3Y ⋅ ρ1Y ,3Y
+ σ32Y
≡ σ12Y
(
b ≡ (− σ
a
− σ22 Y
+ σ32Y
2
3Y
− 2 ⋅ σ1Y ⋅ σ3Y ⋅ ρ1Y ,3Y
+ σ1Y ⋅ σ3Y ⋅ ρ1Y ,3Y
)
)
)
(1)
c ≡ σ32Y − σ22Y
x1, 2 =
− b ± b2 − a ⋅ c
a
However, the volatility σ2Y of the 2Y cashflow is unknown. Gupton et al. 1997
suggest to choose σ2Y as the average of σ1Y and σ3Y which is
σ 2Y =
0.0485% + 0.2375%
= 0.14% .
2
Applying equation (1) yields x=43.1% as the fraction of the first cashflow which
has to be mapped onto the 1Y vertex. A simple test shows that x=43.1% keeps the
volatility of the 2Y cashflow unchanged
0.4312 ⋅ 0.000485 2 + 0.569 2 ⋅ 0.002375 2
+ 2 ⋅ 0.431 ⋅ 0.569 ⋅ 0.000485 ⋅ 0.002375 ⋅ 0.31
= 0.14%
5
σ1Y: volatility of the 1Y Euro cashflow, σ3Y: volatility of the 3Y Euro cashflow, ρ1Y,3Y:
coefficient of correlation between the 1Y and the 3Y Euro cashflow, x: fraction which is
mapped onto the 1Y vertex.
The New Basel Capital Accord
85
Table 3. Volatilities, prices, and coefficients of correlations of the risk factors as of December 11th, 2003
Risk fators
1Y € zerobond
3Y € zerobond
5Y € zerobond
1M $ zerobond
1Y $ zerobond
€/$
DAX
S&P 500
Price Vola ⋅ 1.65
0.9763
0.0485
0.9100
0.2375
0.8303
0.3948
0.9990
0.0017
0.9854
0.0748
0.8189
3’921
1’068
0.9833
1.96
1.19
3Y €
0.31
1
Coefficient of correlation
5Y € 1M $ 1Y $
€/$ DAX
0.27 0.00 0.25 0.27 -0.31
S&P
-0.11
0.98
-0.06
0.61
0.45
-0.43
-0.53
1
-0.05
0.59
0.42
-0.40
-0.55
1
0.30
-0.27
0.03
0.24
1
0.03
-0.33
-0.15
1
-0.49
1
-0.40
0.47
1
Source: RiscMetricsTM dataset, Vola = daily price volatility in %.
Table 4. Mapping of the Bundesanleihe
1Y
2Y
3Y
4Y
5Y
Sum
Discount
rate
2.41
2.79
3.16
3.49
3.74
Cashflow
35’000
35’000
35’000
35’000
1’035’000
1’175’000
PV cash- Weight Mapping
2Y
flow
short maturity
34’176
14’278
33’129
0.4310
31’878
18’851
30’511
0.4907
861’391
991’085
Mapping
4Y
Synthetic
PV
48’454
14’972
65’701
15’538
876’929
991’085
All figures are in €.
For the 4Y cashflow a similar problem arises. Applying the same calculation principles yields a fraction of 49.07% of the 4Y cashflow which has to be mapped
onto the 3Y vertex and the rest onto the 5Y vertex. Table 4 shows the result for
both, the 2Y and the 4Y cashflow mapping. The overall result of the mapping procedure is summarized in the last column. Originally, five cashflows were induced
by the Bundesanleihe position. The number of cashflows has been reduced to
three synthetic cashflows revealing the same risk and the same sum of present
values. The first instrument, the Bundesanleihe, has been split up in a portfolio of
three synthetic cashflows. The same procedure is applied to the three other fixed
income positions. For the bond issued by the state of NRW, again the 2Y Euro and
the 4Y Euro cashflows have to be mapped to the adjacent vertices according to
equation (1). The short position in the double A rated zerobond is mapped on the
3Y Euro and the 5Y Euro grid. Finally, the US T-Bill has to be mapped onto the
86
Claudia Holtorf, Matthias Muck, and Markus Rudolf
1M and the 1Y US-$ vertex after denominating it in €. Table 5 shows, how the
four fixed income positions are expressed as cashflows. The total value of the
fixed income portfolio is 6’231’511 €. Multiplying the synthetic cashflows by
1.65, which is the 95% quantile of the normal probability distribution, and by the
daily volatility yields the VaRs due to daily changes of the risk factors. For instance according to table 5, the maximum potential loss of the fixed income portfolio due to changes of the 5Y Euro interest rate is 8’405 €. Adding all figures in
the last column of table 5 yields 12’803 € which represents the total fixed income
risk position if diversification effects were ignored. The sum of the volatilities is
equal to the volatility of the portfolio only if the correlation would be perfect between all risk factors. Table 3 shows that this is not the case.
Table 5. Summary of the mapping of all fixed income positions
Bundesanleihe
Land
NRW
Zerobond
1Y €
48’454
166’129
3Y €
65’701 1’508’604 -128’136
5Y €
876’929 1’385’125 -132’979
1M $
1Y $
Total
991’085 3’059’858 -261’116
Undiversified VaR
Diversified VaR
Synthetic
cashflow
(CF)
Vola
· 1.65
214’583
1’446’169
2’129’075
1’322’465 1’322’465
1’119’218 1’119’218
2’441’684 6’231’511
0.0485
0.2375
0.3948
0.0017
0.0748
T-Bill
VaR
= CF ·
vola ·
1.65
104
3’435
8’405
22
837
12’803
12’337
All figures are in €, Vola = daily price volatility in %.
Rather, the portfolio of fixed income instruments is a portfolio consisting of five
imperfectly correlated risk factors. Therefore, the risk of the portfolio has to be
calculated with respect to the correlations
0.31
0.27
0.00
 1

1
0.98 − 0.06
 0.31
(104 3'435 8'405 22 837)⋅  0.27 0.98
1
− 0.05
 0.00 − 0.06 − 0.05
1

0.59
0.30
 0.25 0.61
0.25  104 

 
0.61   3'435 
0.59  ⋅  8'405  (2)

 
0.30   22 

 
1   837 
= 12'337
Hence, the diversified VaR is below the undiversified VaR. The next step is to
analyze the risk of the stock portfolio. The VaR is equal to 1.65 times the volatility of the risk factor times the market value of the position times the beta of the
share with respect to the risk factor. For the Microsoft shares, the market value is
2’826’073 € and 1.65 times the daily volatility is 1.19%. The beta is given in Table 1, it is 1.1. From this follows the VaR for the Microsoft holdings in the portfolio
The New Basel Capital Accord
VaR Microsoft =1.65 ⋅ Market Value ⋅ β ⋅ σ S& P500
87
(3)
= 2'826'073 ⋅1.1 ⋅1.19% = 37'033
There are two other instruments which are exposed to stock market risks: The investments in DaimlerChrysler and in the DAX put options. The market value of
the DaimlerChrysler shares is 346’500 €. The beta to the DAX is 1.2. Hence, the
investment into the DaimlerChrysler shares can be compared to 1.2 · 346’500 =
415’800 € invested in the DAX (long DAX position). The DAX put options are
exposed to the DAX as well. Since the delta of the put options is -0.52, they can
be replicated by a short position in the DAX. The total market value of DAX indices underlying the put options 500 ⋅ 0.52 · 3’821 = 993’439 € (short DAX position). As indicated in table 6, adding up the long and the short position yields a total DAX exposure of -577’639 €. Again, three original positions (Microsoft,
DaimlerChrysler shares and DAX put options) are split up into two cashflows:
The first cashflow is exposed to the DAX, the second to the S&P 500. Multiplying
these cashflows by 1.65 and by the daily volatilities shows that the maximum loss
of the portfolio with respect to changes of the DAX is 11’350 € (the sign is not
important due to the assumption of normally distributed returns) and 37’033 €
with respect to changes in the S&P 500. The undiversified VaR which is given by
the sum of both turns out to be 25’683 € (table 6). Again, this would assume perfectly correlated cashflows. According to table 3 the coefficient of correlation of
the two risk factors is 0.47, which is substantially below 1. Therefore, the portfolio
risk principle illustrated in equation (2) needs to be applied. Surprisingly, the diversified VaR in table 6 is higher than the undiversified VaR. Assuming perfect
correlation between the returns on the DAX and the S&P 500 implies a perfect
negative correlation between the net short DAX position and the long position in
the S&P 500. Therefore, assuming perfect correlation assumes having a perfect
hedge for the risk of the DAX investment. Using the actual coefficient of correlation (0.47) instead implies replacing the perfect hedge by an imperfect hedge
which necessarily increases risk.
Table 6. Summary of all stock positions
DaimlerChrysler
Microsoft
DAX Put
DAX
346’500
1’910’460
S&P
2’826’073
Beta
1.2
1.1
Delta
-0.52
Undiversified VaR
Diversified VaR
All figures in €, Vola = daily price volatility in %.
Delta / beta
weighted
cashflow
(CF)
-577’639
3’108’680
Vola
·1.65
VaR
= CF·1.65
· vola
1.96
1.19
-11’350
37’033
25’683
33’254
88
Claudia Holtorf, Matthias Muck, and Markus Rudolf
The final risk category which has to be analyzed is the foreign exchange risk.
There are two foreign exchange positions, the US T-Bill and the Microsoft share.
The market values of these positions in € are 2’426’630 and 2’826’073, respectively. Given a daily volatility of the exchange rate multiplied by 1.65 of 0.9833%
yields a VaR of 51’651 €. This is shown in table 7.
Table 7. Summary of the foreign exchange positions
Risk factors
€/$
VaR
T-Bill
Microsoft
Cashflow
Vola · 1.65
2’426’630
2’826’073
5’252’702
0.9833
VaR
= CF · 1.65 ·
vola
51’651
51’651
All figures in €, vola = daily price volatility in %.
Table 8. VaR and risk factor exposures for the mapped positions
Risk factors
1Y €
3Y €
5Y €
1M $
1Y $
$/€
DAX
S&P 500
VaR in €
104
3’435
8’405
22
837
51’651
-11’350
37’033
Risk exposures:
Building blocks
214’583
1’446’169
2’129’075
1’322’465
1’119’218
5’252’702
-577’639
3’108’680
All figures in €.
The last step is to aggregate the exposures of the eight portfolio risk factors, which
is done in table 8. The original portfolio has been decomposed into eight risk factors. Each risk factor can be described in terms of volatilities and correlation coefficients (table 3). Table 8 shows that the most significant source of risk for the
portfolio is the €/$ exchange rate. The value of the portfolio may go down over the
course of the next day by more than 51’651 € with a confidence level of 95%. The
VaR of the portfolio depends on the coefficients of correlation (table 3) and the
VaR of the single risk factors (table 8). It turns out to be
The New Basel Capital Accord
′
0.31
0.27
0.00
0.25
0.27
  1
104
 

1
0.98 − 0.06 0.61
0.45
 3'435   0.31
 8'405   0.27
0.98
1
0.42
− 0.05 0.59
 

  0.00 − 0.06 − 0.05
 22
1
0.30 − 0.27
 ⋅

837
0
.
25
0
.
61
0
.
59
0
.
30
1
0.03
 

 51'651   0.27
0.45
0.42 − 0.27 0.03
1
 

11
'350
0
.
31
0
.
43
0
.
40
0
.
03
0
.
33
0
.49
−
−
−
−
−
−
 

 37'033   − 0.11 − 0.53 − 0.55 0.24 − 0.15 − 0.40
 

− 0.31
− 0.43
− 0.40
0.03
− 0.33
− 0.49
1
0.47
− 0.11  104


 
− 0.53   3'435 
− 0.55   8'405 

 

0.24   22
⋅

 
− 0.15   837

− 0.40   51'651 

 
0.47   − 11'350


1   37'033 
89
(4)
= 56'033.
This is 0.59% of the total portfolio value (9’470’097 €). Figure 1 depicts the probability distribution of the portfolio and the VaR.
4%
Portfolio value = 9'470'097 EUR
VaR = 56'033 EUR
Probability
3%
2%
1%
VaR
0%
9.35
9.40
9.45
9.50
9.55
9.60
Portfoliovalue after 1 day in Mio. EUR
Fig. 1. One day VaR on a 95% confidence level
Calculating the VaR is the essential prerequisite for determining the capital requirement with respect to market risk. The regulation of most countries, including
the German KWG, is inspired by the 1996 amendments of the Basel Accord. The
capital requirement according to the KWG based on internal models is based on
the VaR with a 99% (one-tailed) confidence level and a 10 day time horizon. In
the case study thus far, the RiskMetricsTM guidelines have been used, i.e., a 95%
confidence level and a 1-day holding period. Fortunately, it is fairly straightfor-
90
Claudia Holtorf, Matthias Muck, and Markus Rudolf
ward to transform the RiskMetricsTM-VaR into a Basel-VaR. Dividing the BaselVaR by the square-root of 10 times 2.33 (which is the 99% quantile of the normal
probability distribution) yields the same value as the division of the RiskMetricsTM-VaR by 1.65.
VaR Basel
2.33 ⋅ 10
=
VaR RiskMetrics
1.65
⇔
4.47 ⋅ VaR RiskMetrics = VaR Basel
(5)
It follows from equations (5) that the RiskMetricsTM-VaR just needs to be multiplied by 4.47 to receive the Basel-VaR, hence
4.47 ⋅ 56'033 = 250'977 = VaR Basel .
(6)
The specification of the significance level implies that there remains a probability
of 1% that the realized loss exceeds this value. In order to protect financial institutions against these unlikely occurrences, the capital requirement is actually set
above the VaR level. According to the KWG - Principle 1 (see section 32), the
capital requirement is based on the maximum of the VaR calculated in (6) and of k
times the average of the VaR for the last 60 days (one quarter, t: today)

1
k ⋅ max VaR t −1;
60

60

i =1

∑ VaR t −1 
(7)
The constant k is a special scaling factor, which is typically set equal to 3 and
which accounts for very unlikely loss events. If the model is not reliable then it
can be even higher. Whether the model is reliable or not is determined by an overrolling back testing procedure: If the 10 days losses for the last 250 trading days (1
year) exceeds the VaR calculated in (6) by at least three times, then the capital requirement multiplier is increased stepwise by values between 0.4 and 2. The multiplier increases the capital requirement significantly. Saunders 1997, states that
the idea of a minimum multiplication factor of 3 is to create an "incentive compatible" scheme6. Specifically, if financial intermediaries using internal models
constantly underestimate the amount of capital they need to meet their market risk
exposure, regulators can punish those banks by raising the multiplication factor.
Additionally a high specific multiplication factor provides an incentive to improve
the internal model’s quality. On the other hand, the scaling factor makes the use of
internal models relatively unattractive, although they capture the real risk exposure more appropriately than the standard model. This issue will be addressed in
greater detail below. The Federal Banking Supervisory Office in Germany
(BAFin) states that for the 15 German banks the supplements on the multiplier
varies from 0.0 up to 1.87. If it is assumed, that the multiplier in the case study
considered here is only 3, then the capital requirement is
Capital requirement = 3 ⋅ 250'977 = 752'931
6
7
See Saunders 1997, p. 172.
See BAFin, annual report 2003, p. 103, http://www.bafin.de.
(8)
The New Basel Capital Accord
91
This is 8.0% of the total market value of the trading book which is 9’470’097 €. In
the next section, this value is compared to the capital requirement according to the
standard model.
We are now in a position to compare the capital requirements of the trading book
according the BIS-standard-model and the VaR approach. Banks have the opportunity to choose between an internal model, as it was outlined, and the standard
approach. The standard approach has different capital requirements for stock, currency, option and fixed income positions. For Germany, the capital requirement
according to the standard model can be found in the Kreditwesengesetz (KWG)
principle 1. Additional to prestige reasons any bank will prefer the standard model
if the capital requirement is lower than for the VaR approach. References to the
standard approach are Crouhy et al. 1998 or Leippold and Jovic 1999. In Germany, only 15 banks use internal models for the calculation of the capital requirement. This is due to the more complex calculation and qualification procedure and due to the fact that the resulting capital requirement is not necessarily
below internal models.
As a first step, the fixed income positions of the trading book (see table 9) are
analyzed. In this case the specific risk of the fixed income positions is zero. The
capital requirement for market risk of fixed income positions in the standard
model is calculated on the basis of the time to maturity of the instrument. The
longer the maturity is, the higher the capital requirement will be. Furthermore, due
to the fact that bonds with low coupons have higher durations (i.e., interest rate
risks), the standard model distinguishes between fixed income positions with coupons above and below 3% of the principal amount. Table 9 shows the different
maturity bands, the risk weights and the open long and short positions of the trading book which has been specified in table 1. The 15 maturity bands are merged
into three maturity zones. Depending on the coupon payment (greater or smaller
than 3%), these zones refer to different maturities of the fixed income instruments.
Table 9 indicates a net long position of 2’426’630 € in the third maturity band
which arises from the US T-Bill. In maturity band 7, the bond of the state of NRW
amounts to 3’046’500 €. In maturity band 8, there is a long position (Bundesanleihe) and a short position (Zerobond AA). Since the risk weight is 2.75%, the
risk-weighted long position is 27’466 € and the risk weighted short position is
7’181 €. The risk-weighted open position in maturity band 8 is therefore the difference between the two figures, specifically 20’285 €. This amount has to be
brought into capital. Furthermore, parts of the closed position have to be weighted
by capital as well. In maturity band 8, there is a closed position of 7’181 €. The
capital requirement according to the standard model is 10% (= 718 €) of that
amount which is the so called vertical disallowance factor.
Maturity
band
Coupon > 3%
1
Years <
1M
3M
6M
12M
1.9Y
2.8Y
3.6Y
4.3Y
5.7Y
7.3Y
9.3Y
10.6Y
12Y
20Y
Coupon < 3%
Years > Years < Years >
1
0
1M
0
2
1M
3M
1M
3
3M
6M
3M
4
6M
12M
6M
2
5
1Y
2Y
1Y
6
2Y
3Y
1.9Y
7
3Y
4Y
2.8Y
3
8
4Y
5Y
3.6Y
9
5Y
7Y
4.3Y
10
7Y
10Y
5.7Y
11
10Y
15Y
7.3Y
12
15Y
20Y
9.3Y
13
20Y
10.6Y
14
12Y
15
20Y
Capital requirement for the open positions
Capital requirement for all positions
Zone
3’046’500
998’760
2’426’630
Long
261’116
Short
Open positions
0.00
0.20
0.40
0.70
1.25
1.75
2.25
2.75
3.25
3.75
4.50
5.25
6.00
8.00
12.50
Risk
weights
in %
Table 9. Capital requirements of the fixed income positions according to the standard model
718
98’538
99’256
68’546
20’285
7’181
68’546
27’466
Open positions in the
bands
9’707
7’181
Short
Vertical
disallowance
10%
9’707
Long
Risk weighted positions
92 Claudia Holtorf, Matthias Muck, and Markus Rudolf
The New Basel Capital Accord
93
The sum of the risk-weighted open positions is 98’538 € which yields a total capital requirement of 99’256 € due to interest rate risk. In addition to the vertical disallowance factor, the standard approach requires three types of horizontal disallowance procedures. The first horizontal disallowance refers to open positions
within the zones. The-risk weighting factors are 40% for the first zone, 30% for
the second zone and 30% for the third zone. The second horizontal disallowance
refers to closed positions between neighboring zones, the risk weight is 40%. And
finally, horizontal disallowance between the zones 1 and 3 requires a capital ratio
of 100%. Since there is only one position in each zone in the example considered
here, horizontal disallowance is not necessary. Only in zone 3, there is a long position (Bundesanleihe) and a short position (Zerobond AA). Since both positions are
in the same maturity band, the capital requirements for the open part of the net position (the minimum out of 27’466 € and 7’181 €) is already covered by the vertical disallowance. The capital requirement of the fixed income positions is 99’256
€ in total. This is shown in table 9.
In addition to the fixed income positions, there are also capital requirements for
the stock, the foreign exchange and the option positions of the portfolio. For stock
positions, there is a capital requirement of 8% of the market value due to (systematic) stock market risk and 4% due to (unsystematic) specific risk. The market
value of the DaimlerChrysler holding is 346’500 €. The capital holding is 12% of
the position which is 41’580 €. The second stock position is invested in Microsoft
shares. The market value of these holdings is 2’826’073 € which results in a capital requirement of 339’129 €. For the foreign exchange positions, a capital ratio of
8% is required. The T-Bill and the Microsoft position are denominated in foreign
currency ($). Therefore, the capital requirement is 8% of 5’252’702 € which is
420’216 €. For the option position, there are different alternatives for calculating
the required capital. In our example the method implies covering the market value
of the position which equals 86’750 €.
Table 10. Comparison of the capital requirement according to the RiskMetricsTM and the
standard approach
Standard model
Stocks and options
Fixed income positions
Foreign exchange positions
Sum (Capital requirement)
467’459
99’256
420’216
986’931
VaR model
VaR
Capital require(RiskMetricsTM)
ments
56’033
752’931
All figures in €.
Adding these figures to the capital requirement for the fixed income positions
yields the situation depicted in table 10. Diversification effects are not taken into
account by the standard model. The sum of the capital requirements for stocks, options, fixed income and foreign exchange positions is 986’931 €. This is a capital
requirement of 10.4% of the trading book due to market risk. The RiskMetrics TM-
94
Claudia Holtorf, Matthias Muck, and Markus Rudolf
based internal model yields a VaR of 250’977 € based on a ten days holding period and a 99% confidence level. If this amount is multiplied by a scaling factor of
3 (which is necessary because the internal model is used to cover specific and general market risks) under the assumption a sufficient back testing procedure the
capital requirement is 752’931 €. The capital ratio is 8.0% and thus substantially
lower than that of the standard model.
The partial use of internal models is also feasible, e.g. a bank may choose to use
internal models only for their fixed income positions. This partial use opens the
door for cherry picking, i.e. banks may compare the different capital requirements
of their portfolios for each method and choose the cheaper option. But this is indeed not a real world problem. The partial use is only an option for the first years
to force the use of internal models for capital requirements. Banks can use the internal models for their main portfolios and use the standard model for small investments in other instruments. The major reason for using internal models for
capital requirement is the reputation of the bank. They want to show, that they use
the most modern technology of risk measurement.
As pointed out earlier, most German banks do not apply internal models in order
to calculate the capital ratio for market risks; they prefer the standard model for
obvious reasons: Internal models require more (costly) know-how. Although the
internal model presented here describes the risk position of the portfolio more appropriately than the standard model and although internal models do regard diversification effects between different asset classes, they do not necessarily lower the
capital requirement. Capital is expensive. Therefore banks want to achieve a capital ratio which is as low as possible. The results of this study show that the frequently used standard framework for market risk management is not appropriate
in order to describe the real risk position of the trading book of a bank but it implies similar or sometimes even lower costs. It is therefore not surprising that a
large number of German banks apply internal models in order to control their
market risk but use the standard framework for the official risk reporting.
Most of the new Basel rules deal with the minimum capital requirement for the
credit risk of the banking book. The simplistic weighting scheme of the first Basel
Accord is replaced by more sophisticated ones. Basically, a bank has the choice
between two general alternatives including the Standardized Approach and the Internal Ratings-Based Approach (IRB Approach). So far, there is only one risk
weight for commercial companies although their credit-worthiness might be substantially different. The overall intention of the Basel Committee is to acknowledge this fact and to lower the capital requirement for those banks with a good
credit portfolio while the opposite is true for banks with a bad one. Therefore, the
new Accord also comprehends detailed rules for a better recognition of credit risk
mitigation techniques (e.g. guarantees, financial collateral, credit derivatives etc.).
Finally, Basel II addresses the issue of securitization in great detail to prevent
The New Basel Capital Accord
95
banks from “capital arbitrage” (i.e. enabling banks to avoid maintaining capital
commensurate to the risks they are exposed to).
Under both the Standardized and the IRB Approach the risk-weighted assets
(RWA) must be determined in a first step. In a second step the new Basel Accord
stipulates that the minimum capital requirement (MCR) satisfies
MRC
(CRM + CRO )⋅12.5 + RWA
= 8.0% ,
(9)
where CRM is the capital requirement for market risks as determined above and
CRO is the capital requirement for operational risk as described bellow8. The sum
of CRM and CRO is multiplied by 12.5, the reciprocal of 8%. The RWA can be
obtained by multiplying the single credit risk exposures with the correct credit
weight and adding up the results9.
The Standardized Approach relies on external ratings from rating agencies like
Standard & Poor’s or Moody’s. Table 11 shows the risk weights for claims on
rated as well as unrated corporates and claims in the retail portfolio, which comprises personal loans, revolving credit lines, small business facilities, etc. If a
company is unrated then a claim enters the RWA with its full amount because the
risk weight is 100%. However, if the company has an A+ rating then only 50% of
the claim must be considered in the RWA. The better the credit rating the smaller
is the corresponding credit weight und thus the additional capital required.
Table 11. Risk weights in the standardized approach for claims on corporates10
Rating
Rating
AAA to
AA-
A+ to A-
Risk
Weight
20%
50%
Corporates
BBB+ to
Below BBBB100%
150%
Retail
Unrated
100%
75%
Alternatively, the IRB-approach can be employed. This approach is characterized
by the fact that banks estimate the probability of default (PD) based on internal
models where PD must be greater or equal to 0.03%. The new Accord distinguishes between the Foundation Approach and the Advanced Approach. In the
Foundation Approach the bank determines the probability that a claim may default
within the next year. Furthermore, it assumes standardized values for the loss
given default (LGD)11, the exposure at default (EAD), and the effective maturity
8
See paragraphs 40 and 44 of the new Basel Accord.
The Basel Committee might require banks using the Internal Ratings Based-approach outlined bellow to apply a scaling factor on the sum currently estimated to be 1.06, see paragraph 44 of the new Basel Accord.
10 See paragraph 50 of the new Basel Accord which follows the notation of Standard &
Poor’s although rating schemes of other external credit assessment institutions could
equally be used.
11 The LGD is the percentage amount of the claim lost upon default.
9
96
Claudia Holtorf, Matthias Muck, and Markus Rudolf
(M). In the Advanced Approach, banks determine all parameters based on their internal models. The variables PD, LGD, EAD and M are plugged in a formula
which returns the contribution to the RWA of a particular claim. Based on this risk
weights can be determined. However, in contrast to the standardized approach this
formula only produces capital requirements for the unexpected losses (UL). Expected losses (EL) are taken into account in such a way that the difference between EL and provisions may be included or must be deducted from regulatory
capital12.
300%
Retail: Residential
Mortgages
Corporates
Risk weight
250%
200%
150%
100%
50%
24%
22%
20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
0%
Probability of default
Fig. 2. Risk weights for claims on corporates as well as for residential mortgages13
Figure 2 shows the risk weights for claims on corporates as well as for residential
mortgages. The calculations assume that LGD = 45%, EAD = 100 €, and M = 2.5
years. Similar graphs can be drawn for other types of claims. Furthermore, table
12 compares the risk weights of the Standardized and the IRB-Approach to each
other. It turns out that the IRB-approach may lead to more favorable risk weights
for debtors with a good creditworthiness while the opposite is true for companies
with a bad credit rating. Therefore, for banks with a good credit portfolio it might
be worthwhile to adopt the IRB-approach. This is especially true for German
banks that have a lot of costumers from the so called Mittelstand (medium sized
companies). Typically, these companies do not have a credit rating. Thus, a credit
weight of 100% would have to be assigned in the Standardized Approach potentially leading to increased interest for corporate loans.
12
The rules for the determination and treatment of EL as well as the provisions under the
IRB-approach are outlined in the paragraphs 43 and 374 – 386 of the new Basel Accord.
13 For the risk-weight formulas see paragraphs 272 and 328 of the new Basel Accord.
The New Basel Capital Accord
97
Table 12. Rating, corresponding statistical default probabilities and risk weights in the
Standardized and the IRB-Approach14
Rating
Stat. Prob.
Stand. Approach
IRB
AAA
0.00%
20.0%
AA
0.00%
20.0%
A
0.05%
50.0%
BBB
0.22%
100.0%
BB
0.94%
100.0%
B
8.38%
100.0%
CCC
21.94%
150.0%
14.4%*
14.4%*
19.6%
46.2%
90.3%
180.7%
242.3%
* Default probability set equal to the minimal probability of 0.03%
The new Basel Accord explicitly addresses operational risk as another important
source of risk. Therefore, a bank needs to hold capital in order to protect against
losses from it. “Operational risk is defined as the risk of loss resulting from inadequate or failed internal processes and systems or from external events. This definition includes legal risk, but excludes strategic and reputational risk.” 15 To measure
operational risk, the new Accord provides three methodologies: The Basic Indicator Approach, the Standardized Approach and the Advanced Measurement Approaches (AMA). Banks using the Basic Indicator Approach are required to hold
capital equal to a fixed percentage of the average annual gross return over the previous three years. The Standardized Approach is similar: The capital charge is determined from the gross income of several business lines of the bank. Finally, the
banks may use their internal systems to calculate the capital requirement (Advanced Approach). The Standardized Approach and Advanced Approach must fulfill several quantitative and qualitative criteria outlined in the Accord 16.
The focus of this chapter is to analyze the risk exposure and the implied capital
requirements for an arbitrary trading book based on two approaches: the internal
RiskMetricsTM-based and the so-called standard approach. It has been shown that
the capital requirements for the standard model may be below those of the internal
model. In cases like this it might be advantageous for a bank to turn to internal
models as an alternative methodology despite the greater resource and know-how
14
The default probabilities are the cumulative default probabilities for a 1-year time horizon 1981 – 2000. Source: Standard & Poor’s.
15 See paragraph 644 of the new Basel Accord.
16 For the qualifying criteria of the Standardized Approach and the Advanced Approach see
paragraphs 660 – 663 and 664 – 679 of the new Basel Accord respectively.
98
Claudia Holtorf, Matthias Muck, and Markus Rudolf
intensity. However, a drawback of internal models is the requirement to release internal bank information to the regulatory authority to ensure compliance.
It has to be kept in mind that risk-taking is still a core competence of banks.
Bank managers are responsible for managing their risk exposure and increasing
shareholder value. In contrast, regulators are responsible for the stability of the financial system but not for the stability of single banking institutes. However, they
have erected regulatory barriers to force banks into using the inferior standard
model (only 15 banks use the standard model in Germany).
The case study considers the market risk of the trading book only. The total
capital requirement of the bank is obtained by taking into account the credit risk
exposure from the banking book and the operational risk as well. The basic methodologies to determine credit risk include the Standardized Approach and the
IRB-Approach. It turns out that it might be worthwhile especially for banks with
good credit portfolios to adopt the IRB-approach. Finally, the assessment of operational risk is briefly outlined. However, no market standard has evolved in this
field yet.
Basel Committee of Banking Supervision (1996) Amendment to the Capital Accord to Incorporate Market Risks
Basel Committee of Banking Supervision (2004) International Convergence of Capital
Measurement and Capital Standards
Crouhy M, Galai D, Mark R (1998) A New 1998 Regulatory Framework for Capital Adequacy: Standardized Approach versus Internal Models. In: Alexander C (ed.) Risk
Management and Analysis. Wiley, Chichster et al., pp. 1-36
Gupton GM, Finger CC, and Bhatia M (1997) CreditMetrics™ Technical Document. New
York, J.P. Morgan Bank
Leippold M, Jovic D (1999) Das Standardverfahren zur Eigenmittelunterlegung: Analyse
der Wahlmöglichkeiten. Finanzmarkt und Portfolio Management 13: 260-290
Markowitz HM (1952) Portfolio Selection. The Journal of Finance 7: 77-91
Saunders A (1997): Financial Institutions Management – a Modern Perspective. McGraw
Hill, Chicago et. al., 2nd edition
Alois Paul Knobloch 1
1
Department of Accounting and Finance, University of Hohenheim,
D-70593 Stuttgart, Germany
Abstract: This article surveys several applicational as well as theoretical aspects
of Value at Risk as a measure of risk. First, we compare different calculation
methods with respect to accuracy, implementational issues as well as suitability
for resource allocation and optimization. We contribute to capital allocation
based on Value at Risk and provide an optimization model. Afterwards, we concentrate on shortcomings of Value at Risk as a measure of risk from a theoretical
point of view. The focus is on the relation to decision theory and to coherent
measures of risk. Alternatives to Value at Risk such as the lower partial moment
one or the tail conditional expectation are included. We give some reasons to prefer the latter as a measure of risk.
JEL classification: G21, G22, M23.
Keywords: Value at Risk, Capital Allocation, Risk Measures, Risk Management, Tail Conditional Expectation
6JG%QPEGRVQH8CNWGCV4KUMCPFKVU4QNGKP%QPVGO
RQTCT[4KUM/CPCIGOGPV
Well-known corporate disasters due to failures in controlling financial activities
have revealed a tremendous lack in assessing financial risks. The answer that was
given by the financial industry as well as regulators to fill the gap was the implementation of sophisticated risk management systems that are mandatory even for
non-financial enterprises in Germany since 19981. Within this framework, a meas*
1
This is a revised version of the article “Value at Risk: Tool for Managing Trading Risks”
published in the first edition together with Wolfgang Eisele. Permission of the former coauthor is granted. Special thanks to Michael Jaeger for his helpful comments.
This is due to § 91 AktG, that was introduced by the “Gesetz zur Kontrolle und Transparenz im Unternehmensbereich” (BGBl. I 1998, 786).
100
Alois Paul Knobloch
ure of risk, the currently popular Value at Risk (VaR), has attained a crucial role,
primarily but not exclusively for financial institutions2. VaR has a variety of applications some of which we will describe at the end of the first section. We will focus on the question of whether VaR meets the requirements imposed explicitly or
implicitly by these applications. Thus, section 2 deals with the accuracy of the calculation methods and their appropriateness for its use as a management tool. We
show how the delta-normal calculation method can be employed as an allocation
and optimization tool (section 3). In the last section, we will provide some criticism on VaR from a theoretical perspective and discuss alternatives, especially the
tail conditional expectation. First of all, we have to introduce the notion of VaR
and state more precisely its applicability to specific risks.
(
)
Given a portfolio whose value P t , r1t , K , rnt is a function of time t and the values of a certain number of risk factors at that time rit (i = 1, K , n ) , then its associated Value at Risk is the negative difference of the future portfolio value and its
present value that will not be exceeded at a confidence level of 1 − α at the end of
a predefined (holding) period of length ∆ t . Formally we define
( (
) (
))
VaRαP, ∆ t = − Pα t + ∆ t , r1t + ∆ t , K , rnt + ∆ t − P t , r1t , K , rnt ,
(
)
where Pα t + ∆ t , r1t + ∆ t ,K, rnt + ∆ t is the lowest future portfolio value that satisfies
(
(
))
Prob P ≤ Pα t + ∆ t , r1t + ∆ t ,K , rnt + ∆ t ≥ α (e.g. Artzner et al. 1999 p 216)3; P
represents the future portfolio value as a random variable. Thus VaRαP, ∆ t is the
positive value of the α -quantile of the associated profit & loss (P&L) distribution
(e.g. Bühler and Schmidt 1998 p 89; Schmidt 1998 pp 38, 39; Guthoff et al. 1998
p 126)4. Since the valuation formula P (.) does not change, the portfolio structure
is assumed to be constant over the holding period. The holding period chosen is
usually one day, extended to ten days by regulators. It should reflect the time
needed to detect losses and close positions. The confidence level 1 − α will usually be 95% or 99%.
Even though the definition of VaR seems to be of great generality, it does not
cover all risks of interest. First, consider the trading book that contains marketable
2
3
For instance, DaimlerChrysler (2003) pp 124-127 employs Value at Risk analyses as part
of its risk management.
They use a definition that may produce different VaR for discrete probabilities. We as-
(
) (
)
sume Pα t + ∆ t , r1t + ∆ t ,K , rnt + ∆ t ≤ P t , r1t ,K , rnt , otherwise we define VaRαP, ∆ t as zero.
4
Sometimes VaR is defined as the loss from the expected value of the portfolio at the end
of the holding period, neglecting the time drift in the VaR-calculation, see Jorion (2001)
p 109; Huschens (1998) p 235.
Value at Risk
101
securities that will be held for only a short period, normally no longer than 90
days, on the bank’s own account. It is a reflection of the institution’s gambling
“with” the financial market. Consequently, there are distinct sources of risk associated with these securities5. First, such securities are subject to changes in market
prices that generate market risk. This comprises risk in interest rates, foreign exchange rates, stocks or commodities’ prices. VaR focuses on this risk. Second, for
securities representing debt there is some (additional) credit risk emerging. Primarily, statistical issues and calculation-related problems are obstacles in capturing credit risk by a VaR calculation. Even if these problems are mitigated in welldiversified portfolios, only one calculation method for VaR, Monte Carlo simulation, can incorporate credit risk adequately. There are risks as liquidity risk, which
arises when market prices are not reliable, legal risk, e.g. from uncertainties of
outstanding verdicts, and operational risk, that is due to human errors, organizational deficiencies, failures in the electronic system and so on6. These risks are
generically erratic and thus difficult, or even impossible to capture by VaR calculation methods. They must be handled by other components of the risk management system7. We now address the question of why VaR should, or even must be
calculated within a risk measurement framework.
#RRNKECVKQPUCPF4GIWNCVQT[$CEMITQWPF
The objective of VaR is primarily management information. It provides a simple
but comprehensive statement about the exposure to market risk on an aggregate
portfolio level. Thus, it may indicate the necessity for decisions to mitigate risks.
Furthermore, the calculated value directly indicates the capital needed to cover the
risks. This makes it amenable to regulatory purposes.
In 1996, the Basle Committee on Banking Supervision made internal models of
risk measurement, and explicitly VaR, acceptable as a basis for capital requirements on market risks (Basle Committee on Banking Supervision 1996 pp 39-50).
The Basle Committee suggests a ten-days holding period and a 99% confidence
level. However, its proposals are merely recommendations. In Germany, the obligation for all financial institutions to provide for market risks derives from the
5
6
7
For a description of the associated risks see Jorion (2001) pp 15-21; Bühler and Schmidt
(1998) pp 75-78; Kropp (1999) pp 7, 48-123. All risks primarily apply to transaction exposure. Nevertheless, in managing these risks we should be aware of the accounting effects of financial decisions and the existing conflicts between the objectives of internal
risk assessment and its reflection within the legal financial accounting system as shown
by Eisele (1997) pp 68-77. As a consequence, translation and transaction risks may not
coincide, as is pointed out in Dufey and Hommel (1996) pp 201, 202.
As far as it does not include model risk, which represents the risk of misspecification or
wrong calibration of financial models. This risk, in turn, is strongly related to the quality
of a VaR calculation.
There are minimal requirements that are prescibed by German Banking Supervision (see
Bundesaufsichtsamt für das Kreditwesen 1995), some are formulated by the Basle Committee for operational risk (see Basle Committee on Banking Supervision 1998, 2003).
102
Alois Paul Knobloch
Sixth Amendment to the German Banking Act and the corresponding Amendment
in Principle I (Grundsatz I, abbr.: Grds. I), both from October 19978. Principle I is
promulgated by the German banking authority (Bundesanstalt für Finanzdienstleistungsaufsicht9). It defines the extent to which credit risks and market risks have
to be backed by equity. For credit risk, eight percent (§ 2 Grds. I) of the credit’s
book value or an equivalent (§ 6 Grds. I) must be covered10. Besides a merely
rough “standard method”11, market risk can be quantified on a VaR basis12, with
holding period (ten days) and confidence level (99%) being the same as in the Basle proposal (§ 34 Grds. I). § 10 KWG defines the capital that can be employed to
cover the risks. It consists of capital stock and retained earnings (tier 1) 13, but also
includes supplementary capital (tier 2), e.g. from certificates of participation or
long-term subordinated debt. Furthermore, short-term subordinated debt (tier 3),
and the actual surplus of the trading book (tier 4) can be included 14.
As a consequence of the capital requirement, a fraction of the bank’s equity and
thus cost of capital may be assigned to VaR. In this regard, VaR represents an instrument to allocate limited resources (Jorion 2001 pp 379-381). Moreover, breaking down VaR to the level of business units or even positions makes it possible to
set limits for traders15. VaR is also used as the denominator for business returns.
(RA)RORAC [(risk adjusted) return on risk adjusted capital] measures are often
based on VaR as the economic capital employed (Lister 1997 pp 208-224;
Schröck 1997 pp 96-106; Poppensieker 1997 pp 48-52). In light of the abovementioned obligation to provide equity for market risks, the expected return is,
obviously, related to the cost of equity. If we adopt this external definition, or em8
Sechstes Gesetz zur Änderung des Gesetzes über das Kreditwesen (6. KWG-Novelle)
10/28/1997 (BGBl. I 1997, 2518) and Bundesaufsichtsamt für das Kreditwesen (1997a).
In Germany, financial institutions have been obliged to quantify market risk since 1995,
see Bundesaufsichtsamt für das Kreditwesen (1995).
9 The formerly responsible “Bundesaufsichtsamt für das Kreditwesen” has been incorporated into the new institution in May, 2002.
10 This is reduced if the debt is supposed to be less risky than a standard one (§ 13 Grds. I),
e.g. if the debtor is a financial institution (20% weight) or a (solvent) state (0%, no risk).
The new standard of the Basle Committee (Basel II) propagates that the weight for risky
assets should depend on an external rating, or on an approved internal rating, see Basle
Committee on Banking Supervision (2003).
11 For further description see Basle Committee on Banking Supervision (1996) pp 10-38;
Bühler and Schmidt (1998) pp 81-86; Johanning (1996) pp 289-291.
12 The VaR has to be multiplied by a factor of three. The factor is even augmented up to
four depending on the grade of inaccuracy the internal model reveals in a backtesting
procedure (§§ 33, 36, 37 Grds. I). § 32 Grds. I allows to use internal models that have
been approved by the supervision authority. This, in turn, calls for a VaR measure, see
Bundesaufsichtsamt für das Kreditwesen (1997b).
13
There are modifications for this definition and the following ones due to § 10 KWG.
14 There are limitations on tier 2, tier 3 and tier 4 capital to be employed depending on the
available tier 1, tier 2 capital respectively.
15 There is, however, no definite way to split the bank-wide VaR. We will make a proposal
for this in section 3.2.
Value at Risk
103
ploy an internally defined economic capital based on VaR, we have to properly
adjust the cost of equity that refers to its market value for our definition 16. In the
context of normally distributed returns, VaR is strongly related to the standard deviation of portfolio returns. Thus, it may be employed to adjust for differential
rates of return, similar to the Sharpe ratio (Jorion 2001 pp 394-398; Dowd 1999a,
1999b). This set of relationships suggests that VaR may serve as part of a performance measurement system.
Of course, the applications cited assume a certain accuracy of the VaR measure.
The calculation methods at hand meet this demand differently. In practice, there
is, however, a trade-off between the accuracy needed for distinct purposes and
some implementational issues. These comprise, for example, the methods’ demand for available data and the time interval between succeeding calculations, as
well as the resources consumed by the application of the methods. We will pick up
these issues in the presentation of the calculation methods we address next.
One may easily imagine that a VaR calculation for a portfolio containing a large
number of stocks, bonds in various currencies and with different maturities, as
well as derivatives, calls for simplification. Since our VaR measure is related to
the future, we have to model the uncertainty imposed on the future portfolio value
as a result of the uncertainty about the relevant risk factors. A first simplification,
therefore, refers to the number of risk factors that will be considered. For an exact
calculation, one would have to take into account all factors that influence the portfolio value and cannot be expressed in relation to each other (Read 1998 p 147).
For the portfolio outlined above, this would necessitate modeling one interest rate
for each maturity and the respective currency (spreads may also be added for
credit risk), considering all stocks, and, possibly, detecting additional risk factors
for the derivatives. This process, however, would be too costly, and the data might
not be available. Thus, we take some standard vertices on which the securities’
cash flows are mapped17.
The selection of appropriate risk factors ri (i = 1,K, n ) is a common feature of
all calculation methods, even though the selection may not be independent of the
model chosen. We consider the delta-normal, delta-gamma method, respectively,
16
We should be aware that the premium due to (value at) risk may only be an add-on to the
risk-free rate of return on the capital employed, see Schierenbeck (1999) pp 66-72; Poppensieker (1997) p 50. The derivation of the cost of equity from capital market theory assumes certain statistical properties of the returns, e.g. a normal distribution. These assumptions may not be met in the generalized framework of VaR. Therefore, employing
VaR can be a pitfall, see Johanning (1998) pp 74-80.
17 Mappings for different financial instruments are described in Read (1998) pp 127-147.
104
Alois Paul Knobloch
(
)
and the historical and Monte Carlo simulation. P t + ∆ t , r1t + ∆ t , K , rnt + ∆ t indicates
the starting point from which to differentiate these models. Basically, there are
two components for which different assumptions are made. The first one is how to
model the evolution of the risk factors between t and t + ∆ t , where we denote the
changes by
∆ ri
= rit + ∆ t − rit (i = 1,K, n ) or by the vector
∆r
. The second compo-
nent is the relationship P (.) that constitutes the portfolio value as a function of the
(future/current) values of the risk factors and that we further assume to have an
analytical representation18. Table 2.1 gives a short survey of the assumptions underlying the methods with respect to both items. We will pick up these issues in
what follows.
Table 2.1. Basic components of VaR models
Risk factor modeling
Functional relationship
Deltanormal
jointly normal distribution
first order Taylor polynomial
Deltagamma
jointly normal distribution
second order Taylor polynomial
∆ ri (i = 1,K, n )
Monte Carlo evolution according to stochastic processes
simulation
historical
changes
Historical
simulation
P (.)
(
P (r + ∆ r ) ≈ T1 P, r, r t
(
P (r + ∆ r ) ≈ T2 P, r, r t
)
)
function P (.) (full valuation)
function P (.) (full valuation)
According to Table 2.1, the delta-normal and the delta-gamma methods assume
jointly normally distributed risk factor changes, and approximate the valuation
function P (.) by first or second order Taylor polynomials.
Thus, the delta-normal method assumes
∆P=
∂P
⋅∆ t +
∂t
n
∂P
∑ ∂ ri ⋅ ∆ ri
i =1
with
∆r
~ N (µ , Σ ) 19.
(1)
18
Some inevitable model risk will remain in the mapping of the risk factors onto the portfolio value. However, we must for instance assume the Black/Scholes formula to be a suitable approximation for option pricing of the market.
19 µ = (µ ,K , µ ) ' indicates the vector of expected factor changes, and Σ = σ
1
n
ij i , j =1,K, n
( )
the variance-covariance matrix for the factor changes. Both refer to the holding period.
N (.) represents the Gaussian distribution function.
Value at Risk
105
The partial derivatives ∂ P ∂ ri (= di ) represent the sensitivities of the portfolio
value with respect to the individual risk factors. They constitute the vector d of
sensitivities. We also write d t for ∂ P ∂ t . As the linear combination of normally
distributed risk factors results in a normally distributed random variable, we get
∆ P ~ N (d t ⋅ ∆ t + d '⋅µ , d ' Σd ) . VaR as the negative α -quantile of this distribution
is therefore given by
(
VaRα = − d t ⋅ ∆ t + d' ⋅µ + zα ⋅ d' Σd
)
(2)
where z α denotes the α -quantile of the standard normal distribution. For a 95%
confidence level we have z α = − 1.65 . We will further assume α ≤ 0.5 ; neglecting the drift yields:
VaRα = zα ⋅ d' Σd
(3)
where µ = 0, d t = 0 .
As eq. (1) suggests, the delta-normal method allows us to model the risk factors
independently. If the risk factors are interest rates for different maturities, we may
model their changes to be stochastically independent, or according to some correlation. This issue generalizes the usual duration concept. Take, for instance, a portfolio consisting of two zerobonds with face values of c1 , c 2 , maturity dates T1 ,
T2 and prices P1 and P2 respectively. The present value of the portfolio is
P = c1 ⋅ e − (T −t )⋅r + c2 ⋅ e − (T −t )⋅r . From eq. (1), assuming d t = 0 , we have
1
1
2
− (T1 − t )⋅r1
2
⋅∆ r1 − (T2 − t ) ⋅ c2 ⋅ e − (T2 − t )⋅r2 ⋅∆ r2 and therefore we yield
∆P
= −(T1 − t ) ⋅ c1 ⋅ e
∆P
= − D1 ⋅ P1⋅∆ r1 − D2 ⋅ P2 ⋅∆ r2 20. Now, we model the continuous interest rate r2
equal to r1 and a constant spread s , so
∆P
= − D1 ⋅ P1⋅∆ r1 − D2 ⋅ P2 ⋅∆ r2 = − D1 ⋅ P1⋅∆ r1 − D2 ⋅ P2 ⋅∆ r1 = − DP ⋅ P⋅∆ r1 ,
which represents the well-known sensitivity of the portfolio value to a parallel
shift in interest rates based on duration in a continuous setting 21.
The delta-normal approach provides a simple calculation method, and it is a
highly adaptable tool for manipulating VaR. This is because eq. (2) [(3)] offers an
analytical representation of the portfolio’s VaR that is amenable to optimization.
Criticism of this approach is based on statistical aspects. First, the normal distribution is usually accepted for interest rate and stock price risk, but it does not
comply with option risk (e.g. Jorion 2001 p 229; Schröder 1996 pp 84, 88;
20
21
D := ∑t t ⋅ b ∑t b defines the duration of a portfolio (security), where bt represents the
t
t
present value of the cash flow at time t and where P = ∑ t bt .
However, the notion gets closer to key rate durations; see Schmidt (1998) pp 29-38. The
derived formula underlies the VaR concept only if Σ consists of identical elements.
106
Alois Paul Knobloch
Deutsche Bundesbank 1998 p 80). But even for the former risks, the normal distribution merely represents an approximation. In particular, perceived distributions
of interest rates show fatter tails compared to a normal distribution22. This means
that in reality extreme values have a higher probability than suggested by a normal
distribution23. Since VaR refers to extreme positions a portfolio can take, it may be
underestimated. But even if risk factor changes underlie a jointly normal distribution, we face a sampling problem. The accuracy for our VaR measure depends extremely upon the size of the sample at hand. Indeed, several methods can serve to
estimate the parameters of the normal distribution from the sample. They may differ in the weighting of the sample elements according to their time of appearance,
and may even account for fat tails24. Thus, a confidence interval for the calculated
VaR can provide useful information about its reliability25. Surely, the simple linearized function leads to further deviation from the true portfolio value. The
deviation increases with the “distance”26 of the vector of risk factor changes from
the evolution point. We may improve the calculation if we take into account a
portfolio’s convexity, i.e. the second derivatives27.
The delta-gamma approach extends the Taylor polynomial to incorporate convexity. Assuming zero drift, we approximate the change in portfolio value by
n
∆ P = ∑ d i ⋅∆ ri +
i =1
1
1 n n ∂ P2
⋅∆ ri ⋅∆ r j = d '∆ r + ⋅∆ r 'Γ ∆ r
⋅ ∑∑
2
2 i =1 j =1 ∂ ri ∂ r j
(4)
where Γ represents the suitable Hessian matrix and ∆ r ~ N (0, Σ ) .
Unfortunately, there is no closed form to calculate the α -quantile of ∆ P ‘s distribution28. Therefore, we may pursue a variation of the method that is referred to
as the delta-gamma maximum loss. As ∆ r 'Σ ∆ r obeys a χ 2 -distribution with n
degrees of freedom, the following optimization problem yields (the negative of)
a(n upper) bound for the (absolute value of the) α -quantile
1
min . d ' ∆ r + ⋅ ∆ r ' Γ ∆ r
2
s.t. ∆ r 'Σ ∆ r ≤ c1−α ,
where c1−α represents the 1 − α -quantile of the χ (2n ) -distribution.
22
See on this issue Duffie and Pan (1997).
For an analysis of various distributions improving tail estimates see Klüppelberg (2002);
see also Smith (2002) on extreme value theory.
24 See for a comparison of methods Davé and Stahl (1998).
25 See Huschens (1998) for confidence levels on equally weighted estimates. Jorion (1996)
p 47 suggests that “VaR should be reported with confidence intervals.”
26
Not necessarily the Euclidean one.
27 The values of trading portfolios with long positions in fixed-income instruments may be
underestimated. Since these securities usually have positive convexity, the linearized
function represents a lower bound for the true value, see Dufey and Giddy (1994) p 168.
28 For a detailed discussion of this topic see Read (1998) pp 63-116.
23
Value at Risk
107
Thus, we calculate the minimum of all changes in portfolio value over a range
(an ellipsoid) of risk factor changes that comprises a percentage according to our
confidence level. Clearly, the VaR cannot be higher than the negative of the calculated value. To solve the problem, however, we must have recourse to numerical methods29. Nonetheless, delta-gamma maximum loss complies with constraints
on the implementation in practice.
Criticism of the delta-gamma method is in some ways similar to that of the
delta-normal method. The fitting of the true function, however, is improved, especially for fixed-income instruments. Nevertheless, deviations from the current
portfolio value due to large changes of risk factor values might not be captured
adequately. Moreover, tests show that Black/Scholes option prices may not be
well approximated by the Taylor series of reasonable degrees (Locarek-Junge
1998 pp 220-225). Estimation error on Σ and the criticism of the appropriateness
of the normal distribution to describe movements of risk factors remain relevant.
These problems might be avoided by the use of simulation techniques.
Both historical and Monte Carlo simulation generate a distribution of portfolio
values from a sample of vectors of risk factor movements ∆ r i (i = 1, K , k ) . The
portfolio values are computed with the exact functional relationship (full valuation) – as far as possible. VaR is then calculated on the basis of the resulting distribution’s α -quantile. The methods differ, however, in creating the sample.
Historical simulation refers to a certain number of risk factor movements in the
past. For instance, a daily VaR could be calculated using the realized vectors of
factor changes ∆ r t −i for the last 250 days (i = 0, K , 249) . It is assumed that the
distributions that underlie the realized factor changes and the factor change for the
next day are identical. This implies stationarity. Since we draw our sample from
the “real” distribution, there is no need for modeling the distribution of risk factors. Thus, historical simulation is not exposed to model risk.
Stationarity is subject to criticism with respect to the sample size, which must be
large enough to guarantee statistical significance. However, stationarity cannot be
assumed for long observation periods, for example several years. It may not even
be valid over a shorter period, if relevant circumstances change (Beder 1995 p 17).
The problem of significance is aggravated by the fact that our focus is on extreme
and therefore low-probability values. Consider, for instance, a sample of 250 values that are ordered according to the associated losses. We take the third biggest
loss as our VaR at a 99% confidence level. Obviously, the result is very sensitive
to abnormal values, or simply to the randomness of the selected value in the interval between the preceding and succeeding ones. And, it will never exceed the
highest loss in the sample. Statistical refinements can mitigate but not eliminate
29
For references see Locarek-Junge (1998) p 217; Read (1998) p 127.
108
Alois Paul Knobloch
this problem30. Historical simulation may be appropriate when statistical distortions can be ruled out. If trends change, it may be misleading31.
Monte Carlo simulation, in contrast, does not assume stationarity of the distribution of risk factor movements. A vector of factor changes ∆ r i is assumed to be
the “result” of a stochastic process32. For this, the holding period is divided into
time intervals of length d t . Repeatedly employing the model that underlies the
process in common for all risk factors T times, with ∆ t = T ⋅ d t , yields one vector of factor changes.
Since the approach allows us to model a variety of relationships, including
spreads for credit risk, Monte Carlo is the most flexible method for calculating
VaR. Moreover, the size of our sample is not limited by an observation period of
the past. Thus, the α -quantile can be approximated by an arbitrary number of realizations as far as enough “real” random numbers can be produced. Furthermore,
the use of implied data may reduce the problem of estimating parameters solely
from the past. However, the application of Monte Carlo simulation implies an
enormous amount of effort, since the method requires the calculation of a path for
each element of the sample. As a consequence, Monte Carlo simulation consumes
several times the resources that other techniques need. These comprise not only
computational33 but also human resources for the specification of the model. Statistical errors may arise from the estimation of the model parameters. Moreover,
the construction of the model itself may be subject to misspecification. Hence,
Monte Carlo simulation is prone to model risk. This is especially true for crashes
that represent strong discontinuities in risk factor movements.
To summarize our description of methods, Table 2.2 presents a synopsis of their
pros and cons. As Table 2.2 reveals, the management information function is best
fulfilled by Monte Carlo simulation, unless the calculation cannot be completed in
time. Thus, for overnight VaR calculation the delta-normal approach may also be
appropriate. This, however, depends on the composition of the portfolio. Strong
option elements require at least an application of the delta-gamma, or the historical simulation. For the latter, we should be aware of the stationarity assumption.
30
See for statistical methods on this subject Ridder (1998) pp 165-173.
See the example provided by Beder (1995), especially p 15.
32 The method accounts for possible correlations between the risk factors.
33 Convergence related aspects of Monte Carlo simulation as well as an approach “to mitigate the computational burden” are presented by Moix (2001) pp 165-195.
31
Value at Risk
109
Table 2.2. Comparison of methods34
Method
Feature
Sources of
inaccuracy
- statistical
- due to
functional
approximation
Application
- implementation/computability
- allocation
to business
units
- optimization
Deltanormal
possibly
severe
- normal
distribution
- estimation
error
- risk factor
grid
- large factor
changes
favorable
- quick and
easy
Deltagamma
possibly
high
- normal
distribution
- estimation
error
- risk factor
grid
- large factor
changes
acceptable
- easy
- possible
- hardly
(separate
calculations)
- not suitable
- suitable
Historical
simulation
often acceptable
- risk factor
grid
Monte Carlo
simulation
least
severe
- model risk
(including
estimation
error)
- risk factor
grid
acceptable
- not too
consuming
(computational/human
resources)
- hardly
(separate
calculations)
- not suitable
costly
- very
consuming
(computational/human
resources)
- hardly
(separate
calculations)
- not suitable
- stationarity
- sample size
Of course, the reliability of the calculated VaR is also an important issue for performance measurement. Thus, Monte Carlo simulation is favorable, at least for the
total portfolio. If, however, the portfolio is to be allocated to business units, or
even lower levels, delta-normal should be used. This is because the other methods
do not allow separately calculated VaR to be added up to the VaR of the whole
portfolio. With delta-normal such a decomposition exists. Delta-normal also provides a basis for optimization. We refer to both items in the next section.
The simulation techniques have turned out to be very helpful instruments to calculate VaR. But, generically, they are not appropriate for rearranging a portfolio
structure. More guidance can be drawn from the analytical representation provided
by the delta-normal method. Hence, we will rely on this method to minimize a
portfolio’s VaR.
34
See also Jorion (2001) p 230.
110
Alois Paul Knobloch
The VaR calculation of delta-normal underlies the assumption of normally distributed portfolio returns. Hence, the well-known portfolio theory applies. Since we
will not assume a perfect capital market, the portfolio selection depends on the
specific circumstances. Thus, we will take the view of a portfolio manager who
seeks to minimize the portfolio’s VaR in a limited framework. Consider n securities the manager can add long or short to the portfolio. She chooses the respective
amounts of the securities so that the expected portfolio return is unchanged35, and
the additional net investment/time drift is zero36.
We assume that the risk factor changes have zero expectation. The prices of the
hedge securities and their sensitivity vectors are denoted by bi , h i respectively
(i = 1, K,n ) 37. The optimal hedge is the solution to the following problem
(P1)
min . z ⋅ d ' Σ d
α
n
∑ k i ⋅ bi = 0,
s.t.
i =1
n
∑ ki ⋅
i =1
∂ bi
= 0,
∂t
where d = d + k1 ⋅ h 1 + K + k n ⋅ h n describes the vector of sensitivities for the
hedged portfolio. The vector k = (k1 , K , k n ) ' represents the amounts of the hedging securities which are added to the portfolio. The solution of problem (P1) is
straightforward; we provide an example.
Consider a bank whose trading portfolio is divided into two parts, each belonging
to a business unit. The sub-portfolios consist of bonds whose cash flows are listed
in Table 3.1. The VaR has to be calculated over a ten-days holding period at a
95% confidence level.
35
We assume that the securities are efficient in a way that further reduction in risk with a
higher expected return will not be possible. The problem could be appropriately extended.
36 We adopt the zero-investment strategy from Sharpe (1994). A simpler hedging problem
is provided by Zöller (1996) pp 127-129. Emmer et al. (2001) investigate a more sophisticated portfolio selection problem that constrains VaR (Capital at Risk, respectively) and
that underlies a dynamic modeling of factor movements; see also Klüppelberg (2002) pp
38-58.
37 The sensitivity vector for the i th security is hi = (∂ b ∂ r ,K ,∂ b ∂ r )' .
i
1
i
n
Value at Risk
111
Table 3.1. Cash flow pattern
Business unit 1
Business unit 2
1 Y.
1
-101
2 Y.
1
4
3 Y.
-99
209
4 Y.
106
Portfolio
-100
5
110
106
The risk factors rt (t = 1,K ,4 ) are given by the continuous interest rates for zerobonds maturing 1, 2, 3 and 4 years ahead. The actual values of interest rates as
well as their correlations and ten-days standard deviations are listed in Table 3.2.
Table 3.2. Risk factor data
1 Y.
4.9
2 Y.
5.0
3 Y.
5.3
4 Y.
5.5
σr
0.069
0.099
0.122
0.140
correlations 1 Y.
2 Y.
3 Y.
4 Y.
1
0.85
0.80
0.75
1
0.90
0.82
1
0.92
1
rt [% ]
t
Let c tP denote the portfolio’s net cash flow at time t . Then, with ∂ P ∂ rt =
− t ⋅ ctP ⋅ e − rt ⋅t , the sensitivity vector before hedging is
− 2 ⋅ c 2P
⋅e
−2⋅r2
, − 3 ⋅ c3P
⋅e
−3⋅r3
, − 4 ⋅ c 4P
⋅e
−4⋅r4
(
d = − c1P ⋅ e − r1 ,
) ' . Hence, for the VaR before the
hedge we have VaR = z 5% ⋅ d' Σ d = 125.5 . Now, we turn to problem (P1) to reduce the portfolio’s VaR. Our hedging instruments are given by zerobonds maturing after 1, 2, 3, and 4 years respectively. Their sensitivity vectors are


i⋅ri
h i =  0, K ,0, −
i ⋅2
e −4
14
3 ,0, K ,0  ' (i = 1, K , 4 ) . The solution of (P1) is straightfori


ward. We will buy or sell short the hedging instruments according to the vector
k = (291.6, − 470.5, 271.2, − 103.8) ' , where the ordering follows maturity. We
might therefore buy zerobonds with maturities of one and three years, financed by
going short on two and four year maturities. Now, the risk measure is
VaR = z5% ⋅ d ' Σ d = 106. Hence, we reduced our VaR by 15% without an addi-
tional investment and without altering the portfolio’s expected rate of return38. □
38
Note the assumption of zero expectation for the risk factor changes. We should be aware
that additional long positions can call for new regulatory capital due to credit risk.
112
Alois Paul Knobloch
Problem (P1) represents just one example of how delta-normal can be employed
to “optimize” VaR. We could modify our problem by including different constraints or by minimizing regulatory capital requirements. Unfortunately, the other
calculation techniques are not amenable to such an optimization. Delta-normal is
also favorable for the allocation of VaR on different business units as it is shown
next.
There are various methods for calculating the VaR of sub-portfolios, and, basically, each method is arbitrary. For example, we can calculate the sub-VaRs separately. However, this procedure is problematic. Since we do not account for a subportfolio’s contribution to the risk reduction due to diversification, the VaRs for
the different sub-portfolios do not add up to the VaR of the overall portfolio. On
the other side, we can calculate an incremental VaR that represents the difference
between the portfolio’s VaR and the VaR calculated for the portfolio without the
securities in the respective business unit. Whereas the sub-VaRs of the former
method sum up to more than the overall VaR39, the latter method produces subVaRs that add up to less than the overall VaR. Therefore, it does not provide a
complete allocation of the overall VaR, e.g. for capital requirements. In both
cases, all calculation methods for VaR apply. Thus, we might employ the most accurate one, the Monte Carlo simulation. However, the allocation might be costly
and the result unsatisfactory. A suitable assignment of VaR to sub-portfolios must
be such that the sub-VaRs add up exactly to the overall VaR. We describe such a
decomposition that has a plausible interpretation40. The decomposition is based on
the delta-normal method.
We calculate the sub-VaR of a business unit e , separate from the sub-VaR for
the remaining portfolio. We will refer to the sub-portfolios by the indices e , re
respectively, and to the total portfolio by P . Once again, we neglect the portfolio’s drift. According to delta-normal, VaR is a function of portfolio variance 41.
Hence, we first consider how the portfolio variance changes with respect to the
sensitivity of risk factor i (∈ {1, K , n}) that is added by the sub-portfolio e .
∂ σ P2
∂ d ie
r
r
= 2 ⋅ 1i ' Σd = 2 ⋅ Cov ( ∆ ri , ∆ P ) , where 1i denotes the n -vector whose i th
component is one, the others being zero. With
∂ σ P2
∂ d ie
= 2 ⋅σ P ⋅
39
∂σ P
∂ d ie
, we describe
This is true except in the case where the positions of all sub-portfolios are perfectly positively correlated.
40 For different allocation rules including the one which serves as a basis for the following
exposition see Albrecht (1998) pp 246-253, especially 247-248.
41
2
2
(
e
VaR = z ⋅ σ P ; σ P = d' Σd = d + d
α
re
)' Σ (d
e
+d
re
).
Value at Risk
113
the relative change in the portfolio’s VaR with respect to the change in the risk
factor’s sensitivity as
r
Cov ( ∆ ri , ∆ P ) 1i ' Σd
∂σ P
∂ VaRP
=
=
=
=: β i .
d ' Σd
VaRP ∂ d ie σ P ⋅ ∂ d ie
σ P2
Note that β i is independent of the (relative) extent to which d is influenced by
the sub-portfolio’s sensitivity d ie . It is merely characteristic for the risk factor i
(Zöller 1996 pp 123, 124). The covariance indicates that β i measures some systematic risk that the risk factor adds to the portfolio. The β i (i = 1,K, n ) constitute the components of the vector β = Σd d ' Σd . If we multiply the subportfolio’s actual sensitivity by β i , we yield
β i ⋅ d ie =
∂ VaRP ⋅ d ie
, which represents the elasticity of the portfolio’s VaR
VaRP ⋅ ∂ d ie
with respect to the business unit’s sensitivity towards risk factor i . We denote the
(
(= d
)
'⋅β ) . We add up both fig-
sum of the elasticities for all risk factors by a e = d e '⋅β . Similarly, the corresponding sum for the remaining portfolio is a
ures and get
(
re
)
re
a e + a re = d e '⋅β + d re '⋅β = d e + d re '⋅β =
d ' Σd
=1.
d ' Σd
Hence, we can interpret a e and a re as (elasticity based) weights to allocate the
portfolio’s VaR. The decomposition is
VaR P = VaR P ⋅ d e '⋅β + VaR P ⋅ d re '⋅β .
142
4 43
4 14
4244
3
=: VaRe
=: VaRre
Unlike a separate VaR calculation for the business unit, the sub-VaR can be negative. In this case, the business unit even reduces the overall (Value at) risk.
Consider the bank from Example 3.1, and its position before the hedge. The decomposition on the business units 1 and 2 yields a 1 = 0.227 and a 2 = 0.773 .
Since VaR P = 125.5 , we get VaR1 = 28.5 and VaR2 = 97 .
The above derivation depends on the assumptions underlying the delta-normal
method. Suppose a portfolio has strong option positions. Consequently, we face a
dilemma of choosing between a sound decomposition of a poor VaR estimate and
a doubtful allocation of a good VaR estimate. Applying the weights from the decomposition based on delta-normal to the VaR calculated by Monte Carlo merely
represents a pragmatic solution.
The discussion so far has focused on practical issues of VaR as a management
tool. We have not questioned whether the VaR concept itself represents an appropriate measure of risk. We turn to this topic now by presenting some shortcomings
114
Alois Paul Knobloch
of VaR where we concentrate on decision theoretical issues. Furthermore, we discuss other measures of risk as alternatives to VaR.
We take the relation between VaR and expected utility as a starting point for criticism of VaR42. Rational behavior is assumed to follow Bernoulli’s principle (e.g.
Schneeweiß 1966 pp 32, 77, 78). Hence, decisions based on VaR should be in accordance with expected utility (maximization).
First, consider two alternative portfolios P and G for the same investment.
The changes in portfolio value are described by the random variables w P and wG
defined on appropriate probability spaces. For convenience we take each one
based on the interval [c ,d ] and furnished with the probability measure Q P or QG ,
respectively. The distribution functions are denoted by FP (.) and FG (.) . We say
P dominates G according to first order stochastic dominance (P f1 G ) , if
FP ( x ) ≤ FG (x ) ∀x ∈ [c , d ] and ∃x : FP ( x ) < FG ( x ) . This complies with expected
utility in the following sense:
P f 1 G ⇒ Pf UB1 G , where PfUB1 G means
EQ (u (x )) > EQ (u (x )) ∀u ∈ U1 = {u u ' (x ) > 0 ∀x} . Thus, if the probability distriP
G
bution of returns on portfolio P dominates the corresponding probability distribution for portfolio G , every rational investor with positive marginal utility prefers P to G . As P f 1 G also implies VaRαG ≥ VaRαP (∀α ) , i.e. according to our
VaR measure we do not prefer G to P
(Pf VaR G ) , VaR does not contradict ra-
43
tional behavior . Of course, positive marginal utility can usually be assumed for
all investors. However, situations where one portfolio dominates the other by first
order stochastic dominance will be very rare. Thus, the compliance we have
shown does not provide sufficient evidence to pursue VaR. Therefore, let us now
consider investors with positive and decreasing marginal utility, i.e. where
u ∈U 2 = {u u ∈U 1 ∧ u ' ' (x ) < 0 ∀x}. The utility functions of U 2 imply a riskaverse behavior of investors. This is a well-known assumption from capital market
theory. The new setting enables us to combine the notion of expected utility with
an ordering for a broader range of probability distributions. We introduce second
order stochastic dominance between the portfolios P and G as: P f 2 G if
x
x
∫c FP (s ) d s ≤ ∫c FG (s ) d s ∀x
42
x
x
c
c
and ∃x : ∫ FP (s ) d s < ∫ FG (s ) d s . It is well
We adopt the following statements from Bawa (1975) and the description of Guthoff et
al. (1998); Johanning (1998) pp 54-62; Kaduff (1996) pp 13-46.
43 Note that we may be indifferent with respect to VaR, even when one portfolio is dominated by the other one. Consequently, we might choose the dominated one.
Value at Risk
115
known that second order stochastic dominance complies with expected utility for
every investor represented by u ∈U 2 , i.e. P f 2 G ⇒ Pf UB 2 G . But now we have
P f2 G ⇒
/ P fVaRG . This means that if one probability distribution dominates the
other by second order stochastic dominance, VaR might recommend choosing
(strictly) the dominated portfolio, contrary to rational behavior. We provide an example for discrete probability distributions.
'ZCORNG
The portfolios P and G yield identical possible outcomes with different probabilities. The changes of portfolio value and the probabilities are given in Table
4.144. We add a portfolio H which we will refer to later.
Table 4.1. Portfolio value changes and probability weights
∆
P/ ∆ G/ ∆ H
Prob( ∆ P )
Prob( ∆ G )
Prob( ∆ H )
-5
-4
-3
-2
0
8
0.5%
2.5%
2.0%
3.0%
42%
50%
1.0%
2.0%
1.5%
3.5%
42%
50%
0.5%
3.0%
2.0%
2.0%
42%
50.5%
At a 95% confidence level, VaR for portfolio P equals 3, whereas portfolio G
has a VaR of 2. Thus, VaR suggests to prefer G . P , however, dominates G according to second order stochastic dominance. So, every risk-averse investor will
prefer P to G . Take, for instance, an investor whose utility function is
u( x ) = ln (2⋅ ∆ X + 11) + u 0 , where ∆ X denotes the change in portfolio value and
u 0 is a constant representing components that are equal for P and G . The expected utilities are E QP (u ( ∆ P )) = 2.773 + u 0 and E QG (u ( ∆ G )) = 2.769 + u 0 . □
The shortcoming exhibited emerges because VaR does not take into account the
extent to which the low-probability changes in portfolio value are below the
(negative) VaR45. This can be captured by a more general shortfall measure.
LPM n ( y ) =
y
∫c ( y − x )
n
⋅ f ( x ) d x defines a lower partial moment n . If for n = 0
we take y = −VaR and thus yield LPM 0 (−VaR ) = α , we are close to the VaR
measure46. If instead, we choose n = 1 , the LPM1 includes the differences from
negative changes in portfolio value and some (arbitrarily) chosen cut-off point y .
44
We adapt this part of the example from Guthoff et al. (1998) pp 122-125.
This issue turns out to be of practical relevance, see Berkowitz and O’Brien (2002).
46 For discrete probability distributions we have LPM (−VaR ) ≥ α .
0
45
116
Alois Paul Knobloch
For the same y we will then choose between two portfolios consistently with
Bernoulli’s principle, because P f 2 G ⇒ P f LPM 1 G .
The lower partial moment one extends the compatibility of risk measure and rational behavior to a broader range of probability distributions. Nonetheless, we
might take wrong decisions when we use LPM1 instead of VaR. This is because
the ordering induced by second order stochastic dominance is not complete. Consider, for instance, portfolio H of Example 4.1. Suppose the portfolio results
from adding a new security to portfolio P . The new security increases the expected portfolio return from EQP ( x ) = 3.76 to EQH ( x ) = 3.8 , but also assigns
higher probabilities to extreme values, i.e. the portfolio is riskier. Now stochastic
dominance does not apply to our new situation. Whereas the VaR at a 95% confidence level is still the same VaR5P% = VaR5H% = 3 , the LPM1 with
(
)
(
)
(− 3) = 0.035 < 0.04 = LPM 1H (− 3) 47. Thus, we
y = −3 =
yields
strictly prefer P to H according to LPM1, i.e. we do not buy the new security,
whereas our VaR does not reject altering the portfolio. The latter, however, is favorable for our investor, since expected utility is higher for H than for P
EQH (u ( ∆ H )) = 2.776 + u0 > 2.773 + u0 = EQP (u ( ∆ P )) . Hence, in a generalized
(
−VaR5P%
LPM 1P
)
LPM-framework like that of the (well-known) portfolio and capital market theory,
the trade-off between risk and return can only be treated under supplementary assumptions. These concern the probability distributions of portfolio returns or the
shape of investors’ utility functions48. If we simplify the situation by neglecting
expected changes of portfolio values and assuming normally distributed returns,
VaR and LPM1 can be used exchangeably49. Both lead to rational decisions.
LPM1, however, is favorable when strong option positions determine the extent of
adverse portfolio movements in low-probability areas. Since these positions might
be created intentionally, as regulation relies on VaR, the introduction of LPM 1 can
avoid such a manipulation towards higher risk (Johanning 1996 pp 297, 298).
However, it is hard to furnish this recommendation with so general a theoretical
underpinning.
Another criticism of VaR is based on general properties that a reasonable (“coherent”) measure of risk should exhibit50. One of them is sub-additivity, i.e. that
the risk measure applied separately to two sub-portfolios should add up to more
(not less) than when applied to the portfolio as a total. This means that the measure should account for risk reduction by diversification. VaR does not meet this
requirement in a general setting. For normally distributed returns, however, the requirement is fulfilled51. Coherent measures of risk can be based on a common sce47
The LPM definition for discrete probabilities is straightforward.
For portfolio decisions in a LPM-framework see e.g. Schröder (1996); Kaduff (1996).
49 The confidence level has to be chosen 1 −α ≥ 0.5 .
50 Such a set of properties is provided by Artzner et al. (1999).
51 See Artzner et al. (1999) pp 216, 217 on both issues. For the former one they provide an
example.
48
Value at Risk
117
nario for all portfolios. Of course, the appropriateness of the scenario-based risk
measure will crucially depend on the choice of the scenario that is more or less arbitrary. Another example is WCE := − inf E QP ( ∆ P A) Q P ( A) ≥ α , the worst
{
conditional expectation, which is close to TCE := − E QP
(
}
∆
P
∆P
)
≤ −VaRαP , the
52
tail conditional expectation, which in turn refers to VaR . We restrict ourselves to
those cases where WCE is equal to TCE. Therefore, we assume
AP =
{P
∆
∆P
≤ −VARαP
} . Contrary to VaR, TCE does take into account to
53
what extent losses exceed a specified limit. Thus, it is similar to LPM 1. Whereas
the latter specifies the cut-off point as an absolute or relative value, TCE prescribes a probability that determines the events to be considered.
Suppose we choose LPM1 to calculate a provision for risk. The problem is how
to specify the cut-off point y . This point might be determined by the portfolio
value multiplied by some internally or externally prescribed “hurdle” rate of return
rrH in order to account for the portfolio’s size54. If regulation required an amount
of equity of LPM 1 ( y = rrH ⋅ P ) , what would this mean? Equity would cover an
“expected loss”, where “loss” has a special meaning and is considered as exceeding − y , and where – appropriately defined – “gains” are ignored. What kind of
capital apart from equity would, for instance, cover losses of − y with 0 > y > y ,
if we chose rrH < 0 ? We might easily imagine a portfolio for which losses will
not exceed − y , but for which losses are nonetheless possible in the range of
[0,− y ] .
Then LPM1 would yield zero and thus ignore these losses. Even if
rrH ≥ 0 , LPM1 does not represent a weighted loss that can uniquely be attributed
to equity with its full or debt with its limited liability. Thus, a standard based on
LPM1 will not, however, have the appealing interpretation of VaR. Instead, a more
intuitive interpretation is given by the TCE. Immediately, we can regard TCE as
an amount of capital to be held for losses of a specified probability and conditional
on the occurrence of such a loss. Contrary to VaR, it takes into account the size of
these losses.
Further, we suggest that TCE – and WCE as far as the above assumption is valid
– is compatible with expected utility for u ∈ U 2 for continuous density functions
52
See Artzner et al. (1999) pp 223, where we have substituted the strict inequality in the
definition of WCE to comply with our definition of VaR.
53
AX ⊆ [c ,d ] is the event determining the infimum of WCE with respect to portfolio X .
54
For internal use, we have to be aware that the correspondence between LPM1 and expected utility can be lost for different portfolio sizes. Hence, we should compare two
portfolios with the same y as an absolute change in portfolio value.
118
Alois Paul Knobloch
or if Q X ( A X ) = α ∀X 55 for discrete ones56. Thereby, we mean that, if portfolio
P dominates portfolio G according to second order stochastic dominance, and
thus is preferred by all risk averse investors57, TCE does not contradict the induced
ordering. We concentrate on portfolios with the same expected return.
2TQRQUKVKQP
Suppose the changes in the values of portfolios P and G have continuous densities f P (.) and f G (.) on the domain [c , d ] such that they have the same expected
return and the former portfolio dominates the latter one according to second order
stochastic dominance, then TCE P ≤ TCE G , i.e. portfolio P is not rejected by the
tail conditional expectation criterion.
Proof. If portfolio P dominates G according to second order stochastic dominance, then f G (.) can be constructed from f P (.) by adding Mean Preserving
Spreads (MPS), which leads to the representation f G = f P + s (5) (Rothschild
and Stiglitz 1970, especially pp 231-232). (We denote by FP (.) and FG (.) the respective distribution functions.) As s(.) is mean-preserving and merely represents
∫c ts(t )dt = ∫c s(t )dt = 0 . Then, according to
Rothschild and Stiglitz (1970) the spread function s(.) has the following propera shift in probability weights we have
d
d
x
ties58: With S ( x ) = ∫ s(t ) dt we have
c
S (c ) = S (d ) = 0
(6)
∃z : S (x ) ≥ 0 if x ≤ z and S (x ) ≤ 0 if x > z
and
y
and for T ( y ) = ∫ S ( x ) dx they show that: T ( y ) ≥ 0, y ∈ [c, d ), T (d ) = 0.
c
Now, TCE P / G = −
1
FP G
equivalent to TCE P / G = −
55
−VaRαP G
(− VaR P G ) ∫c
α
1
α
−v P / G
∫c
(7)
(8)
xf P G (x ) dx which by assumption is
xf P / G (x ) dx where we use v P / G := VaRαP / G .
The condition means that for all portfolios the probability for A X is always equal to α .
We omit the proof of this case and refer to Eisele and Knobloch (2000) pp 174-175.
Pflug (2000) shows the compliance of the Conditional Value at Risk (CVaR) with expected utility for u ∈ U 2 differently from our proof and for portfolios that may have different expected returns.
57 For the “equivalence” of second order stochastic dominance and expected utility for all
such investors, see Rothschild and Stiglitz (1970); Kaduff (1996) p 24.
58 Without further impact on our result, they use the domain [0,1] instead of [c , d ] .
56
Value at Risk
119
Thus, we have to show that
1
TCEG − TCE P = − 
α

−vG
∫

xf P (x ) dx  ≥ 0


−v P
∫
xf G (x ) dx −
c
c
−vG
−v P
c
c
∫
⇔ ∆G − P := − xf G (x ) dx +
(9)
∫ xf (x ) dx ≥ 0.
P
We take condition (7 ) as a starting point.
( )
1. − v P ≤ z : As S − v P ≥ 0 the α -quantile of portfolio G’s P&L distribution
cannot be greater than − v P , i.e. − v G ≤ −v P . Eq. (5) and integration by parts
yield
∆G − P = −
−vG
−v P
c
−v G
∫ xs(x )dx +
(
−vG
−v P
c
−vG
) ∫ S (x ) dx + ∫ xf (x ) dx.
G
G
∫ xf P (x ) dx = v S − v +
P
−v P
G
∫−vG f P (x )dx = S (− v )
Further, loosely speaking,
(10)
represents the probability the
P&L-distribution of P has to catch up after the point − v G or, alternatively, it has
lost against G till that point. So, we have FP − v P − FP − vG = S − vG
( )
( )
( )
(11) . Therewith, we get from (10) :
⇔ α − S − vG = FP − vG
G−P
∆
G
(
= v S −v
=v
G
G
( ) ( )
)+ T (− v )+ xF
G
P
S (− v ) + T (− v ) − v
G
G
P
(x )
FP
−v P
−v P
−vG
−
∫ FP (x ) dx
−vG
(− v )+ v
P
G
FP
−v P
(− v )− ∫ F
G
P
(x ) dx
−vG
(
=T −v
G
)+ α (v
G
−v
P
−v P
)− ∫ F
P
(x ) dx.
−vG
(
)
Now 0 ≤ FP ( x ) ≤ α , x ≤ −v P , and (8) yield ∆G − P ≥ T − v G ≥ 0 .
(
)
2. − v P > z : Now, S − v P ≤ 0 the α -quantile of portfolio G’s P&L-distribution
cannot be lower than − v , i.e. − v G ≥ −v P . From eqs. (5) and (9 ) we have
G−P
∆
P
−v P
−vG
c
−v P
= − ∫ xs(x ) dx −
−v P
−vG
c
−v P
∫ xf (x ) dx = v S (− v )+ ∫ S (x ) dx − ∫ xf (x ) dx.
G
P
P
G
(12)
120
Alois Paul Knobloch
Applying the above argument, we write
(
)
(
)
−vG
∫−v P
(
( )
f G ( x )dx = − S − v P as well as
)
(
)
(
)
FG − v G − FG − v P = − S − v P ⇔ α + S − v P = FG − v P .
(13)
Eq. (12 ) yields together with eq. (13) :
( )
∆G − P = v P S − v P +
=
−v P
−v P
−vG
∫ S (x ) dx − xFG (x ) −v P +
c
(
)
G
P
∫ S (x ) dx + α v − v +
c
(
G
=α v −v
(
P
)+
−vG
−vG
∫ FG (x ) dx
−v P
−vG
∫PFG (x ) dx
−v
−vG
x )) dx
(x2
) −4S (43
∫ S (x ) dx + ∫P (1FG44
c
) (
−v
)
= α vG − v P + T − vG +
−v
= FP ( x )
G
∫ FP (x ) dx
−v P
( )
With FP ( x ) ≥ α , x ≥ −v P , and (8) we get again ∆G − P ≥ T − vG ≥ 0 . □
Since there is a tendency to create a unified framework for market and for credit
risk, we will finally show some shortcomings of VaR as a measure of credit risk.
Consider a portfolio of securities representing debt. The portfolio value may suffer
not only from defaults but also from devaluations following rating downgrades. In
either case, the effect on portfolio value are jumps that are difficult to predict over
a short range of time. It is easy to imagine that the α -quantile does not change
with the amount of devaluation that may reach the total value of the security less a
recovery rate. Then, VaR is invariant to the extent of a loss. Diversification will
reduce this effect. For a well-diversified portfolio59 the P&L distribution due to
credit risk will be much smoother than for an undiversified portfolio. Thus, VaR
will be affected by some but not all possible downgrades, respectively defaults.
Consider a portfolio of 50 securities of the same value V s , each with a 0.4% probability of (complete) default and mutually independent default risks60. The future
portfolio value P obeys a binomial distribution B (50, 0.004) . At a 99% confidence level, we yield a VaR of 2 ⋅Vs , which we also yield for confidence levels in
59
A well-diversified portfolio, for us, has the connotation that the risks are not or at least
not completely positive correlated.
60 For a similar example and the following conclusion, see Artzner et al. (1999) pp 217-218,
where the authors refer to Albanese (1997).
Value at Risk
121
[0.98274, 0.99891] . If instead our portfolio consists of only two securities, each of
which is worth Vs = 25 ⋅Vs , the given confidence level leads to a VaR of 0 , unchanged in the broader range [0, 0.99202] of confidence levels. VaR suggests to
prefer a less diversified portfolio, ignoring concentration risk, and it does not account for the exposure 2 ⋅Vs . But even for a diversified portfolio, we have to be
aware that credit risk typically creates strong asymmetric distributions. Thus, the
simple calculation methods do not apply.
In a perfect world with risk described by normal distributions, VaR is a useful instrument for several tasks that a risk management system has to fulfill. In particular the delta-normal method provides a tool for calculation, allocation, and optimization with respect to a portfolio’s exposure to risk. Even in such a setting a
source of inaccuracy is given by sampling errors, the effects of which sometimes
may be mitigated by statistical refinements. However, if we have to deal with options, the problem of inaccuracy is aggravated. Although it is the most accurate
calculation method, Monte Carlo simulation nonetheless does not offer a tool for a
sound allocation of risk capital or even for optimization. In this world, we must
also be aware of some pitfalls that are inherent in the concept of VaR. The criticism focuses on the fact that VaR does not take into account the extent to which
losses exceed this critical point. The calculation of a tail conditional expectation
can avoid this problem and complies with decision theory for portfolios that
dominate each other according to second order stochastic dominance. In contrast
to lower partial moments, it maintains the intuitive appeal of VaR with regard to
the predefined probability for losses to be considered.
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John F. O. Bilson1
1
Melbourne Business School, The University of Melbourne, 200 Leicester Street,
Carlton, VIC 3053, Australia
Abstract: The standard approach to the risk analysis of fixed income portfolios involves a mapping of exposures into representative duration buckets. This approach does not provide a transparent description of the portfolio risk in the case
of leveraged portfolios, particularly in the case of portfolios whose primary intent
is to trade convexity. In this paper, an alternative approach, based upon Level,
Slope and Curvature yield curve factors, is described. The alternative approach
offers a linear model of non-linear trading strategies.
Keywords: Value at Risk, Fixed Income Strategy, Duration, Convexity
The objective of Value at Risk (VaR) analysis is to provide a report to senior
management that describes the prospective risks inherent in a portfolio of financial
instruments. In its ultimate manifestation, VaR is reduced to a single number:
“There is an X% probability that the portfolio will lose more than Y dollars over a
Z day time horizon.” In a complex portfolio, the VaR number may reflect valuations of thousands of different instruments held in different locations around the
globe. VaR applications like JP Morgan’s RiskMetrics and 4.15 or Bankers
Trust’s RAROC 2020 include hundreds of risk factors. Each of these risk factors
must be modeled with an estimated volatility and correlations with all other risk
factors. With 500 risk factors, there are 125,250 variance and covariance terms to
be estimates. Furthermore, all of these estimates need to be consistent with each
other in order to prevent singularities in the resulting covariance matrix. As a result, most of the commercially available VaR models rely upon extremely simple
126
John F. O. Bilson
volatility calculators. It is not surprising, then, that different models can give very
different estimates of the Value at Risk.
In this paper, I propose that the variance covariance matrix be reduced in dimension by describing the yield curve in terms of Level, Slope and Curvature
(LSC) factors. The characterization adopted in the paper is based upon the yield
curve model developed in Nelson and Siegel (1987) and extended by Wilmer
(1996). The LSC model also provides senior management with a more intuitive
breakdown of the component risks in the portfolio. The LSC model is an alternative to the key rate duration (KRD) approach described in the RiskMetrics Technical Document (1995). In the KRD model, the yield curve is described by a series
of discrete points along the curve. All cash flows are then shredded and allocated
to a particular duration using either duration or volatility matching principles.
Based upon the covariance matrix of the key rates, we can then estimate the VaR.
The difficulty with this approach is that the components of the risk portfolio is
hard to define. The 10-year duration risk cannot be easily distinguished from the 5
year or the 15 year risk. The LSC model is more parsimonious in representing the
yield curve and it provides a more transparent description of the components of
risk than the alternative.
Throughout the paper, I will illustrate the different concepts with a simple example. I assume that an investment management firm has hired to fixed income managers for its government bond fund. The only restrictions imposed on the managers is that the duration of the portfolio be less than 10 years and that the leverage
of the portfolio, defined as the ratio of the sum of the absolute value of the positions to the total portfolio value, be less than 20. Both traders have an initial capital of $10 million. To simplify matters, I also assume that the only instruments
available to the traders are zero coupon bonds paying $1 million at maturity. The
objective of the exercise is to create a risk management report for senior management describing the activities of the two traders. The original, Pre-VaR, reports
are presented below.1
1
The positions are based upon the U.S. spot yield curve on November 30, 2002.
Parsimonious Value at Risk for Fixed Income Portfolios
127
Table 1.1. Trader Risk Management Report
Trader
Maturity
0.5
1
3
5
10
20
30
ABC
Spot
1.28%
1.46%
2.50%
3.26%
4.21%
4.78%
5.05%
Leverage
Convexity
20
-1211
Trader
Maturity
0.5
1
3
5
10
20
30
XYZ
Spot
1.28%
1.46%
2.50%
3.26%
4.21%
4.78%
5.05%
Leverage
Convexity
20
+1850
Position
-76
0
0
+75
+63
+1
-90
Duration
Yield
Position
+ 68
0
0
-39
- 53
- 70
+170
Duration
Yield
CashFlow
1000
1000
1000
1000
1000
1000
1000
Total
Present
Value
-75,144
0
0
+63,682
+41.051
+207
-19,856
+10,000
10
18.51%
CashFlow
1000
1000
1000
1000
1000
1000
1000
Total
Present
Value
+67,700
0
0
-33,256
-34,804
-26,925
+31,300
+10,000
10
-10.86%
Notes to Table 1.1
Maturity – the term to maturity of the zero coupon bond,
Spot – the continuously compounded yield to maturity,
Position – number of bonds purchased (+) or sold (-),
CashFlow – amount paid at maturity on each bond held,
Present Value – Present discounted value of the positon.
In this traditional presentation, the risk of the portfolio is described in terms of
portfolio leverage, duration, convexity and yield. The leverage of the portfolio is
the ratio of the sum of the absolute value of the present value of the position to the
sum of the actual values of the positions. Both traders have leverage statistics of
20, which means that they have taken positions with an absolute value of $200
million on the foundation of their capital of $10 million. The duration of the portfolio is the value weighted average yield to maturity. Duration is typically used as
a measure of the sensitivity of the portfolio to changes in interest rates. Both traders have a portfolio duration of 10 years, which means that a full basis point parallel shift in the yield curve will reduce the value of these portfolios by 10%. The
convexity of the portfolio is the value weighted sensitivity of the portfolio to the
squared change in yield. The trader ABC has selected a portfolio with strong negative convexity while the trader XYZ has selected a portfolio with strong positive
128
John F. O. Bilson
convexity. The differing levels of portfolio convexity are reflected in the yields on
the two portfolios. ABC has a positive portfolio yield of 18.51%%, relative to the
10 year spot yield of 4.21%, while XYZ has a negative yield of 10.86%. ABC is
buying yield by selling convexity, while XYZ is selling yield to buy convexity.
From a risk management perspective, XYZ has the greatest risk when rates are
stable, while ABC has the greatest risk when rates are volatile.
Trading on the second moments of the distribution is an increasingly common
strategy in hedge funds and proprietary trading groups. As risk management
groups have become more sophisticated, they have increased their ability to detect
large direct or linear exposures to prices. Furthermore, experience has demonstrated that it is very difficult to forecast the direction of prices and interest rates.
Traders consequently find that their ability to take linear bets – bets on the delta of
an equity portfolio or the duration of a fixed income portfolio – is generally
closely controlled by clients or management. The modern trader focuses on second moments – volatility and correlation – both because these moments are more
predictable and because standard risk management tools are less effective in detecting hidden risks from these activities.
To demonstrate this point, we begin with a simple Value at Risk (VaR) calculation based upon duration and convexity. The analysis begins with the continuously compounded present value formula:
Vi ,t = K i e
− ri ,t (Ti −t )
(1.1)
A second order Taylor series expansion of this formula expresses the change in
value as it relates to the duration, convexity and yield of the instrument.
dVi ,t = −(Ti − t )Vi ,t dri ,t +
(Ti − t )2
Vi ,t dri 2,t + ri ,tVi ,t dt
2
(1.2)
Standard fixed income terminology refers to the parameters of this expansion as
the duration, convexity and yield of the portfolio. Specifically:
Di = (Ti − t ) and
Ci = (Ti − t ) 2
(1.3)
If we are willing to make the heroic assumption that all movements in yield are
identical, then the change in the value of the portfolio can be written as:
1
dV p = − D pV p dr + C pV p dr 2 + rpVp dt
2
where D p =
Vi
∑V
i
p
Di ; C p = ∑
i
Vi
Ci
Vp
and
rp = ∑
i
(1.4)
Vi
ri . The duration,
Vp
convexity and yield of the portfolio are then simply value weighted averages of
the duration, convexity and yield of the underlying instruments.
The objective of the VaR analysis is to estimate the Value at Risk at the 5%
confidence level and the one month horizon. Since the target portfolio duration is
Parsimonious Value at Risk for Fixed Income Portfolios
129
10 years, the analysis will be based upon the variability in the 10 year spot rate.
Following industry practice, we will assume that the percentage change in the spot
rate is normally distributed.
drp
rp
→ N (0, σ 2 )
(1.5)
The assumption of normality is very questionable for changes in discount rates
since the observed distribution of these changes tends to be lepto-kurtotic relative
to the normal. The kurtosis typically means that the VaR is underestimated in the
tails of the distribution and over-estimated in the normal range around plus or minus one standard deviation. The actual and the normal tend to coincide at the 5%
confidence level. For confidence levels outside 5%, more sophisticated distributional assumptions should be employed.
Over the one month horizon, the percentage change in the 10 year spot yield has
a standard deviation of around 4.0%. If the level of the rate is around 5%, the standard deviation of the change in the rate is approximately 20 basis points.
Consequently, a 1.64 standard deviation move, corresponding to the 5% confidence level, is around 33 basis points. Since the portfolio is long duration, the
portfolio will lose value if rates rise. We therefore estimate the change in the portfolio for a 33 basis point increase in rates.
Table 1.2. Elementary Value at Risk Analysis
10 Year Volatility
10 Year Yield
Volatility of Change (1 Month)
Critical Value (5%)
Duration VaR
Convexity VaR
Yield VaR
Total
Data
4.00%
4.21%
0.20%
0.33%
ABC
XYZ
18.51%%
-10.86%%
-328
- 65
+ 154
-239
-328
+100
- 91
-319
Legend:
10 Year Volatility = Standard deviation of percentage change in 10 year yield,
10 Year Yield = yield on 10 Year zero coupon bond, or trader portfolio.
Volatility of Change = Standard deviation of change in yield over 1 month.
Critical Value = Change in yield corresponding to 5% confidence level.
Duration VaR = Value at Risk due to duration factor.
Convexity VaR = Value at Risk due to convexity factor.
Yield VaR = Change in portfolio value due to yield over 1 month horizon.
If yields rose by 33 basis points, both traders would lose $328 due to the duration
factor. ABC would lose an additional $65 due to the negative convexity of the
portfolio but would gain $154 because of the high portfolio yield. XYZ, in contrast, would gain $100 due to convexity but would lose $91 because of the negative portoflio yield. In aggregate, ABC has a Value at Risk at the one month hori-
130
John F. O. Bilson
zon and the 5% confidence level of -$239 which is less than the -$319 VaR of
XYZ. The simple model clearly indicates the tradeoff between convexity and
yield. In this particular example, ABC’s higher yield compensates for the loss due
to negative convexity.
The elementary model is primarily concerned with the exposure of the portfolio
to parallel shifts in the yield curve. While this approach may be justified in some
circumstances, it is clearly inappropriate when the portfolio consists of a series of
leveraged bets across the yield curve. In the case under consideration, exposure to
twists and bends (slope and curvature) in the yield curve may be more important
than exposure to parallel shifts. The elementary model is also inadequate in its assumption that risk can be measured in terms of the standard deviation of the ten
year yield. In the next section, a portfolio VaR will be developed based upon the
Key Rate Duration (KRD) approach. This approach will then be compared with a
similar model based upon Level, Slope and Curvature (LSC) factors.
A portfolio of fixed income instruments will typically have cash flows that mature
at many different times in the future. It is computationally impossible to model
the covariance between all such points and it is consequently necessary to reduce
the dimensionality of the problem. The accepted approach to this problem was
originally published by JP Morgan in the RiskMetrics Technical Document in
1995. The RiskMetrics approach begins by defining a series of key rate durations
and then develops a methodology for mapping non-standard cash flows into the
appropriate cash flow buckets. In the case considered here, the key rate durations
will be the seven standard maturity vertices used by the traders. This approach
avoids the necessity of discussing mapping issues. The change in the value of the
position at any point along the curve can be approximated by:
dVi ,t = δ i ,t
dri ,t
ri ,t
+
γ i ,t dri ,t 2
(
) + ri ,tVi ,t dt
2 ri ,t
(2.1)
where δ i ,t = −(Ti − t )Vi ,t ri ,t measures the sensitivity of the change in value of
the cash flow to a percentage change in the yield to maturity. The sensitivity depends upon the term to maturity, the present value of the instrument and whether it
2
2
is long or short, and the spot yield to maturity; γ i ,t = (Ti − t ) Vi ,t ri ,t measures
the convexity of the cash flow or, equivalently, the sensitivity of the present value
to the squared change in the yield. If the percentage change is normally distributed, the squared change will be chi-squared with one degree of freedom. It is
consequently not possible to treat equation (2.1) as a standard linear combination
of normally distributed random variables.
Of the various approaches that have been developed to deal with this problem,
the model developed by Tom Wilson (1996) is best suited for the purposes of this
Parsimonious Value at Risk for Fixed Income Portfolios
131
paper because it provides important analytic content that is absent from some
other approaches. Wilson begins be defining the VaR as the largest lost occurring
within the constraint of a given confidence elipse. Specifically, the VaR is the outcome of the following constrained optimization problem:
min dV p = ∑ dVi
i
 dr 
 dr 
subject to   ' Ω −1   ≤ α 2
 r 
 r 
(2.2)
In this equation, Ω represents the covariance matrix of the percentage changes in
yield and α is the critical value associated with the confidence interval used in the
VaR calculation. The basic idea behind Wilson’s method is to find the vector of
proportional yield changes that is associated with the greatest portfolio loss subject to the constraint that the vector lie within the confidence ellipse associated
with the VaR critical value. The procedure leads to the familiar VaR result when
the payoff functions are linear. When there is sufficient non-linearity so that the
maximum loss occurs within the confidence ellipse, then the VaR is actually the
maximum loss on the portfolio rather than the loss that will be exceeded x% of the
time. However, this quibble is a minor issue given the higher level of analytic
transparency resulting from the methodology.
This transparency arises from the concept of the “critical scenario” which is the
vector of proportional yield changes associated with the Value at Risk calculation.
The critical scenario is important in its own right, because it is a way of informing
management of the types of outcomes that will be associated with the VaR loss. In
many cases, including the specific example that we are studying, the critical scenario can be more informative to management that the components of the VaR
calculation. However, the two are closely related since VaR components can be
calculated directly from the critical scenario. Specifically:
CompVaR = δ i ,t
dri*,t
ri*,t
γ i ,t  dri*,t
+

2  ri*,t
2

 + ri ,tVi ,t dt


(2.3)
In this equation, r* denotes the value of the yield variable associated with the solution to the optimization problem. These estimates of the component VaR are
identical to Mark Garman’s estimation procedure when the payoff functions are
linear and they share the property of Garman’s estimator that the components sum
up to the total VaR.2 The approach is more flexible, however, in that it is able to
handle the second order effects in the payoff functions.
Table 1.3 contains a VaR report based upon the key rate duration model. In this
table, the trader’s position is summarized under the account title. The position is
measured in contracts, based upon management preferences. Alternative methodologies could emphasize the notional size of the position or the present value of
the position. The scenario is the yield curve outcome associated with the Value at
Risk. It should be compared with the description of the current spot yield curve in
2
Garman (1997)
132
John F. O. Bilson
the second column of the table. The VaR column gives the estimated VaR for each
maturity bucket and the total VaR of the portfolio.
Table 1.3. KRD Value at Risk Analysis
Maturity
.5 Years
1 Year
3 Year
5 Year
10 Year
20 Year
30 Year
Total
Spot
1.28%
1.46%
2.50%
3.26%
4.21%
4.78%
5.05%
ABC
-76
0
0
75
63
1
-90
Scenario
1.35%
1.59%
2.74%
3.54%
4.46%
4.97%
5.20%
VaR
-53
0
0
-716
-872
-7
+774
-875
XYZ
+68
0
0
-39
-53
-70
+170
Scenario
1.23%
1.39%
2.37%
3.11%
4.12%
4.76%
5.07%
VaR
+90
0
0
-340
-454
-201
-121
-1026
In the case of ABC, it is clear that the predominant source of portfolio risk lies in
the maturity spectrum from 5 years to 10 years. The trader is long these maturities
and consequently the critical scenario involves an increase in these rates relative to
current values of between 20 and 30 basis points. The 30 year VaR is positive
which indicates that this position is a hedge against the long mid-curve positions.
It is particularly noticeable that the critical scenario involves an increase of 15 basis points on this position and that this increase in approximately half the increase
in the mid-curve yields. The critical scenario consequently consists of a pronounced increase in the curvature of the yield curve. The total VaR of the ABC
position is -875 or a loss of 8.75% relative to the notional committed capital.
In the case of XYZ, the position consists of a long position of 170 contracts in
the 30 year offset by short positions in the 5 to 20 year maturities. The critical scenario involves a steepening of the yield curve with short rates falling while long
rates are relatively stable. Because the portfolio is short the mid-curve, the steepening results in losses on these positions that are not offset by gains on the long
positions at the 30 year maturity. Indeed, the 30 year bond rate actually increases
slightly so that there is also a small loss on this position. Overall, XYZ has a VaR
of $1026 or 10.26% of notional committed capital. Once again, the position with
long convexity is considered to be riskier than the short convexity position.
The question that must be addressed is whether this report provides the most
transparent and informative description of the trading activities of the two accounts. The KRD approach is certainly an improvement over the elementary duration/convexity model because it allows for changes in the slope and curvature of
the yield curve. In both accounts, the critical scenario involves an increase in the
curvature of the yield curve. Because it allows for non-parallel shifts, the key rate
duration model also predicts a larger potential loss than the elementary model and
this is also consistent with historical simulation of the positions. There are, however, some aspects of the report that senior management may find confusing. The
practice of defining exposures in terms of duration buckets is often a source of
management concern since it is difficult to explain how a 10 year exposure is dif-
Parsimonious Value at Risk for Fixed Income Portfolios
133
ferent from a 20 year exposure when long yields tend to move quite closely together. It is also difficult to explain the critical scenario concept within the confines of the key rate duration approach. The risk manager must explain why this
particular scenario is associated with the VaR point. Management will often comment that the particular scenario is “unrealistic,” even though this would appear to
be a desirable characteristic of a situation that is expected to occur less than 5% of
the time.
The most important limitation of the KRD is that it does not offer an analytic interpretation of the risks taken by the two traders. In Table 1.1, we observed that
ABC was long yield and short convexity while XYZ had the opposite position.
Positions with short convexity are meant to be riskier than positions with long
convexity and yet ABC is found to have a total VaR that is smaller than XYZ. Is
this because of a problem with the statistical methodology or is it an actual reflection of the risks of the two portfolios? The LSC methodology, which will be developed in the next section of the paper, is an attempt to deal with these issues.
Statistical decompositions of the variation in the yield curve demonstrate that
there are three principal sources of variation in the curve: level, slope and curve.
The level of the curve is typically defined as the longest maturity yield, the slope
as the difference between the long and short yields, and the curve as the mid-curve
yield relative to an average of the long and short yields. While changes in the level
account for the major part of variation in the yield curve, level changes may not be
a significant source of changes in value for positions that involve spreads across
maturities. These positions, which are often referred to as ‘barbells’ and ‘bullets’
by fixed income traders, are important features of modern fixed income trading.
First, clients often specify a target or benchmark duration for their portfolio and
this limits the ability of the trader to take bets on the direction of the overall
movement in the yield curve. Second, changes in the slope and curvature are often
considered to be more predictable than the level of the curve. While amateurs bet
on the direction of rates, longer term professional traders focus on the slope of the
curve and shorter term professional traders focus on curvature. The reason is that
the slope and curvature characteristics tend to exhibit predictable mean reversion.
The slope characteristic, which is generally a reflection of the state of the economy, has a cycle which reflects the cycle in the economy. The curve characteristic,
which reflects the volatility of interest rates, tend to exhibit greater volatility and
faster mean reversion.
While there are many models of the yield curve, value at risk analysis demands
transparency and linearity. Transparency means that the model should be easy to
relate to the underlying sources of yield curve variation – level, slope and curve –
and linearity is important because it enables exposures to be summed across positions. The Nelson-Siegel (1987) model is a simple approach to the yield curve that
embodies these two important characteristics. The Nelson-Siegel model begins
with a description of the forward curve:
134
John F. O. Bilson
R (m) = L + S exp(−
m
m
m
)+C
exp(−
)
m*
m*
m*
(3.1)
In this equation, R(m) is the instantaneous forward rate for maturity “m”, “m*” is a
“location parameter” representing the point of greatest curvature and L, S, and C
are parameters representing the level, slope and curvature of the curve respectively. L is the yield on the long end, S represents the difference between the short
rate and the long rate, and C represents the degree of curvature in the curve. If
equation (3.1) represents the forward curve, Nelson and Siegel demonstrate that
the spot curve is defined by the equation:
m  m
m

) /
)
− C exp(−
r (m) = L + ( S + C ) 1 − exp(−
m*  m*
m*

= L + S f 1(m) + C f 2(m)
(3.2)
The yield curve factors, f1(m) and f2(m), are plotted in Figure 3.1 against the maturity value. For the purposes of these calculations, the location parameter is assumed to be 3 years.3 Figure 3.1 demonstrates that the slope factor starts at unity
and declines asymptotically towards zero. The curve factor starts at zero, rises to a
maximum, and then declines towards zero.
1,2
Level
1
0,8
Slope
0,6
0,4
0,2
Cur ve
0
0
5
10
15
20
25
30
Fig. 3.1. Yield Curve Factors
3
The results are not particularly sensitive to the value of the location parameter. For the
U.S. data, normal values are between 3 and 5 years. For less developed markets or markets with high or variable inflation, the location parameter can be considerably shorter.
Parsimonious Value at Risk for Fixed Income Portfolios
135
Using the standardized factors, the shape of the yield curve at any point in time
can be represented by the parameters L, S, and C, representing the level, slope and
curvature characteristics of the curve. In Table 1.4, the procedure is illustrated
with U.S. spot yield data for November 30, 2002. At this time, the U.S. yield
curve was steeply sloped in reflection of the depressed short-term economic
conditions and the market expectation that the recession would be relative short
lived.
Table 1.4. LSC Curve Fitting
Maturity
.5 Years
1 Year
3 Year
5 Year
10 Year
20 Year
30 Year
Regression
Spot
1.28%
1.46%
2.50%
3.26%
4.21%
4.78%
5.05%
Analysis
Fitted
1.27%
1.58%
2.55%
3.21%
4.14%
4.78%
5.06%
Coeff
StdError
R-Sq.
f0(m)
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Level
5.55%
0.10%
0.9987
f1(m)
0.92
0.85
0.63
0.49
0.29
0.15
0.10
Slope
-4.62%
0.10%
f2(m)
0.07
0.13
0.26
0.30
0.25
0.15
0.10
Curve
-0.30%
0.36%
In Table 1.3, the two factors f1(m) and f2(m) are calculated using the formulae in
equation (3.2). The third factor, f0(m), corresponds to the level factor. The three
factors are related to spot curve through a simple linear regression. 4 The regression
coefficients correspond to the Level, Slope and Curve characteristics of the curve
at each point in time. The constant term in the regression is 5.55%. This is an estimate of the spot yield to maturity on a hypothetical infinite maturity zero coupon
bond.5 The second coefficient, -4.62%, is an estimate of the slope of the yield
curve. At this point, the short rate is estimated to be 4.62% below the long rate of
5.55%. Finally, the curvature coefficient is effectively zero at this point. This
means that the degree of curvature is adequately represented by the non-linear
shape of the slope function.
4
5
Wilmer (1996) suggests weighting the observations by the duration of the instrument.
This procedure results in a better fit for the longer maturities.
This is an hypothetical construct because the true yield to maturity on an infinite maturity
zero coupon bond is presumably undefined.
136
John F. O. Bilson
10,00%
8,00%
Level
6,00%
4,00%
Curve
2,00%
0,00%
-2,00%991
92
19
1.
1
1.
0
2.
0
-4,00%
0
0
2.
93
94
19
1.
.0
02
95
19
1.
0
2.
0
96
19
1.
0
0
2.
97
19
1.
0
0
2.
0
99
98
19
1.
0
2.
0
00
19
1.
19
1.
0
2.
0
0
2.
01
20
1.
0
0
2.
02
20
1.
0
2.
0
20
1.
0
2.
0
Slope
-6,00%
-8,00%
Fig. 3.2. Yield Curve Factors
The LSC model was estimated using end of month data over the period from
January, 1991 to November, 1992 for the United States.6 The estimated values of
the parameters are plotted in Figure 3.2. Over this period, the Level factor has
drifted down from around 8% to 6%. It is clearly difficult to predict the direction
of this characteristic of the yield curve. The Slope factor has ranged from a value
of -6% in the 1992 recession to slightly positive at the height of the boom in 2000.
The Slope factor does appear to exhibit some mean reversion but the cycle length
is quite long. Finally, the Curve factor is the most volatile of the three characteristics and it also exhibits the most rapid mean reversion. It is consequently not surprising that active traders tend to focus on this characteristic of the yield curve.
For risk management purposes, the value of the LSC approach lies in its ability
to capture changes in the shape of the yield curve that cannot be captured easily
within the KRD methodology. The LSC model also has the advantage that it is not
necessary to map cash flows into standardized duration buckets because the factors effectively model each point along the curve. These advantages would be of
little value, however, if the model was not an accurate statistical predictor of the
changes in yields. To explore this issue, the following table explores the ability of
the model to predict changes in yields at various points along the curve. The forecasting equation is described in equation (3.3):
∆r ( m ) ∆ L
∆S
∆C
=
+ f 1(m)
+ f 2(m)
+ ε (m)
L
L
L
L
= ∆l + f 1(m) ∆s + f 2(m) ∆c + ε (m)
6
The spot yield data was taken from Datastream.
(3.3)
Parsimonious Value at Risk for Fixed Income Portfolios
137
If the LSC model is an accurate depiction of the yield curve, then a regression of
the change in yield, normalized by the level of the curve, should be closely related
to changes in the three (normalized) characteristics. Furthermore, the regression
coefficients should be close to the hypothesized values from the Nelson-Siegel
model.
Table 1.5. LSC Forecasting
Maturity
.5 Years
1 Year
3 Year
5 Year
10 Year
20 Year
30 Year
∆L/L
0.95
1.05
1.02
0.94
1.01
1.00
0.99
∆S/L
0.88
0.88
0.66
0.46
0.26
0.15
0.10
∆C/L
0.05
0.15
0.27
0.30
0.22
0.15
0.10
R-Square
0.97
0.97
0.99
0.99
0.98
0.99
0.99
StdError
0.006
0.007
0.004
0.003
0.005
0.003
0.002
The values reported in Table 1.4 are estimated regression coefficients derived
from estimating equation (3.3) for each maturity using monthly data over the period from 1991 to 2002. The R-squared statistics indicate that the model does a
very respectable job of accounting for the changes in the yield at each maturity.
Furthermore, the regression coefficients are very consistent with the predicted
values from the Nelson-Siegel model. These results therefore provide some support for the model as a risk analysis mechanism.
1.4 LSC Risk Analysis
The LSC risk analysis model begins with the valuation equation previously described in equation (1.2).
dVm = − DmVm L
drm 1
dr 2
+ CmVm L2 2m + rmVm dt
L 2
L
(4.1)
Equation (4.1) describes the change in the value of a cash flow of maturity “m” in
terms of the duration, convexity and yield of the maturity. In order to make the
model consistent with the previous exposition of the LSC, the changes in yield are
expressed relative to the Level characteristic of the curve. Abstracting from the error term, the proportional change in yield can then be written as:
drm
= dl + f 1(m) ds + f 2(m) dc
L
(4.2)
138
John F. O. Bilson
Similarly:
 drm drn 

 = (dl + f 1(m) ds + f 2(m) dc)(dl + f 1(n) ds + f 2(n) dc)
 L L 
(4.3)
While it is possible to use equation (4.3) to develop a full LSC duration and convexity model, this is typically unnecessary in most situations because the three
factor duration model provides a sufficient description of the risks in the portfolio.
The additional information contained in Level, Slope and Curve convexities is
small and of little practical value. If the second order effects are ignored, the
change in the value of the portfolio can be written as:
dV p = ∑ dVm = δ l dl + δ s ds + δ c dc + δ t dt
(4.4)
where
δ l = −∑ DmVm L; δ s = −∑ DmVm L f 1(m); δ c = −∑ DmVm L f 2(m)
m
and δ t =
∑r V
m
m m
m
m
. These parameters, which are closely related to Wilmer’s defi-
nitions of LSC durations, describe the sensitivity of the portfolio to proportional
changes in Level, Slope and Curvature. After the Level, Slope, Curvature and
Yield delta’s have been computed, it is a simple matter to create the VaR report as
a standard linear combination of normally distributed random variables. As with
all financial time series, the normality assumption is highly questionable. As demonstrated in the following table, the yield curve factors tend to have fat tails relative to the normal. The issues of mean reversion in the characteristics of the yield
curve must also be ignored in this paper.
Table 1.6. LSC Risk Characteristics
Maturity
Average
StdDev
Maximum
Minimum
Correlation
∆S/L
∆C/L
∆L/L
-0.24%
3.01%
7.64%
-9.61%
1.00
-0.60
0.37
∆S/L
-0.42%
5.62%
10.76%
-35.66%
∆C/L
0.03%
12.39%
48.61%
-33.76%
1.00
-0.31
1.00
As one would expect from Figure 3.2, the curvature characteristic is the most
volatile of the three factors. The slope characteristic has the largest kurtosis and
downward skew but this may simply reflect the speed with which the 2001 recession hit the American economy after September 11 and the burst of the dot.com
bubble. One important feature of the risk characteristics is the strong negative correlation between the Level and Slope characteristics of the yield curve. When the
Parsimonious Value at Risk for Fixed Income Portfolios
139
level of the curve increases, there is a tendency for the slope of the curve to flatten.
Table 1.7. LSC Value at Risk
Trader
Level
Slope
Curve
Yield
Total
ABC
Delta
-5524
-9959
-7582
10000
Scenario
0.91%
1.64%
17.70%
18.51%
VaR
-50
-163
-1342
154
-1401
XYZ
Delta
-5524
6586
5712
10000
Scenario
0.91%
-3.29%
-15.43%
-10.86%
VaR
-50
-217
-882
-91
-1239
In this report, the Delta’s represent the weighted value exposures to the three risk
factors and the yield. ABC is clearly short Level, Slope and Curvature while XYZ
is short Level and long Slope and Curvature. The scenario is the vector of outcomes that is associated with the VaR point. Since the Curve characteristic is the
most volatile component of the three characteristics of the yield curve, the ABC
scenario involves a 17.70% proportional increase in curvature while the XYZ
portfolio involves a 15.43% proportional decrease. The component VaR calculations clearly indicate that both portfolios are taking their major bets on changes in
curvature. Curvature exposure represents 95% of ABC’s risk budget and 71% of
XYZ’s risk budget. As a consequence, ABC has an estimated total VaR of $1,401
while XYZ has a total VaR of $1,239. Relative to the KRD approach, the LSC
model predicts a higher level of risk for both traders and, more significantly, predicts that ABC will have a greater downside risk than XYZ.
The reason for this result lies in the correlation between the risk factors. Given
that both traders are taking their most significant bets on the curvature of the
curve, it is important to recognize that they are also operating under the duration
constraint. The duration constraint corresponds to a Level exposure in LSC terminology. In the case of ABC, the trader is short Level and short Curve. Since Level
and Curve are positively correlated factors, the Level exposure extends the risk resulting from the Curve position. On the other hand, XYZ is short Level and long
Curve thereby resulting in some diversification of the position.
At the time when these positions were constructed, the Curve characteristic was
in a neutral area relative to its historical range. This means that it is as likely to
rise as it is to fall. Senior management should consequently question why both
traders are undertaking such aggressive positions in this characteristic when the
predictable evolution of the factor is so uncertain. On the other hand, the Slope
characteristic is very close to the bottom of its range because the U.S. economy
was in a recession at the time that the positions were taken. ABC is short Slope,
which suggests that the trader anticipates that the slope of the curve will flatten,
while XYZ is taking the opposite bet. Senior management should explore the reasons behind the differing strategies regarding slope exposure.
140
John F. O. Bilson
The fundamental argument in favor of the LSC model for Value at Risk analysis
is that it provides a more transparent perspective on the risks taken by traders. In
some instances, senior managers are primarily interested in the total risk exposure
and are uninterested in the strategies behind the positions. Traditional VaR is perfectly adequate under these circumstances. On the other hand, when management
is focused on strategy, the LSC model does provide an overview of the exposure
of the portfolio to the primary characteristics of the yield curve.
1.5 Conclusion
In the developmental stage of Value at Risk analysis, positions taken by traders
were taken as given. In part, this assumption reflected the status of traders relative
to risk managers. In part, it reflected the minimalist requirements of senior management for risk oversight. As the art of risk management has developed, however, the status of risk managers has increased and the requirements of senior
management have become more sophisticated. Modern management wants to
know why positions are taken and whether positions are appropriate in the light of
economic and historical conditions. This is particularly the case in hedge funds
and proprietary trading where senior managers are typically only a few years past
the trading desk. Under these circumstances, risk managers and traders have a
symbiotic relationship which, if successful, combines to promote the overall interests of the corporation. Risk management reports that offer a transparent focus on
the trading strategy are an important component of this relationship. The purpose
of this chapter is to develop a prototype VaR report for a fixed income trading
desk. By defining the component VaR in terms of factor exposures, the approach
developed clearly delineated the trading strategies behind the positions taken.7
The examples developed in this paper demonstrate that the LSC model provides a
different and more informative report on risk than more traditional methodologies.
4GHGTGPEGU
Bilson JFO (2002) The Shadow of the Smile. Working Paper, Illinois Institute of Technology
Dowd K (1998) Beyond Value at Risk: The New Science of Risk Management. John Wiley
and Sons
JP Morgan (1995) RiskMetrics. Technical Document. JP Morgan, New York
Garman M (1997) The End of the Search for Component VaR. Financial Engineering Associates, Berkeley
7
I have developed a similar model for equity option trading in Bilson (2002).
Parsimonious Value at Risk for Fixed Income Portfolios
141
Jorion P (2000) Value at Risk .2nd ed. McGraw Hill
Nelson CR, Siegel AF (1987) Parsimonious Modeling of the Yield Curve. Journal of Business 60:4
Wilmer R (1996) A New Tool for Portfolio Managers: Level, Slope and Curvature Durations. Journal of Fixed Income
Wilson TC (1996) Calculating Risk Capital. In Alexander C (ed) The Handbook of Risk
Management and Analysis. John Wiley and Sons
4KUM$WFIGVKPIYKVJ8CNWGCV4KUM.KOKVU
Robert Härtl1 and Lutz Johanning2
1
Ludwig-Maximilians-University Munich, Institute for Capital Market Research
and Finance, Schackstr. 4, D-80539 Munich, Germany,
www.kmf.bwl.uni-muenchen.de.
2
European Business School, International University, Schloß Reichartshausen,
Endowed Chair for Asset Management, D-65375 Oestrich-Winkel, Germany,
www.amebs.de.
Abstract: Our analysis focuses on the risk budgeting process for banks using value
at risk limits. In this context, we investigate three major practical problems: a)
differences in time horizons between the bank’s total risk budget and the trading
divisions’ activities; b) adjustment for accumulated profit and losses to risk budgets, and c) incorporation of correlations between assets into the risk budgeting
process. To analyze these practical problems, we use Monte Carlo simulation.
Thereby, it can be shown that differences in time horizons among risk budgets and
trading units can be adjusted by the square root of time rule. Three types of limits
are proposed for the adjustment of accumulated profit and losses: the fixed, stop
loss and dynamic limits. While the two latter restrict the maximum loss to the ex
ante specified limit and show a symmetric profit and loss distribution, the dynamic
limit’s distribution is skewed to the right. We further illustrate that the average
usage of total risk capital is only 31.45 % for a trading division with thirty independently deciding traders. This shortfall is due to diversification effects. This setting is compared with a benchmark model in which total risk capital is always
used at the full capacity of 100 %. The comparison shows that the average profit
in the former model is only 33.13 % of the generated profit in the benchmark
model. The results may have interesting organizational implications on the banking sector.
JEL classification: G11; G20; G21; G31
Keywords: Value at Risk Limits, Risk Capital, Capital Allocation, Correlation
144
Robert Härtl and Lutz Johanning
+PVTQFWEVKQP
Although the theoretical deficits of the value at risk concept are well known
(Artzner et al. 1997; Artzner et al. 1999), value at risk has become the most popular risk measurement tool in the financial industry in recent years. While researchers have focused their investigations on approaches to compute value at risk (see
for an overview for market risks Knobloch; Bilson as well as Overbeck; Frerichs
and Wahrenburg for credit risks in this book), the risk capital allocation process
has hardly been covered so far. This fact is surprising, as the 1996 Basle Committee on Banking Supervision amendment to the capital accord for the incorporation
of market risks requires banks to install bank-wide value at risk limits to control
the traders’ risk takings. The common risk budgeting process in banks is a top
down allocation of capital from the top management down to the single business
units. In this paper, a couple of important and unsolved risk budgeting issues will
be further analyzed.
The first aspect to be discussed is the difference between time horizons for risk
capital in the context of the banking business. Risk capital is allocated top down
on a regular basis e.g. two or four times a year, whereas the time horizon of the
business units, namely the trading divisions, is short, e.g. a couple of minutes
only. Thus, the long time horizon for the capital allocation has to be transformed
into an appropriate short time horizon. The next issue raised is the commonly used
limit adjustment for realized profits and losses as a risk budgeting practice in
banks. And the third most challenging problem is the incorporation of correlations
between the exposures of business units and risk factors. It is a well known fact
that asset correlations smaller than one require a bank with multiple businesses to
hold less risk capital than would be required for the sum of these businesses on a
stand-alone basis (Saita 1999). For instance, the diversified risk capital of a New
York based investment bank with twenty trading businesses is only 29.8 % of the
sum of the stand alone risk capital of all units (Perold 2001). In order to use the
risk capital at full capacity, correlations have to be incorporated into the top down
allocation process. We address these three aspects by presenting the key approaches and results of the simulation studies of Beeck et al. 1999 and Dresel et
al. 2002.
In chapter 2 we briefly outline the theory behind value at risk limits and risk
capital allocation. In chapter 3 we describe the simulation model, which will be
used in the following chapters. The approaches for adjusting for differences in
time horizons and for profits and losses will be considered in chapter 4. The incorporation of correlations among traders’ exposures will be discussed in chapter
5. Finally, we conclude our analysis in chapter 6.
Risk Budgeting with Value at Risk Limits
145
&GHKPKVKQPQH8CNWGCV4KUM.KOKVU
Value at risk is defined to be an €-amount of loss. Real losses of the trading position or portfolio can only be larger with a small probability p, e.g. 1 %, at the end
of a short holding period H, e.g. one day. For normally distributed profits and
losses, the value at risk (VaR) is defined as:
VaR = −( µ ∆V + L( p ) ⋅ σ ∆V ) .
(1)
Following this definition, VaR is a positive number. µ∆V is the expected profit or
loss for a given holding period, and σ∆V is the corresponding standard deviation.
L(p) is the quantile of the standard normal distribution, which is determined by the
probability p, e.g. for p=1 % (5 %), L(p) is -2.33 (-1.64).
Accordingly, we define risk capital as the ex ante assigned value at risk limit,
e.g. 1 mill. €. Applying straightforward this definition of VaR, more than the limit
can be lost with probability p at the end of the holding period H. A value at risk
limit of 1 mill. € allows a trader to take risk at a maximum of 1 mill. €. Of course,
he might not use the limit at full capacity. For normally distributed profits and
losses, the following linear risk restriction can be derived:
µ ∆V ≥ −VaR - Limit − L ( p ) ⋅ σ ∆V .
(2)
The trader is allowed to select portfolios that have less risk than or equal risk to
the limit. These portfolios lie above or on the linear line (2) in a µ∆V,σ∆V –diagram
(see for a graphical illustration figure 2.1.).
µ∆V
VaR-limit = 0,8 mill., slope: 1,64
VaR-limit =1 mill., slope: 2,33
A
VaR-limit = 1 mill., slope: 1,64
B
efficient frontier
-0,8 Mio.
σ∆V
-1 Mio.
Fig. 2.1. Portfolio selection and shortfall constraints (value at risk limits)
The concave line represents the Markowitz efficient frontier and shows the universe of the trader’s efficient portfolios. The three linear lines have three different
146
Robert Härtl and Lutz Johanning
VaR-limits. The line with a limit of 1 mill. € and a slope coefficient of 1.64 gives
a value of –1 mill. at p=5 % and for σ∆V =0. It intersects the efficient frontier in
point B. The trader is free to select from all portfolios left (or above) the limit line.
Point B is the efficient portfolio which has a VaR of exactly 1 mill. €. If p = 1 %,
the slope of the line changes to 2.33. Note, that for the given efficient frontier, a
risk limit of 0.8 mill. € at p= 5 % (slope of 1.64) would yield exactly the same
maximum risk exposures (point A). One important conclusion is that a risk limit
operates via the combined setting of size and confidence level (1-p). Since this
kind of risk budgeting restricts the probability of a shortfall, this approach is
known as portfolio selection with shortfall constraints (Leibowitz and Henriksson
1989, Leibowitz and Kogelman 1991).
The analytics of risk budgets are not limited to the normal distribution. Via
Tschebyschew’s inequality equation, the limit line can be derived for any distributions to be:
µ ∆V ≥ − VaR - Limit +
1
p
⋅ σ ∆V .
(3)
The slope coefficient is 1/√p, which is, at a given p, always larger than the slope
coefficient in equation (2).
Before we show how to use value at risk limits for the daily management of security traders, we briefly review the key literature about risk budgeting. Commonly, information asymmetries between the management and business units are
assumed. Beeck et al. 1999 and Dresel et al. 2002 believe that the traders have superior skills to forecast securities’ returns. Bühler and Birn 2001 model the allocation process for a two layered hierarchy. Because of unknown correlations, the
management has to increase the overall and individual unit’s risk capital substantially. In a principal agent model, Froot and Stein 1998 assume that it is costly to
raise external funds due to uncertain investment payoffs and a potential cash shortfall for which penalties have to be paid. Risk management arises as an endogenous
consequence to avoid an adverse selection problem in this line-up. Their key objective is to derive an investment specific hurdle rate. The difficulty is that the cost
of risk capital is a function of the covariance of a business unit’s profit with firmwide profits. But if business units decide independently about the size and direction (long or short) of their exposures, this covariance remains unknown ex ante.
Therefore, Froot and Stein 1998 derive the hurdle rate for the limit case in a scenario when the size of new investments is small. In that case, the effect of a new
investment on risk capital of other business units is small as well and can be neglected. In contrast, if big investments are assumed, interdependencies between
the investments arise and an optimal decision making in the risk budget allocation
can only be reached by a central decision authority. Stoughton and Zechner 1999
extend the Froot and Stein model. Their incentive model focuses on capital budgeting decisions of banks with multiple business units under the consideration of
asymmetric information. The main purpose is to derive an optimal capital allocation mechanism in order to achieve overall value maximization from a shareholder’s perspective. Stoughton and Zechner 1999 derive the optimal mechanism
Risk Budgeting with Value at Risk Limits
147
for only two divisions. Perold 2001 models a firm that has to provide guarantees
for its performance on customer contracts. Due to these guarantees, the firm has to
suffer deadweight costs. Only uncorrelated profits between business units are considered. In other words, risk effects of one investment decision on other business
units are ignored. To conclude, the capital allocation process can only be solved
for two business units or under extremely simplified assumptions, e.g. through the
inauguration of a central decision authority. However, a centralized decisionmaking does not seem to be a practical approach. Costs and delays need to be considered that are associated with the transmission of new information to headquarters whenever an investment is made. This may have prohibitive effects on the
business.
6JG5VTWEVWTGQHVJG5KOWNCVKQP/QFGNU
Our objective is to develop a practical risk budgeting approach. This is illustrated
in a simulation model (see for details Beeck et al. 1999 and Dresel et al. 2003).
We assume a decentralized organization, i.e. the bank’s management delegates its
trading decisions to the trading division. In the first step, there is only a single
trader. In the second step, the trading department consists of thirty traders. Each
trader is only allowed to trade a single security or risk factor.
The bank wants to maximize the expected profit, which is subject to a given risk
constraint because of e.g. external capital requirements. The key proposition can
be summarized in the management’s objective to maximize the total return of the
trading division without breaching neither the bank’s total limit nor each trader’s
individual limit. Since it is the risk capital and not liquidity that is scarce in the
banking industry, the optimization problem is stated as follows:
max µPF,t s.t. VaRPF,t≤VaR-LimitPF,t and VaRi,t≤VaR-Limiti,t for i = 1 to 30.
(4)
µPF,t represents the expected profit of the trading portfolio at time t. VaRPF,t is the
value at risk and VaR-LimitPF,t the value at risk limit for the total trading portfolio
at time t. Accordingly, VaRi,t stands for the value at risk and VaR-Limiti,t represents the value at risk limit for the exposure of a single trader i at time t. If there is
only a single trader, then VaRi,t = VaRPF,t. If there are thirty traders, value at risk
for the whole trading portfolio is
VaRPF ,t = VaR T ⋅ R ⋅ VaR
,
(5)
where VaR is the vector of the single value at risk numbers, and R is the correlation matrix. For all VaR calculations a confidence level of 99 % is applied. Because the holding period is always one day, the VaR is determined by ignoring the
expected returns:
VaR = − L( p) ⋅ σ ∆V .
(6)
148
Robert Härtl and Lutz Johanning
As value at risk fails to measure coherent and increasing risk (Artzner et al. 1997,
Artzner et al. 1999), we assume normally distributed stock returns in order to
guarantee a solution to our optimization approach.
We assume that the correlation matrix as well as the return vectors and standard
deviations of the traders’ stocks are exogenous and stable over time. Each trader
can choose individually whether to invest long or short, however the value at risk
limit needs to be fully exploited. It is supposed that the stocks are arbitrarily separable. Central to our study is that, like in Stoughton and Zechner 1999 trader decisions are independent of each other. All positions are opened in the morning of
one day and closed the next morning. There is no intraday-trading. With a 55 %
chance, the traders correctly anticipate the direction of the next day’s pricemovements. Consequently, if the trader predicts a price increase (price decrease)
he will decide to invest long (short).
The model neglects any principal-agent problems between the traders and topmanagement. Thus, there is no strategic trading implied in the model, i.e. a trader
does not invest strategically against the exposure of other traders, but only follows
his forecasted price movement.
#FLWUVKPI4KUM.KOKVUHQT6KOG*QTK\QPUCPF2TQHKVU
CPF.QUUGU
In the first step, we assume there is only a single trader dealing one security. The
trader’s risk budget for one year is 1 mill. €, while the trader’s investment horizon
is only one day. We assume a year has T=250 trading days and transform the annual limit AL in a daily limit DL by using the square root of time rule:
DL = AL / T .
(7)
Knowing his daily limit and the standard deviation σ of returns, the trader can derive the maximum exposure Vmax he is allowed to invest by:
V max = − DL / L( p ) ⋅ σ .
(8)
In the following, we consider three different types of limit systems. If the trader
receives the same size of the limit each day given by equation (6), we define it to
be the fixed limit. This risk capital is independent of the profits and losses accumulated before. If the annual limit is decreased by the accumulated losses in that
year, we define it to be the stop loss limit. Daily risk limit at day t is then given by
DLt = ( AL − accumulated losses in that year at day t)/ 250 .
(9)
However note that in this case the annual limit does not increase when accumulated profits and losses are positive. The third limit is defined to be the dynamic
limit. Like for the stop loss limit, accumulated losses decrease the annual limit, but
accumulated profits will as well increase the annual limit. Successful traders are
Risk Budgeting with Value at Risk Limits
149
rewarded by receiving a larger limit. Note that for the fixed limits breaches of annual value at risk limits are possible. Since the stop loss and the dynamic limit are
self-exhausting, breaches of annual limits can only happen with extreme small
probability. We find some German banks using the fixed and especially the stop
loss limit but not the dynamic limit for the capital allocation in their trading divisions.
Using Monte Carlo simulation, we generate a return process with a drift rate of
7 % p.a. and a standard deviation of 24 % p.a. for 5,000 years with 250 trading
days each year. The daily profits and losses (P&L) are accumulated to an annual
profit and loss. Figure 4.1. and table 4.1. show the distribution of the annual P&L
for the 5,000 simulated years. The average P&L for the fixed limit (FL) is 616
thousand € and is thereby larger than the average P&L for the stop loss limit (SL),
even despite a lower standard deviation (494 thousand €). The minimum P&L is –
1,151 thousand €, which exceeds the ex ante risk limit. But this is the only limit
breach observed within the 5,000 years. Therefore, the observed frequency of limit
breaches is much smaller than the expected probability of p=1 %. For the stop loss
and dynamic limit, the minimum P&L is both 725 thousand €. The annual average
P&L for the dynamic limit (DL) is larger with a value of 855 thousand €, but with
a much higher standard deviation (977 thousand €). Figure 4.1. shows that the
P&L distribution for the dynamic limit is skewed to the right, and therefore gives
an expected P&L that is larger than for the fixed and stop loss limit. However, the
probability of a loss is larger as well.
Table 4.1. Annual profits and losses in thousand € for 5,000 simulated years
limit system
Average std. median
FL
SL
DL
616
594
855
494
504
977
25%-quantile
615
600
642
75%-quantile
277
242
164
960
949
1,318
max
min
2,437
2,436
9,052
-1,151
-725
-725
std.=standard deviation, max=maximum profit and loss, min=minimum profit and loss,
FL=fixed limit, SL=stopp loss limit, DL=dynamic limit
fixed limit
stop loss limit
250
250
200
200
dynamic limit
180
150
100
140
frequency
frequency
frequency
160
150
100
50
50
0
0
120
100
80
60
40
in T€
Fig. 4.1. Profit and loss distribution for 5,000 simulated years
in T€
3.000
2.750
2.500
2.250
2.000
1.750
1.500
750
1.250
500
1.000
250
-250
-500
-750
3.000
in T€
-1.000
2.750
2.500
2.250
2.000
1.750
1.500
750
1.250
500
1.000
250
-250
-500
-750
3.000
0
-1.000
2.750
2.500
2.250
2.000
1.750
1.500
750
1.250
500
1.000
250
-250
-500
-750
-1.000
20
150
Robert Härtl and Lutz Johanning
To draw a first conclusion, the P&L distribution of the - in practice mostly used stop loss limit, restricts the maximum loss to the ex ante specified risk capital.
However, it is shown that it is in fact dominated by the symmetric distribution of
the fixed limit. The dynamic limit offers higher expected P&L for a larger loss
probability. These results have the practical implication to either use the fixed or
the dynamic limit for risk budgeting.
+PEQTRQTCVKPI#UUGV%QTTGNCVKQPU+PVQ4KUM$WFIGVU
In addition to the described problems of adjusting for different time horizons and
profit and losses, the more severe problem is to incorporate the correlations among
assets into the risk budgeting process. Neglecting these correlations may lead to an
insufficient utilization of risk capital. Perold 2001 reports that a New York based
investment bank with twenty trading businesses uses on average only 29.8 % of
the stand-alone risk capital.
As this diversification effect is essential for our analysis, we want to analyze its
genesis in greater detail (Dresel et al. 2003). The problem of unused risk capital
emerges when business units decide independently about size and direction (long
or short) of their investments. Consider an example with two traders 1 and 2
(trader 1 can be seen as the first business unit and trader 2 as the second business
unit). Each trader deals only one stock. Suppose that the stocks A and B have a
given and commonly known correlation ρAB and covariance σAB. Both traders always invest independently a maximum exposure of VA and VB , which is consistent
with their individual budget constraint (e.g. a value at risk limit). Then the correlation between the payoffs of the traders’ exposures ρΤ1,Τ2 only depends on the direction (long or short) of the investments; ρΤ1,Τ2 equals either ρAB or –ρAB.
We adopt this simple setting to our simulation model with thirty traders each
dealing a single security. Again, each trader has a forecasting ability of 55 % (see
chapter 3 for the model description). These traders decide independently about the
direction (long or short) of their exposures. The correlations, standard deviations
and expected daily returns are stable and given (see appendix 1).
The total daily value at risk limit for the trading division equals 3 mill. €. We
assign a value at risk sub-limit to each individual trader. These sub-limits are assigned in a way that each trader can invest the same market value Vi considering
the exogenous given standard deviation of his stock. The detailed calculation for
the traders’ limits is documented in appendix 2. As in our special case, all correlations between the thirty stocks are positive, the value at risk of the portfolio will
reach its maximum, when all traders invest long at the same time (or short at the
same time). Although this constellation rarely happens because of the supposed
independency of trading decisions, we have to calculate the sub-limits on the basis
of this unlikely scenario to make sure that the total value at risk is never exceeded.
Table 5.1. presents the resulting value at risk sub-limits for all thirty traders. The
individual value at risk limits range from 108,391 € (trader 19) to 315,735 €
(trader 26). The nominal sum of the sub-limits equals 5,043,514 €, whereas the
Risk Budgeting with Value at Risk Limits
151
aggregated limit accounting for the underlying stock correlations is exactly 3 mill.
€.
Table 5.1. Value at risk limits for the thirty traders in €
trader 1
160,745
trader 7
144,072
trader 13
217,744
trader 19
108,391
trader 25
131,501
trader 2
151,974
trader 8
167,734
trader 14
127,159
trader 20
131,304
trader 26
315,735
trader 3
134,048
trader 9
145,099
trader 15
220,535
trader 21
251,019
trader 27
134,115
trader 4
154,401
trader 10
183,360
trader 16
158,131
trader 22
129,265
trader 28
203,129
trader 5
162,087
trader 11
182,060
trader 17
138,132
trader 23
147,245
trader 29
148,867
trader 6
162,302
trader 12
136,092
trader 18
273,877
trader 24
167,952
trader 30
155,437
Whenever trading decisions of multiple traders are independent, at least to some
degree, the above optimization approach cannot be solved analytically. Therefore
we run a Monte Carlo simulation and simulate 20,000 trading days with the above
described data as input parameters. Cholesky factorization is applied to incorporate correlations into the vectors of identical and independent distributed standard
normal random variables (Hull 2003). The traders know their individual value at
risk sub-limit. A trader’s exposure Vi is derived via the historically estimated (250
days) standard deviation. The results of the simulation are presented in table 5.2.
Table 5.2. Results for 20,000 simulated trading days – basic model
VaR of the trading
division in €
Utilization of available limit (in %)
Total profit of trading division in €
25%quantile
75%quantile
maximum
mean
std.dev.
Median
943,479
221,050
883,440 787,679
1,040,672 550,490
2,432,677
31.45
--
29.45
34.69
18.35
81.09
180,714
417,278
173,848 -82,149
435,199
-2,139,179
3,019,759
26.26
minimum
The value at risk is shown in line 2. On average total “used” value at risk amounts
to 943,479 €, which is only 31.45 % (line 3) of the “allocated” risk capital of 3
mill. €. Even the maximum value at risk with 2,432,677 € is far below the “allocated” risk capital limit amount (81.09 %). These figures show the tremendous diversification effect. Line 4 in table 2 gives information about the profit of the trading division. The average profit is 180,714 €.
As on average only 31.45 % of the risk capital of 3 mill. € is used, the trading
profit should obviously be far below its optimum. Looking for the best limit allocation process, we develop a benchmark scenario in a way that the value at risk
limit of 3 mill. € is fully used on each trading day. This can be achieved by using
the correlation structure between the exposures of the traders. Each trader has to
152
Robert Härtl and Lutz Johanning
report his trading decision (whether he invests short or long) to a central authority
(the risk controlling division)– similar to the Stoughton / Zechner (1999) assumption. Knowing all thirty trading directions, the authority determines the size of the
trader’s exposure. In accordance to the basic model, the exposures are calculated
so as to assure each trader can invest the same market value Vi. The simulation’s
results are shown in table 5.3. The same random numbers are used as in the basic
model before. Therefore, all differences in the results are due to model specification.
Table 5.3. Results for 20,000 simulated trading days – benchmark scenario
Sum of all
30 traders
VaR divided
by 30 in €
Total profit
of all traders
in €
mean
std.dev.
Median
25%quantile
75%quantile
minimum
maximum
559,923
110,250
571,312
484,241
641,512
207,152
920,133
545,443
1,277,300
578,576
-282,873
1,403,039
-5,525,159 5,589,909
By assumption, the total value at risk of the trading division is always 3 mill. €.
The average value at risk of a single trader (the sum of individual values at risk
divided by 30 – without taking correlation effects into account) is 559,923 € with
a standard deviation of 110,250 € (line 2). The average profit for the trading division more than triples compared to the basic scenario (from 180,714 € to 545,443
€) but the range of profits increases as well (see the standard deviation of
1,277,300 € or maximum and minimum values).
Of course, this model does not seem to be a practical application to real trading
world. However it illustrates a benchmark: the traders’ forecasting abilities are always fully exploited and the total value at risk limit is never exceeded. Obviously,
this should result in the highest possible profit for the trading division assuming
the given forecasting ability. This result also documents the cost of not taking diversification effects into account. These costs are equal to the large shortfall in average daily profits of 364,729 € (545,443 € minus 180,714 €).
Since the benchmark model is not applicable for real trading situations, Dresel
et al. 2003 developed a treasurer model to solve this problem. First, the treasurer
determines the value at risk for the trading division. Knowing the amount of unused risk capital, he buys or sells an equally weighted stock index consisting of
the thirty stocks. In the treasurer model I, the treasury always fills up the total
value at risk to the maximum of 3 mill. €. Applying this procedure, the average
profit of the trading division more than doubles compared to the basic model. The
P&L standard deviation is comparable to the benchmark model. Since the treasurer has no forecasting skills, the utilization is still not at the optimum. In the
treasurer model II, the individual traders’ limits are scaled by the factor of 2.5. Instead of filling up, the treasurer has to scale down the total risk limit when the risk
limit of all thirty traders exceeds 3 mill. €. In that model the average P&L can be
Risk Budgeting with Value at Risk Limits
153
further increased significantly without further increasing the P&L standard deviation. But still, the average P&L in the benchmark model remains larger (see for
details Dresel et al. 2003).
To conclude, the incorporation of correlations between assets into the risk budgeting process can optimize the utilization of risk limits and can be achieved in an
easy and practical way by installing a treasurer.
%QPENWUKQPCPF2TCEVKECN+ORNKECVKQPU
We investigate the risk budgeting process for banks and analyze how to adjust risk
limits for accumulated profits and losses and differences in time horizon for the
total and single trading divisions’ risk budget. By simulating 5,000 years, we show
that the fixed limit – in which the trader has the same daily limit independent of
the accumulated P&L – yields a symmetric P&L distribution, whereas the
distribution of the dynamic model is skewed to the right. Therefore, in the
dynamic limit, the expected P&L as well as the loss probability are larger as
compared to the results for the fixed and stop loss limits.
Modeling a trading division of thirty traders, we show that due to independent
trading decisions, the utilization of the total risk limit is only 31.45 %. We compare this setting with a benchmark model, in which a central authority receives the
trading signal and then determines the traders’ exposures. In a simulation with
20,000 trading days, the average profit as well as the standard deviation more than
triple. Since in real trading situations there is no time to communicate via a central
decision authority, we suggest that a treasurer takes the residual risk. Depending
on the treasurer’s field of responsibility, the P&L of the whole bank can be increased significantly.
The organizational implication of our simulation results might be of interest.
Since trading requires some market specific skills, banks organize their trading
business according to countries or industries. For example, in many financial institutions one trading unit trades short- and long-term European bonds and another
trades long- and short-term US bonds. This happens although the correlation between long- and short-term bonds is often close to zero, however between longterm European and US-Bonds the correlations are higher. From the perspective of
an optimal risk budgeting process, all assets with high correlations should be managed in one business unit. However, the correlation between different trading units
should be close to zero. If the risks of the business units are independent, single
trader’s decisions would only slightly affect the other traders’ risk. Since the correlation between long-term European and US bonds, and between short-term
European and US bonds are positive, it would be advantageous to trade all longterm bonds in one unit and all short-term bonds in the other. We recommend to
consider at least these correlation aspects as well as the market specific skills. Of
course, this kind of organization requires a detailed analysis of the structure and
stability of long- and short-term correlations.
154
Robert Härtl and Lutz Johanning
#RRGPFKZ
Appendix 1: Correlation Matrix Used in the Simulation Model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1
1.0000
0.1921
0.4327
0.3231
0.3798
0.4081
0.3243
0.3458
0.2950
0.3472
0.4360
0.1757
0.1300
0.2125
0.3081
0.1121
0.1570
0.2906
0.4196
0.1987
0.2000
0.3465
0.2199
0.4236
0.1386
0.2130
0.0698
0.3234
0.4641
0.4222
2
0.1921
1.0000
0.4627
0.3683
0.5334
0.3895
0.5297
0.4802
0.3486
0.5878
0.4882
0.2591
0.4006
0.4716
0.3123
0.2021
0.1251
0.3143
0.3852
0.2939
0.2703
0.4221
0.7783
0.4215
0.4619
0.3723
0.3463
0.4871
0.3011
0.3985
3
0.4327
0.4627
1.0000
0.6936
0.4898
0.5060
0.5168
0.5201
0.5075
0.5568
0.5641
0.1525
0.3416
0.3352
0.3960
0.1815
0.1498
0.2747
0.4605
0.2886
0.2606
0.4399
0.4410
0.4333
0.3327
0.3269
0.2804
0.4751
0.4624
0.5585
4
0.3231
0.3683
0.6936
1.0000
0.3566
0.4159
0.3744
0.3621
0.4386
0.4057
0.3641
0.1156
0.3602
0.2934
0.3475
0.1890
0.1008
0.2666
0.3978
0.2777
0.1673
0.3656
0.3357
0.2977
0.2725
0.2627
0.2523
0.4109
0.3540
0.3880
5
0.3798
0.5334
0.4898
0.3566
1.0000
0.4884
0.5899
0.5248
0.2326
0.6248
0.3756
0.2863
0.3977
0.3031
0.3325
0.0567
0.2042
0.4184
0.3402
0.2432
0.3058
0.2630
0.5097
0.4096
0.2300
0.3132
0.2631
0.4953
0.3451
0.5688
6
0.4081
0.3895
0.5060
0.4159
0.4884
1.0000
0.4141
0.5618
0.4855
0.4870
0.4787
0.2262
0.3147
0.2723
0.3073
0.2141
0.1339
0.3363
0.4196
0.2486
0.2422
0.3553
0.3939
0.4490
0.1901
0.2546
0.2266
0.4440
0.4549
0.6666
7
0.3243
0.5297
0.5168
0.3744
0.5899
0.4141
1.0000
0.4867
0.4045
0.6860
0.5130
0.1559
0.2974
0.3026
0.4056
0.1552
0.2266
0.3609
0.4794
0.2991
0.3263
0.3634
0.5316
0.5347
0.2962
0.3191
0.2835
0.4611
0.3939
0.4889
8
0.3458
0.4802
0.5201
0.3621
0.5248
0.5618
0.4867
1.0000
0.3618
0.5904
0.5294
0.1537
0.4544
0.2942
0.4300
0.1711
0.1267
0.4424
0.3719
0.3352
0.3072
0.3948
0.4666
0.4062
0.3018
0.4222
0.2194
0.6081
0.4181
0.6106
9
0.2950
0.3486
0.5075
0.4386
0.2326
0.4855
0.4045
0.3618
1.0000
0.4174
0.3837
0.0084
0.2163
0.2656
0.3382
0.2199
0.2052
0.1800
0.4438
0.2716
0.1547
0.4836
0.3103
0.3960
0.2812
0.2206
0.2594
0.3808
0.3484
0.4132
10
0.3472
0.5878
0.5568
0.4057
0.6248
0.4870
0.6860
0.5904
0.4174
1.0000
0.5064
0.2336
0.5539
0.3330
0.5396
0.1174
0.2130
0.5249
0.4355
0.3275
0.4667
0.3959
0.5896
0.4749
0.3276
0.5450
0.3247
0.6808
0.4663
0.5616
11
0.4360
0.4882
0.5641
0.3641
0.3756
0.4787
0.5130
0.5294
0.3837
0.5064
1.0000
0.1556
0.2677
0.2741
0.4091
0.1680
0.1285
0.3338
0.4484
0.2310
0.2830
0.4080
0.5106
0.4694
0.2451
0.2613
0.1573
0.4932
0.3424
0.5111
12
0.1757
0.2591
0.1525
0.1156
0.2863
0.2262
0.1559
0.1537
0.0084
0.2336
0.1556
1.0000
0.0948
0.2970
0.1376
0.0400
0.1163
0.1172
0.1294
0.0433
0.1452
0.1029
0.2716
0.2320
0.2251
0.1441
0.1454
0.1739
0.2979
0.2092
13
0.1300
0.4006
0.3416
0.3602
0.3977
0.3147
0.2974
0.4544
0.2163
0.5539
0.2677
0.0948
1.0000
0.2669
0.5611
0.0250
0.1244
0.5057
0.2130
0.2402
0.3501
0.2164
0.3615
0.1925
0.1983
0.4529
0.2310
0.6365
0.2574
0.3353
14
0.2125
0.4716
0.3352
0.2934
0.3031
0.2723
0.3026
0.2942
0.2656
0.3330
0.2741
0.2970
0.2669
1.0000
0.1396
0.2138
0.1814
0.1104
0.3408
0.2364
0.1498
0.4221
0.4695
0.2368
0.6316
0.1755
0.3380
0.3451
0.2876
0.1798
15
0.3081
0.3123
0.3960
0.3475
0.3325
0.3073
0.4056
0.4300
0.3382
0.5396
0.4091
0.1376
0.5611
0.1396
1.0000
0.2013
0.1542
0.5559
0.2872
0.2736
0.3965
0.2671
0.2649
0.3041
0.1740
0.4491
0.1502
0.7100
0.3167
0.4241
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
16
0.1121
0.2021
0.1815
0.1890
0.0567
0.2141
0.1552
0.1711
0.2199
0.1174
0.1680
0.0400
0.0250
0.2138
0.2013
1.0000
0.0716
0.0326
0.1328
0.1768
0.1217
0.2845
0.1915
0.1942
0.1572
0.0648
0.1764
0.1529
0.1349
0.0994
17
0.1570
0.1251
0.1498
0.1008
0.2042
0.1339
0.2266
0.1267
0.2052
0.2130
0.1285
0.1163
0.1244
0.1814
0.1542
0.0716
1.0000
0.1124
0.1889
0.0520
0.0513
0.2113
0.1607
0.1767
0.1323
0.0863
0.2592
0.1712
0.1331
0.1932
18
0.2906
0.3143
0.2747
0.2666
0.4184
0.3363
0.3609
0.4424
0.1800
0.5249
0.3338
0.1172
0.5057
0.1104
0.5559
0.0326
0.1124
1.0000
0.2047
0.2824
0.2650
0.1658
0.2658
0.3426
0.1390
0.5068
0.2485
0.6552
0.2832
0.3938
19
0.4196
0.3852
0.4605
0.3978
0.3402
0.4196
0.4794
0.3719
0.4438
0.4355
0.4484
0.1294
0.2130
0.3408
0.2872
0.1328
0.1889
0.2047
1.0000
0.3712
0.2191
0.4988
0.4045
0.5288
0.3108
0.1379
0.2218
0.3883
0.4873
0.4997
20
0.1987
0.2939
0.2886
0.2777
0.2432
0.2486
0.2991
0.3352
0.2716
0.3275
0.2310
0.0433
0.2402
0.2364
0.2736
0.1768
0.0520
0.2824
0.3712
1.0000
0.0895
0.3294
0.2230
0.3506
0.3161
0.2798
0.1606
0.3929
0.3578
0.2268
21
0.2000
0.2703
0.2606
0.1673
0.3058
0.2422
0.3263
0.3072
0.1547
0.4667
0.2830
0.1452
0.3501
0.1498
0.3965
0.1217
0.0513
0.2650
0.2191
0.0895
1.0000
0.2222
0.2872
0.2930
0.1410
0.3335
0.2145
0.4218
0.2757
0.2937
22
0.3465
0.4221
0.4399
0.3656
0.2630
0.3553
0.3634
0.3948
0.4836
0.3959
0.4080
0.1029
0.2164
0.4221
0.2671
0.2845
0.2113
0.1658
0.4988
0.3294
0.2222
1.0000
0.4362
0.3826
0.4417
0.2505
0.3340
0.3473
0.3629
0.3176
23
0.2199
0.7783
0.4410
0.3357
0.5097
0.3939
0.5316
0.4666
0.3103
0.5896
0.5106
0.2716
0.3615
0.4695
0.2649
0.1915
0.1607
0.2658
0.4045
0.2230
0.2872
0.4362
1.0000
0.3922
0.4140
0.3281
0.3603
0.4241
0.2726
0.4163
24
0.4236
0.4215
0.4333
0.2977
0.4096
0.4490
0.5347
0.4062
0.3960
0.4749
0.4694
0.2320
0.1925
0.2368
0.3041
0.1942
0.1767
0.3426
0.5288
0.3506
0.2930
0.3826
0.3922
1.0000
0.2147
0.2240
0.2475
0.4238
0.5175
0.4675
25
0.1386
0.4619
0.3327
0.2725
0.2300
0.1901
0.2962
0.3018
0.2812
0.3276
0.2451
0.2251
0.1983
0.6316
0.1740
0.1572
0.1323
0.1390
0.3108
0.3161
0.1410
0.4417
0.4140
0.2147
1.0000
0.2406
0.3985
0.3549
0.1993
0.2094
26
0.2130
0.3723
0.3269
0.2627
0.3132
0.2546
0.3191
0.4222
0.2206
0.5450
0.2613
0.1441
0.4529
0.1755
0.4491
0.0648
0.0863
0.5068
0.1379
0.2798
0.3335
0.2505
0.3281
0.2240
0.2406
1.0000
0.2503
0.5734
0.2860
0.3703
27
0.0698
0.3463
0.2804
0.2523
0.2631
0.2266
0.2835
0.2194
0.2594
0.3247
0.1573
0.1454
0.2310
0.3380
0.1502
0.1764
0.2592
0.2485
0.2218
0.1606
0.2145
0.3340
0.3603
0.2475
0.3985
0.2503
1.0000
0.2812
0.1631
0.1974
28
0.3234
0.4871
0.4751
0.4109
0.4953
0.4440
0.4611
0.6081
0.3808
0.6808
0.4932
0.1739
0.6365
0.3451
0.7100
0.1529
0.1712
0.6552
0.3883
0.3929
0.4218
0.3473
0.4241
0.4238
0.3549
0.5734
0.2812
1.0000
0.4593
0.5509
29
0.4641
0.3011
0.4624
0.3540
0.3451
0.4549
0.3939
0.4181
0.3484
0.4663
0.3424
0.2979
0.2574
0.2876
0.3167
0.1349
0.1331
0.2832
0.4873
0.3578
0.2757
0.3629
0.2726
0.5175
0.1993
0.2860
0.1631
0.4593
1.0000
0.4418
30
0.4222
0.3985
0.5585
0.3880
0.5688
0.6666
0.4889
0.6106
0.4132
0.5616
0.5111
0.2092
0.3353
0.1798
0.4241
0.0994
0.1932
0.3938
0.4997
0.2268
0.2937
0.3176
0.4163
0.4675
0.2094
0.3703
0.1974
0.5509
0.4418
1.0000
Risk Budgeting with Value at Risk Limits
155
Annualized expected returns of the thirty stocks (in percent)
Stock 1
Stock 2
Stock 3
Stock 4
Stock 5
Stock 6
Stock 7
Stock 8
Stock 9
13.50
-31.34
-4.74
-3.07
-46.31
-2.15
-52.97
-7.96
-13.67
Stock
10
-32.09
Stock
11
-57.37
Stock
12
-23.63
Stock
13
-59.41
Stock
14
-2.68
Stock
15
-40.94
Stock
16
-22.71
Stock
17
3.38
Stock
18
-68.15
Stock
19
-9.96
Stock
20
-42.29
Stock
21
-49.55
Stock
22
-20.42
Stock
23
-12.31
Stock
24
-20.45
Stock
25
-0.43
Stock
26
0.48
Stock
27
-6.60
Stock
28
-31.41
Stock
29
-23.39
Stock
30
-22.22
Annualized standard deviations of the thirty stock returns (in percent)
Stock 1
Stock 2
Stock 3
Stock 4
Stock 5
Stock 6
Stock 7
Stock 8
Stock 9
38.81
36.70
32.37
37.28
39.14
39.19
34.79
40.50
35.04
Stock
10
44.28
Stock
11
43.96
Stock
12
32.86
Stock
13
52.58
Stock
14
30.71
Stock
15
53.25
Stock
16
38.18
Stock
17
33.36
Stock
18
66.13
Stock
19
26.17
Stock
20
31.71
Stock
21
60.61
Stock
22
31.21
Stock
23
35.56
Stock
24
40.56
Stock
25
31.75
Stock
26
76.24
Stock
27
32.39
Stock
28
49.05
Stock
29
35.95
Stock
30
37.53
Appendix 2: Calculation of traders’ individual value at risk limits
From equation (6) it follows
VaRi = −Vi ⋅ L( p ) ⋅ σ i
with Vi as the market value of trader i and σi the known and stable standard deviation. Equation (5) gives:
2
VaRPF
= VaR T ⋅ R ⋅ VaR .
Assuming identical positions for all traders leads to:
n
n
2
VaRPF
= V ⋅ L( p) 2 ⋅ ∑∑ σ i⋅σ j ⋅ ρ ij .
2
i =1 j =1
Since the maximum position V is identical for all traders, the index i is dropped. It
can be derived by (note that the correlation ρ ij needs to be multiplied with –1 if
one of the traders i and j is short):
2
VaRPF
V=
L( p) 2 ⋅
n
n
∑∑
i =1 j =1
.
σ i ⋅ σ j ⋅ ρ ij
We derive the value at risk limit for trader i taking into account the stock’s standard deviation σi:
VaR − Limit = −V ⋅ L( p) ⋅ σ i
156
Robert Härtl and Lutz Johanning
~
Given this limit, the traders derive their exposure Vi with the estimated (250 days)
~
historical standard deviation σ~i . Note that the average value of Vi equals V.
VaR − Limit i
~
.
Vi = −
L( p) ⋅ σ~i
Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent Measures of Risk. Mathematical
Finance 9: 203-228
Artzner P, Delbaen F, Eber JM, Heath D (1997) Thinking Coherently. Risk 10: 68-71
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Beeck H, Johanning L, Rudolph B (1999) Value-at-Risk-Limitstrukturen zur Steuerung und
Begrenzung von Marktrisiken im Aktienbereich. OR Spektrum 21: 259-286
Berger AN, Herring RJ, Szegö GP (1995) The Role of Capital in Financial Institutions.
Journal of Banking and Finance 19: 393-430
Bühler W, Birn M (2001) Steuerung von Preis- und Kreditrisiken bei dezentraler Organisation, Working Paper No. 01-05, University of Mannheim, August 2001.
Dresel T, Härtl R, Johanning L (2002) Risk Capital Allocation using Value at Risk Limits
if Correlations between Traders’ Exposures are unpredictable. European Investment
Review 1: 57-61
Dresel T, Härtl R, Johanning L (2003) Risk Capital Allocation using Value at Risk Limits:
Incorporating unpredictable Correlations between Traders’ Exposures, Working Paper,
Ludwig-Maximilians-University Munich, Institute for Capital Market Research and
Finance, www.kmf.bwl.uni-muenchen.de
Froot KA, Stein JC (1998) Risk Management, Capital Budgeting, and Capital Structure
Policy for Financial Institutions: An Integrated Approach. Journal of Financial Economics 47: 55-82
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Leibowitz ML, Henriksson RD (1989) Portfolio Optimization with Shortfall Constraints: A
Confidence-Limit Approach to Managing Downside Risk. Financial Analysts Journal
45: 34-41
Leibowitz ML, Kogelman S (1991) Asset Allocation under Shortfall Constraints: Journal of
Portfolio Management 17: 18-23
Perold AF (2001) Capital Allocation in Financial Firms, Harvard Business School, Working Paper 98-072, February 2001
Risk Budgeting with Value at Risk Limits
157
Rothschild M, Stiglitz JE (1970) Increasing Risk: I. A Definition. Journal of Economic
Theory 2: 225-243
Saita F (1999) Allocation of Risk Capital in Financial Institutions. Financial Management
28: 95-111
Stoughton NM, Zechner J (1999) Optimal Capital Allocation Using RAROC and EVA,
Working Paper, UC Irvine and University of Vienna, January 1999.
Jack E. Wahl1 and Udo Broll2
1
University of Dortmund, Department of Finance, 44221 Dortmund, Germany
University of Dresden, Department of Business Management and Economics
01062 Dresden, Germany, broll@iwb-dresden.de
2
Abstract: We study the implications of the value at risk (VaR) concept for the optimum amount of equity capital of a banking firm in the presence of credit risk. As
a risk management tool VaR allows to control for the probability of bankruptcy. It
is shown that the required amount of equity capital depends upon managerial and
market factors, and that equity and asset/liability management has to be addressed simultaneously.
JEL Classification: G21
Keywords: Equity Capital, Value at Risk (VaR), Banking, Risk Management, Asset/Liability
Management, Credit Risk, Risk Regulations
+PVTQFWEVKQP
In recent years, value at risk (VaR) has become a heavily used risk management
tool in the banking sector. Roughly speaking, the value at risk of a portfolio is the
loss in market value over a risk horizon that is exceeded with a small probability.
Bank management can apply the value at risk concept to set capital requirements
because VaR models allow for an estimate of capital loss due to market and credit
risk (Duffie and Pan 1997; Jackson et al. 1997; Jorion 1997; Saunders 1999;
Simons 2000; Broll and Wahl 2003). The aim of our study is to answer the question what is the optimum amount of equity capital of a banking firm using the
value at risk concept in the presence of credit risk?
We model a banking firm, in which a risk averse or risk neutral bank management has to decide on the assets and liabilities of the bank given a competitive financial market. An excellent discussion of bank management can be found in
Greenbaum and Thakor 1995; Bessis 2002; for the analysis of banking firm models see, e.g., Freixas and Rochet 1997; Wong 1997; Broll and Wahl 2003. We as-
160
Jack E. Wahl and Udo Broll
sume the return on the bank’s portfolio of loans to be uncertain. The banking firm
is exposed to credit risk and may not be able to meet its debt obligations. Instead
of coping with the exposure of the banking firm to financial risk by using hedging
instruments such as futures and options (Broll et al. 1995), we incorporate as a risk
management tool the value at risk approach in order to address bankruptcy risk.
As depicted in figure 1 the bank faces a loss distribution. Given a confidence
level of 99 percent, to the equity holder there is 1 percent chance of losing VaR or
more in value. Hence, if VaR determines the optimum amount of bank equity
capital, then the confidence level gives the probability that the bank will be able to
meet its debt obligations.
Probability
1 percent
99 percent
0
VaR
Loss
Fig. 1. Loss distribution, VaR and bank equity
The study proceeds as follows: Section 2 presents the banking firm model in a
competitive market environment under default risk of loans. The uncertain market
value of loans is assumed to be lognormally distributed. The value at risk concept
is formulated for this stochastic setting. In section 3 we investigate how optimum
volume of equity capital is affected by value at risk. It is demonstrated that managerial and market factors determine optimal asset/liability and equity management
of the banking firm and that the probability of bankruptcy has a complex impact
upon the risk management process and organization. Section 4 discusses the case
of value maximization and reports a distinct relationship between optimum equity
and VaR. Final section 5 concludes the paper.
#$CPMKPI(KTO
In this section we study how a risk averse bank management, acting in a competitive financial market, can use the value at risk approach to deal with credit risk.
The main questions of the investigation are: What is the optimum amount of equity capital of a bank using the value at risk concept? What is the optimum interaction between the bank’s assets and liabilities?
Value at Risk, Bank Equity and Credit Risk
161
Loans granted by the banking firm, L , exhibit some default risk. At the beginning
of the period the return of the bank’s portfolio of loans, r%L , is random. The loan
portfolio is financed by issuing deposits, D , and equity capital, E . The intermediation costs compounded to the end of the period are determined by C ( L + D ) .
Hence intermediation costs are modelled to depend upon the sum of the bank’s financial market activities. The cost function C (⋅) has properties C '(⋅) > 0 and
C ''(⋅) > 0 , i.e., marginal costs are positive and increasing.
The balance sheet of the banking firm is at each point in time:
Bank balance sheet
Loans L
Deposits D
Equity E
Equity is held by shareholders and necessarily E = L − D . Optimum decision
making of bank’s management has to satisfy this balance sheet constraint. The
debt/equity ratio of the banking firm is endogenous and follows from the market
structure, preferences and risk regulations embedded in the model. Given that the
bank’s loans have risky returns, bankruptcy of the banking firm occurs if the bank
cannot meet its debt obligations. Value at risk is a risk management tool which allows to cope with this kind of bankruptcy risk.
If the bank’s loss in market value does not exceed equity capital at some confidence level 1 − α , then value at risk, VaR α , measures the maximum size of that
loss in the next period. That is to say, E ≥ VaR α implies that the firm is able to
meet its debt obligations with probability 1 − α . Therefore α measures the
maximum probability of bankruptcy of the banking firm.
In the literature, value at risk is discussed as an indicator for minimum capital
requirements regarding the solvency of banks (Jackson et al. 1997). Our paper derives the optimum amount of equity capital and the optimum amount of loans and
deposits using the value at risk concept. Note that from the balance sheet identity
bank management has to take into account only two of the three magnitudes.
We consider a competitive bank which faces the risky return r%L on its loans and
% , can be
the given riskless market rate of deposits, rD . The bank’s risky profit, Π
stated as follows:
% = r% L − r D − C ( L + D ).
Π
L
D
Probability distributions of returns are ranked by applying the expected utility hypotheses. For expository reasons let us assume that a quadratic utility function
represents, in the relevant range of profits with positive marginal utility, the risk
attitude of bank management. Furthermore, let us assume that 1 + r%L is lognor-
162
Jack E. Wahl and Udo Broll
mally distributed with a given expected value and a given variance. Quadratic
preferences imply the well-known straightforward mean-variance objective function below.
Risk management of the bank management determines the amount of equity
capital, E , by maximizing expected utility of profits:
max U ( µ ,σ ) = µ − a( µ 2 + σ 2 ); a > 0,
subject to
L=D+E,
VaR α ≤ E ,
where µ denotes expected profits and σ 2 denotes the variance of profits.
The goal of the competitive banking firm is to establish the optimum amount of
equity capital, E ∗ , satisfying the balance sheet identity and the value at risk constraint. Provided that the expected interest margin is positive, in the optimum the
required amount of equity capital is positive.
6JG5VQEJCUVKE5GVVKPI
Risk management of the bank has to take into account bankruptcy risk. The risk of
bankruptcy means that the bank may not be able to meet its financial obligations
vis-a-vis its depositors without further contributions by the owners. If the owners
are not able or not willing to contribute the bank goes bankrupt. Value at risk is a
risk management tool which allows to control for the probability of bankruptcy.
The bankruptcy risk of a banking firm can be defined as the positive probability
that a decline in market value of the bank’s loans is greater than equity capital (see
Appendix A1). In order to control for bankruptcy risk bank management has to adjust assets and liabilities. Thereby the probability of extensive losses is limited to a
given probability of insolvency. This leads to the solvency condition
Probability(Loss ≤ Equity) ≥ Confidence Level,
where the confidence level is determined by 1 − the probability of insolvency
α . Statistically speaking, the value loss that leads the banking firm to bankruptcy
has a maximum of α percent probability of occurring at the end of the period.
The solvency condition can be reformulated to its deterministic equivalent (see
Appendix A2)
Maximum acceptable amount of Loss ≤ Equity.
The maximum acceptable amount of loss is given by rα L , where rα represents
unit value at risk, i.e., value at risk of a risky loan of 1 Euro, meeting the target
Value at Risk, Bank Equity and Credit Risk
163
confidence level 1 − α . Value at risk of the banking firm loans is then determined
by multiplying unit value at risk with the amount of risky loans for the target confidence level:
VaR α = rα L.
Bank management restricts the size of loss in the value of the bank by taking into
account risk regulations regarding the confidence level, which in turn determines
bankruptcy risk and solvency of the banking firm.
8CNWGCV4KUMCPFVJG$CPMŏU2TQHKV
Taking into account the bank’s balance sheet identity, the value at risk constraint
and the deterministic equivalent of the solvency condition, uncertain profit of the
banking firm is determined by:


% =  1 r% L − 1 − rα r  E − C  2 − rα E  .
Π
D


r
rα
 rα

 α

Hence the bank’s random profit is a function of the bank’s equity capital, which
has to be chosen by bank management in order to maximize expected utility of
profits.
1RVKOCN%CRKVCN4GSWKTGOGPV
In the following we derive the implications of the value at risk concept to optimum equity and asset/liability management of the banking firm. The amount of
equity capital which maximizes expected utility of profits depends upon the target
confidence level. This level may be a result of risk regulations or risk policy of the
banking firm or both. As in the case of preferences, for expository reasons let us
assume that inter-mediation costs can be represented by a quadratic cost function:
C ( L + D ) = θ ( L + D ) 2 / 2; θ > 0 where θ is a positive parameter of the operation cost function.
We obtain our first result. Risk management by value at risk implies that optimum amount of equity capital depends upon (i) managerial factors such as the target confidence level, the expectations about the return on risky loans and the degree of risk aversion, and (ii) market factors such as the rate of deposits and the
intermediation costs.
Maximizing the mean-variance function with respect to equity E and thereby
using the bank’s profit equation leads to:
164
Jack E. Wahl and Udo Broll
E ∗ = rα
mL − rD (1 − rα )
,
R∗ sL2 + θ (2 − rα ) 2
where absolute risk aversion R∗ = (1/a − µ ∗ ) −1 and mL and sL2 denote expected
return on loans and variance of the return on loans, respectively.
Our first result reveals that optimum equity E ∗ is determined implicitly. This
demonstrates that deriving a distinct relationship between the probability of bankruptcy and the required endowment of equity capital is not an easy task as many
textbooks suggest.
Furthermore, the result shows that the optimally required amount of equity capital cannot be derived from value at risk considerations alone but has to be determined simultaneously with the bank’s asset/liability policy.
An important question is how the bank’s debt/equity ratio is affected by value at
risk. First, optimum debt/equity ratio is determined by unit value at risk and does
not depend upon the amount of loans. We have D∗ /E ∗ = (1 − rα ) /rα . Second, if
bank management has to set a higher confidence level because of risk regulations,
then optimum debt/equity ratio will decrease. This result follows from the fact that
unit value at risk rα is a decreasing function of the probability of bankruptcy α .
Note that enforcing a higher confidence level by bank regulation does not necessarily imply that the bank’s optimum volume of deposits will increase.
To sum up, the implicit form of the optimum equity equation shows that the optimum amount of bank equity capital using the value at risk concept depends upon
several key factors. The comparative statics of the different input parameters are,
in general, not determinable. This points out the complex interaction between equity, expected value and variance of the return on risky loans, market rate of deposits, probability of bankruptcy, level of risk aversion and intermediation costs.
8CNWG/CZKOK\CVKQPCPF$CPM'SWKV[
Let us now investigate risk management which maximizes the value of the banking firm. Hence, bank management has to choose the amount of equity capital E
which maximizes expected profits:
max µ
subject to
L=D+E,
VaR α ≤ E ,
where the bank’s risky profit is defined in section 2.1.
Value at Risk, Bank Equity and Credit Risk
165
Since the bank’s random profit is still represented by the profit equation in section 2.3, optimum amount of equity capital for the banking firm reads:
E ∗ = rα
mL − rD (1 − rα )
.
θ (2 − rα ) 2
Therefore we obtain the following result. The value at risk concept under value
maximization implies that optimum amount of equity capital depends upon (i)
managerial factors such as the confidence level and expectations about the return
on risky loans, and (ii) market factors such as the rate of deposits and the intermediation costs.
The probability of bankruptcy affects equity management under value maximization via unit value at risk. In spite of the value maximization behaviour of bank
management optimum equity depends upon the volatility of the loans’ return. The
reason is that the value at risk measure is influenced by the variance sL2 of the
risky return.
With a positive expected interest margin mL − rD (1 − rα ) the optimum amount
of equity capital is positive. It follows that a banking firm that maximizes its value
will decrease its amount of equity capital if target confidence level is lowered.
Under value maximization this effect of confidence level is unambiguous as figure
2 shows.
E*
0
α
Fig. 2. Optimum equity and the probability of bankruptcy
Note that unit value at risk increases when the volatility of the loans’ return increases. Hence a higher volatility will induce an increase of the optimum amount
of equity capital, other things being equal. This effect is also unambiguous.
166
Jack E. Wahl and Udo Broll
%QPENWUKQP
Models of value at risk (VaR) have become a widespread risk management approach in many different types of organizations. This paper uses the value at risk
concept to analyze optimum equity capital requirements for a competitive banking
firm in the presence of lognormal credit risk and given risk regulations.
In the framework of risk management VaR-models achieve a control of the solvency of the banking firm with a certain probability (section 2). Expected utility
maximization and value maximization lead to the result that the optimum endowment of a bank in equity depends upon managerial and market factors. Especially
the target confidence level set by bank management to account for risk regulations
or the bank’s risk policy has a complex impact on the optimum amount of equity
capital. This also holds for the optimum asset/liability management (sections 3 and
4).
For the case of maximizing the value of the banking firm (section 4), however,
we derive a distinct relationship between the optimally required amount of equity
capital and the target confidence level on one hand, and the volatility of the loans’
return on the other hand. Increasing the confidence level or return risk will call for
a higher endowment of equity.
Value at Risk, Bank Equity and Credit Risk
167
#RRGPFKZGU
#
(Bankruptcy Condition):
The bank is exposed to insolvency risk if and only if
Prob((1 + r% L ) L − (1 + rD ) D < 0) > 0.
Using the balance sheet identity this condition implies
% > E ) > 0,
Prob(− rL
where r% = (1 + r% L ) / (1 + rD ) − 1 (Broll and Wahl 2003, p. 134).
#
(Solvency Condition):
In analytical terms the solvency condition reads
% ≤ E) ≥ 1 − α ,
Prob(− rL
which can be transformed to
Prob(1 + r% ≥ 1 − E /L ) ≥ 1 − α .
Given that the log-return ln(1 + r% L ) is normally distributed, the random variable
ln(1 + r% ) has also a normal distribution with expected value m and variance s 2 .
It follows that (see, e.g., Fisz 1977)
1 − ( E /L)α ≤ exp(m + uα s),
where uα denotes the α -fractile of the unit normal distribution. Defining
rα = 1 − exp(m + uα s) we obtain the deterministic equivalent of the solvency
condition:
rα L ≤ E .
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Jack E. Wahl and Udo Broll
4GHGTGPEGU
Bessis J (2002) Risk management in banking. 2nd ed. Chichester
Broll U, Wahl JE (2003) Value at risk and bank equity. Jahrbücher für Nationalökonomie
und Statistik 223: 129-135
Broll U, Wahl JE, Zilcha I (1995) Indirect hedging of exchange rate risk. Journal of International Money and Finance 14: 667-678
Duffie D, Pan J (1997) An overview of value at risk. The Journal of Derivatives 4: 7-49
Fisz M (1977) Wahrscheinlichkeitsrechnung und mathematische Statistik. 7. Aufl., Berlin
Freixas X, Rochet J-C (1997) Microeconomics of banking. Cambridge (MASS) et al.
Greenbaum SI, Thakor AV (1995) Contemporary financial intermediation. Forth Worth
(TX) et al.
Jackson P, Maude DJ, Perraudin W (1997) Bank capital and value at risk. The Journal of
Derivatives 4: 73-90.
Jorion P (1997) Value at risk: the new benchmark for controlling market risk. New York et
al.
Saunders A (1999) Credit risk measurement. New York et al.
Simons K (2000) The use of value at risk by institutional investors. New England Economic Review Nov./Dec.: 21-30
Wong K (1997) On the determinants of bank interest margins under credit and interest rate
risk. Journal of Banking and Finance 21: 251-271
Wolfgang Drobetz1 and Daniel Hoechle2
1
Department of Corporate Finance (WWZ), University of Basel, Petersgraben 51,
Switzerland, Phone: +41-61-267 33 29, Mail: wolfgang.drobetz@unibas.ch
2
Department of Finance (WWZ), University of Basel, Holbeinstrasse 12, Switzerland, Phone: +41-61-267 32 43, Mail: daniel.hoechle@unibas.ch
Abstract: This paper explores different specifications of conditional return expectations. We compare the most common specification, linear least squares, with
nonparametric techniques. Our results indicate that nonparametric regressions
capture some nonlinearities in financial data. In-sample forecasts of international
stock market returns are improved with nonparametric techniques. However,
there is very little out-of-sample prediction power for both linear and nonparametric specifications of conditional expectations. If an asset manager relies on a
simple instrumental variable regression framework to forecast stock returns, our
results suggest that linear conditional return expectations are a reasonable approximation.
JEL classification: G12, G14, C14, C53
Keywords: Conditional Expectation, Predictability, Linear Least Squares, Nonparametric
Regression
170
Wolfgang Drobetz and Daniel Hoechle
Following Fama’s (1970) dictum, asset markets are said to be efficient if security
prices fully reflect all available information. This strong form of efficiency has
been subject to considerable critique from two main strands. Grossman and
Stiglitz (1980) argue that efficient markets should reflect relevant information
only to the point where the marginal benefits of acting on information exceed the
marginal costs. If markets are perfectly efficient, the return on getting information
is nil. There would be little reason to trade and markets would eventually collapse.
The degree of market inefficiency determines the effort investors are willing to
expend to gather and trade on information. Hence, a non-degenerate market equilibrium will arise only when there are sufficient profit opportunities, i.e., inefficiencies to compensate investors for the cost of trading and information gathering.
Second, for a long time financial economists used to think that market efficiency
is intricately linked to the random walk model of stock prices. In fact, the market
efficiency-constant expected return model performed well in the early literature.
However, the argument has come under attack both theoretically as well as empirically in recent years. In particular, it is only under the strong assumption of
constant expected returns that the random walk model is directly implied by the
traditional efficient market hypothesis. Contradicting empirical evidence by Fama
and French (1988a) and Poterba and Summers (1988) suggested that stock returns
measured at long horizons exhibit strong negative autocorrelation. More recent
work shows that returns are predictable on the basis of instrumental variables related to the business cycle, such as term spreads, default spreads, industrial production, and the dividend yield, among others. As Lo and MacKinlay (1999) put
it, “[...] financial markets are predictable to some degree, but far from being a
symptom of inefficiency or irrationality, predictability is the oil that lubricates the
gears of capitalism.”
Two explanations have been asserted in the literature. First, predictability could
be evidence for market inefficiency due to some form of general or limited irrationality, such as fads, speculative bubbles, or noise trading. In this case, return
predictability reflects irrational deviations of prices from their fundamental values.
Shiller (1981) and Campbell and Shiller (1988) argued that stock returns exhibit
too much variation to be rationally explained. Lakonishok, Shleifer, and Vishny
(1994) found that value strategies yield higher returns because these strategies exploit suboptimal behavior of investors and not because they are fundamentally
riskier. They argued that investors extrapolate past growth rates too far into the future, without accounting for mean reversion in growth rates. However, although
the literature on “irrationality” is constantly growing, there is still no parsimonious
and empirically robust general equilibrium model.1
1
For example, see Barberis, Shleifer, and Vishny (1998).
Parametric and Nonparametric Estimation of Conditional Return Expectations
171
Second, a large number of empirical studies posit that expected returns vary
with the business cycle. Fama (1991) argued that “[…] if the variation through
time in expected returns is rational, driven by shocks to taste or technology, then
the variation in expected returns should be related to variation in consumption, investment, and savings.” Ferson and Harvey (1993) documented that the time
variation in economic risk premiums explains a substantial fraction of return predictability. In addition, intertemporal asset pricing models posit that predictability
is perfectly consistent with the concept of market efficiency. Balvers, Cosimano,
and McDonald (1990), Checcetti, Lam, and Mark (1990) and Drobetz and Wegmann (2002) showed that stock prices need not follow a random walk. They argued that equilibrium stock returns can be predicted to the extent that there is predictability in the endowment process. For example, evidence of trend reversion in
aggregate output was documented by Cochrane (1988), and predictability follows
almost mechanically from the standard intertemporal pricing restriction.
Our analysis does not directly focus on asset pricing issues. Instead, an ad-hoc
attempt is taken to deepen our understanding of the link between time variation in
expected returns and business cycle conditions. In particular, we shed new light on
simple instrumental forecasting models. Economic theory tells us how to link conditional expectations with conditional risk and reward, but unfortunately it remains silent on how the conditional expectations are generated. Clearly, a correct
specification of conditional expectations is key to any empirical analysis. Virtually
all previous studies, the exception being Harvey (2001), used linear regression
models to represent conditional expectations.2 In contrast, we compare conditional
expectations produced from both linear and nonparametric regression models for
international stock market data. Our results show that forecasts of stock returns are
only marginally improved with nonparametric techniques. We report high insample explanatory power, but similar to linear models there is very little out-ofsample predictability.
The empirical observation that stock returns are hard to predict is neither astonishing nor should it induce quantitative investors to abandon regression-based prediction models in general. Even if only a small degree of predictability can be
maintained for a large number of independent bets, this information is valuable. In
fact, this is the notion behind the “Law of Active Management”, which was originally proposed by Grinold (1989). Using realistic values from our empirical analysis, we give a simple example of an application of the law.
The remainder is as follows. Section 2 presents a unifying framework incorporating both parametric and nonparametric regression analysis. We give an intuitive
introduction to kernel regression models. Section 3 contains a brief data description. Section 4 presents our empirical results and demonstrates how they are re-
2
Stambaugh (1999) showed that simple instrumental variable models using OLS tend to
produce biased coefficient estimates, because the regression disturbances are often correlated with the changes in the instrument. See also Johnston and Di Nardo (1997), p.153f.
We nevertheless use these simple models to compare them with the results of nonparametric regressions.
172
Wolfgang Drobetz and Daniel Hoechle
lated to the framework presented in Grinold and Kahn (2000). Finally, section 5
provides a summary and some conclusions.
In this section we present a unifying framework which integrates both parametric
and nonparametric regression models. We exploit the econometric notion of loss
and apply weighted least squares (WLS) estimation. To keep things simple, our
analysis focuses on regression models with a single explanatory variable. However, we briefly outline how to generalize our framework to the multivariate case.
Let X and Y be two random variables. While Y is assumed to depend on X, the
latter is supposed to be exogenous. X is also referred to as the independent or explanatory variable. Denoting specific outcomes of random variables by lowercase
letters, the conditional mean of Y given some particular outcome X=x is given as:
E(Y X = x ) =
+∞
∫ yf
YX
(y x )dy =g(x ) .
(1)
−∞
The conditional density, denoted as fY|X(y|x), describes the distribution of Y given
that X takes on a particular value x. Note that the conditional mean, g(x), only depends on the independent variable X, but not on Y. Because regression theory
concentrates on the conditional mean specified in (1), the functional relationship
between Y and X can be modeled as:
y = g(x ) + ε ,
(2)
where g(x) is denoted as the regression function, and ε is called error or disturbance term. Taking conditional expectations on both sides in (2) gives:3
E (ε X = x ) = 0 .
(3)
Because the conditional mean cannot be observed directly, it has to be estimated
from the data. But the econometrician faces a specification problem: Is it appro3
E(Y|X=x) = E(g(x)|X=x) + E(ε|X=x) = g(x) + E( ε|X=x) ⇔ E(ε|X=x) = 0, where equivalence results from applying (1).
Parametric and Nonparametric Estimation of Conditional Return Expectations
173
priate to assume a particular functional form for g(x)? This is the procedure in
(traditional) parametric regression analysis. Alternatively, we could let the data
determine the relationship between X and Y. Loosely speaking, this is what is
done in nonparametric regression analysis. At first glance, one is tempted to prefer
the latter approach. Nevertheless, additional opportunities usually come at costs.
Specifically, the main problem with nonparametric regression analysis is that it
only holds as an asymptotic theory. If the sample is (too) small, the estimated conditional mean, ĝ(x), is biased and unstable.4
In the empirical part of this paper, we compare both parametric and nonparametric specifications of g(x). Popular parametric specifications are the following:
(global) constant:
(global) linear:
(global) polynomial:
g(x ) = a
(4)
g(x ) = a + bx
(5)
g(x ) = a + b1 x + b 2 x 2 + ...
(6)
Note that all parameters a, b, and bj in (4)-(6) are constants and independent of x.
This is no longer the case if one specifies the conditional mean in a nonparametric
way. As the word “nonparametric” already implies, there are no “global“ or constant parameters. However, it is possible to specify the conditional mean similar to
~
~
the parametric case. But now the “parameters” ~
a , b , and b j are just “local”, that
means, they depend on x:5
local constant:
g(x ) = ~
a
(7)
local linear:
local polynomial:
~
g(x ) = ~
a + b~
x
(8)
~
~ 2
g (x ) = ~
a + b1~
x + b2 ~
x + ...
(9)
Note that in the “local” specifications of g(x) the ~
x variables on the right hand
side in (7)-(9) merely represent transformed X values. We will come back to this
issue in section 2.2.
As already mentioned above, the regression function g(x) cannot be observed directly – it has to be estimated from the data. But how can we assess if an estimate
4
5
More technically speaking, the rate of convergence in nonparametric econometrics is
much slower than in the parametric framework. See Pagan and Ullah (1999) and Hagmann (2003).
To minimize notational burden in (7)-(9), we omit explicitly writing the parameter’s de~
~
pendency on x. Clearly, ~
a (x) , b (x) , and b j (x) , respectively, would be a more correct
notation.
174
Wolfgang Drobetz and Daniel Hoechle
for g(x) fits the true data generating process in an adequate and reliable way? It is
common in econometrics to measure the “goodness” of an estimate in terms of a
loss function, denoted as L(y,g(x)). Usually, loss functions can simply be specified
as L(ε), where we use the fact that the error or disturbance term ε is given by ε=y–
g(x). Loss functions represent a numerical value indicating how concerned we are
if the estimated conditional mean is off by a certain amount. Therefore, it is often
assumed that L(ε) is non-decreasing in the absolute value of ε, and an estimate of
the conditional mean is said to be optimal if it minimizes the value of the loss
function. The most popular choice for L(ε) is quadratic loss, or squared-error loss.
The corresponding functional form is L(ε) = ε2.
Transferring the concept of quadratic loss minimization to the estimation of the
conditional mean implies that g(x) must be estimated such that the weighted sum
of squared residuals, denoted as wRSS, is minimized. Formally, given a sample
containing n pooled outcomes of Y and X, {(Yi,Xi), i=1,2,…n}, the weighted least
squares (WLS) criterion is:
n
gˆ(x) = argmin wRSS = ∑ εi2 ⋅ φi2 ,
gˆ(x)
(10)
i =1
where φi denotes the weight assigned to the i-th residual, εi, and ĝ(x) is the
weighted least squares estimate of g(x). In parametric regression analysis, the
WLS technique is widely used to correct for heteroscedasticity in the residuals.
But as will be shown in section 2.3, weighted least squares estimation may be applied in a more general setup, which conveniently allows relating parametric and
nonparametric regression models to each other.
6JG2CTCOGVTKE#RRTQCEJ#P7PWUWCN4GRTGUGPVCVKQPQH1.5
In this section we consider parametric estimation of the conditional mean. Although there are several methods applicable, we focus on ordinary least squares
(OLS) regression analysis. From (10) we see that OLS is just a special case of
WLS. Specifically, we simply set φi equal to unity for each i (i=1,…,n) and assume that the conditional mean is the particular ĝ(x) which minimizes the sum of
squared residuals. In what follows, we derive the estimators for g(x) specified in
(4) and (5), respectively.
Consider the specification of the regression function g(x) in (4), i.e., assume that
g(x)=a is constant and independent of x. With this choice of g(x), we estimate the
conditional mean of Y given X=x using the ordinary least squares criterion. Setting all the φi’s equal to unity, the following optimization problem must be solved:
Parametric and Nonparametric Estimation of Conditional Return Expectations
n
n
i =1
i =1
min RSS = ∑ ε i2 = ∑ (Yi − α ) 2 ,
α
175
(11)
where α denotes an estimate of the true parameter a, and RSS is the sum of
squared residuals. The optimal value for the conditional mean is found by setting
the first derivative of (11) with respect to α equal to zero. This gives:
n
ĝ(x) = α * = ∑ Yi /n = y ,
i =1
(12)
where an asterisk (*) denotes an “optimal estimate” and y is the sample average
of {Yi, i=1,…,n}. Obviously, because g(x) is specified as a global constant, it does
not depend on x. Note however, that fitting a “local” constant with weighted least
squares does not reveal this property anymore. We will come back to this issue in
section 2.3.
Next, consider a linear specification of g(x). As already expressed in (5), the functional form for the conditional mean is given by g(x)=a+bx in this case. To find
the optimal estimate of the conditional mean, ĝ(x), we have to minimize the sum
of squared residuals, RSS, with respect to α and β:
n
n
i =1
i =1
min RSS = ∑ ε i2 = ∑ (Yi − α − βX i ) 2 .
α,β
(13)
Setting the partial derivatives of (13) equal to zero and solving the resulting system for α and β gives the well-known optimal estimates (again marked by asterisks) of the true parameters a and b:
α* = y − β *x
β* =
∑i=1 (X i − x )(Yi − y ) ,
2
n
∑i=1 (X i − x )
(14)
n
(15)
where x and y denote the sample averages of X and Y, respectively. Note that ĝ(x)
varies with x, and the estimate of the conditional mean is given by
ĝ(x) = α * + β * x .
(16)
To understand the link between parametric and nonparametric regression models,
it is important to see that in expressions (13)-(15) we implicitly fix the y-axis at
xY=0, where xY denotes the intersection point of the y-axis on the x-axis. Accordingly, all the observed Xis (i=1,…,n) are values which provide a measure on how
much they differ from zero. But why should we restrict ourselves to measure an
176
Wolfgang Drobetz and Daniel Hoechle
Xi’s distance from zero? Alternatively, all the X-values might be transformed in
the following way:
~
X i = (X i − x Y ) .
(17)
Hence, an alternative in (13) is to minimize the sum of squared residuals with all
~
~
the Xis replaced by the corresponding X i s . According to (17), X i and Xi are
identical for xY=0 by construction. However, the key in this framework is that we
allow the y-axis to move along the x-axis. To be more specific, to compare parametric and nonparametric regression estimators, we let the intersection of the yaxis with the x-axis coincide with the point where we want to evaluate the conditional mean of Y given X=x, i.e., we choose the y-axis to cross the x-axis at x Y=x.
This procedure is repeated for each point where the conditional mean has to be estimated. As will be shown in section 2.3, this idea of letting the y-axis move along
the x-axis is crucial to any nonparametric regression estimator.
~
According to (17) all the X i s are linearly transformed values of the original Xis,
hence, the resulting estimators of a and b in (13) may differ from those derived in
~
(14) and (15). We denote the new optimal estimators by α~ * and β * . For xY=x the
estimated conditional mean of Y given X=x now is:
~
~
~
ĝ(x) = α~ * + β * ~
x = α~ * + β * (x − x Y ) = α~ * + β * (x − x) = α~ * .
(18)
Since (17) is merely a linear transformation of the (exogenous) X-values, we expect the conditional mean in (18) to equal the conditional mean in (16), hence:
α~* = α * + β * x = y − β * (x − x) .
(19)
From (19) it is obvious that although we are still in a parametric setup, α~* depends
on x. Hence, applying (17) converts the “global” estimator α* of expression (14)
~
into the “local” estimator α~ * derived in (19). In contrast, note that β * = β *
holds.6 Obviously, the slope is still a “global” parameter, but this has no direct impact on the magnitude of the estimated conditional mean in (18).
0QPRCTCOGVTKE4GITGUUKQP#PCN[UKU
In ordinary least squares regressions all the weights φi are set equal to unity.
Therefore, observations of the exogenous variable X which are far off from a particular value x are assumed to explain the conditional mean of Y given X=x
equally well as an observation which is very close to x. Although this assumption
is correct if the relationship between Y and X is linear, it may fail when there are
nonlinearities in the relationship between Y and X. Hence, it seems reasonable to
6
Given that the Y-axis is fixed at xY=x, it holds for x' ≠ x that ĝ(x' ) = α * + β * x' =
~
~
~
= α~ * + β * (x'− x) ⇔ α * + β * x' = α * + β * x + β * (x'− x) ⇔ β * = β * .
Parametric and Nonparametric Estimation of Conditional Return Expectations
177
give more weight to observations close to x and less weight to those observations
which are far off from x. In fact, this is already the notion of nonparametric regression analysis.
To illustrate this concept in more detail, we start by introducing the kernel function and show how it may be used as an alternative weighting scheme in the WLS
criterion expressed in (10). We then use this kernel function weighting scheme
and the methodology of a moving y-axis presented in the last part of section 2.2 to
derive a nonparametric regression estimator which fits a local constant. 7 This is
the well-known Nadaraya-Watson estimator originally proposed by Nadaraya
(1964) and Watson (1964). We then continue by fitting a local regression line to
receive the so-called local linear estimator, which was introduced by Stone (1977)
and Cleveland (1979).
6JG-GTPGN(WPEVKQPCUCP#NVGTPCVKXG9GKIJVKPI5EJGOG
As an alternative to choosing φi=1 (for i=1,…,n), we could let the weights φi depend on where the conditional mean of Y is evaluated. In addition, if we want to
estimate the conditional mean of Y given that the exogenous variable X takes on a
particular value x, it seems reasonable to let the weight assigned to observation i
(i=1,…,n) depend on the magnitude of the absolute difference between Xi and x.
Although such a weighting function could be defined in many different ways, it
is convenient to assume some probability density function-like relationship between the squared weight assigned to observation i and the scaled difference between Xi and x. In this setting, the probability density function is called “kernel
function”, or simply “kernel”, and it is denoted by K(u). Its properties are: 8
+∞
K (u ) ≥ 0
and
∫ K(u ) = 1 ,
where
u=
−∞
X−x
.
h
(20)
Here u measures the scaled difference between X and x, and h is the scaling factor. Since h>0 determines the spread of the kernel, it is usually referred to as the
smoothing parameter or the bandwidth. There are many kernel functions which
satisfy the conditions required in (20). The most popular are the following: 9
Epanechnikov kernel:
7
8
9
3
4
(1 − u 2 ) ⋅ I(| u |≤ 1)
(21)
To be exact, this type of a nonparametric regression estimator is called “kernel regression
estimator” because it relies on a kernel function weighting scheme.
Note that although K(u) is a (standard parametric) probability function, it does not play a
probabilistic role here. Explicitly, we do not assume that X is distributed according to
K(u) (which again would be a parametric assumption). It is just a convenient way to determine the weight that is assigned to observation i (i=1,…,n).
In the empirical part of this paper, we will just consider the Gaussian or normal kernel,
because this kernel has convenient numerical properties which are particularly useful if
the multivariate local linear estimator in (38) is used to estimate the conditional mean.
178
Wolfgang Drobetz and Daniel Hoechle
Triangular kernel:
Gaussian or normal kernel:
(1− | u |) ⋅ I(| u |≤ 1)
(22)
(2π)−1/ 2 exp(− 12 u 2 )
(23)
In (21) and (22), I(|u|≤1) is the indicator function which is one if the absolute
value of u, |u|, does not exceed one and zero otherwise. As already mentioned
above, the kernel function may be used as an alternative weighting scheme in the
weighted least squares criterion in (10). Assuming that φi denotes the weight assigned to observation Xi (i=1,…,n), the kernel weighting scheme is defined as:
(
)
φ i = K (u i ) = K Xi − x ,
h
i=1,…,n.
(24)
Note that all the weights φi in (24) depend on x, implying that these weights are
local parameters. In addition, they also depend on h in an intuitive way. Loosely
speaking, a larger bandwidth will scale down the difference between Xi and x,
which leads to more similar weights throughout the sample than if the smoothing
parameter is chosen to be small.
(KVVKPIC.QECN%QPUVCPVŌ6JG0CFCTC[C9CVUQP'UVKOCVQT
We now have all ingredients to introduce the simplest nonparametric regression
estimator. We specify the conditional mean of Y given X=x according to equation
(7), i.e., we assume g(x) to be a “local” constant. By the term “local” we mean that
the constant varies with x, indicating that the constant depends on where along the
x-axis the conditional mean is evaluated. Similar to the last part of section 2.2, we
do not fix the y-axis at xY=0, but instead assume that for the estimation of g(x) the
y-axis intersects the x-axis at xY=x. Using the weighted least squares criterion expressed in (10) together with the kernel weighting scheme in (24), the goal is to
a (x ) ≡ ~
a according to:
find an estimate α~ (x ) ≡ α~ of the true local constant ~
X −x
min
wRSS = ∑ ε i2 ⋅ φ i2 = ∑ (Yi − ~
α ) 2 ⋅ K i  .
~
α
 h 
i =1
i =1
n
n
(25)
To find the optimal estimate for the conditional mean, we set the first derivative of
(25) with respect to α~ equal to zero and solve for α~ to obtain the famous Nada*
:
raya-Watson estimator, labeled as α~NW
X −x 
∑i=1 Yi ⋅ K ih
n
*
α~NW
=
X −x 
∑i=1 K ih
n

.
(26)


It can readily be seen that the Nadaraya-Watson estimator collapses to the global
constant estimator derived in (12) if the bandwidth h is chosen to be large:
Parametric and Nonparametric Estimation of Conditional Return Expectations
X −x 
∑i =1 Yi ⋅ K ih
n
*
lim α~NW
h→∞
= lim
h →∞
179
 K(0) ⋅ n Y
∑i =1 i = n Y /n = α * .
=
∑ i
X −x
n
n ⋅ K(0)
i =1
∑i=1 K ih 
(27)
Given the assumptions of a nonstochastic X variable and white noise disturbances
εi (i=1,…,n) with mean 0 and variance σ2 (and some additional properties), it can
be shown that the bias and the variance of the Nadaraya-Watson estimator approximately are:10
*
Bias(α~NW
)=
∞
h2
g (2) (x) ⋅ f(x) + 2f (1) (x) ⋅ g (1) (x) ⋅ ∫ u 2 K(u)du ,
2f(x)
−∞
(
*
Var( α~NW
)=
)
∞
σ2
K(u) 2 du .
nh ⋅ f(x) −∫∞
(28)
(29)
In (28) and (29) the i-th derivative of a function with respect to x is denoted by a
bracketed superscript i. Note that the bias contains a component which depends on
the product of the slope of the regression function, g(1)(x), and the slope, f(1)(x), of
the (true) probability density function of X, f(x). As will be shown below, this
term vanishes for the local linear estimator. From (29) it is also clear that the variance of the Nadaraya-Watson estimator is high if f(x) takes on a small value. This
implies that the variance of the Nadaraya-Watson estimator is high where only a
few observations are available. Accordingly, estimating the conditional mean for
outlier values with the Nadaraya-Waston estimator is a difficult task. In our empirical analysis we therefore delete outliers from the data sample. Finally, from
(28) and (29) it is also evident that there is a tradeoff between bias and variance.
Choosing a large (small) bandwidth reduces (increases) the variance of the Nadaraya-Watson estimator, but it simultaneously increases (decreases) the bias.
(KVVKPIC.QECN4GITGUUKQP.KPGŌ6JG.QECN.KPGCT'UVKOCVQT
~
a + b~
x , where
Let us now specify g(x) according to (8), i.e. we assume that g(x ) = ~
~
~
both a and b are local “parameters” which depend on x. Again, we do not fix
the y-axis at xY=0, but instead let the y-axis intersect the x-axis at x Y=x in order to
estimate g(x). Given the kernel weighting scheme in (24), the weighted least
squares optimization problem becomes:
n
n
2
2
2
~ ~
 Xi − x  .
min
~ wRSS = ∑ ε i ⋅ φ i = ∑ [Yi − α − β (X i − x)] ⋅ K 
~
α, β
 h 
i =1
i =1
10
(30)
We neglect higher order terms. For a derivation of the bias and the variance of the Nadaraya-Watson estimator in the univariate case, see Pagan and Ullah (1999) pp. 96-104.
180
Wolfgang Drobetz and Daniel Hoechle
~
Setting the partial derivatives of (30) with respect to α~ and β equal to zero and
solving the resulting system of equations, we finally receive the local linear esti*
mator ( α~LL
) of the conditional mean of Y given X=x:
~*
*
α~LL
= ψ − β LL
ξ,
~*
=
β LL
(31)
∑i=1 [((X i − x) − ξ )⋅ (Yi − ψ )]⋅ K ( ih
2
n
X −x
∑i=1 ((X i − x) − ξ ) ⋅ K ( ih )
n
X −x
)
(32)
~
where ψ and ξ denote the weighted means of Y and X , respectively. They are
defined as follows:
∑i =1 Yi ⋅ K ( ih
X −x
n
∑i =1 K ( ih )
n
ψ≡
X −x
)
∑i =1 (X i − x) ⋅ K (
X −x
n
∑i =1 K ( ih )
n
and
ξ≡
Xi −x
h
).
(33)
Because we let the y-axis intersect the x-axis exactly at those points where the
conditional mean of Y given X=x is evaluated, the local linear estimator of g(x) is
*
. The derivation of this result is similar to (18), hence, it is not repeated here.
α~LL
It is also important to note that the local linear estimator may be written as follows:
~*
~*
*
*
α~LL
= ψ − β LL
ξ = α~NW
− β LL
ξ
(34)
Expression (34) shows, that the local linear estimator corresponds to the difference
between the Nadaraya-Watson estimator and an additional term. In fact, the local
linear estimator not only considers the intercept of a local regression (as does the
Nadaraya-Watson estimator), but it also accounts for the local pattern around the
point X=x where the conditional mean is evaluated. This is similar to the ordinary
least squares regression estimator in (14). Therefore, one could expect that the local linear estimator is more consistent than the Nadaraya-Watson estimator. Indeed, it can be shown that although the variance of the local linear estimator is
similar to that of the Nadaraya-Watson estimator, its bias no longer depends on the
distribution of X:11
*
Bias(α~LL
)=
h 2 g (2) (x) ∞ 2
⋅ ∫ u K(u)du
2
−∞
(35)
From (35) it is obvious that the local linear estimator has no bias for truly linear
regression functions because g(2)(x) = 0 in this case. This is in contrast to the Nadaraya-Watson estimator which contains a bias component which depends on the
slope of the regression function.
11
We again abstract from higher order terms. For a detailed analysis about the properties of
the local linear estimator see Fan and Gijbels (1992).
Parametric and Nonparametric Estimation of Conditional Return Expectations
181
Looking at the choice of the bandwidth h, we see from expressions (35) and (29)
that there is still a tradeoff between the bias and the variance of the estimator.
Note that for h → ∞ the local linear estimator collapses to the ordinary least
squares estimator derived in (19). It can be shown that if h gets large, the weighted
~
~*
means of Y and X approach their arithmetic means, and β LL
converges to the ordinary least squares slope coefficient β * derived in (15).
6JG/WNVKXCTKCVG%CUG
Using matrix algebra, the multivariate versions of the estimators presented in sections 2.2 and 2.3 can be derived in a similar way. We do not present the derivations, but merely provide short explanations of the multivariate formulas.
We begin with the multivariate ordinary least squares estimate of g(x), where x
now denotes the 1×k row vector containing the point where the conditional mean
of Y is evaluated. Note that if there is an intercept in the regression, the first column in the n×k independent variables matrix X and the first element in the x vector are ones. In addition, if y is the n×1 vector containing the dependent variable’s
observations, and an inverted comma denotes the transpose of a matrix, then the
set of estimated regression parameters may be shown to be β=(X’X)-1X’y. Therefore, the multivariate OLS estimator of the conditional mean is given by: 12
g(x) = xβ = x(X’X)-1X’y.
(36)
The multivariate Nadaraya-Watson estimator is similar to the one in (26), but the
univariate Kernel K(u) has to be replaced by its multivariate counterpart. Although
fully general multivariate kernels exist, they are not very convenient to work with.
Therefore, so-called product kernels are used in practice. These kernels are simply
the product of univariate kernels as the ones described in expressions (21)-(23). If
K(uij) is a univariate kernel with entry uij = (Xij-xi)/hi, then the product kernel is
K (u j ) = ∏ik=1 K (u ij ) , where k again denotes the number of independent variables.
Using product kernels and letting K(u) be an n×1 vector containing K(uj) as its j-th
element, the multivariate Nadaraya-Watson estimator is: 13
−1
*
α~NW
= (ι ' K (u) ) y ' K (u) .
(37)
Here, ι is an n×1 column vector consisting of ones. Denoting with e1 a column
vector of dimension k×1 with unity as its first element and zero elsewhere, the
multivariate local linear estimator can be shown to be:14
12
See for example Johnston and DiNardo (1997).
See for example Pagan and Ullah (1999). Expression (37) complies with their formula
(3.14), but it is transformed to matrix notation.
14 Of course, since there is also a local constant, the first entry in the x vector is equal to one
and the first column of the X matrix consists entirely of ones. See Pagan and Ullah
13
182
Wolfgang Drobetz and Daniel Hoechle
−1
*
α~LL
= e1′ [( X − ιx )' ω( X − ιx )] [( X − ιx )′ωy ] ,
(38)
where ω = diag(K (u )) is a diagonal matrix with ωj,j = K(uj) and zero elsewhere.
$CPFYKFVJ5GNGEVKQPHQT0QPRCTCOGVTKE4GITGUUKQP'UVKOCVQTU
As already noted above, the choice of the bandwidth h has an important impact on
the results of nonparametric regressions. In addition, there is a tradeoff between
the variance and the bias of an estimate. Although one could find a visually satisfying bandwidth by trial and error in the two variable regression case, this approach is not feasible for multivariate regression models. Therefore, a technical
method to select the “optimal” bandwidth is preferable.
From a theoretical point of view, the cross validation approach to find the suitable bandwidth is probably the most appealing one. This method chooses h such
as to minimize the estimated prediction error.15 However, the cross validation approach has its limitations in practical applications. The problem is that if there are
several identical observations in the independent variables, the optimal bandwidth
choice is h=0. Obviously, this choice is not useful.16
Therefore, following Harvey (2001), we choose the bandwidth according to the
Silverman rule, which was actually developed for nonparametric density estimation. For the univariate density estimation (combined with the Gaussian kernel),
Silverman (1986) showed that h should be chosen as:
h = 1.06 σˆ X n
−1 5
,
(39)
where n denotes again the number of observations and σ̂ X is the standard deviation estimated from the data. For the multivariate case, however, the use of a single bandwidth may be inappropriate if the independent variables are not drawn
from similar distributions. Therefore, we apply the multivariate version of the
Silverman rule, which is also known as the multivariate normal reference rule.17
Specifically, for the exogenous variable i the bandwidth is chosen as:
 4 
hi = 

 k + 2
1 (4 + k )
( )
σˆ Xi n −1 4+ k ,
(40)
where k is again the number of conditioning variables. Note that this is an objective bandwidth selection procedure which reflects the volatility of each explanatory variable, denoted as σ̂ Xi for conditioning variable i.
(1999), p. 93, for the local linear estimator and Johnston and DiNardo (1997), p. 171, for
a matrix version of the weighted least squares estimator.
15 See for example Pagan and Ullah (1999), p. 119.
16 The problem is related to the specification of the “leave one out estimator” and the resulting “quasi-likelihood-function” to be optimized. See Hagmann (2003).
17 See Scott (1992), p. 152.
Parametric and Nonparametric Estimation of Conditional Return Expectations
183
&CVC&GUETKRVKQP
The variables of interest for our empirical analysis are the excess returns on the
MSCI total return indices for Germany, the United Kingdom, and the United
States. We use monthly continuous returns from January 31, 1983 to July 31, 2003
(247 observations). To compute excess returns, we subtract the 1-month Eurocurrency interest rate in Euro, U.K. pound, and U.S. dollar, respectively. Some summary statistics of these excess returns are given in panel A of table 3.1. Since
these return data are widely used in the literature, we do not further comment on
them.
Table 3.1. Summary statistics (1983.01 – 2003.07)
Variable:
Mean
Germany
UK
USA
0.36%
0.35%
0.55%
Default spread
TED spread
Dividend yield
Germany
UK
USA
Term spread
Germany
UK
USA
0.010
0.006
Std. dev.
Max.
Min.
ρ1
ρ2
ρ3
ρ12
Panel A: Excess market returns
0.060
0.152 -0.248
0.074 0.037 0.034 0.049
0.049
0.129 -0.305
0.039 -0.074 -0.043 -0.004
0.046
0.120 -0.239
0.017 -0.052 -0.016 0.000
Panel B: Instrumental variables
0.003
0.023 0.005
0.911 0.844 0.775 0.615
0.005
0.026 0.001
0.839 0.737 0.660 0.516
0.021
0.039
0.027
0.005
0.008
0.012
0.034
0.058
0.053
0.012
0.023
0.010
0.940
0.964
0.985
0.872
0.929
0.970
0.807
0.900
0.956
0.427
0.720
0.857
0.012
0.001
0.013
0.013
0.017
0.012
0.034
0.039
0.037
-0.019
-0.046
-0.013
0.972
0.959
0.947
0.942
0.910
0.900
0.915
0.860
0.852
0.527
0.481
0.357
This table shows summary statistics of the input data. Panel A describes the excess market
returns, panel B the instrumental variables. The dividend yield and the term spread are local
variables, the default spread and the TED spread refer to U.S. data. ρi denotes the
autocorrelation at lag i.
The instrumental variables (which are all lagged by one month) we use include:
the local dividend yields, the spread between the yield on local government bonds
with a maturity longer than 10 years and the respective 1-month Eurocurrency interest rate (term spread), the spread between Moody’s Baa- and Aaa-rated U.S.
corporate bonds (default spread), and the spread between the 3-month Eurodollar
rate and the 90-day yield on the U.S. treasury bill (TED spread). Summary statistics are shown in panel B of table 3.1. Note that all information variables are
highly autocorrelated.18 This imposes problems for inferences in long-horizon regressions, leading to inflated t-statistics and R-squares (e.g., see Valkanov (2002)
18
However, results from Dickey-Fuller unit root tests show that all instrumental are stationary. The null hypothesis of a unit root is rejected at least at the 10 percent level.
184
Wolfgang Drobetz and Daniel Hoechle
for a recent discussion). A related problem is that the error terms will contain a
moving-average process due to overlapping forecast horizons.19 Since we test a
regression setup with non-overlapping monthly returns, we do not further pay attention to these issues.20 We now give a brief description of each instrumental
variable.
Dividend yield: The informational content of the dividend yield can be inferred
from well known present value relations. Such models posit that – ignoring dividend growth – stock prices are low relative to dividends when discount rates are
high. The dividend yield varies positively with expected stock returns. However,
assuming that expected stock returns are time varying, things are complicated by
the fact that the relation between prices and returns becomes nonlinear. Campbell
and Shiller (1988) developed a loglinear framework to show that high prices relative to dividends must be associated with high expected future dividends, low expected future returns, or some combination of the two. In fact, their approach is a
dynamic version of the simple Gordon growth model and should be seen as a pure
accounting identity. A high current stock price must be followed by high future
dividend growth, and/or a low expected return in the future. Equivalently, log
dividend yield is high if expected dividend growth is low and/or expected return is
high.21 Most important, dividend price ratios can only vary if they forecast changing dividend growth and/or changing expected returns. Otherwise, the dividend
yield would have to be constant.
Term spread: The economic story to motivate the term spread as an appropriate
instrument variable builds on the life-cycle hypothesis of Modigliani and Brumberg (1954) and the permanent income hypothesis of Friedman (1957). Investors
prefer a smooth consumption stream rather than very high consumption at one
stage of the business cycle and very low consumption at another stage. Hence,
consumption smoothing drives the demand for insurance (hedging). A natural way
to do so is to substitute bonds of different maturities. Harvey (1991) argued that if
the economy is in a growth stage, but a general slowdown is expected, people will
hedge by buying assets that deliver a high payoff during the economic downturn.
For example, an investor could purchase long-term government bonds and simultaneously sell short-term bonds to hedge. If many investors follow this strategy,
the price of long term bonds increases, implying a decreasing yield. On the other
hand, the selling pressure for short term bonds will drive prices down, i.e., increase the yield. As a result, the term structure becomes flat or even inverted. In
other words, the term spread decreases or becomes even negative. Supporting the
consumption smoothing argument empirically, Harvey (1991) found that the term
spread is an excellent predictor of the business cycle. Similarly, Chen (1991) re19
For example, Fama and French (1988b) adjust for moving-average errors using NeweyWest (1987) standard errors in their long-run regressions.
20
Ferson, Sarkissian, and Simin (2003) show that there may be spurious regression problems even if the rates of return are not highly persistent. If the underlying expected returns are persistent time series, there is a risk of finding a spurious relation between returns and an independent, highly autocorrelated lagged variable.
21 Note that this interpretation rules out the existence of price bubbles.
Parametric and Nonparametric Estimation of Conditional Return Expectations
185
ported that an above average term spread forecasts that the gross national product
will continue to increase over the next four to six quarters, and vice versa. Finally,
Fama and French (1989) showed empirical evidence that the term spread is one of
the best variables to forecast stock returns as well as bond returns. The term
spread is low near business cycle peaks and high near business cycle troughs. Specifically, it steeply rises at the bottom of recessions and is inverted at the top of a
boom.
Default spread: The default spread is calculated as the difference between the
yield on U.S. Baa low-grade corporate bonds and the yield on U.S. Aaa highgrade bonds with the same maturity. The U.S. default spread is a legitimate proxy
for global default risk. While the term spread reflects anticipation of future health
of the economy, the default spread can be taken as an indicator of its current state.
In times of a slowdown or a recession investors will demand a higher return premium for investing in low-grade corporate bonds, implying a larger default premium. Keim and Stambaugh (1986), Fama and French (1989), and Evans (1994)
used the U.S. default premium to explain time-variation in expected U.S. stock
and bond returns.
TED spread: The TED spread is the difference between the 3-month Eurodollar
rate and the 90-day yield on the U.S. treasury bill. The TED spread is affected by
three major factors: (i) world political stability, (ii) balance of trade, and (iii) fiscal
policy in the United States. When political uncertainty is high and the risk of disruption in the global financial system increases, the yield differential widens.
When the balance of trade is worsening, the TED spread should rise as well.
Therefore, the TED spread can be assumed to be another indicator of the current
health of the economy. The yield differential should be higher during times of
economic recessions, and smaller in expansionary periods. Ferson and Harvey
(1993) used this variable to explain international equity premia.
'ORKTKECN4GUWNVU
This chapter presents our empirical results. Following Harvey (2001), we start
with in-sample regression results for excess returns in Germany, the United Kingdom, and the United States. We then extend the analysis and explore the out-ofsample properties of parametric and nonparametric forecasts of excess returns.
+PUCORNG4GUWNVU
In this section we look at in-sample regression results. The sample period runs
from January 31, 1983 to July 31, 2003 (247 months). Table 2 presents a summary
of the in-sample conditional mean analysis. We regress instrumental variables
available at t on excess returns to be realized at t+1. Hence, the instrumental variables are available at the beginning of a month. With our four conditioning variables, there are six bivariate regression specifications for each country. We restrict
186
Wolfgang Drobetz and Daniel Hoechle
ourselves to the case of two regressor variables because the nonparametric technique will only perform well on low dimensional regression specifications given
our small sample.22 To derive realistic estimates of the conditional mean excess returns, we delete extreme values in the instrumental variables. We define an outlier
as being three standard deviations above or below the mean of a particular series.
To compare the nonparametric estimates with their parametric counterparts, we
look at the R-squares of pseudo-predictive regressions. Specifically, we regress
the realized excess returns Xi,t+1 on their expectations from the different specifications. Computing the “predicted” excess return for each month as the fitted value
from the respective instrumental variables regression, we run the following auxiliary regression:
X i , t +1 = a + b ⋅ X̂ i , t + ε t +1 ,
(41)
where X̂ i, t denotes the predicted excess return (available at t) over the period
from t to t+1. Our results in table 2 show that the local linear estimator exhibits the
highest “R-squares”, some of them are even above 20%. The R-squares of the
OLS regressions are much lower – with one exception they are never above 5%.
Overall, our analysis indicates that the specification of a linear conditional mean
model can significantly be improved by accounting for the nonlinear relationship
between excess returns and the instrumental variables. This is in contrast to the
findings in Harvey (2001), who cannot detect significant increases of the insample explanatory power.
Expression (41) can also be used to test whether the estimates are unbiased.
Clearly, the null hypothesis of such a test implies that the computed coefficients in
(41) obey the following joint condition:
H 0 : â = 0 and b̂ = 1.
(42)
This joint restriction can easily be tested using a Wald test. The corresponding pvalues are shown in brackets underneath the R-squares in table 4.1. Note that in
the case of OLS regressions these restrictions are satisfied by construction and we
do not report the corresponding p-values. While the null hypothesis cannot be rejected for any of the local linear kernel regressions at conventional significance
levels, the respective p-values of the Nadaraya-Watson estimator often indicate
significance. But this result should not come as a surprise, because the local linear
estimator has a bias independent of the slope of the regression function.23
22
This problem in nonparametric econometrics is also referred to as the “curse of dimensionality”. For a detailed discussion see Scott (1992), chapter 7.
23 See equations (28) and (35), respectively.
Parametric and Nonparametric Estimation of Conditional Return Expectations
187
Table 4.1. Bivariate regressions (in-sample)
Div. yield
TED spread
Germany:
OLS
0.0480
0.1812
NW
(0.0068)
0.2202
LL
(0.5603)
United Kingdom:
OLS
0.0533
0.1678
NW
(0.0514)
0.2039
LL
(0.2119)
United States:
OLS
0.04341
0.1551
NW
(0.0055)
0.2050
LL
(0.1353)
Regressor variables in bivariate regressions:
Div. yield
Div. yield
TED spread
TED spread
Default sprd
Term spread
Default sprd
Term spread
Term spread
Default sprd
0.0185
0.1432
(0.0091)
0.1878
(0.1717)
0.0251
0.1399
(0.0261)
0.1653
(0.1873)
0.0395
0.095
(0.2415)
0.1145
(0.5556)
0.0391
0.1041
(0.1660)
0.1296
(0.2077)
0.0082
0.0589
(0.1216)
0.0962
(0.0676)
0.0226
0.1270
(0.0407)
0.1738
(0.0788)
0.0254
0.1280
(0.0347)
0.1785
(0.1527)
0.0410
0.1230
(0.2742)
0.1653
(0.2351)
0.0464
0.1608
(0.0335)
0.2002
(0.2056)
0.0048
0.0929
(0.1852)
0.1270
(0.2100)
0.0100
0.1155
(0.0411)
0.1696
(0.0805)
0.0159
0.1345
(0.0056)
0.2005
(0.1289)
0.0398
0.1278
(0.1434)
0.1612
(0.2935)
0.0341
0.1183
(0.0468)
0.1826
(0.1261)
0.0060
0.1139
(0.0192)
0.1324
(0.3168)
This table shows the R-squares of regressions of monthly excess returns on two instrumental variables over the sample period from January 31, 1983 to July 31, 2003. NW is the
Nadaraya-Watson estimator and LL the local linear estimator. The dividend yield and the
term spread are local variables, the default spread and the TED spread refer to U.S. data.
The numbers in brackets show the p-values of chi-square distributed Wald test statistics for
the null hypothesis that the estimates are unbiased, as specified in (42).
Figure 1 plots the models’ fitted values of the bivariate regressions for Germany.
Harvey (2001) reported that the nonparametric fitted values had a smaller range
and were much smoother than the OLS fitted values. We cannot detect this pattern
in our sample. In contrast, the unconditional variance of the nonparametric fitted
values is significantly larger and accounts for the higher R-squares in table 4.1.
One might object, however, that the nonparametric fitted values are too variable to
plausibly reflect investors’ expectations. This problem also relates to the problem
that our data sample is very small even for only two regressors. Intuitively, this is
because only a small part of the sample is used at any point of interest. To increase
the sample size, we replicated all results using weekly data. However, given that
our model is based on fundamental variables, prediction power is expected to increase with the time horizon. In fact, this is what we observe; with weekly data the
model generates even less explanatory power. Therefore, weighting increased estimation accuracy against economic intuition, we decided to emphasis the latter.
Nevertheless, our results must be interpreted with due care.
188
Wolfgang Drobetz and Daniel Hoechle
Dividend yield & TED spread
Dividend yield & default spread
0.2
0.15
0.1
0.10
0.0
0.05
-0.1
0.00
-0.2
-0.3
-0.05
84
86
88
90
OLS
92
94
96
98
00
02
-0.10
84
86
Local linear estimator
88
90
OLS
Dividend yield & term spread
0.15
92
94
96
98
00
02
Local linear estimator
TED spread & default spread
0.08
0.06
0.10
0.04
0.05
0.02
0.00
-0.05
0.00
84
86
88
90
OLS
92
94
96
98
00
02
-0.02
84
86
Local linear estimator
88
90
OLS
TED spread & term spread
92
94
96
98
00
02
Local linear estimator
Term spread & default spread
0.05
0.08
0.00
0.06
-0.05
0.04
-0.10
0.02
-0.15
0.00
-0.20
-0.25
84
86
88
90
OLS
92
94
96
98
00
Local linear estimator
02
-0.02
84
86
88
90
OLS
92
94
96
98
00
Local linear estimator
Fig. 1. Comparison of in-sample conditional mean excess returns for Germany
02
Parametric and Nonparametric Estimation of Conditional Return Expectations
189
Another finding in Harvey’s (2001) study was that the OLS model produced a
large number of negative expected excess returns. From a theoretical point of
view, this is clearly a troubling aspect. For example, in the specification containing the dividend yield and the TED spread (upper-left panel in figure 1) the conditionally expected excess return for Germany is negative in 58 of 247 months
(23%). Unfortunately, using the fitted values from nonparametric regressions cannot help. For example, there are 57 negative expected excess returns using the local linear estimator. In other models, the nonparametric fitted values produce an
even larger number of negative excess return forecasts than the OLS specification.
1WVQHUCORNG4GUWNVU
While we have been careful in selecting the instrumental variables on the basis of
economic theory, our in-sample results are clearly not free from data mining problems. Given that many researchers previously used the same variables in similar
studies, it is conceivable that our inferences in table 2 are upward biased and suffer from data mining problems. In his classical article, Lovell (1983) shows that
exaggerated t-statistics are likely to be generated by intensive search over candidate explanatory variables.24
A simple possibility to check if the models have explanatory power is to conduct out-of-sample tests, i.e., to generate forecasts outside the estimation window.
Note also that this has great practical implications for whether any of the regression specifications can be used in practical asset management applications. The
exact procedure is as follows:
• First, we estimate the model using a calibration window of 180 months from
January 31, 1983 to December 31, 1997.
• Second, using the parameter estimates and the level of the conditioning variables as of December 31, 1997, we compute the return forecast for the next
month (i.e., the excess return to be realized on January 31, 1998) as the fitted
value.25
• We then shift the sample one month forward and re-estimate the model from
February 28, 1983 to January 31, 1998. Using the new coefficient estimates, we
get an out-of-sample forecast for February 1998.
• With monthly sliding windows of 180 months, this procedure ultimately results
in 67 out-of-sample forecasts for each market.
Table 3 reports the out-of-sample results. The R-squares are computed by running
the regression specified in (41), but where the forecasts now relate to a month outside the estimation window. The regression contains 67 observations. Again, the
numbers in brackets denote the p-values of Wald tests specified in (42).
Obviously, the explanatory power of almost all specifications deteriorates dramatically. This is especially the case for the nonparametric regression models,
24
In fact, Lovell (1983) provides a rule of thumb for deflating the exaggerated claims of
significance generated by data mining activity.
25 Strictly speaking, there are no coefficient estimates in the nonparametric case.
190
Wolfgang Drobetz and Daniel Hoechle
whose forecasting power is extremely low. It is interesting to observe that it is
generally even lower than in the OLS specifications, whose R-squares are sometimes even higher out-of-sample than in-sample. In all countries, the “best” specifications involve the TED spread. Overall, however, our results are clearly disappointing and indicate substantial “overfitting” problems, especially for kernel
regression estimators. An important implication for practical asset management
applications is that if an asset manager relies on instrumental variable models to
produce excess return forecasts, simple linear specifications are sufficient.
Table 4.2. Bivariate regressions (out-of-sample)
Div. yield
TED spread
Germany:
0.0557
OLS
(0.2284)
0.0130
NW
(0.4537)
0.0502
LL
(0.5588)
United Kingdom:
0.0479
OLS
(0.4592)
0.0142
NW
(0.6851)
0.0042
LL
(0.1937)
United States:
0.0306
OLS
(0.1718)
0.0531
NW
(0.3664)
0.0427
LL
(0.1064)
Regressor variables in bivariate regressions:
Div. yield
Div. yield
TED spread
TED spread
Default sprd
Term spread
Default sprd
Term spread
Term spread
Default sprd
0.0219
(0.3746)
0.0016
(0.2595)
0.000
(0.2851)
0.0011
(0.4370)
0.0003
(0.2179)
0.0010
(0.0310)
0.0491
(0.5396)
0.0289
(0.7089)
0.0168
(0.3300)
0.0468
(0.3119)
0.0435
(0.6507)
0.0513
(0.8015)
0.0346
(0.2607)
0.0011
(0.3942)
0.0007
(0.2536)
0.0105
(0.4896)
0.0153
(0.5223)
0.0005
(0.0899)
0.0013
(0.3694)
0.0047
(0.2043)
0.0321
(0.0048)
0.0252
(0.4765)
0.0000
(0.0973)
0.0003
(0.0602)
0.0166
(0.5408)
0.0000
(0.1582)
0.0001
(0.1184)
0.0003
(0.4225)
0.0065
(0.0596)
0.0013
(0.0523)
0.0150
(0.2130)
0.0233
(0.1732)
0.0381
(0.0985)
0.0108
(0.2794)
0.0224
(0.2855)
0.0316
(0.2394)
0.0171
(0.4543)
0.0025
(0.2464)
0.0016
(0.0704)
0.0201
(0.3942)
0.0000
(0.2748)
0.0140
(0.4425)
0.0037
(0.2227)
0.0007
(0.1506)
0.0003
(0.0424)
This table shows the R-squares of bivariate out-of-sample regressions. There are 67 out-ofsample forecasts from January 1998 to July 2003. NW is the Nadaraya-Watson estimator
and LL the local linear estimator. The dividend yield and the term spread are local
variables, the default spread and the TED spread refer to U.S. data. The numbers in
brackets show the p-values of chi-square distributed Wald test statistics for the null
hypothesis that the estimates are unbiased, as specified in (42).
However, one should not discard instrumental variable models too early. The fact
that such models are useful in practice even with very low forecasting power can
be demonstrated on the basis of the “Law of Active Management” proposed by
Grinold (1989) and Grinold and Khan (2000). Their framework builds on the information ratio, denoted as IR, which is defined as the average excess return relative to some benchmark (i.e., the alpha) per unit of volatility in excess returns. Intuitively, the information ratio measures the quality of a manager’s information
discounted by the residual risk in the betting process. Using a mean-variance
Parametric and Nonparametric Estimation of Conditional Return Expectations
191
framework for residual returns, Grinold and Kahn (2000) forcefully demonstrated
that the active manager’s goal must be to maximize the information ratio, and the
“Law of Active Management” provides a (very) rough “recipe”.
It starts with the definition of conditional expectation and – after several assumptions and crude approximations – finally posits:
IR = IC ⋅ m ,
(43)
where m denotes the number of independent forecasts and the information coefficient, IC, measures the quality of these forecasts (i.e., the correlation between the
realized and the forecasted returns).26 Intuitively, the law can be interpreted as follows: “You have to play often (high m) and to play well (high IC)”. In other
words, to achieve a high information ratio the managers can either make more accurate forecasts or cover more securities or forecast the same securities more frequently.
Table 4.3. Hit-rates of the direction of excess return forecast
Div. yield
TED spread
Germany:
OLS
0.522
NW
0.493
LL
0.537
United Kingdom:
OLS
0.597
NW
0.537
LL
0.537
United States:
OLS
0.537
NW
0.552
LL
0.537
Regressor variables in bivariate regressions:
Div. yield
Div. yield
TED spread
TED spread
Default sprd
Term spread
Default sprd
Term spread
Term spread
Default sprd
0.478
0.463
0.478
0.448
0.522
0.522
0.507
0.537
0.522
0.493
0.597
0.582
0.478
0.493
0.478
0.507
0.522
0.487
0.463
0.4448
0.4448
0.537
0.522
0.493
0.567
0.463
0.493
0.478
0.493
0.493
0.507
0.522
0.507
0.537
0.522
0.567
0.507
0.507
0.493
0.493
0.478
0.478
0.478
0.493
0.463
This table shows the hit-rates of the direction of excess return forecasts in bivariate out-ofsample regressions. There are 67 out-of-sample forecasts from January 1998 to July 2003.
NW is the Nadaraya-Watson estimator and LL the local linear estimator.
To give an example on how the law works, table 4.3 shows the hit-rates of our
out-of-sample forecasts. The hit-rates indicate the percentage of correct predictions of the direction or sign (and not the absolute level) of excess market returns.
On the one hand, it must be noted that many hit-rates are below 0.5, implying that
even a random guessing of the direction is superior. On the other hand, it does not
require a very high hit-rate to add value in the framework of the law in (43). Following Grinold and Kahn (2000), assume a binary variable x that takes the value
+1 if the excess return is positive and –1 if it is negative. Given that positive ex26
It should be noted that the IR in (43) is an ex ante theoretical concept. It has no direct
correspondence with the ex post information ratio except (roughly) as an upper bound.
192
Wolfgang Drobetz and Daniel Hoechle
cess returns are equally likely as negative excess returns, the expected value is 0
and the standard deviation is 1. The manager’s predictions of the direction of excess market returns can also take the values +1 and –1, with expected return 0 and
standard deviation 1. In this case, the information coefficient, or hit-rate, is:
IC = Cov(x t , y t ) =
1 n
∑ xtyt ,
N t=1
(44)
where N denotes the number of forecasts. Note that the correlation coefficient
equals the covariance under the specific assumptions. If the manager predicts the
direction correct in N1 cases (x=y), and if her predictions are incorrect in N−N1
cases (x=−y), (44) becomes:
IC =
1
(N1 − (N − N1 )) = 2  N1  − 1 .
N
 N
(45)
Assume the manager produces predictions for the direction of excess returns in 30
stock and 30 bond markets at the beginning of each month.27 Then, for the number
of forecasts per year, m, we have:
m=
(30 + 30) ⋅ 12 =
720 = 26.8 .
(46)
Goodwin (1998) analyzed 212 U.S. funds and found that over all fund categories,
except for small-cap funds, an information ratio of 0.5 puts a manager into the upper quintile. For example, to get a highly respectable information ratio of 0.54, the
manager needs a hit-rate of 0.51. Inserting into (45) we have:
IR = (2 ⋅ 0.51 − 1) ⋅ 26.8 = 0.02 ⋅ 26.8 = 0.54.
(47)
Note that many of our specifications in table 4.3 exhibit hit-rates above 0.51, indicating that they could in fact be applied in practice to improve portfolio performance. This is a surprising result, because ex ante one would not expect that such a
tiny prediction power can actually add value and deliver a high information ratio.
But this is exactly the intuition behind the law of active management; even small
advantages can be exploited, but a manager has to act on new information as often
as possible.
We conclude that, despite their very low out-of-sample prediction power, instrumental variable regressions can be a useful tool in the asset allocation process.
Clearly, although not all specifications and estimation techniques are equally
suited, some of them can deliver significant value. A prerequisite is that the manager has a good understanding of her model and good confidence in its predictions.
However, there is one caveat to mention. The forecasts for stock and bond markets are not independent. But independence of the predictions is one crucial assumption of the law. In addition, the prediction power will not be equal for all
27
This seems a realistic scenario. It roughly covers the MSCI (developed markets) and IFC
(emerging markets) universe of countries.
Parametric and Nonparametric Estimation of Conditional Return Expectations
193
markets, which also implicitly underlies the law. In the end, the law is not an identity and cannot directly be put into practice. Ultimately it is an empirical question
how good it works.
%QPENWUKQP
The goal of the paper was to explore the specification of conditional expectations.
Empirical asset pricing tests crucially depend on the correct specification of conditional expectations. Most previous studies used linear least squares models, conditioning on instrumental variables which are related to the business cycle. This approach is popular because it is relatively easy to implement empirically. However,
one could expect that the relationship between expected returns and conditioning
variables is nonlinear. In fact, in a more general setting conditional expectations
can be obtained with nonparametric regression analysis. Given enough data, this
method allows to estimate the conditional mean in a data-driven way.
We compare the linear least squares specification with nonparametric techniques for German, U.K., and U.S. stock market data. Using carefully selected
conditioning variables, our results indicate that nonparametric regressions capture
some nonlinearities in financial data. In-sample forecasts of international stock
market returns are improved with the nonparametric techniques. However, there is
very little out-of-sample prediction power for both linear and nonlinear specifications. This indicates data mining problems. Nevertheless, we also show that surprisingly small prediction power is required to add value. Using the framework
suggested by Grinold and Kahn (2000), we argue that a hit-ratio of forecasts for
the direction of excess returns as low as 0.51 can produce a highly respected portfolio performance. However, it is necessary to maintain the prediction power over
a large number of forecasts (and possibly over a large number of assets).
If an asset manager relies on a simple instrumental variables regression framework to forecast stock returns, our results suggest that linear conditional expectations are a reasonable approximation.
#EMPQYNGFIGOGPV
Financial support from the National Centre of Competence in Research “Financial
Valuation and Risk Management” (NCCR FINRISK) is gratefully acknowledged.
The NCCR FINRISK is a research program supported by the Swiss National Science Foundation. We thank Matthias Hagmann and David Rey for valuable comments and suggestions.
194
Wolfgang Drobetz and Daniel Hoechle
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%TGFKV4KUM2QTVHQNKQ/QFGNKPI#P1XGTXKGY
Ludger Overbeck1
1
Justus-Liebig University Giessen, Germany
2WTRQUGQH%TGFKV4KUM/QFGNKPI
Modeling Credit Risk has two main objectives:
The first objective is the analysis of each single counterparty and transaction in the
portfolio of a financial institution or investor. This leads to a thorough analysis of
the creditworthiness of the counterparty combined with the structure of the transaction.
The second one is the analysis of the entire universe of transactions and counterparties from a portfolio point of view. This should include all classical lending
products as well as all investment bank transactions and traded products. The purpose of this analysis is to assess whether the portfolio is consistent with the risk
appetite of the bank. An important aspect of the portfolio analysis is the assessment of new transactions within the context of the existing portfolio and the decision whether they fit into the risk profile of the institution.
In the current paper we are mainly concerned with the second objective. Naturally, the analysis of the entire portfolio requires a careful study of all single transactions and counterparties. We will therefore also present all aspects of the single
counterparty analysis, which are required for the portfolio modeling.
Further to the use of credit portfolio models in Risk Management, these models
are utilized for the valuation of financial products, which depend on a portfolio of
transactions. These products include Basket Credit Derivatives, Asset-BackedSecurities and Collaterized Debt Obligations. In this overview we will not give details on the valuation of these products but refer to the books by Bluhm et al.
(2003) and Bluhm and Overbeck (2004). In general, a more detailed treatment of
the questions addressed in this paper can be found in Bluhm et al. (2003).
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A financial institution should measure all its risk in a consistent manner. This
means it has to identify all risk types and quantify the cost of taking these different
risks consistently. The purpose of this is of course to avoid internal arbitrage. A
business unit running a risk which is not yet identified or measured or measured
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Ludger Overbeck
inconsistently might produce a high return without contributing adequately to the
risk profile and consequently to the risk costs- of the firm.
To the decision makers in the institution a business unit like this might appear
highly profitable leading to a decision to expand this business activity. Clearly, if
this decision is based on an incorrect risk assessment the impact of the decision
could be damaging for the firm. A typical example for this is the 8%- percent rule
for regulatory capital. Usually, low-rated customers are willing to pay a higher
margin on their loan then highly rated customers. If both customers are corporates
then the regulatory capital charge is identical for both, and hence to a bank, which
assesses its risk based on regulatory capital charges the lowly rated customer segment appears more profitable. Banks, however, will not only apply the regulatory
risk costs but take into account the experience that lending to lowly rated customers generates higher credit losses. There will therefore be an adjustment of the decision based on the credit rating. The current restatement of the regulatory capital
requirements under the Basel II initiative will lead to a more risk-adequate regulatory framework, which takes the creditworthiness of individual borrowers into account.
The same argument is applicable to entire business lines. For example, investment banking might be viewed as a relatively profitable business line, since the
counterparty credit risk in OTC-transactions is potentially underestimated compared to credit risk assessment of classical loan products, or even not considered at
all in the internal credit capital calculation.
These arguments should make clear that an enterprise-wide risk measurement is
necessary for sound management of a financial institution.
From a measurement point of view there are at least two important items of information the management of a bank wants to obtain from the enterprise-wide
Risk Measurement process. The first one the total risk the institution has and the
second one how much the single business lines or even transaction have contributed to this risk. The first one is usually called Economic Capital and the second
one Capital Allocation.
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Economic Capital is a figure in which the total risk of a firm is measured and reported. It should contain all different risk types in all business lines of the bank
and should take into account all diversification benefits but also all concentration
risks in the institution. Economic Capital is the demand side of the capital. If the
economic capital equals 110 Million Euro, the financial institution needs 110 Million Euro of capital to support its business. This paper does not consider the supply side of capital but assumes that the institution has at least invested this amount
in riskless and liquid assets, such that in case of emergency it can pay out the
amount calculated for the Economic Capital, in the example 110 Million Euro,
immediately.
In this concept the Economic Capital is to some extend lost, since it can not be
used to make more money out of entrepreneurial activities, which by their very nature require some risk taking. The difference between the riskfree rate earning by
Credit Risk Portfolio Modeling: An Overview
199
the riskfree investment of Economic Capital and the return required on the supplied capital by the shareholder, has to be earned by the businesses. In that sense
the investment of the Economic Capital Amount in no-risk assets produces some
costs. This cost has to be earned usually be those who have caused this cost. It is
therefore unavoidable to answer the question how much each business contributes
to the cost of (economic) capital.
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From a modeling point of view this question is answered by the capital allocation
algorithm that is applied by the bank. The capital allocation algorithm answers the
question how much of the overall Economic Capital is caused by a given business
unit or even a single transaction. For the clearness of exposition we assume that
the bank has three business units A, B and C. The allocation process should then
find a sound procedure to calculate figures CEC(A), CEC(B) and CEC(C), Contributory Economic Capital (CEC) of the business units, such that.
EC=CEC(A)+CEC(B)+CEC(C)
This formula implies that the business units should all benefit from diversification.
The same capital amount is reported on all different hierarchy levels of the institution. This is in contrast to Market Risk, where frequently the Value-at-Risk (VaR)
is applied to determine the economic capital. Since VaR is non-additive, the group
wide VaR will be different – typically smaller – then the sum of the VaR measures
for the business units. From an enterprise wide risk management point of view this
saving of capital by diversification and the question who will benefit from this diversification should be avoided. In smaller units like trading desks hierarchies this
approach is still valid, since the trader should know the market in the first place
and not the entire portfolio of the bank. The head of trading, and his performance
however should be linked to the overall risk profile of the institution.
To summarize the discussion on enterprise wide risk management from a measurement and modeling point of view, a modeling approach should always be able
to provide an overall risk figure, the economic capital EC and the contributory
economic capital figures CEC(BU) of a business unit BU. The latter should sum
up to the overall capital.
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Starting from an Enterprise Wide Risk Management point of view it is clear that
the different risk types, namely Market, Credit, Operational and other risks, should
be measured in an integrated way. The current models usually measure each risk
type separately. Usually, the bank either simply adds Economic Capital figures for
the different risk types or combines them at the top level by employing a dependency assumption, often based on a normal copula approach. It should be understood that this approach is based on the assumption that it is possible to separate
the analysis of risk types on the single transaction level. This is more a historically
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Ludger Overbeck
grown concept than economic reality. In particular the distinction between market
and credit risk does no longer exist in the modern capital markets. For a product
bearing credit and market risk in the traditional sense like a corporate bond, it is
almost impossible to define and model exactly what is market and credit risk.
Although it is obvious that an integrated view to “Risk” is necessary we will
concentrate our exposition on credit risk. We will take some market risk components into account later when considering rating migration modeling and the modeling of stochastic credit exposure.
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In the current state of discussion it is unquestionable that a prerequisite of any risk
measure is a loss distribution (or, in the case of market risk, the left hand-side of a
profit loss distribution).
Mathematically, a loss distribution is the cumulative distribution function, of the
random variable L describing the potential losses the bank might suffer. Quite often the loss variable L is also called loss distribution itself. Here, one basic structure of “risk” is apparent. Risk only occurs since the future is random and not deterministic. Therefore every modeling of risk has to start from the concepts of
probability theory and statistics and the concept of a random variable is at the
heart of these mathematical disciplines.
In order to go further it is obvious that one has to fix a future point in time T, at
which the institution wants to analysis the accumulated losses. To be specific we
assume that this time horizon is one year.
The loss L is therefore the accumulated loss during the time from today until
T(=one year). Since these losses appear in future they depend on the future “stateof-the worlds” ω. The loss L is indeed a function L(ω). Since all randomness is
captured in the scenarios ω, once a scenario is chosen then the loss is known. This
is like going back in history where in each past year a scenario ω was realized and
the loss in that scenario was observed.
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The most common risk measure is the so-called VaR. It is the amount of money,
which will be lost by an institution in an extremely adverse scenario. “Extremely
adverse” scenarios are specified in terms of the quantile of the loss distribution. If
the risk measure is the 99%-VaR and it equals 10 Million Euro, then this implies
that the probability of loosing more than 10 Million Euro is 1%.
From a conceptual point of view, however, the Value-at-Risk lacks some important properties a reasonable risk measure should have. Most importantly, Value-atRisk does not recognize diversification. A portfolio A consisting of two subportfolio B and C might have the property that the Value-at-Risk of portfolio A is larger
than the sum of the two Value-at-Risk figures of portfolio B and portfolio C. The
axiomatic approach to risk measures was introduced to finance in the paper Artz-
Credit Risk Portfolio Modeling: An Overview
201
ner et al. (1997). Their so-called coherency axioms are generally satisfied by an
alternative risk measure, known as Expected Shortfall. This risk measure quantifies the expected loss above the quantile. It therefore considers the magnitude of
losses beyond the quantile.
We will consider this risk measure in the context of credit portfolio models
later.
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In the present article we concentrate on the use of portfolio models in the risk
management area. However, portfolio models become more and more important
also in the valuation and pricing routines of financial institutions, since transactions depending on the creditworthiness of an entire portfolio of loans or other
credit related instruments are gaining ground. These products known under the
names of ABS (Asset Backed Securities), CL(B)(D)O (Collaterized Loan
(Bond)(Debt) Obligations), CMBS etc) require the modeling of the joint default
behavior of the assets in the underlying reference pool of assets. Additionally, in
this context it is important to consider the timing of default and the cash flow
analysis in each default scenario. We will not consider these extensions here but
refer to Bluhm et al. (2003) or (Credit Metrics) and references therein.
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As set-up in Section 1 credit portfolio models consist of the analysis of a single
transaction and the concept of the joint default. First we will put together the necessary input parameters.
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A single transaction is described by two components the exposure and the loss
given default. The counterpart is parameterized by its default probability and the
portfolio by a dependency concept.
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The exposure is the outstanding amount, which will be lost in the case of default.
It is therefore called the Exposure-at-Default, EAD. Although we consider the defaults accumulated until time 1, we may assume for simplicity that the amount
outstanding at time 1 is considered. Another view on this is to assume that defaults
happen exactly at time 1. Whereas in most implemented models the EAD is a
fixed non-random parameter in reality it should be modeled as a random variable.
For traded products it is obvious that one cannot ignore the value of the product at
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Ludger Overbeck
the time of default. But even in the case of traditional lending products, like commitments, the outstanding at default is usually not known with certainty.
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The loss given default, LGD, is the percentage of the EAD which is lost after the
work-out and recovery process has taken place. Again, it should be modeled as a
random variable. Sometimes a Beta distribution is used to incorporate the randomness of the LGD. If independence of the LGD is assumed the effect of stochastic LGD will impact the risk measures only slightly in large portfolios, since
in a large portfolio the law of large numbers will come into effect. If one aims to
model a random LGD one should try to incorporate a dependency between LGD
and the other variables in the model. It is known from empirical analysis that default rates and loss given default rates are positively correlated.
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The default probability measures the likelihood of a default of the counterparty. It
is necessary to specify a time horizon, like 1 year. Then the default probability
quantifies the probability that the counterparty will default in 1 year from today.
This specification is the industry standard and also used in the risk weight functions proposed for the capital rules under the Basel II framework.
For more advanced models, which take into account transaction maturing before
1 year or those with maturities beyond the one year time horizon, an entire term
structure of default probabilities are required as input.
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Portfolios of loans usually contain a large number of counterparties. If they all
were independent, by almost perfect diversification, the loss would always be
close to the expected loss1, implying almost no risk. Economically this follows
from classical portfolio theory identifying risk with standard deviations. Mathematically the result follows from the Law of Large Numbers for independent random variables.
However, the empirically observed default rates exhibit a different pattern. For
many years the observed loss is far below the expected loss whereas in recessions
the actual loss is a multiple of the expected loss. This implies dependency of default events. If we want to analyze the question whether a given counterparty A
defaults and we know that many other similar counterparties do actually default,
then we would infer a higher probability of A defaulting than in the case where
many other counterparties do not default.
Probabilistically this means that the default event of counterparty is not independent from the default event of other counterparties. In other words the empiri-
1
Here expected loss is defined as default probability times loss given default.
Credit Risk Portfolio Modeling: An Overview
203
cal behavior of default rates implies that there is a positive default correlation between obligors.
However, estimating the default correlation from the fluctuation of default rates
over time, usually results in a very low correlation, in many cases smaller than
5%. This could lead to the temptation to ignore them. But looking at some underlying variables driving the default events shows that this is dangerous and dependence is strongly underestimated.
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Let us assume that default happens if the ability-to-pay of a counterparty at year 1
is deteriorating, or to be more precise, that the ability-to-pay is below a certain
threshold value. If the ability-to-pay (APP) is modeled as real-value random variable A with distribution function FA , then the threshold C is implied by the default
probability via 1- FA(C)=DP. Furthermore, if the distribution of the ability-to-pay
is the standard normal distribution (or any multivariate family of distribution
whose dependency is specified by correlation parameters only), then it only remains to identify the correlation of all pairs of ability-to-pay variables. By their
very nature the ability-to-pay changes continuously over time. Therefore the ability-to-pay correlation can be analyzed and calibrated by studying the correlation of
time-series. To see the effect of this approach, which can always be postulated assuming a latent variable A, we show a table relating default correlation and underlying ability-to-pay correlations.
Table 1. Default correlations depend on APP-Correlation and PD
PD1 = 133bp
PD2
0.0002
0.0003
0.0016
0.0016
0.0026
0.0026
APP-correlation
0.48
0.65
0.48
0.65
0.48
0.65
Default correlation
0.038
0.087
0.074
0.149
0.081
0.169
JDP
0.00006
0.00018
0.00039
0.00070
0.00052
0.00087
To explain the table, we consider counterparty 2 with probability of default of
0.0002 and an ability-to-pay correlation of 48% with counterparty 1 having
PD=0.0133. The default correlation then equals the number in the third column in
the first row of the table, namely 3.8%.
The values of the ability-to-pay correlations are much closer to our intuition of
dependent entities than the values of the default (event) correlation. Hence, already from a communication point of view, it is much more reasonable to address
the question of dependency from an ability-to-pay, or more generally time-series,
point of view. Naturally in Financial Theory the ability-to-pay is nothing else than
an abstraction from Merton’s asset-value model. The term “ability-to-pay” only
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Ludger Overbeck
liberates Merton’s concept from the actual analysis of the valuation of a firms assets.
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The aggregated output of the portfolio models are based on the loss variable L
which is the sum of the single transaction losses L(i).
L=L(1)+…+L(m)
The first statistics of interest is the expected loss, the mean of L. Since the mean is
additive the expected loss of the portfolio is just the sum of the expected losses of
the single transactions
E[L]=E[L(1)]+…+E[L(m)]
Most financial institutions do not view the expected loss as a risk measure, it is
viewed as a cost and should not be covered by capital, but rather by margin income.
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The amount of capital, which is required as a buffer to survive also very severe
losses, in our case, caused by credit events, is called Economic Capital, EC. EC is
a required amount of capital, similar to Regulatory Capital. The subtleties lie in
the specification “to survive very severe losses”.
There are at least two mathematical specifications of this notion, namely Valueat-Risk and Expected Shortfall.
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Severe losses means that still larger losses occur only with a very small probability. Therefore Value-at-Risk requires a probability, in that context called a confidence level, as a parameter. If the 99%-Value-at-Risk turns out to be 10 Million
Euro, then the probability that accumulated portfolio losses are higher than 10
Million is 1%. In mathematical notation
P[L> α−VaR]=1- α.
The Value-at-Risk can be obtained from the inverse of the distribution function of
the portfolio loss variable.
Typical examples of α are 99% and 95% for Market Risk VaR in the trading
book with a time horizon of 1, 3 or 10 days. In credit risk usually with a time horizon of 1 year, 99.98%, 99.97%, 99.95% and 99.9% are widespread. Since credit
risk still makes up most of the total risk of commercial banks, these confidence
levels are related to the default probability of the firm and in light of the credit
agency rating it reflects the intended rating of the bank. A bank basing its EC on
Credit Risk Portfolio Modeling: An Overview
205
99.98% confidence level would intend to have AA+ rating, since a default probability of 2 BP is associated with that rating.
Also the regulator usually bases its capital requirement on a confidence level. In
the Basel II approach the 99.9%-quantile of a loss distribution is implicitly used.
As mentioned in Section 1, the main disadvantage of Value-at-Risk, lies in the
fact that it does not reward diversification. It might be the case that aggregating
two portfolios together into one portfolio results in a higher Value-at-Risk than the
sum of the two single portfolios. In mathematical terms Value-at-Risk lacks subadditivity as a function on “Portfolios”.
Another critique on VaR comes from its 0-1 character. VaR ignores how large
the losses are beyond the quantile. Two portfolios with the same VaR can still
have different distributions of their potential losses beyond the VaR, in the socalled tails of the distribution. Loss distributions, which have fatter tails are of
course more risky than those with tighter tails, even if they have the same VaR.
A very simple measure, which takes also the tails beyond the VaR into account,
is the Expected Shortfall.
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Expected Shortfall measures the average loss given that the loss is higher than a
quantile. It is the conditional expected value of L given that L is larger than a
quantile of L. Therefore it also requires a confidence level. “Severe losses” here
means the set of large losses – more precisely the interval of losses between infinite and a loss - which has a probability of exactly α. “To survive” means to survive in average.
ES(L,α)=E[L|L>q(L,α)]
From the external view to a bank there is no default probability anticipated. To
show how Value-at-Risk for some distributions is related to expected shortfall we
refer to the table below:
Volatility
99%-Quantile
ES
Increase
Student(3)
1.73
4.54
6.99
54%
N(0,1)
1
2.33
2.66
14%
Lognormal(0,1)
2.16
8.56
13.57
58%
Weil(1,1)
1
4.6
5.6
28%
The distributions in this table are the Student-t distribution with three degrees of
freedom the Normal distribution with mean 0 and standard deviation of 1 the LogNormal distribution with parameters zero and 1and the Weilbull distribution with
parameters 1 and 1. The increase denotes the increase if one switches the capital
definition from 99%-Quantile to the Expected Shortfall above this quantile.
In addition to the fact that ES rewards diversification it moreover satisfies all
axioms of a coherent risk measure. These are presented in the next section.
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Ludger Overbeck
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Mathematically, the loss variable might be seen as bounded real value random
variable on a probability space (Ω,F,P). A coherent risk measure is a mapping r
defined on L(Ω,F,P), the space of bounded real valued random variables with values in R, such that the following properties hold
(i) Subadditivity: r(X+Y)< r(X)+r(Y)
(ii) Monotonicity: X<Y implies r(X)<r(Y)
(iii) Positive homogeneity: For a positive real number k: r(kX)=kr(X)
(iv) Translation invariance: For all real numbers y : r(X+y)=r(X)+y
The most important property is (i), which is equivalent to assuming that diversification does not increase risk, meaning that in most cases diversification improves
the risk profile.
The second axiom is also intuitively clear. A loss function, which is greater or
equal to another loss function in all states of the world Ω, should also be more
risky. Property (iii) means that a linear non-random increase also increases the risk
by the same amount2. The last axiom can be interpreted such that a non-random
(deterministic) additional loss of y increases the risk by the amount y.
It is well established that Value-at-Risk does not fulfill axiom (i), i.e. it does not
always reward diversification. Expected shortfall, when properly defined (some
problems are caused by discontinuities of random variables), satisfies all coherency axioms (see Artzner et al. (1997), Tasche (1999)).
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Once a financial institution has agreed on a capital definition, the next question is
how to allocate the capital to the different activities of the institution. The basic
question is to determine how much of the portfolio risk is generated by a single
transaction or by a business unit responsible for a certain set of transactions.
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A widespread technique to allocate capital is based on risk contributions RC(i). By
definition RC(i) is the sensitivity of the standard deviation of the loss variable
with respect to a change of exposure to counterparty i. It can be easily verified that
the RC(i) is the covariance between the exposure to counterparty i and the portfolio loss L. This resembles the Beta of a single stock with respect to the market index as known from the CAPM.
Since the sum of the weighted RC(i) over all counterparties gives the portfolio
standard deviation, the Risk Contributions have to be scaled up by a factor, the socalled Capital Multiplier CM to arrive at the capital. The Capital Multiplier is defined as the Value-at-Risk divided by the standard deviation. For a normal distribution the CM, for the capital defined at the 99%-quantile equals 2.33. For the
2
This might be debatable in the case of name concentrations!
Credit Risk Portfolio Modeling: An Overview
207
normal distribution and other elliptic distribution the concept of Capital Multiplier
makes sense since the quotient CM does not depend on the portfolio weights! For
other distribution assumptions the contribution to Value-at-Risk must be calculated directly without using the concept of Capital Multiplier and Risk Contribution.
The Value-at-Risk contribution is studied for example in (Tasche 1999).
We do not detail this approach here since the Value-at-Risk is not coherent, in
particular there is no allocation rule associated with Value-at-Risk which by its
construction ensures that the capital allocated to a sub-portfolio is smaller or equal
to the capital of that portfolio seen as an independent (stand-alone) from a larger
portfolio.
This property is strongly related to the sub-additivity axiom of coherent risk
measures. Again it is clear that is a natural consequence of diversification that the
capital of a portfolio viewed as a sub-portfolio is smaller or equal the capital of
that portfolio seen as stand-alone.
A formal approach to capital allocation in the spirit of coherency might be found
in (Kalkbrener 2003), (Tasche 1999) or (Kalkbrener et al. 2004). In the latter reference the computational issues coming with expected shortfall contribution are
also solved.
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Due to L=L(1)+…L(m) the natural decomposition of Expected Shortfall also
gives the correct allocation rule
ES(L,α)=E[L|L>q(L,α)]= E[L(1)|L>q(L,α)]+ ..+ E[L(m)|L>q(L,α)]=
ESC(1)+..+ESC(m).
The conditional expectation of the loss in transaction i given that the entire portfolio loss is larger than the quantile also has an obvious causal interpretation. Since a
financial institution holds its risk capital against exactly those events “loss large
than quantile” the risk measure ESC(i) really measures the impact or contribution
of transaction i to those large losses.
Concrete examples, comparison and calculations in the credit risk model described below can be found in (Kalkbrener et al. 2004).
2QTVHQNKQ/QFGNU
In this section the currently used models for the analysis of portfolio credit risk are
presented. The models are concerned with the structure underlying the loss variable L and more importantly with the single loss variables L(i). In mathematical
terms the multivariate distribution of the vector (L(1),..L(m)) has to be specified.
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Ludger Overbeck
This is not a trivial task since m might be very large3. In the actuarial approach the
emphasis lies in the modeling of the portfolio loss, neglecting the structure of single losses, whereas the second approach mainly based on Merton’s asset value
concept has its origin in the attempt to model accurately also single defaults.
#EVWCTKCN#RRTQCEJ
The standard approach for loss distributions in insurances assumes that the total
loss accumulated over a given time horizon can be modeled using the following
concept. First the number of loss events during this time horizon is specified. This
is called the frequency distribution. In a second step the distribution of the actual
loss in a loss event is specified, the severity distribution. The total loss is therefore
the sum of the single severity distributions and the length of the sum is random
and distributed according to the frequency distribution. The probabilistic model is
therefore specified by the multivariate distribution of the infinite dimensional vector (N, S(1),S(2),S(3),…), where N denotes the arbitrary (this means every natural
number can be realized) large number of possible events and S(i) is the severity in
the i-th event. The loss variable in state-of-the-world, or scenario, ω can then be
written as
L(ω)=S(1)(ω)+…+S(N(ω))(ω)
If for example the scenario ω corresponds to 7 loss events in the next year, N=7, it
remains to specify in that scenario the seven severities S(1)(ω),..,S(7)(ω). The total loss equals the sum of the seven severities.
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The main advantage of this type of model is its analytic tractability in some important cases. A widespread assumption for the frequency distribution of default
events is the negative-binomial distribution. It can be constructed as a Poisson distribution with a parameter l, which is itself random with a Gamma distribution.
Hence the specification of the number of losses in each future loss scenario is a
two-step process in itself. First the parameter l of the Poisson distribution is chosen according to a Gamma distribution. In economic terms this means that a default intensity is realized in year 1. In the second step of the scenario, the number
of defaults is specified in accordance with a Poisson distribution. This gives N(w)
for each scenario w. It remains to determine the loss amount S. It is usually assumed that the distribution of the vector (S(1),S(2),…) is that of an identically, independent sequence. All S(i) have the same distribution and they are independent
among themselves and independent of the frequency N.
3
Some banks have counterparties of 100.000 and more or after aggregation at least ten
thousand risk segments!
Credit Risk Portfolio Modeling: An Overview
209
In actuarial mathematics there are numerous possibilities to specify the severity
distribution S. In the case of Credit Risk+4 a popular instance of this model type,
the severity distribution corresponds to the empirical distribution of the exposures
in the portfolio. To illustrate this let the portfolio have 20% of exposures between
1 and 2 Million Euro, 40% between 2 and 3 Million, 10% between 3 and 4 Million, 20% between 4 and 5 Million, and 10% between again between 5 and 10
Million. The empirical distribution approximating this exposure profile might take
the value 1.5 Million with probability 0.2, 2.5 Million with probability 0.4 and so
on. Once the number of defaults N is chosen the loss amount in each event is then
chosen according to this exposure profile. It is clear that the total amount in this
case can only be a multiple of 0.5 Million.
The theoretical property that the empirical distribution is always designed in a
way that it takes only finitely many values allows applying an inversion algorithm
to obtain the probability distribution of the portfolio loss variable. One procedure
of this type is the Panjeer algorithm which is detailed in (Bluhm et al. 2003) and
(Credit Risk+).
There are other procedures to invert probability generating function which do
not assume a discrete severity distribution.
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The dependencies are modeled through the random intensities of the frequency
distribution. The intensity of a counterparty with 50% weight in automotive and
high tech for example has the form
λ = 0.5 * λ aut + 0.5 * λhit
where the two intensities are the intensity of automotive and of high-tech. If there
is an additional idiosyncratic risk then this is simply added. More details on the
dependency modeling can be found in the original technical document (Credit
Risk+).
Here also lies the main disadvantage of the model. The lack of a structure underlying defaults hinders the explicit incorporation of default correlation into the
model. On the other hand this underlying structure is readily given in the structural
approaches.
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There are several extensions of the presented actuarial model. However the advantage of having a closed form solution is often lost if more advanced approaches,
like rating migration modeling, are pursued.
4
Credit Risk+, a software and concept, introduced by Credit Suisse Financial Product into
Finance.
210
Ludger Overbeck
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As the term “Structural” indicates this approach assumes a structure underlying
the default event. The classical formulation goes back to the Nobel Prize winning
paper “Pricing of corporate debt”, 1974 by Robert Merton.
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A firm is defaulting if the asset-values fall below the value of its liabilities. The
shareholders are not able to pay the firms debt by liquidating the assets of the firm.
In a literal application of this concept it is necessary that the corporate, whose default behavior is analyzed, is a firm issuing traded equities and debts. Many borrowers of banks do not fall into that category.
However the structural concept that default can be identified with the event that
the ability-to-pay of the borrower is not strong enough to pay back the debts can
be applied to all customers. The actual calibration of this “ability-to-pay” concept
will of course differ between different borrower types. From a modeling point of
view, we assume that in the structural approach there is for each borrower i a variable A(i), the ability-to-pay and a (fixed) level c(i), the default threshold, such that
default at horizon happens if the ability-to-pay deteriorates below the default
threshold, in formula
D(i)={A(i)< c(i)}.
In most models only the ability-to-pay is a random variable and the threshold is
fixed.
In addition to the indicator variable for the default event one also has to specify
the loss given default parameter LGD(i) and the Exposure at default EAD(i). For
simplicity we assume these to be fixed and denote their product by l(i), the actual
amount which the lender looses in case of default.
The total loss of the portfolio can then be written as the sum of the single losses
m
L = ∑ l (i)1{A(i )≤c ( i )}
i =1
In contrast to the actuarial approach an analytic form for the loss distribution is
difficult to achieve. One can of course write down a formula for the distribution
F(x)
P[ L ≤ x] =
m
∑{ ∑
} {
∑ P[A(i ) ≤ c(i ),.., A(i ) ≤ c(i ), A(i) > c(i), i ∉ {i ,..i }]
k =1 i1 ,..ik ⊆ 1,..,n} l ( i1 ) +..l ( ik )≤ x
1
1
k
k
1
k
If the multivariate distribution of the ability-to-pay variables is given then the distribution function is in principle given and the inverse can then be found also numerically.
Credit Risk Portfolio Modeling: An Overview
211
However a simple “closed form solution” is not known.
One usually applies a Monte-Carlo-Simulation to obtain an estimate of the risk
measures of interest. The procedure for this simulation is exhibited in Figure 1.
&GRGPFGPEKGU
Since the joint default events are specified by the event that all counterparties fall
below their respective default threshold at horizon, all dependencies are modeled
through dependencies between the ability-to-pay variables at horizon. Therefore
the joint distribution of the vector of the ability-to-pay variable, i.e. the distribution of
r
A = ( A(1),..., A(n) )
has to be specified.
The basic dependence model can be explained in the following figure, which
shows the joint default behavior of two obligors.
APP Obligor 2
APP Obligor 1
Default Threshold 2
211Threshold 2
Default Threshold 1
Fig. 1. Joint default behavior of two obligors
From a portfolio modeling point of view it is efficient to decouple the single default probabilities from the joint default behavior. Quite often it is assumed that
the default thresholds are given by exogenous default probabilities.
212
Ludger Overbeck
Independently of the source model from which the default probability is obtained the default event is always caused by the ability to pay variable crossing the
default threshold in structural models.
In this setting any marginal distribution for the ability-to-pay can be used as
long as the distribution and the default threshold are consistent in the sense that
the probability mass of the distribution between negative infinity and the threshold
equals the default probability, i.e. if F(i) denotes the distribution function of the
ability-to-pay for the i-th counterparty and PD(i) is its default probability we only
require
F(i)(C(i))=P[A(i)<C(i)]=PD(i).
The standard assumption is to use a standard normal distribution for the marginal
distribution of the ability to pay. The joint distribution of the ability to pay variables of all obligors in the portfolio is multivariate normal.
This assumption implies that only the correlation structure of the multivariate
distribution has to be determined to fully specify the model.
Usually this is done by deriving the paired correlations between obligors from a
factor model which ascribes the dependencies between the individual ability-topay variables to a set of systematic economic or market factors.
A widely used approach that is implemented by the MKMV5 and CreditMetrics
structural models is to use a combination of regional and industry specific factors
as the systematic variables in the factor model. The correlations between the systematic factors are determined from correlations between equity or asset value indices.
Once the covariance structure of the systematic factors is known the correlation
of two obligors depends on two types of parameters. First the so-called R-squared
parameter, which specifies how much risk in the ability-to-pay of the counterparty, is explained by the systematic factors and secondly the industry and country
weights indicating the dependency of the obligors’ economic state from the industries and regions.
If we denote the covariance matrix of the factors by Σ, the weight vector by of
counterparty i by w(i) and the R-squared parameter by R2(i), then the correlation
between counterparty i and j is given by the product of the vector w(i) times the
covariance matrix times the vector w(j) transposed times the products of the normalized R-squared, in formula:
Corr(A(i),A(j))=Cov(A(i),A(j))=R(i)/(σ(w(i)*Φ))*R(j)/(σ(w(j)*Φ))*w(i)*Σ*w(j)T
Here Φ denotes the vector of all factors. Typical values for this parameter are
• R-squared for large industrial companies and banks between 50%
and 70%
• Typical only two or three industries weights different from zero
• R-squared for retail customers between 5% and 15%
5
MKMV = Moody’s KMV, Consulting firm specialized on Credit Risk.
Credit Risk Portfolio Modeling: An Overview
213
•
Weights for retail customer usually according to country of residence
and employment
• R-squared for SME (small and medium sized enterprises) between
12% and 30%.
• Average asset correlation for large portfolio around 10%-15%
The number of different factors depends on the business structure of the financial
institution. A global operation will need at least 20 regional factors and approximately the same number of industry factors.
.QUU&KUVTKDWVKQP
Usually the loss distribution is derived from a Monte-Carlo-Simulation, however
for a qualitative picture of the portfolio loss distribution an approximation technique can be used.
Generation of asset
return for all
counterparties
correlated via the
factor model
empirical loss distribution
N
∑1
k =1
No default
Next
counterparty
Default
[ 0, x ]

 m k
 ∑ l i 1{ APPi <Ci } 

 i =1
Add exposure of
counterparty to loss
after last
counterparty
Portfolio loss
in simulation
Next
simulation
after last
simulation
Loss distribution
Example: Portfolio of 2.000 middle market loans
Fig. 2. Derivation of the loss distribution by Monte-Carlo Simulation
Monte-Carlo-Simulation
In a Monte-Carlo simulation we generate a large number of ability-to-pay scenarios for all obligors in the portfolio and calculate the corresponding losses.
In each scenario k, the calculation of the portfolio loss requires the following
steps.
• Generate a multivariate normal random vector, representing a realization of
the factor model, Φ(k).
• Generate for each obligor i an additional normal random variable ε(i,k).
214
•
Ludger Overbeck
Obtain the ability-to-pay A(i,k) of each counterparty by adding the systematic risk component
R 2 (i) *w(i)∗Φ(k) and the idiosyncratic part (1-
R 2 (i) )ε(i,k).
•
Collect all obligors with an ability-to-pay below their default threshold C(i)
A(i,k)< C(i) and add up all their exposures times their loss given default.
• This last sum gives the portfolio loss in scenario k.
This algorithm is exhibited schematically in the following figure.
Analytic approximation
The best known analytic approximation is the infinite granular homogeneous portfolio approach, developed by Vasicek [VAS], and later by Gordy (2001) in the
regulatory capital framework.
Its main advantage is that the notion of default probability p and ability-to-pay
or asset correlation ρ is maintained in the limit.
In the next figure we plot the limiting portfolio loss distribution for different
values of the correlation parameter. It is obvious that the different correlation assumptions, which represent different business segments of a financial institution,
lead to very different portfolio loss distributions.
High systematic risk
Low systematic risk
Systematic risk
Low
1%
Average
10 %
High
30 %
99.98%-quantil
0.81
4.30
16.68
EC
0.51
4.00
16.38
UL
0.09
0.35
0.86
Cap. Mult.
5.67
11.43
19.05
Average systematic risk
.
Fig. 3. Limiting portfolio loss distribution for different values of the correlation parameter6
The low systematic risk corresponds to Retail banking the high systematic risk to
investment and large corporates banking and the medium to middle market or a
mixture of all business types
6
UL=Unexpected Loss = Volatility of loss distribution, CM=Capital Multiplier = EC/UL.
Credit Risk Portfolio Modeling: An Overview
215
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The advantage of the structural model is its flexibility to incorporate model extensions. This flexibility is due to the adaptability of the Monte-Carlo simulation. All
additional randomness can easily be incorporated into the implementation of the
Monte Carlo simulation. Additional research might still be necessary to develop
the parameter calibration for the model extensions.
The most important extensions are
• Credit Migration:
o Here the technical extension is to add to the one threshold triggering default additional thresholds for the asset-values at horizon
triggering a rating migration. The additional rating transition
thresholds are easily calibrated using a rating migration matrix.
This extension is implemented in the CreditMetrics model. The
basic implementation concept is exhibited in the following figure.
Compared to the first figure 1, more thresholds are added to distinguish the rating classes into which the obligor has migrated at
horizon. In the scenario indicated in Figure 4 the blue counterparty has migrated to CC-class.
AAA
AA
A
BBB
BB
B
CCC
Default
Fig. 4. Migration Modeling in the asset-value approach
•
Volatile Loss Given Default:
o It is obvious from the data on Loss Given Defaults that the realized quotas fluctuate significantly around the mean. Therefore a
random LGD is essential. Even more important is the dependency
between LGD and default events. This is more difficult to calibrate and implement and therefore even the most advanced models currently assume a random LGD, which is independent of
other risk factors.
216
•
•
Ludger Overbeck
Volatile Exposure:
o This is an important extension if one also wants to represent counterparty risk from trading businesses like swap exposures within a
credit portfolio model. This problem is an important catalyst in
the bottom-up integration of Credit and Market Risk.
Multi-period models:
o If the interest lies in the performance of a portfolio for a longer
time horizon than one year (which is assumed in most ECmodels), one has to perform a multistep simulation. A one step
simulation until a long time horizon, e.g. 5 years would imply a
too strong assumption regarding the stability of the portfolio and
homogeneity of the cash flows. For an accurate modeling of CDO
structures a multistep approach is a definite necessity. Details on
this can be found in (Bluhm et al. 2002), (Bluhm and Overbeck
2004).
5WOOCT[
In this paper an overview of current credit risk modeling approaches is given.
Starting from the need of risk modeling in an enterprise wide risk management
framework, it is shown how credit risk models fit into a consistent measurement of
risk and capital allocation. The basic inputs and outputs for a credit portfolio
model are specified. The two most widespread approaches are discussed: On the
one hand the actuarial Credit Risk+ framework, which also fits into the reduced
form credit models and on the other hand the structural model approach, which
goes back to the seminal work of Merton. It is shown how the basic features of
these models work and how they can be implemented. In particular the question of
capital allocation and dependency modeling is emphasized. Some possible extensions, a few of which are already available in some specific models, are also discussed.
Credit Risk Portfolio Modeling: An Overview
217
4GHGTGPEGU
Artzner P, Delbaen F, Eber J, Heath D (1997) Thinking Coherently. RISK 10 (11)
Bluhm C, Overbeck L, Wagner C (2003) An Introduction to Credit Risk Modeling.
Chapman Hall/CRC
Bluhm C, Overbeck L (2004) CDO-Modeling. Draft
Credit Risk+. Technical Document. Credit Suisse Financial Products.
Credit Metrics. Technical Document. Risk Metrics Group.
Gordy M (2001) A Risk Factor Model Foundation for Ratings-Based Bank Capital Rules.
Draft. February
Kalkbrener M (2003) Axiomatic Approach to Capital Allocation. Preprint
Kalkbrener M, Lotter H, Overbeck L (2004) Sensible and efficient capital allocation for
credit portfolios. RISK 1, 2004
Merton R (1974) On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.
The Journal of Finance 29, pp. 449-470
Tasche D (1999) Risk Contributions and Performance Measures.
http://www.ma.tum.de/stat/
Vasicek OA (1987,1991) Probability of Loss on Loan Portfolio. KMV Corporation
Hergen Frerichs and Mark Wahrenburg1
1
Chair of Banking and Finance, University of Frankfurt/Main, P.O. Box 111 932,
60054 Frankfurt/Main, Germany
Abstract: The problem how to evaluate and monitor the quality of credit risk models has recently received much attention. The discussions about the inclusion of
internal models in the Basel Capital Accord highlight this fact. Basel II does not
allow the use of full-scale credit portfolio risk models for regulatory capital calculation because regulators are concerned that model quality cannot be validated
accurately enough. However, banks are allowed to use internal credit rating systems although it is by far not clear how accurately their quality may be evaluated.
This paper discusses the current state-of-the-art concerning methods and empirical results for validating both credit portfolio risk models and internal credit rating systems. In order to allow for a meaningful assessment of the scope and limits
of model validation we closely follow and compare our results to the existing literature on validating market risk models.
JEL classification: G2, G21, G28, C52
Keywords: Credit Risk, Model Validation, Bank Regulation, Basel II
The evaluation of credit risk models is an important topic. The current discussion
about the revision of the Basel Capital Accord (Basel II) highlights this fact.
In its 1999 report “Credit Risk Modelling: Current Practices and Applications”,
the Basel Committee on Banking Supervision concludes that credit portfolio risk
models, that is, models that quantify potential losses from holding a portfolio of
risky debt, are not likely to be used in the process of setting regulatory capital requirements in the near future due to data limitations and problems in model
validation.
220
Hergen Frerichs and Mark Wahrenburg
This report decisively influenced the Basel II proposals. As a matter of fact, in
the proposal’s latest version (Basel Committee 2003), the most advanced banks
will be allowed to use own estimates of individual default probabilities, loss given
default, and exposure at default, but will not be allowed to use full-scale internal
credit portfolio risk models. Instead, bank regulators will prescribe the type of
credit portfolio risk model to be used as well as determine the level of correlation
between individual default risks.
The issues of data limitations and model validation are intimately related. Data
limitations concern three points: first, most credit instruments are not marked to
market, and comprehensive records of historical prices are not available; second,
defaults are rare events; and third, the risk horizon in credit risk is much longer
than in market risk. It follows from these points that model validation is a difficult
task. Regulatory backtesting procedures used in market risk cannot simply be
transferred to credit risk. The importance of credit risk for most commercial banks
and the illiquidity of most credit instruments make high confidence levels and
long holding periods necessary. The existence of credit cycles aggravates the data
problem as data spanning several credit cycles might be necessary for model validation.
Despite these problems in model validation, credit portfolio risk models are
widely used in the market. Not only do banks try to measure credit portfolio risk
as part of their enterprise-wide risk management, they also actively sell credit
portfolios in the market place as collateralized debt obligations (CDOs).
Prices for CDO tranches are usually based on a tranche’s external credit rating.
In order to decide on a credit rating for a CDO tranche, credit rating agencies like
Moody’s or Standard & Poor’s need to apply credit portfolio risk models. These
models will be implicitly accepted by Basel II as investors in CDOs are required
to hold regulatory capital based on external credit ratings. This shows the Basel
Committee’s ambivalence with respect to the admittance of credit portfolio risk
models.
When it comes to internal credit rating systems, the Basel Committee seems to
be more confident about their use for regulatory capital calculation. An internal
credit rating is a summary indicator of the risk inherent in an individual credit determined by the lending financial institution. Under Basel II, there will be two internal ratings-based approaches. Banks that qualify for the foundation approach
will be allowed to derive individual default probabilities of borrowers internally,
while banks that qualify for the advanced approach will also be allowed to estimate loss given default and exposure at default internally.
Bank regulators will have to evaluate the eligibility of internal credit rating system for the internal ratings-based approach. This evaluation is not trivial. In the
Basel Committee’s 2000 report “Range of Practice in Banks’ Internal Rating Systems”, only ten of thirty G10-financial institutions surveyed claim to perform
some degree of backtesting. This is all the more astonishing as only those institutions were chosen for the study that are believed to have a well-developed internal
credit rating system. Even institutions that perform some kind of backtesting were
not able or willing to provide detailed information on backtesting procedures.
Evaluating Credit Risk Models
221
The same data limitations that make it difficult to evaluate credit portfolio risk
models complicate the evaluation of internal credit rating systems. The evaluation
of rating systems is further complicated by the fact that there will be a three-year
transition period after the implementation of Basel II. At the beginning of the transition period, banks will need to have only two years of historical credit loss data.
This requirement increases by one year for each of the three transition years. Under these circumstances, an accurate model evaluation seems almost impossible.
Research on the evaluation of credit risk models has just started. None of the recent textbooks treats this subject (see for example Duffie and Singleton 2003). A
promising approach is to investigate whether successful model validation techniques used in market risk can be transferred to credit risk. Research in the validation of market risk models has considerably grown over the last decade. Advances
made in this area led the Basel Committee to allow internal market risk models for
regulatory capital calculation (Basel Committee 1996). By today, the issue of
backtesting market risk models has reached a certain degree of maturity, which
can be seen at the fact that the area’s major textbooks provide for self-contained
discussions of the subject (e.g. Jorion 2001; Dowd 2002).
(Dowd 2002) differentiates between three different kinds of backtesting methods: 1. Backtests based on the frequency of tail losses; 2. Backtests based on loss
density forecasts; and 3. Forecast evaluation approaches to backtesting. We will
use this structure to discuss the applicability of the different approaches for the
validation of credit risk models.
Section 2 and 3 are concerned with the evaluation of credit portfolio risk models. Section 2 treats traditional backtests based on the frequency of tail losses. The
regulatory backtesting of market risk models is based on this sort of backtest. We
show that this class of backtests is not sufficiently powerful in credit risk. In section 3, we discuss backtests based on loss density forecasts. We show that under
suitable assumptions this kind of backtest is powerful for the evaluation of credit
portfolio risk models. In section 4, we discuss forecast evaluation approaches to
backtesting and show an implementation of these approaches for the evaluation of
internal credit rating systems. Section 5 concludes.
Although most risk models produce a complete loss distribution at the risk horizon, the primary model output is generally seen to be the value-at-risk (VaR). The
value-at-risk is the maximum loss that a portfolio incurs with a given probability
over the prespecified risk horizon. The most intuitive backtest to test the accuracy
of the value-at-risk is based on the frequency of value-at-risk violations.
(Kupiec 1995) examines the performance of two approaches: first, a test based
on the time until the first violation; and second, a test based on the relative frequency of violations over an observation period. Both tests are based on the assumption that risk forecasts are efficient, which means that they incorporate all information known at the time of the forecast. As a consequence, risk forecasts will
222
Hergen Frerichs and Mark Wahrenburg
be independently distributed over time. Further distributional assumptions are not
needed.
In the following, we apply these tests to a typical credit risk setting. We proceed
in this way for two reasons: First, we want to show that these tests are not very
powerful in a credit risk setting. Second, knowing that this result is not surprising,
we want to obtain specific power estimates in order to be able to evaluate power
improvements, which we achieve when implementing more sophisticated backtesting procedures.
We consider a credit portfolio risk model that predicts the 99%-VaR over a oneyear risk horizon. We look at two types of commercial banks: The first type of
bank has access to historical loss data over twenty years, while the second type of
bank has access to historical loss data over only ten years. The first type of bank
might be a US commercial bank whose borrowers are either externally rated or
who is able to map its internal ratings to a rating scale of an external credit rating
agency such that it is able to draw on the external rating agency’s loss database,
which is assumed to comprise twenty years of yearly loss data (like the one of
Standard & Poor’s). The second type of bank might be a non-US commercial bank
whose borrowers are not externally rated and for whose borrowers it is also not
feasible to map internal ratings to external ones. Therefore, the bank has to resort
to its own loss database, which we assume to comprise ten years of yearly portfolio default rates (knowing that only the most advanced banks may have such long
internal default histories).
Tests are carried out using a 10%-significance level. A test’s significance level
is equivalent to its type-I error, which is the probability that a correct model will
be rejected. Usually tests use a significance level of 1% or 5%. Yet, as we face an
environment that makes it difficult to reject models at all, we are willing to take a
larger type-I error into account.
The first test (Kupiec 1995) proposes is based on the time until the first VaRviolation. Under the independence assumption, the number of observations T=t
until the first violation is a geometric random variable with parameter p. The
probability mass function is given by
P{T = t} = (1 − p ) p,
t−1
t = 1,2,...
Table 2.1. Cumulative probabilities of time until first violation
T (in years)
1
2
3
4
5
6
…
298
p*=1%
1.0%
2.0%
3.0%
3.9%
4.9%
5.9%
…
95.0%
p=2%
2.0%
4.0%
5.9%
7.8%
9.6%
11.4%
…
99.8%
p=13%
13.0%
24.3%
34.1%
42.7%
50.2%
56.6%
…
100.0%
(1)
Evaluating Credit Risk Models
223
Table 2.1. shows cumulative probabilities of the geometric distribution under the
true VaR-level of p*=1%, and two alternative VaR-levels. Based on a two-sided
test of the 99%-VaR at a significance level of 10%, a model would be rejected if it
produces its next violation within the next five years or only after 298 years. Obviously, models overestimating the VaR cannot be reasonably identified. A model
underestimating the VaR such that the probability of a violation is not 1% but 2%
equals 10% within the first five years, which is also very low. The VaR-model
would have to underestimate the VaR such that the probability of a violation
equals 13% each year in order to obtain a rejection rate of more than 50%. For this
test, it makes no difference if there are ten or twenty years of historical default
data.
What do these results mean economically? If we assume that the P/L-distribution is normal, then the 99%-VaR can be expressed as 2.33 times the distribution’s
standard deviation. If a bank underestimates the VaR and reports the 98%-VaR as
the 99%-VaR, then the reported VaR will be 2.05 times the standard deviation,
which is equivalent to an underestimation of 12%. Reporting the 87%-VaR as the
99%-VaR already leads to an underestimation of 1-1.13/2.33 = 52%.
The P/L-distribution of credit portfolios is usually strongly negatively skewed,
which might lead to even larger effects. In a two-state single-factor asset value
model, which will be described in more detail later on, with a 1% unconditional
default probability and a uniform asset correlation of 5%, the underestimations
equal 13% and 48%, respectively, which is quite similar to the normal distribution. If the asset correlation is increased to 20% the underestimations increase to
23% and 72%, respectively.
The second test (Kupiec 1995) proposes takes more information into account as
it not only relies on the number of observations until the first violation, but on the
relative frequency of violations over an observation period. Under the independence assumption, the number of violations X=x given n observations follows a binomial distribution and equals
 n
n− x
P{ X = x} =  (1 − p ) p x ,
 x
x = 1,2,...
(2)
Table 2.2. Cumulative probabilities of number of VaR-violations
Observations
in years
10
10
10
20
20
20
Violations
p*=1%
p=2%
p=17%
p=9%
0
1
2
0
1
2
90.4%
99.6%
100.0%
81.8%
98.3%
99.9%
81.7%
98.4%
99.9%
66.8%
94.0%
99.3%
15.5%
47.3%
76.6%
-
15.2%
45.2%
73.3%
224
Hergen Frerichs and Mark Wahrenburg
Table 2.2. shows cumulative probabilities of the binomial distribution under the
true VaR-level of p*=1%, and two alternative VaR-levels for default histories of
ten and twenty years.
Based on a two-sided test of the 99%-VaR at a significance level of 10%, any
model generating more than one violation would be rejected irrespective of the
length of the default history. Due to the discreteness of the binomial distribution
the actual type-I error would be very small at less than 0.5% for a ten-year history
and 1.7% for a twenty-year history. Again, models overestimating the VaR cannot
be detected. The probability that a model underestimating the VaR such that the
probability of a violation is not 1% but 2% is correctly rejected equals 1.6% with
ten years of data and 6.0% with twenty years of data. A model’s actual violation
probability would have to equal at least 17% with ten years of data and 9% with
twenty years of data, in order to be correctly rejected with a probability of more
than 50%. For the simple credit risk model introduced above (1% default probability, 5% asset correlation), these numbers correspond to VaR-underestimations of
53% and 41%, respectively.
The current regulatory market risk backtesting procedure (Basel Committee
1996) also relies on the frequency of VaR-violations. The only difference is that
regulators are not interested in risk models that overestimate the VaR. Therefore,
their objective function is more accurately reflected by a one-sided rather than a
two-sided test. With ten years of data, models with any violations will be rejected.
A model underestimating the VaR by 37% is rejected with a probability of more
than 50%. With twenty years of data, models with more than one violation will be
rejected. Here, a model needs to underestimate the VaR by 41%, in order to obtain
a rejection probability of at least 50%.
In market risk, regulatory backtesting procedures are much more powerful. With
250 observations and a 99%-VaR, a model maybe sanctioned if there are at least
five violations. In this case the type-I error equals 11%. A model underestimating
the VaR by 12% is already identified with a probability of more than 50% if returns are normally distributed. Even if returns were t-distributed with only five
degrees of freedom, the degree of underestimation necessary to identify an incorrect system with sufficient power would increase to only 18%.
As backtests based on the frequency of tail losses do not work well in the typical
credit risk environment, (Lopez and Saidenberg 2001) propose cross-sectional resampling techniques to make them more powerful. The idea is simple: Credit risk
models are not only evaluated with respect to the accuracy of VaR-forecasts over
time, but also with respect to forecast accuracy at a given point in time for simulated credit portfolios. For example, for each year of a ten-year default history,
1,000 subportfolios are randomly drawn, each comprising 20% of all borrowers.
For each subportfolio, the value-at-risk is calculated and compared with the actual
portfolio loss. If the portfolio loss is larger than the value-at-risk, a violation is recorded. With this procedure, the number of available forecasts and observed outcomes can be arbitrarily scaled up. In our example, there are now 10,000 instead
of ten observations.
(Lopez and Saidenberg 2001) propose the binomial method discussed above as
one way to perform a statistical test on the resampled data. The main drawback of
Evaluating Credit Risk Models
225
this approach is, and this was shown by (Frerichs and Löffler 2002), that resampling cross-sectionally does not change anything with respect to the amount of information present in the time series dimension. Most current credit portfolio risk
models assume that credit portfolio losses are primarily caused by the common
dependence of individual default risks on the change of a few systematic risk factors. These systematic risk factors are thought to represent the position of the
economy in the business cycle. If the economy moves into a recession, portfolio
losses will increase (possibly with a time lag). If the economy recovers, portfolio
losses will decrease. If it is additionally assumed that given the systematic factors
defaults are independent, then cross-sectional resampling does not improve data
quality in any way. In a bad year, there might be a VaR-violation for the overall
portfolio. If 1,000 subsamples are drawn for this year, then the probability is high
that there will be violations in each of the subportfolios. The same reasoning holds
for good years: if there is no violation in the overall portfolio, there probably will
not be a violation in any of the subportfolios. Consequently, randomly drawn subsamples are not independent for a given year, and standard tests do not work.
Backtests based on the frequency of VaR-violations are closest in spirit to the
VaR-concept. Yet, the last section showed that available information needs to be
used more efficiently to successfully validate credit portfolio risk models. Unfortunately, there is a trade-off that needs to be kept in mind in the following discussion (see also Dowd 2002, pp 188-9). Using additional information, for example,
on the quality of loss density forecasts for other parts of the loss distribution than
the tails, might increase the power of validation procedures, but might not increase
confidence in the risk model. The risk model might be accurate in the middle of
the distribution and therefore pass the backtest although it predicts the VaR or
large losses unsatisfactorily.
To somewhat mitigate this problem, (Dowd 2002) proposes to compare the distribution of empirical tail losses against the tail-loss distribution predicted by a
bank’s risk model. This procedure goes beyond the VaR-concept, but is still solely
concerned with that part of the P/L-distribution, which is most interesting for risk
managers. Yet, with credit loss data covering ten or twenty years and a 99%-VaR,
the probability of tail loss events is such low that the test’s power will not be significantly larger than the traditional binomial test.
(Crnkovic and Drachman 1995, 1996) propose not only to evaluate differences
between empirical and theoretical loss densities in the tails of the P/L-distribution,
but to use the entire P/L-distribution instead. Apart from the assumption that
model evaluators are actually interested in differences in densities across the complete P/L distribution, this approach is based only on the independence assumption
made in the last section and the assumption that the risk model is actually able to
forecast a complete P/L-distribution. Else, this approach is non-parametric and
does not even need a stationarity assumption.
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Hergen Frerichs and Mark Wahrenburg
The main idea is that each loss observation can be assigned to a particular quantile of the predicted P/L-distribution for the time period, in which the loss happens. For example, the cumulative predicted probability of a credit loss of ten million Euros in 1993 might equal 45% given a bank’s credit portfolio risk model for
this time period, and a loss of twenty million Euros in 1994 might have a cumulative probability of 95%. If the correct risk model is used for each past time period,
then we expect to obtain each quantile (from 1% to 100%) with equal probability.
This means that the credit loss history is transformed into a time series of iid
U(0,1)-random variables if the correct risk model is used. This property can be
tested.
There are different tests to evaluate the difference between an empirical and a
theoretical distribution. The most well-known test is the Kolmogorov-Smirnov
test, which is based on the maximum value of the absolute difference between the
two distribution functions. Crnkovic and Drachman propose the Kuiper’s test,
which is based on the sum of the maximum amount by which each distribution exceeds the other. As the Kuiper’s test weights all deviations of the empirical distribution function from the theoretical distribution function equally, they suggest to
introduce a U-shaped worry function that increases the weights of differences in
the tails of the distributions.
Unfortunately, the performance of this test procedure is reported to be low for
less than 500 observations, which obviously would make it worthless for validating credit risk models. Yet, there are two ways to improve the test’s performance:
first, there are more powerful tests than the Kuiper’s test. (Berkowitz 2001) addresses this point by suggesting an additional transformation of the loss history.
Second, further assumptions may be imposed. (Frerichs and Löffler 2003) implement the Berkowitz approach for credit portfolio risk models and show that this
approach is quite powerful for a large class of credit portfolio risk models, if it is
assumed that there is no model risk.
Like Crnkovic and Drachman, Berkowitz first proposes to transform the history
of observed losses into a series of independently uniformly distributed random
variables under the null hypothesis that the correct credit risk model is used:
xt = Fˆ ( yt ) =
∫
yt
−∞
fˆ (u )du ,
(3)
where yt are observed losses and fˆ (u ) is the forecasted probability of a loss of u.
Instead of testing on uniformity, Berkowitz suggests to apply another transformation using the inverse of the standard normal distribution function Φ(.)
zt = Φ −1 ( xt )
(4)
The resulting series of transformed observations zt will be iid N(0,1) if the risk
model is correct for each year of the observation period. Berkowitz provides for a
short proof that differences between the empirical and theoretical distribution
function are preserved by this two-step transformation. He recommends likelihood
ratio tests for testing whether zt is actually distributed iid N(0,1):
Evaluating Credit Risk Models
(
LR = −2 L(0,1,0) − L( µˆ , σˆ 2 , ρˆ
)
227
(5)
with

µ 
 zt −

2
 σ  
1 − ρ 
1
1
−
L = − log(2π ) − log 
2
2σ 2
2
2
1 − ρ 
1− ρ 2
−
T −1
log(σ 2 ) −
2
∑
2
(6)
 (zt − µ − ρzt −1 )2 


t =2 

2σ 2


T
Under the null hypothesis this test has a chi-square distribution with three degrees
of freedom. Berkowitz points to the fact that this likelihood ratio test only tests for
serial independence and for the correct specification of the first two conditional
moments. It does not test for normality itself.
(De Raaij and Raunig 2002) show that important misspecifications of market
risk models are not detected if one exclusively relies on this likelihood ratio test.
For example, variance/covariance models that falsely assume that portfolio returns
follow a conditional normal distribution will not be identified as incorrect as long
as the volatility dynamics is adequately modeled. Even worse, if the portfolio
variance is falsely assumed to be constant, this will not be detected by the likelihood ratio test as long as the unconditional variance is correct. Finally, models
based on historical simulation, that is models that use the unconditional portfolio
return distribution, will not be identified as false even if volatility is time-varying
as long as the unconditional distribution is stationary or only slowly changing.
Additional diagnostic tests are required to detect these misspecifications.
For the credit portfolio risk models we consider, these shortcomings are not crucial as we only examine alternative models that have an influence on the volatility
of the P/L-distribution.
Berkowitz reports that his testing approach is powerful in typical market risk
applications with sample sizes that are as small as 100 observations. Unfortunately, this is by far not sufficient for credit risk applications.
(Frerichs and Löffler 2003) propose to restrict one’s attention to a particular
class of credit portfolio risk models, namely, asset value models, and to implement
the Berkowitz approach in order to identify parameter misspecifications within
this model class. This proposal involves a trade-off between increased model risk
by focusing on a particular class of models, and increased power of model validation. If asset value models are inaccurate in describing reality, then the increased
power of model validation procedures is misleading. Faced with the severe data
problems that complicate the validation of credit portfolio risk models, it seems to
be a good starting point to restrict the validation problem to the class of asset
value models.
Asset value models are based on the seminal work of (Merton 1974). Correlations in credit events are modeled through latent variables that are thought of as
228
Hergen Frerichs and Mark Wahrenburg
the firm’s asset values. In the option-theoretic approach of Merton, a firm defaults
if its asset value falls below a critical threshold defined by the value of liabilities.
Asset value correlations thus translate into correlations of credit quality changes.
The asset value approach to modeling credit portfolio risk underlies the risk
weight function of Basel II as well as industry models such as CreditMetrics and
KMV PortfolioManager. It has been shown by (Finger 1998; Koyluoglu and
Hickman 1998; and Gordy 2000) that in a two-state world (default and no default),
available credit portfolio risk models like CreditRisk+, CreditMetrics, KMV PortfolioManager or CreditPortfolioView are similar in structure and produce almost
identical outputs when parameterized consistently. For this reason, there is some
confidence that the results of Frerichs and Löffler are applicable to a broad range
of credit risk models.
(Frerichs and Löffler 2003) derive their basic result for a single-factor two-state
asset value model with uniform factor loadings. The base case setup is summarized in Table 3.1.
Table 3.1. Base case setup
Parameter
Number of possible states
Recovery in case of default
Number of borrowers (N) in portfolio
Constant unconditional 1-year default probability (p)
Uniform asset correlation in true data-generating model (w2)
Asset value distribution
Serial correlation of systematic factor
Forecast horizon (years)
Length of credit loss history (T years)
Test size / Type-I error
Value
2
0
10,000
1%
5%
N(0,1)
None
1
10
10%
~
~
Asset value changes ∆Ai depend on one systematic factor Z and idiosyncratic
factors ε~i for borrowers i=1,2,…:
~
~
∆Ai = wZ + 1 − w2 ε~i ,
(7)
~ ~
where ∆Ai , Z and ε~i are iid N(0,1). A borrower defaults whenever
~
∆Ai < Φ −1 ( pi ) ,
(8)
where pi is the unconditional default probability and Φ(.) denotes the cumulative
standard normal distribution function. For a given realization of the systematic
factor Z the conditional default probability pi|Z equals
Evaluating Credit Risk Models
 −1



Φ −1 ( pi ) − wZ 

 Φ ( pi ) − wZ 
=
Φ
pi | Z = P ε i ≤




1 − w2
1 − w2 




229
(9)
Conditional on the realization of the systematic factor Z, the number of defaults is
binomially distributed. The asset correlation is equal to w2 for all pairs of borrowers.
In a simulation exercise, Frerichs and Löffler determine the power of the Berkowitz approach in detecting misspecifications of the asset correlation parameter.
For the true data-generating process, an unconditional default probability of 1%
for each obligor and a uniform asset correlation of w2=5% for all pairs of borrowers is chosen. Both values are consistent with a random effects probit analysis of
Standard & Poor’s rating data from 1982-1999. In alternative models, the asset
correlation is changed from the base case, and for each alternative model the
power of the Berkowitz approach is determined.
In the base case, there is no serial correlation neither in the true data-generating
process nor under the alternative specifications, so that the following likelihood
ratio test can be used:
(
LR = −2 L(0,1) − L( µˆ , σˆ 2
)
(10)
with
L=−
2
T  (z − µ ) 
T
T
log(2π ) − log(σ 2 ) − ∑t =1  t 2 

 2σ
2
2


(11)
The test statistic refers to the chi-square distribution with two degrees of freedom.
Simulation results are shown in Figure 3.1. The test’s power increases faster if
the asset correlation is underestimated than if it is overestimated. The power is
larger than 50% if the assumed asset correlation is below 2.5% or above 10.5%. If
the null hypothesis posits a zero asset correlation, it is rejected in 100% of all
cases. If the null hypothesis coincides with the true model, the power is slightly
higher than the nominal significance level of 10%, which is due to the small sample size.
Let us compare these results with those for backtests that are based on the frequency of tail losses reported in section 2. For once, the binomial test is not able to
detect any overestimations. And then, a power of at least 50% is only reached if
the VaR is underestimated by at least 53% with ten years of data and 41% with
twenty years of data.
With the Berkowitz approach, it is possible to detect both under- and overestimations. With ten years of data, the test’s power is larger than 50% if the VaR is
underestimated by 26%-31%, which is considerably lower than 53%. Overestimations are more difficult to detect. The power reaches more than 50% if the VaR is
overestimated by about 55%. With twenty years of data, VaR-underestimations of
20%-26% (rather than 41% with the binomial approach) and overestimations of
230
Hergen Frerichs and Mark Wahrenburg
Power
33%-38% are detected with a power of more than 50%. Compared with results for
regulatory backtesting of market risk models in the previous sections (VaRunderestimation of 12%-18% depending on distribution type to achieve power of
at least 50%), we see that misspecifications still need to be relatively large in order
to be detected. Yet, the Berkowitz approach represents a clear improvement over
the binomial approach.1
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
0%
3%
5%
8%
10%
13%
15%
18%
20%
Correlation assumption under H0
Transformed losses
Significance level (10%)
Fig. 3.1. Power of Berkowitz test in base case
Frerichs and Löffler show that this result is robust to the level of the asset correlation underlying the true data-generating process, the number of borrowers, the
choice of the significance level, and to heterogeneity and noise in default probabilities. If the asset correlation is misspecified within a multi-state model which
includes migration and recovery risk, the power is lower than in the case of twostate models which forecast only the risk of default. Frerichs and Löffler also examine whether errors referring to other assumptions of the asset value model can
be detected by the test such as the normality of the asset value distribution and the
serial independence of the systematic factor. One important result is that fat tails
in the asset value distribution, which are of particular concern to risk managers,
can be detected by the Berkowitz approach. If asset values follow a t-distribution
with ten degrees of freedom, the probability of rejecting the normal assumption is
above 90%. Finally, the authors show how to exploit information contained in the
cross-section of credit losses.
1
Remember that the regulatory test is a one-sided test. For a two-sided test the respective
underestimations would be 14%-21% instead of 12%-18%.
Evaluating Credit Risk Models
231
An important application of the Berkowitz test procedure could, for example, be
to validate the assumptions underlying Basel II. While the Basel Committee will
not allow banks to use internal credit portfolio risk models due to data and validation problems, Frerichs and Löffler show that the most advanced banks with sufficient loss records could use the Berkowitz approach to test whether the Basel assumptions are consistent with them. If not, these banks could be allowed to change
the parameters which determine capital requirements. Such a procedure would
provide for the right incentives to develop adequate loss databases and improve internal credit risk models.
(QTGECUV'XCNWCVKQP#RRTQCEJGUVQ$CEMVGUVKPI
The main idea of statistical backtesting methods covered in the last sections is that
a model is rejected if the pattern of outcomes it predicts is so different from the
pattern of observed outcomes, that the difference cannot be reasonably explained
by bad luck. We have seen that with the amount of data we typically have at hand
when evaluating credit risk models it is difficult to exclude bad luck if a model
performs badly.
(Lopez 1998) proposes forecast evaluation procedures as a complement to statistical backtesting procedures for the evaluation of VaR-models. The general idea
of these procedures is that the evaluator defines an adequate loss function and applies it to the models he wishes to evaluate. If losses are larger than a predefined
barrier, then a model is judged to be unsatisfactory. For example, (Lopez 1998)
proposes simply to add the number of VaR-violations over an observation period.
The higher the number, the higher is the “loss” for the regulator. Yet, simply
counting the number of violations does not give any hint about the barrier, which
separates satisfactory from unsatisfactory models. (Lopez 1998, Blanco and Ihle
1998, and Dowd 2002) propose other loss functions that additionally depend on
the size of the violation.
(Lopez 1999) introduces the quadratic probability score (QPS), also called Brier
score, as an example for a regulatory loss function. The QPS for a model m over a
sample of size T is defined as
QPSm =
2 T
∑ ( Pt m − Rt+1 ) 2 ,
T t =1
(12)
where Pt m is the probability forecast of model m in t for an event in period t+1,
and Rt+1 is the indicator variable for the event occurring in t+1. Smaller values of
the QPS indicate more accurate forecasts. The optimal QPS of zero is obtained if
the event is correctly forecasted with certainty.
The QPS rewards both calibration and sharpness. A perfectly calibrated system,
which awards the correct average probability forecast to each observation, will
have a lower QPS than any other system, which awards an incorrect average probability forecast to all observations. A perfectly calibrated system with average
232
Hergen Frerichs and Mark Wahrenburg
probability forecasts close to zero or one, thus exhibiting a high degree of sharpness, will receive a lower QPS than another perfectly calibrated system with average probability forecasts close to one half (Winkler 1994). Another important
property of the QPS is that it belongs to the class of strictly proper scoring rules,
which means that the expected QPS given a risk model m is always lower if the
bank reports its probability forecasts truthfully. Formally,
E QPS Pt m | m < E (QPS (Pt ) | m)
(
( ) )
∀ Pt ≠ Pt m
(13)
The QPS is easy to apply if two systems are to be compared based on the same
data. The system with the lower QPS is the better one, which can be tested using a
statistical test of equality of two QPS values (Bloch 1990).
It is more difficult to compare QPS values that are calculated with respect to different datasets. In this case, the scores will not only depend on forecasting ability,
but also on the nature of the forecasting situation. The influence of the nature of
the forecasting situation can be easily seen at an extreme example: if an event has
zero probability of occurring, then a system, which simply always predicts a zero
probability, will receive a lower QPS than a system, which does the same in an
environment in which event probabilities are higher. So-called (symmetrical or
asymmetrical) skill scores were developed to neutralize the influence of the forecasting situation on the score. Yet, these skill scores might raise other concerns.
They might not be strictly proper any more (Winkler 1994), or might be more difficult to interpret economically (Frerichs and Wahrenburg 2003).
(Lopez 1999) examines the QPS for three different kinds of probability forecasts: first, the probability of a VaR-violation in the next period; second, the probability of a portfolio loss in the next period exceeding a fixed percentage amount
of initial portfolio value; and third, the probability that aggregated portfolio losses
over a given time period exceed the amount of economic capital. Simulation results are that the QPS is not better at identifying inferior value-at-risk models
compared with statistical approaches discussed in earlier sections. Lopez concludes that the QPS represents a useful complement to statistical methods because
it makes it possible to directly incorporate regulatory loss functions into model
evaluations.
While the QPS seems to be of limited use for the evaluation of value-at-risk
models, (Frerichs and Wahrenburg 2003) show that it might represent a powerful
tool in the evaluation of internal credit rating systems. As the output of internal
credit rating systems are default probability forecasts, a scoring rule measuring errors in these probability forecasts is obviously useful.
Simply define Pi m as the default probability forecast for borrower i=1, 2, …, n
in t for period t+1 and Ri as the indicator variable for a default in t+1, and we obtain
QPSm =
2 n
∑ ( Pi m − Ri )2
n i =1
(14)
Evaluating Credit Risk Models
233
The QPS is a particularly appropriate measure to evaluate the quality of internal
credit rating systems as it is closely aligned with the regulator’s objective. Regulatory capital requirements directly depend on default probability estimates, and
therefore the error in default probability estimates ought to be the primary measure
for system validation purposes.
(Frerichs and Wahrenburg 2003) evaluate the quality of the QPS in a simulation
study based on a large Deutsche Bundesbank dataset containing annual accounts
and default data of about 24,000 medium-sized and large German companies in
the time period 1994-99. The primary research question is whether it is possible to
identify low-quality internal credit rating systems based on quantitative measures.
The empirical approach proceeds as follows: First, the size of credit portfolios
and the level of portfolio default rates are defined. There are four different credit
portfolio sizes and three different levels of portfolio default rate. The grid of resulting in-sample (1994-98) and out-of-time (1999) defaults is summarized in Table 4.1. For each bank size / portfolio default rate combination, 1,000 random
credit portfolio compositions are drawn from the database.
Table 4.1. Overview of number of defaults resulting from bank size / portfolio default rate
combinations
0.85% (low)
Portfolio default rate
1.70% (medium)
1,875 (small)
16
In-sample
32
3,750 (medium I)
32
64
128
7,500 (medium II)
64
128
255
15,000 (large)
128
255
512
Bank size (# observations)
3.40% (high)
64
Out-of-time
375 (small)
3
6
13
750 (medium I)
6
13
26
1,500 (medium II)
13
26
51
3,000 (large)
26
51
102
Second, for each randomly drawn credit portfolio, banks have the choice between
six kinds of internal credit rating systems (ordered in presumed ascending quality): the trivial system, the optimized Altman system, the Z-score system, the
stepwise system, the benchmark variables system, and the pooled system. The
characteristics of these systems are summarized in Table 4.2.
Third, based on in-sample credit scores, each bank designs an internal credit rating system with eight rating classes of equal size. Rating class default probability
estimates are set equal to the average in-sample rating class default rate, which is
in accordance with current practice of most banks. Given this rating system, each
234
Hergen Frerichs and Mark Wahrenburg
borrower in the 1999 out-of-time sample receives a rating and a default probability
estimate.
Finally, the out-of-time QPS is calculated for each credit portfolio.
Given the out-of-time QPS values for all simulated credit portfolios, (Frerichs
and Wahrenburg 2003) propose two different procedures to identify inferior internal credit rating systems. In the first procedure, systems are classified as inferior if
their QPS is worse than a predefined threshold. Banks submit their QPS statistics
to the bank regulator, who in turn evaluates the system. The regulator does not
have to publish neither the threshold nor any information about the way thresholds
are derived. The second procedure is more complex. The QPS needs to be calculated for the bank’s own system and the regulator’s benchmark system both based
on the bank’s credit portfolio. Bank regulators set a lower threshold on the p-value
of the test of equality of the two statistics such that all banks whose system performs worse than the benchmark system and whose p-value falls below the threshold are classified as inferior systems.
Table 4.2. Overview rating system types
Name
Trivial
Optimized
Altman
Z-score
Stepwise
Benchmark
variables
Pooled
Description
Bank randomly draws one financial variable from set of 49 variables, and
derives optimal logistic credit scoring function based on its own data.
Bank takes financial variables of Altman’s Z’’-score calibrated on US data, and derives optimal logistic credit scoring function based on its own
data.
Bank applies logistic credit scoring function derived on a sample of the
39 largest defaulters in the Deutsche Bundesbank database (revenues > 50
million Euros) and 39 randomly drawn non-defaulters of the same size.
No reference to bank’s own data.
Bank selects financial variables by logistic stepwise selection procedure,
and derives optimal logistic credit scoring function based on its own data.
Bank uses a set of six financial variables that work well for the complete
dataset, and derives optimal logistic credit scoring function based on its
own data.
Logistic credit scoring function derived on the complete learning sample.
Serves as benchmark function to evaluate all other systems.
As the second procedure is more difficult to put into practice, the first procedure
would be preferred if both procedures are equally powerful. As Tables 4.3. and
4.4. show, this is actually the case.
Evaluating Credit Risk Models
235
Table 4.3. Relative frequencies of identifying inferior systems using critical QPS thresholds (in %)
# out-oftime
defaults
3
6
13
26
51
102
Trivial
Optimized
Altman
Z-score
Stepwise
Benchmark
variables
26
47
77
93
99
100
18
22
33
54
85
100
13
14
22
40
72
100
24
23
26
17
12
9
14
14
14
14
15
23
The table shows the relative frequency that the QPS of a given system is higher than the
90%-quantile of the QPS distribution of the pooled system depending on portfolio default
rate and portfolio size (Thresholds: Low default rate: 0.848-0.842-0.837-0.833 (from small
to large bank size), Medium: 1.664-1.646-1.634-1.626, High: 3.18-3.14-3.11-3.09).
In Table 4.3., results are shown for the first procedure. For each randomly drawn
credit portfolio, the QPS of each of the alternative rating systems is compared with
the 90%-quantile of the QPS distribution of the pooled system, which serves as a
benchmark. The table shows the relative frequency that the QPS of a given system
type is higher than this quantile. The trivial system is identified as inferior with a
probability of at least 50% if there are at least thirteen out-of-time defaults. This
means either that the credit portfolio is relatively large, or that the portfolio default
rate is not too small (cf. Table 4.2.). The optimized Altman system is identified
with high probability only if the credit portfolio is large and the portfolio default
rate is high. The other systems are not identified as inferior with sufficient power.
The overall result is that strongly inferior internal credit rating systems are identified based on the QPS if the number of out-of-time defaults is not too small.
Results for the second procedure are shown in Table 4.4. To make results from
Table 4.3. and 4.4. comparable, p-value thresholds in Table 4.4. are set such that
the power of the test for the Altman optimized system is roughly equal to the one
displayed in Table 4.3. It can be seen that the performance of both procedures is
rather similar. Differences of some percentage points may be due to simulation
noise.
(Frerichs and Wahrenburg 2003) show that forecast evaluation approaches, specifically the quadratic probability score for the evaluation of internal credit rating
systems, can be implemented in a way such that statistical statements can be
made. The principal difference to statistical backtesting procedures presented in
former sections is that statistical statements are made with respect to a predefined
benchmark model.
236
Hergen Frerichs and Mark Wahrenburg
Table 4.4. Relative frequencies of identifying inferior systems using critical p-values
(in %)
# out-oftime
defaults
3
6
13
26
51
102
Trivial
Optimized
Altman
Z-score
Stepwise
Benchmark
variables
27
38
61
88
97
100
18
22
33
54
78
99
13
15
27
50
76
98
19
20
25
23
12
2
12
10
12
17
19
14
The table shows the relative frequency that a p-value is smaller than a threshold value. The
p-value results from a test that the QPS of a bank’s rating system is equal to that of the
pooled system both based on a bank’s own dataset. Threshold values are set depending on
the number of out-of-time defaults such that the test’s power for the Altman optimized
system is equal to the one reported in Table 4.3.
%QPENWUKQP
Due to difficulties in validating credit portfolio risk models, bank regulators will
not allow banks to use internal credit portfolio risk models for regulatory capital
calculation. In this article, the performance of various backtesting approaches is
compared. Backtests based on the frequency of tail losses are not sufficiently
powerful due to the very limited number of historical observations available in
credit risk. Backtests based on loss density forecasts and restricted to a particular
class of credit portfolio risk models, namely asset value models, are much more
powerful, although they still perform worse than the regulatory backtest for internal market risk models. In our opinion, bank regulators could still allow the use of
these tests for the most advanced banks to provide for the right incentives to develop better databases and credit risk models. With respect to internal credit rating
systems, bank regulators seem to have more confidence in evaluation procedures.
We show that the quantitative evaluation of internal credit rating systems is by far
not trivial. Only strongly inferior systems are identified with sufficient power. In
the future, the power of evaluation procedures for credit risk models will increase
as loss histories become longer and credit instruments become more liquid.
Evaluating Credit Risk Models
237
4GHGTGPEG
Basel Committee on Banking Supervision (1996) Amendment to the Capital Accord to incorporate market risks. Basel
Basel Committee on Banking Supervision (1999) Credit risk modeling: current practices
and applications. Basel Committee Publication No. 49, Basel
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Stefan Huschens, Konstantin Vogl, and Robert Wania1
1
Technische Universität Dresden, Department of Business Management and Economics, Germany
Abstract: This paper provides estimators for the default probability and default
correlation for a portfolio of obligors. Analogously to rating classes, homogeneous groups of obligors are considered. The estimations are made in a general
Bernoulli mixture model with a minimum of assumptions and in a single-factor
model. The first case is treated with linear distribution-free estimators and the
second case with the maximum-likelihood method. All problems are viewed from
different points of origin to address a variety of practical questions.
The estimation of default probabilities has become more and more popular since
credit derivatives, structured products, and last but not least banking supervision
make it necessary to use sound estimates. Moreover credit risk models are gaining
popularity in the light of the Basel II accord, where the foundations of the future
banking supervision are discussed.
The pioneers of estimation of default probabilities are rating agencies like
Moody’s and S&P. They started to publish not only the ratings of companies but
also their estimated default probabilities. These estimates are produced from historical data.1
Due to the limitations of historical data, models were developed which should
describe real world activities via assumptions on explaining variables. The estimates are produced within the context of the underlying model. Threshold models
or latent-factor models can be named as examples.
Furthermore, there are scoring systems used in banks to describe their large
portfolios of usually non-rated obligors. Quantitative scoring systems use a set of
describing variables such as data from balance sheets and account management for
corporates or income, age etc. for retail clients to produce an individual score
which is then mapped to a rating scale. Logit- or Probit-models are used to assign
1
For further reading on the estimation methods of the rating agencies see Carty (1997),
Kavvathas (2000).
240
Stefan Huschens, Konstantin Vogl, and Robert Wania
default probabilities to individuals via their estimated scores.2 In the context of
credit portfolio analysis usually rating classes are used. Therefore in this contribution the problem is addressed how to assign default probabilities to rating classes
rather than to individuals.
Additional to the default probability the default correlation has to be estimated
as input parameter for credit risk models. Large banks already use credit risk
models to manage their activities in lending and trading. Rather than monitoring
each individual obligor, these models permit a portfolio view because they produce a probability distribution of overall losses. Popular names of credit risk
models3 are Credit-Metrics (JP Morgan), CreditPortfolioView (McKinsey), PortfolioManager (KMV) and CreditRisk+ (CreditSuisse). In these models, the loss
distribution is very sensitive for correlation.
The topic of this article is to provide estimators for the default probability and
default correlation from default data. This is accomplished through a distributionfree setting as well as through discussion of the maximum-likelihood method in
the case of a single-factor model. Therefore some modelling assumptions have to
be introduced. For estimation purposes it is sufficient to consider groups of obligors rather than every individual. Rating classes with homogeneous groups of obligors are therefore the focus of the following sections.
This section shows the way from the raw data to the default probabilities. In the
following, n obligors are considered. For simplicity, each obligor should only
have one liability.
First in section 2.1, a homogeneous group is considered, where all obligors have
the same default probability. This can be understood as picking all obligors with
the same rating from a portfolio. In section 2.2 this approach is extended to several periods which raises the new problem of weighting the data of different time
periods. In section 2.3 the estimation problem is discussed in the context of several homogeneous groups, where the information on correlation between groups is
used for the estimation of default probabilities.
To estimate the one-year default probability historical data is used. The i-th obligor is characterized by a Bernoulli distributed random variable Yi. The outcome of
the default variable Yi is 1 if the i-th obligor defaults and 0 if not. The default
probability is denoted by
2
3
See e.g. Thomas et al. (2002).
For a brief overview of current credit risk models see Crouhy et al. (2000), Gordy (2000).
Estimation of Default Probabilities and Default Correlations
241
π = Pr (Yi = 1) .
If all obligors or more precisely all default variables are stochastically independent, then one can use the number of defaulted obligors divided by the number of
all obligors as an estimator for the unknown default probability. This is referred
to as relative default frequency which is denoted by
F=
n
H
with H = Yi .
n
i =1
∑
In the single-period case the relative default frequency represents the linear minimum-variance estimator as well as the maximum-likelihood estimator. This follows from the maximum-likelihood estimation where the likelihood-function
n
L(π ) = ∏ π y i (1 − π )1− y i = π h (1 − π )n − h
i =1
of the realizations yi of Yi is maximized with respect to π. Thereby, the number of
observed defaults is denominated by h. Since the sum of independent and identically distributed Bernoulli variables is a binomially distributed variable, the variance of F is measured by
Var(F ) =
π(1 − π )
.
n
In this case F is an unbiased and consistent estimator.
Following the scientific discussions there is a wide consensus that the case of independence is not representative for credit risk purposes. For estimation of default probabilities one has to take into account that dependence affects the estimation. For example, a high relative default frequency can result either from a high
default probability with independence or from a low default probability with a
high correlation between the default variables.
Relaxing the independence condition the next step is to consider the case of
equicorrelation. All default variables are treated as pairwise correlated with the
same positive correlation r. Using the relative default frequency as an estimator in
the dependent case, the variance of the estimator is4
Var( F ) =
π (1 − π ) n − 1
+
r π(1 − π ) .
n
n
The asymptotic behaviour is more unpleasant since
4
See e.g. Huschens and Locarek-Junge (2002, p. 108).
(1)
242
Stefan Huschens, Konstantin Vogl, and Robert Wania
lim Var(F ) = r π(1 − π )
n→ ∞
is not zero for positive correlation. Therefore the relative default frequency is not
a consistent estimator of the default probability. Nevertheless, F is the best linear
unbiased estimator in the single-period case. The correlation r is likely to be positive, since r fulfils the lower boundary condition
r≥−
1
n −1
and ensures the positivity of (1). Asymptotically this lower boundary converges
to zero, hence a negative equicorrelation in arbitrarily large portfolios is not possible.
Fig. 1. Quantiles of the asymptotical distribution of the relative default frequency for an asset correlation of 10%. Source: Höse and Huschens (2003a)
Working in the single-factor model5 from Basel II which is described in the next
section, there are probability intervals for the default frequency. We cannot expect that the default frequencies are very close to the default probabilities even for
5
These models are also called latent-factor models.
Estimation of Default Probabilities and Default Correlations
243
very large numbers of obligors. Figure 1 shows the asymptotical probability intervals for a given asset correlation6 ρ of 10%. In the context of the single-factor
model the default correlation depends on the asset correlation as well as on the default probability.7 One can see clearly which realization of the relative default frequency is compliant with certain values of the default probability.
Starting with the estimation of default probabilities, the case of a given correlation
is assumed. This is reasonable for portfolios where the correlation has already
been estimated8 for branches or rating classes and assumed constant over time or
where the correlation is given otherwise like in the Basel II accord,9 which postulates a correlation structure for the Internal Ratings-Based Approach via a functional relation between default probability and asset correlation. This relation can
be used to reduce the dimension of the estimation problem from two unknown parameters to only one.
The core point of the Internal Ratings-Based Approach is a single-factor model10
for the continuous solvency variable
Bi = ρ Z + 1 − ρ U i
(2)
for individual i. The systematic factor Z is assumed to be standard normally distributed and influences all individuals in the same way. The idiosyncratic factor
Ui does only affect the solvency variable of individual i. It is also standard normally distributed and Z,U1,...,Un are independent. How much the systematic factor affects the solvency variable is controlled by ρ. In terms of the Merton-model
framework, Bi can be thought as the asset value of company i.
To model the default variable, a threshold is used. If the solvency variable is
less than the threshold, the individual will not be able to meet its obligations. For
6
The asset correlation ρ describes the dependence between two solvency variables of individuals, whereas the default correlation r describes the dependence between the default
variables of two individuals. Generally, the default correlation is lower than the asset
correlation. One can build the connection between these correlations within the Basel II
single-factor model, see Gersbach and Lipponer (2000) or Höse and Huschens (2003c).
7 The impact of the asset correlation on the default correlation is demonstrated in Erlenmaier and Gersbach (2002), Gersbach and Lipponer (2000). The impact of the asset correlation on the estimation of default probabilities is discussed in Höse and Huschens
(2003a), Höse and Huschens (2003b).
8 E.g. the CreditMetrics model estimates correlation from historical data.
9 For the New Basel Capital Accord see Basel Committee on Banking Supervision (2003).
10 For discussion of the main topics of the Basel II model see also Schönbucher (2000),
Huschens and Vogl (2002).
244
Stefan Huschens, Konstantin Vogl, and Robert Wania
companies the threshold may be the total amount of debt and a default event occurs if the company's asset value is less than the threshold.
This means the outcome of the default variable Yi is 1 if the solvency variable Bi
is less than the threshold Φ-1(π) and 0 otherwise, where Φ-1 denotes the inverse of
the standard normal distribution function Φ. The threshold Φ-1(π) ensures that the
default probability is π.
The default variables Yi are identically distributed but dependent with the pairwise default correlation r which is determined by the asset correlation ρ.11 To employ the maximum-likelihood estimation method it is useful to apply the concept
of conditional independence since for a given realization z of the systematic factor
Z all default variables are independent and the conditional probability can then be
written as
 Φ −1 (π ) − ρ z 
.
p(z; π, ρ ) := Pr (Yi = 1 | Z = z ) = Φ


−
1
ρ


(3)
On condition that Z realizes as z, the default variables are independent Bernoulli
variables with parameter p(z; π,ρ).
Recalling that Z has a standard normal distribution with density function φ, one
derives the likelihood-function of the default probability as a mixture of the Bernoulli distribution with a standard normal distribution
L(π ) =
∞
h
n−h
∫ p(z; π, ρ ) [1 − p(z; π, ρ )] φ (z )dz .
(4)
−∞
It should be remarked that for a given correlation the relative default frequency is
not a maximum-likelihood estimator for the default probability.12 In the special
case of the Basel II model the correlation is given as deterministic function of the
unknown default probability.13 This function shall be denoted as ρ(π). The problem of maximum-likelihood estimation reduces to maximizing (4) for π, where
p(z; π, ρ) is replaced by
 Φ −1 (π ) − ρ (π )z 
.
p ( z ; π ) = Φ


1
ρ
(
π
)
−


(5)
There is no analytical solution to this problem but standard statistical software
should be able to deal with this function numerically.14 This provides a maximum11
For the formal connection between r and ρ see (18) and (19).
For further details see Höse and Huschens (2003c).
13
For details on the correlation function see (Basel Committee on Banking Supervision,
2003, paragraph 241, 242, 252) for corporates, sovereign and bank exposures and (Basel
Committee on Banking Supervision, 2003, paragraph 298, 299, 301) for retail exposures.
14 E.g. using the software package GAUSS the applications MAXLIK and FASTMAX are
able to solve this maximization problem.
12
Estimation of Default Probabilities and Default Correlations
245
likelihood estimator for the default probability from data of one period which is
compliant with the Basel II accord.
Example: To illustrate the maximum-likelihood estimation a homogeneous
portfolio of 1000 obligors is assumed. The number of observed defaults in a time
period is 10 in the first and 100 in the second case. Thus, the relative default frequencies for these two cases are 1% and 10%. Applying the maximum-likelihood
estimation in the Basel II model with correlation function ρ(π) means to maximize
the likelihood-function with respect to π. Figure 2 shows scaled likelihoodfunctions for both cases, where the likelihood-function is divided by the likelihood
of the relative default frequency. Within the figure the dotted lines indicate the
relative default frequencies, whereas the dashed lines show the maximum of the
likelihood-function. The maximum-likelihood estimates are 1.88% for the default
probability in the first case and 11.48% in the second.15 More generally for different numbers of defaults one finds out that correlation drives the estimates of default probabilities from the default frequencies towards 50%.
Fig. 2. Scaled likelihood-functions L(π) / L(h/n) for h = 10 and h = 100 with n = 1000
15
For a graphical illustration of the likelihood-function (4) for different values of ρ see
Höse and Huschens (2003c).
246
Stefan Huschens, Konstantin Vogl, and Robert Wania
/WNVK2GTKQF%CUG
Given the availability of data, it is useful to extend the estimation over periods of
time to improve the quality of estimation. For several time periods the estimation
method has to allow for the correlation of the default variables to achieve sound
estimates for the default probability.
/QFGN(TGG#RRTQCEJ
It is assumed that the expectation of each default variable is π and the correlation
of two different default variables is r. Further, it is assumed that the default variables are independent between time periods and that the default probability π as
well as the default correlation r are constant over time. The different variables are
indexed by t = 1,…,T to indicate the different time periods.
For given relative default frequencies F1,…,FT an unbiased estimator for π is
T
∑ wt Ft
t =1
provided that the sum of the weights wt equals one. The following two cases are
intuitive, but do not take into account the correlation structure:
(a) the relative default frequencies for T periods are weighted equally,16
1
,
T
wt =
(b) all default variables are weighted equally,17
wt =
nt
T
n
s =1 s
∑
.
The best linear unbiased estimator for a given correlation r (Aitken-estimator)
leads to the following weights18
(c)
wt =
T

nt
ns


∑
1 + (nt − 1) r  s =1 1 + (n s − 1) r 
−1
.
In the special case of n1 =…= nT the weights in (b) and (c) reduce to case (a).
This is also the limiting case for (c) if nt approaches infinity for all t. A closer
look at the weighting factor (c) reveals that in the independence case (r = 0) one
16
See Bluhm et al. (2003, p. 23).
See also de Servigny and Renault (2003).
18 This can be shown via a Lagrange approach. See Vogl and Wania (2003) or Pafka and
Kondor (2003, p. 3, eq. 1).
17
Estimation of Default Probabilities and Default Correlations
247
obtains the weighting factor (b). On the other hand in the dependence case with r
= 1 one obtains (a) as weight for each Ft. This is reasonable since within every period there is only one useful observation because the high correlation forces all
observations to be the same.
5KPING(CEVQT/QFGN
Due to the distributional assumptions underlying the single-factor model, it is possible to derive the maximum-likelihood estimator additionally to the distributionfree estimator already given by (c).
Since the independence of time periods is assumed, the likelihood-function is
simply represented by the product of the likelihood-functions of all periods. Referring to equation (4) one obtains
L(π ) =
T
∞
∏ ∫ p(zt ; π, ρ )h [1 − p(zt ; π, ρ )]n −h φ (zt )dzt
t
t
t
(6)
t =1 −∞
as likelihood-function over time periods. Thereby, ht is the number of defaults observed in period t.
Substituting the correlation by the Basel II correlation function ρ(π) one can derive the maximum-likelihood estimator for the default probability by numerical
solution of the maximization problem.
/WNVK)TQWR%CUG
In this section K homogeneous groups of obligors are considered. This means that
all obligors of each group k = 1,…,K are assumed to have the same default probability πk. Furthermore, the homogeneity within the groups of obligors causes a
correlation structure where the correlation between any two different default variables of group k and l is equal and denoted by rkl.
/QFGN(TGG#RRTQCEJ
The derivation of the minimum variance Aitken-estimator which is presented next
uses no other assumptions than the homogeneity just mentioned. The Aitkenestimator is proposed as a favourable multi-period estimator for default probabilities of several homogeneous groups of obligors.19
The column vector of the K default probabilities is denoted by π, and the covariance matrix of the relative default frequencies in period t is denoted by Vt. The
covariance-matrix Vt has off-diagonal elements
rkl π k (1 − π k ) π l (1 − π l )
19
For a more detailed description see Vogl and Wania (2003).
(7)
248
Stefan Huschens, Konstantin Vogl, and Robert Wania
and diagonal elements
π k (1 − π k ) ntk − 1
+
rkk π k (1 − π k ) ,
ntk
ntk
(8)
analogously to (1), where ntk denotes the number of obligors in group k during period t. The best linear unbiased estimator for the vector π is the Aitken-estimator
 T −1 
 V 
∑ t 
 t =1

−1
T
∑ Vt−1 Ft ,
(9)
t =1
where Ft is the column vector of the K relative default frequencies Ft1,…,FtK for
time t.
The matrices Vt in (9) depend on the unknown variances πk(1–πk) of the default
variables. Therefore the optimal weights of the relative default frequencies are not
known beforehand. It is necessary to apply a two-step method analogously to the
two-step-least-squares (2SLS) estimation known from econometric theory.20 In the
first step πk is estimated by
1 T
∑ Ftk
T t =1
neglecting the dependency structure. Using these estimates, covariance matrices
according to (7) and (8) can be computed with knowledge of the default correlations. The second step uses the estimated covariance matrices instead of the true
but unknown covariance matrices Vt to compute the Aitken-estimator in (9).
Equation (9) also represents an important theoretical result since the following
conclusions can be drawn immediately.
• For a single period t, the Aitken-estimator for π is simply Ft. This is remarkable, since it means that even knowing the correlation between the default variables the relative default frequency is the best linear unbiased estimator.
• For a single homogeneous group k, the weights of the estimator (9) simplify
to case (c) in section 2.2.
• If the numbers of obligors in each group k are constant over time, then the
covariance matrices Vt are equal and the weights of the estimator reduce to
1/T which corresponds to case (a) of section 2.2.
• If for all t the number of obligors is very large in all groups, then the covariance-matrices Vt are approximately equal and the weights of the estimator
reduce to 1/T which corresponds to case (a) of section 2.2.
Remark: Though the last point proposes the weights 1/T for arbitrarily large
groups, it is reasonable to apply the weights of (c) to each group instead. This is
justified, since in the asymptotic case the estimator for π k only depends on the
20
See Judge et al. (1985, pp. 596).
Estimation of Default Probabilities and Default Correlations
249
relative default frequencies Ftk and not on the relative default frequencies of other
groups. Therefore, the different groups can be treated separately. The weights of
(c) were proven as weights of the minimum-variance estimator of a single group
and hence are the better choice.
5KPING(CEVQT/QFGN
Staying in the framework of the previous sections the single-factor model is assumed and the index k is added to denote the group of obligors,
Btki = ρ k Z t + 1 − ρ k U tki
for i = 1, K , ntk .
(10)
Assuming that all Zt and Utki are independent and standard normally distributed it
follows that all solvency variables Btki are conditionally independent given the
joint factor Zt. One can see that equation (6) expands to
L(π ) =
T
∞ K
∏ ∫ ∏ p(z t ; π k , ρ k )h [1 − p(z t ; π k , ρ k )]n
tk
t =1 −∞ k =1
tk − htk
φ (z t )dz t
for the case of K homogeneous groups where ρk denotes the asset correlation
within group k. The estimate for the vector of default probabilities is found by the
maximization of the likelihood-function with respect to π.
'UVKOCVKQPQH&GHCWNV%QTTGNCVKQP
%QPEGRVUQH&GRGPFGPV&GHCWNVU
In scientific discussions many terms are used to characterize dependency, e.g. default correlation, default covariance, conditional default probability and simultaneous default probability. All of these terms express aspects of dependence between two variables.
In the framework of Bernoulli distributed default variables with known default
probability21 these concepts are equivalent as shown in the following. Consider
two obligors with the same default probability π and default correlation r. Firstly,
the simultaneous default probability
Pr (Y1 = 1, Y2 = 1) = E(Y1Y2 )
is the probability that two obligors default at the same time. This equals the expectation of the product of the Bernoulli variables which is π² in the case of independence. Further, the conditional default probability is written as
21
E.g. provided by a rating agency or a scoring system.
250
Stefan Huschens, Konstantin Vogl, and Robert Wania
Pr (Y1 = 1 | Y2 = 1) =
Pr (Y1 = 1, Y2 = 1)
.
π
This is the probability for the default of the obligor i given the fact that obligor j
has defaulted. The default covariance can also be expressed in terms of the simultaneous default probability since
Cov(Y1 , Y2 ) = Pr (Y1 = 1, Y2 = 1) − π 2 .
The default correlation
r=
Cov(Y1 , Y2 )
Var(Y1 ) Var(Y2 )
=
Pr (Y1 = 1, Y2 = 1) − π 2
π(1 − π )
is the default covariance scaled by the variances, and thus this term is also determined by the simultaneous default probability. It can easily be seen that, given the
default probabilities, these four aspects of dependence can be computed if only
one of them is known.
To illustrate the order of magnitude, the default probability of π = 1% is assumed. Further, let the default correlation between the default variables of the obligors be r = 2%. Then the simultaneous default probability of these two obligors
is given by22
Pr (Y1 = 1, Y2 = 1) = r π(1 − π ) + π 2 = 0.0298% .
To find out the effect of this value it is useful to determine the conditional default
probability
Pr (Y1 = 1 | Y2 = 1) = 2.98% .
This means knowing that one of two obligors has defaulted triples the default
probability of the other. Thinking in terms of Moody's and S&P ratings this could
equal a downgrade of two rating classes just by knowing a correlated obligor has
defaulted. Concluding, it is emphasized that correlations of the same magnitude
as the default probabilities have to be regarded as high correlations due to their effects discussed just above.
'UVKOCVKQPKPC)GPGTCN$GTPQWNNK/KZVWTG/QFGN
The Bernoulli mixture model23 is presented in the following. It will be shown that
the single-factor model is a special case of the Bernoulli mixture model.
22
Usually the knowledge of the marginal distribution and the correlation is not sufficient to
derive the joint distribution. However, in the case of two Bernoulli variables one can
solve this problem uniquely.
23 For detailed information to the Bernoulli mixture model see Joe (1997, pp. 211).
Estimation of Default Probabilities and Default Correlations
251
5KPING)TQWR%CUG
In period t, the default variables of a specific homogeneous group of nt obligors
are Bernoulli variables which are modeled here with a stochastic Bernoulli parameter Pt. Conditionally on a realization pt of Pt the default variables are assumed to be independent and to have the (conditional) default probability pt.
Thus, dependency of defaults is modeled by the fact that all obligors of the group
share a common default probability in each period t.
Drawing a specific default probability reflects that a certain state of the world
has been realized. For instance, if the economic cycle is in a recession state this
affects the default probability of all individuals. Usually this has a negative effect
which results in a higher default probability for all individuals. This describes the
impact of correlation. Given this default probability, all individuals are even
likely to default independently. This is modeled by the conditionally Bernoulli
distributed default variables
Yti | Pt = pt ~ Ber( pt ) , for i = 1, K , nt ,
where Pt is the stochastic default probability in period t. The random variables Pt
for t = 1,…,T are assumed to be stochastically independent and identically distributed. The independence over time of the random default probabilities induces the
independence of default variables of different time periods.
It follows that the expectation of the stochastic default probability Pt equals the
expectation of Yti which is π. The variance of the stochastic default probability Pt
is denominated by γ. As a further consequence of the Bernoulli mixture model γ
is equal to the default covariance rπ(1–π) which is the covariance between the default variables of two different obligors.
In the following the problem of estimation of the default correlation r is considered for a given default probability π. In this setting the task is equivalent to the
estimation of γ. The latter can be estimated by
g=
1
T
T
∑ (Ft − π )2 ,
t =1
which is asymptotically unbiased if all numbers nt tend to infinity. The expectation of g is
E(g ) = γ + [π(1 − π ) − γ ]ε ,
where ε is defined by
ε=
1
T
T
1
∑ nt
t =1
While rearranging (11), it is easy to derive
.
(11)
252
Stefan Huschens, Konstantin Vogl, and Robert Wania
g − [π(1 − π )]ε
1− ε
(12)
as unbiased estimator for γ. However, (12) can lead to negative estimates for the
positive parameter γ.
The model is expanded for the case of K homogeneous groups. As before, the index k = 1,…,K is attached to the variables indicating the group of obligors a variable belongs to. In the expanded Bernoulli mixture model, the conditional independence of the default variables is assumed according to
i.i.d .
Ytki | Pt = p t ~ Ber( ptk )
(13)
where Pt denotes the vector of the stochastic default probabilities Pt1,…,PtK in period t. Within this setting the vector π of default probabilities satisfies
E(Pt ) = π .
(14)
The variance-covariance matrix of Pt is denoted by
Var(Pt ) = Γ
(15)
with elements γkl. It follows from (13) that
γ kl = rkl π k (1 − π k ) π l (1 − π l )
(16)
holds. This means that γkl represents, on the one hand, the covariance between the
stochastic default probabilities Ptk and Ptl but it also represents, on the other hand,
the covariance between two different default variables (one in group k, the other in
group l).
For a given vector π of default probabilities the covariance matrix Γ can be estimated by the matrix
G=
1 T
∑ (Ft − π)(Ft − π)′
T t =1
for multi-period data, where Ft denotes the vector of the relative default frequencies in period t. This is an unbiased estimator for the off-diagonal elements since
E(G ) = Γ + D ,
(17)
where D is a diagonal matrix with the diagonal elements
d kk = [π k (1 − π k ) − γ kk ]
1
T
T
1
∑ ntk
t =1
.
Estimation of Default Probabilities and Default Correlations
253
For large values of ntk the matrix D reduces to zero. Hence the estimator G is asymptotically unbiased. In order to get an unbiased estimator for the diagonal elements the procedure analogously to (12) can be applied. However this can lead to
disadvantageous consequences as will be explained later at the end of section 4.1.
'UVKOCVKQPKPC5KPING(CEVQT/QFGN
Returning to the single-factor model described by (2) and (10) the estimation of
the correlation is discussed in the following. The default probability is assumed to
be given.
In the case of a single homogeneous group one can use the likelihood-function
L(ρ) which is given by the right side of equation (6) as a function of the asset correlation ρ. For a given π the maximization is solved with respect to ρ and the resulting maximum likelihood estimate for the asset correlation is denoted by ρML.
Looking for the default correlation, this is not yet the desired parameter. But in
the single-factor model the asset correlation determines the simultaneous default
probability and this leads to the default correlation as explained in the following
paragraphs.
Recall that the solvency variables of two obligors are modelled by standard
normally distributed variables with correlation ρ. First, one likes to know the
probability that both obligors default at the same time, which means in the singlefactor model, that both obligors hit their individual default threshold simultaneously. Given the asset correlation the simultaneous default probability of two obligors can be expressed by24
(
)
Φ 2 Φ −1 (π ), Φ −1 (π ) ; ρ ,
(18)
where Φ2 denotes the bivariate standard normal distribution function. Following
the achievements of 3.1 one derives the estimate
r ML =
(
)
Φ 2 Φ −1 (π ), Φ −1 (π ) ; ρ ML − π 2
π(1 − π )
(19)
of the default correlation r by using the estimate ρML for the asset correlation ρ.
Now, the estimation of the default correlation is extended for K homogeneous
groups of obligors (e.g. a rating system). Remaining in the single-factor model
framework the default probabilities of K groups are denoted by π1,…,πK. The de24
In the homogeneous group only obligors with the same default probability are considered.
254
Stefan Huschens, Konstantin Vogl, and Robert Wania
fault correlation between obligors in group k is denominated by rkk (within-group
default correlation) whereas rkl denotes the default correlation between obligors of
group k and group l (between-group default correlation) with
k,l = 1,…,K.
First, the vector ρ of the within-group asset correlations ρ1,…,ρK has to be estimated. Using the maximum-likelihood method again the function
Lt ( ρ) =
∞ K
h
n
∫ ∏ p(zt ; π k , ρ k ) [1 − p(zt ; π k , ρ k )]
tk − htk
tk
−∞ k =1
φ ( z t )dz t
is derived for the single period t, where the conditional default probability from
(3) is enriched by the group index. This expresses that all solvency variables are
affected by the realization of the systematic factor Zt in period t. How much the
individual default probability is affected by Zt is controlled by the within-group
asset correlation of the specific obligor.
Given the default probabilities for all obligors one can again use the considerations of Binomial distributed random variables due to the conditional independence. Either in the single-period case with Lt(ρ) or in the multi-period case with
T
∏ L (ρ)
t =1
t
the maximization with respect to ρ leads to an maximum-likelihood estimate for ρ
with components
ρ1ML ,K, ρ KML .
One result of the single-factor model is that if the within-group asset correlation
between two solvency variables is ρk, then the between-group asset correlation can
be expressed by
ρ kl =
ρ k ρl .
This leads to
ρ klML = ρ kML ρ lML .
Hence, the estimates of the within-group asset correlation are sufficient in order to
compute the default correlation via
rklML =
(
)
ML
Φ 2 Φ −1 (π k ), Φ −1 (π l ) ; ρ kl
− π k πl
π k (1 − π k ) π l (1 − π l )
.
For k = l results the within-group default correlation, and for k ≠ l the betweengroup default correlation.
Estimation of Default Probabilities and Default Correlations
255
In the case of unknown correlation it is necessary to combine the estimation of default probability and default correlation.
Credit portfolio risk models make use of the default probabilities πk and the default correlations rkl as input parameters. Hence, it is important to have good estimates for them. Recall that in the general Bernoulli mixture model (13) the elements of the variance-covariance matrix Γ are given by (16). The parameters on
the right side of (14) and (15) are to be estimated from the realizations of the stochastic independent random vectors F1,…,FT of the relative default frequencies.
The mean
π DF =
1
T
T
∑ Ft
t =1
of the relative default frequencies over time is an unbiased distribution-free estimator for π which was considered first in case (a) of section 2.2 for each group
separately. In the discussion following (9) it was pointed out that πDF minimizes
the variance if the numbers ntk of obligors are constant over time for each k and
that πDF is approximately a minimum variance estimator provided that the numbers ntk of obligors in the groups are very large.
The covariance matrix Γ can be estimated by the distribution-free estimator
Γ DF =
(
)(
)
′
1 T
Ft − π DF Ft − π DF
∑
T − 1 t =1
(20)
which in turn has the same expectation as G in (17). Thus ΓDF is unbiased for the
off-diagonal elements and asymptotically unbiased for the diagonal elements. It is
possible to modify the diagonal elements of ΓDF analogously to (12) in order to
eliminate their bias, but it is not recommended. This is because Γ DF has the preferable property that the estimate is always a positive semi-definite matrix. Since
the unexpected credit portfolio loss25 is approximately26
x ′ Γ x estimated by
25
26
See Bluhm et al. (2003, p. 28).
For perfect diversified portfolios.
x ′ Γ DF x ,
256
Stefan Huschens, Konstantin Vogl, and Robert Wania
where x denotes the vector of loss contributions,27 losing the property of positive
semi-definiteness of ΓDF could result in a negative estimate for the unexpected
portfolio loss.
Returning to the likelihood-function in (6) the simultaneous estimation for the two
parameters can be applied. The maximizing of the likelihood-function simultaneously for ρ and π can only be done with numerical methods28 and results in estimates ρML and πML. The estimate of the default correlation rML is obtained by (19)
substituting π by πML.
As example Table 1 shows the estimation results for different observations.
There are 1000 obligors considered in each of three periods. The number of defaults is varied to show the sensitivity of the results. Under the assumption of time
stable default probabilities and default correlations, the simultaneous estimates
depend strongly on the variation of the default frequencies over time.
Table 1. Examples for the simultaneous estimation of the default probability πML, the asset
correlation ρML, and the default correlation rML
Case
1
2
3
4
Number of defaults in
period 1
period 2
period 3
20
20
20
40
0
20
15
26
19
18
22
20
πML
0.020
0.029
0.020
0.020
ρML
0
0.4115
4.5·10-4
<10-8
rML
0
0.1251
5.4·10-5
<10-8
Case 1 illustrates a sample with constant default frequencies which results in rML
= 0. In contrast, case 2 shows a high correlation estimate because the number of
defaults varies considerably. The third case represents a typical portfolio with
moderate variation of the number of defaults. The estimated default correlation is
small but is still influencing the loss distribution in credit risk models.29 In the last
case, the estimated correlations are negligible. The modest variation in the observations is more likely caused by randomness than by correlation.
27
The loss contribution of group k is given in the simplest case as the sum of the products
of exposures at default and losses given default of the individual obligors.
28 The simultaneous estimation for π and ρ from single-period data results in the estimates
ρML = 0 and πML = h/n, see Höse and Huschens (2003c). Hence, it is necessary to use
multi-period data for the estimation of correlation.
29 For more information on portfolio models see Overbeck (2004).
Estimation of Default Probabilities and Default Correlations
257
The default probability and the default correlation are closely connected. The estimation procedure has to take into account this relation. This article presents estimation methods for both, the general case of a Bernoulli mixture model and the
special case of the single-factor model which is used in the Internal Ratings-Based
Approach of Basel II. Linear estimators are analysed in the general case and the
method of maximum-likelihood is applied in the special case.
The distribution-free estimates in the general case are given in a closed form. The
maximum-likelihood estimates have to be derived by numerical maximization of
the likelihood-functions provided.
To address a variety of practical problems the estimates are obtained for single
groups or several groups as well as single periods or multiple periods. On the one
hand, the estimation of default probabilities can be supported by given correlations
and on the other hand the estimation of correlation can be considered with known
default probabilities. In a third case, the default probabilities and default correlations have to be estimated simultaneously if neither is known beforehand.
Basel Committee on Banking Supervision (2003) The New Basel Capital Accord: 3rd Consultative Document.
Bluhm C, Overbeck L, Wagner C (2003) An Introduction to Credit Risk Modeling. Chapman & Hall, New York.
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Moody´s Investors Service.
Crouhy M, Galai D, Mark R (2000) A Comparative Analysis of Current Credit Risk Models. Journal of Banking and Finance 24:59–117.
de Servigny A, Renault O (2003) Correlation Evidence. Risk 16(7):90–94.
Erlenmaier U, Gersbach H (2002) Default Probabilities and Default Correlations. Working
Paper.
Frerichs H, Löffler G (2002) Evaluating Credit Risk Models: A Critique and a Proposal.
Working Paper.
Frey R, McNeil A (2003) Dependent Defaults in Models of Portfolio Credit Risk. Journal
of Risk 6(1):52-92
Gersbach H, Lipponer A (2000) The Correlation Effect. Working Paper.
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Gordy M (2000) A Comparative Anatomy of Credit Risk Models. Journal of Banking and
Finance 24:119–149.
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Höse S, Huschens S (2003a) Estimation of Default Probabilities in a Single-Factor Model.
In: Schwaiger M, Opitz O (eds) Between Data Science and Applied Data Analysis.
Springer, Heidelberg, pp 546–554
Höse S, Huschens S (2003b) Simultaneous Confidence Intervals for Default Probabilities.
In: Schwaiger M, Opitz O (eds) Between Data Science and Applied Data Analysis.
Springer, Heidelberg, pp 555–560
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für Betriebswirtschaft, 73:139–168
Höse S, Huschens S, Wania R (2002) Rating Migrations. In: Härdle W, Kleinow T, Stahl
G (eds) Applied Quantitative Finance. Springer, Heidelberg, pp 87–110
Huschens S, Locarek-Junge H (2002) Konzeptionelle und statistische Grundlagen der portfolioorientierten Kreditrisikomessung. In: Oehler A (ed) Kreditrisikomanagement Kernbereiche, Aufsicht und Entwicklungstendenzen. Schäffer-Poeschel, Stuttgart, 2nd
edition, pp 89–114
Huschens S, Vogl K (2002) Kreditrisikomodellierung im IRB-Ansatz von Basel II. In: Oehler A (ed) Kreditrisikomanagement - Kernbereiche, Aufsicht und Entwicklungstendenzen. Schäffer-Poeschel, Stuttgart, 2nd edition, pp 279–295
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Judge GG, Griffiths WE, Hill RC, Lütkepohl H, Lee TC (1985) The Theory and Practice of
Econometrics. Wiley & Sons, 2nd edition, New York
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Probability of Default and Asset Size. www.frbsf.org
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Overbeck L (2004) Credit Risk Modeling: An Overview. In: Frenkel M, Hommel U, Rudolf M (eds) Risk Management – Challenge and Opportunity. Springer, Heidelberg,
2nd edition, pp 197-217
Pafka S, Kondor I (2003) Estimated Correlation Matrices and Portfolio Optimization.
Working Paper, http://arxiv.org/abs/cond-mat/0305475
Schönbucher P (2000) Factor Models for Portfolio Credit Risk. Working Paper.
Thomas LC, Edelman DB, Crook JN (2002) Credit Scoring and Its Applications. Society
for Industrial and Applied Mathematics, Philadelphia.
Vogl K, Wania R (2003) BLUEs for Default Probabilities. Working Paper, www.tudresden.de/wwqvs/publ.
/CPCIKPI+PXGUVOGPV4KUMUQH+PUVKVWVKQPCN
2TKXCVG'SWKV[+PXGUVQTUŌ6JG%JCNNGPIGQH
+NNKSWKFKV[
Christoph Kaserer1,2, Niklas Wagner1,2 and
Ann-Kristin Achleitner1,3
1
Center for Entrepreneurial and Financial Studies (CEFS), TUM Business School,
Technische Universität München, 80333 München, Germany
2
Department for Financial Management and Capital Markets, TUM Business
School, Technische Universität München, 80333 München, Germany
3
KfW-Chair in Entrepreneurial Finance, TUM Business School, Technische Universität München, 80333 München, Germany
Abstract: Since private equity investments are not publicly traded, a key issue in
measuring investment risks of institutional private equity investors arises from a
careful measurement of investment returns in the first place. Prices of private equity investments are typically observed at low frequency and are determined by
transactions under low liquidity. This contribution highlights useful approaches to
the problem of return measurement under conditions of illiquidity. Then, specific
risk management issues, including asset allocation issues, are discussed.
JEL classification: G1, G2
Keywords: Private Equity, Risk/Return Measurement, Net Asset Values, Cash Flows, Illiquidity, Stale Pricing, Risk Management, Asset Allocation
Private equity has become an increasingly important alternative asset class for institutional investors as it may offer return as well as diversification benefits relative to traditional stock and bond market investments. In fact, the market for private equity investments has grown dramatically over the 1998 to 2000 period.
However, the economic downturn during 2001 to 2003 had a strong negative im-
260
Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
pact on the funds raised by the private equity industry. Nevertheless, it is common
wisdom that private equity will again become an important source of corporate financing and, thereby, an important driver in economic prosperity.
From an economic point of view, one of the most important advantages of private equity compared to public equity is to overcome the free-rider problem in
corporate control. While dispersed ownership as the typical ownership structure in
public equity markets does not generate sufficient incentives to undertake costly
control activities, private equity markets typically go along with concentrated
ownership in portfolio companies. This is because the private equity investor normally holds a large part of equity in his portfolio company. For that reason he exercises a continuous monitoring activity. The private equity investor is typically
by itself a fund where a given number of private or institutional investors, called
limited partners, have paid in their capital. The fund is run by a management team
called general partner. Of course, a conflict of interests between the general and
the limited partners could arise. Normally, however, this problem will be avoided
as the number of limited partners is not too high and the general partner has either
invested his own money in the fund or is paid according to some purposeful incentive scheme. Whether these problems may become more serious in the case of a
fund of funds construction, may be left open here. In such cases the outside investor has only a contractual relationship with the management team of the fund of
funds; the allocation of capital to different private equity funds is made by the
management team.
From an investors point of view it is important to note that several empirical results, which are available particularly for venture capital as a special segment of
private equity focused on financing high risk start-up firms, indicate that this kind
of alternative investment may indeed offer desirable risk-return and particularly
diversification characteristics.1
Despite these potential benefits it is important to point out that the lack of an organised secondary market for alternative investments comes along with low liquidity or even illiquidity in the transfer of alternative asset ownership. Hence, a
major drawback of the private equity asset class is its liquidity risk. The latter can
manifest itself with the impossibility to transact at a given point in time and/or
with the occurrence of substantial transaction cost.2
There are two major consequences of the lack of an organised secondary market. First, liquidity –jointly with investment risk and return– will play a major role
in a fund manager’s decision to include private equity in her managed fund of assets. Second, we argue here that liquidity has an additional indirect impact on the
decision to invest in that it has an important impact on the measurement of returns
of relatively illiquid assets. As risk is statistically derived from return observa1
2
See e.g. Schilit (1993), Cochrane (2001), Chen et al. (2002), Emery (2003), and the literature given therein. For an overview of venture capital see also for example Gompers
and Lerner (1999).
Such transaction costs are for example given by high market impact costs, the cost of
searching potential buyers and sellers as well as potential agency costs related to changes
in the ownership structure given a desired transfer of assets.
Managing Investment Risks of Institutional Private Equity Investors
261
tions, liquidity will also play a key role in an accurate measurement of private equity investment risk.
In the following section we discuss useful approaches to the problem of liquidity related return and risk measurement. Hereby, we present two important methods how return characteristics can be measured in the context of illiquid markets.
The first method presented in Section 2.1 relies on reported asset values, while the
second method presented in Section 2.2 is based on observable, although infrequent cash flows. Given that private equity returns were determined, Section 3
then discusses the consequences for risk management as well as asset allocation
issues. Obviously, most of the risk management and asset allocation methods that
apply to public equity will also apply to private equity. However, there are some
issues, especially related to the problem of illiquidity and measurement biases,
which are specific in risk management and asset allocation of private equity investments. These issues are discussed in Sections 3.1 and 3.2. Section 4 contains a
brief conclusion and gives a topic outlook.
It has already been mentioned that a private equity investment can be undertaken
directly or indirectly via a so-called private equity fund. Therefore, risk-/return
characteristics of private equity investments can basically be defined from two different perspectives. Either one is interested in assessing the return distribution of
an investment in a single company seeking for equity financing or in assessing the
return distribution of an investment in a private equity fund. As far as risk management issues are concerned the first perspective is especially relevant from the
viewpoint of a general partner, as he is supposed to make congruent decisions with
respect to the allocation of capital provided by limited partners to portfolio firms.
The second perspective is relevant for a private or institutional investor considering acting as a limited partner, i.e. to invest money in a private equity fund. 3
Hence, when talking about return distributions one should make clear as to what
kind of return processes he is referring to: returns generated at the level of a single
portfolio firm, labelled as transaction level, or returns generated at the level of a
private equity fund, labelled as the fund level.
As this article deals with risk management issues of institutional investors we
are focussing on return distribution at the fund level. However, much of the methodological issues raised here could safely be applied to return distributions at the
transaction level as well. Hence, these different perspectives are not that important
for what follows here.
From an economic point of view, the most important characteristic of private
equity investments are missing or highly imperfect secondary markets. As a consequence, for any single fund investment there are only a few points in time for
which transaction prices can be observed: when limited partners pay in their capi3
Obviously, this perspective is also relevant for a fund of funds manager.
262
Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
tal and when the investment is liquidated. Usually, such transactions do not happen very frequently. Over a fund’s lifetime, normally 5 to 10 years, one would observe not more than a handful of cash flow transactions between the fund and its
limited partners. Moreover, even if cash flows would arise more frequently, the
lack of reliable information with respect to the market value of a particular private
equity fund will not be offset. As a consequence, no intermediate series of historical returns is available. Therefore, realized returns of private equity investments
can only be observed by looking at the cash flow stream generated over a fund’s
lifetime. However, one should be careful in comparing cash flow based internal
rates of return (IRRs) with return figures derived from the market value observation of a public equity investment. As an alternative, one could calculate stock
based private equity returns on the basis of reported asset values, such as net asset
values (NAV). However, these values do not represent market transactions and
may be subject to rigidity due to smoothing activities undertaken by the fund
management and/or subject to observational noise.
The problem of assessing return distributions of assets traded on markets subject
to liquidity constraints has been first analyzed in the context of the asynchronous
trading literature in finance. Scholes and Williams (1977), Roll (1981) and Cohen
et al. (1983) consider the estimation of asset betas of relatively illiquid small capitalisation stocks. A more elaborated version of this approach has been presented
by Lo and MacKinlay (1990). As they rely on the assumption that market prices of
assets can be observed at least at some points in time, an extension of this approach to the issues in question here is not possible.4 More recently and more appropriate, Getmansky et al. (2003) derive a related econometric time series model
which considers return smoothing as a result of illiquidity in investment portfolios.
This will be considered in more detail here.
Peng (2001) proposes an extension to repeat sales regression which was used in
the real estate finance literature. The method is based on estimating time series returns of a portfolio of infrequently traded assets based on a cross-section of observed transaction prices for a subset of assets.
Other approaches to illiquidity include Longstaff (1995) who uses optionpricing theory in order to assess the maximum value of the ability to trade immediately in a liquid market. The model derives an upper bound for the value by assuming a trader with perfect foresight. In case the trader wants to sell, in a perfectly liquid market he may realize the maximum asset value governing at a given
time period and, hence, realize the value of a lookback option. This option value
then can be considered as an upper bound for the value of marketability to an investor with imperfect foresight.
While the Longstaff (1995) approach may be useful to derive upper bounds on
discounts for private equity asset values, it does not relate to asset price dynamics
and the measurement of returns and risk. As we do not assume to have a
representative cross-section of assets, the approach by Peng (2001) seems to have
too high data requirements. In the following we will therefore first focus on the
approach of Getmansky et al. (2003). Thereafter, we will discuss the implications
4
For a more detailed description of this approach cf. Campbell et al. (1997, p. 84 n.).
Managing Investment Risks of Institutional Private Equity Investors
263
Getmansky et al. (2003). Thereafter, we will discuss the implications of measuring
cash flow based returns instead of asset value based returns.
In an ideal economy with frictionless and informational efficient markets, i.e.
when transaction costs can be neglected and all information is immediately incorporated into market prices, classical finance models apply. In these models asset
prices Pt fluctuate randomly and returns Rt = Pt/Pt–1 − 1 are hence independent.5
In discrete time, i.e. for t = 1, ..., T, such ideal conditions can be formalised by
assuming that the return Rt is independently and identically distributed (iid) with a
common distribution function F. For our purposes we can think of Rt as the true
period-t asset return of an investment in a private equity fund.
Under illiquidity, the true period-t asset return Rt will –by definition– be unobservable. Think, for instance, at a private equity investment in a single company.
In this case, the value of this investment can be observed on the vintage date, i.e.
the day the capital is injected into the company, and on the exit date, i.e. the day
the private equity investor sells its stocks on the market. Frequently there are intermediate transactions, for instance, when there are additional financing rounds or
when the private equity investors makes a partial sell-off from his stake. Hence,
with respect to the value of a single portfolio company investment we have a
handful of points in time where asset prices can be observed on the basis of transaction related prices. If, however, we think of a private equity investment, even
this kind of restricted revelation of market values may break down. As the fund is
typically invested in more than one company, cash flows between the fund and its
investors can be regarded as pooled cash flows between the fund and all its portfolio companies. In this case the fund’s asset values can be observed on the vintage
date and on the liquidation date. However, as long as there are some intermediate
cash flows between the fund and the investors putting these two values into relation does not reveal anything about the true return generated for the investors.
One way out of this problem is to check as to what extend intermediate true returns can be inferred from accounting based asset values. Specifically, private equity funds regularly disclose NAV on a semi-annual or even quarterly basis. These
values are calculated along valuation guidelines developed in some kind of selfregulation context and are supposed to reflect the fair value of the whole investment portfolio. Moreover, under some circumstances the fair value should be derived within a marking-to-market framework. Nevertheless, it is quite obvious that
NAV would only occasionally reflect the true market price, i.e. the price at which a
fund’s assets could be sold in an open market transaction. This may either be due
to unavoidable valuation errors made by the general partner or due to a strategic
disclosure policy followed by the latter.
5
In some models it is assumed that not discrete but continuous returns are independent.
For our purposes here, this difference is of minor importance. Actually, we assume discrete returns to be independent.
264
Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
However, it can be shown that at least under some circumstances it may be possible to derive true returns from observed NAV. Lets assume that we can, in fact,
observe some intermediate proxy return Qt. This, of course, implies a perceivable
loss of information about the underlying distribution F of the true returns Rt. In
particular, we loose information about the risk inherent in these returns. In order to
gather at least some information on F one has to assume some model based relationship between the true returns Rt and the observed returns Qt.
Figure 2.1 illustrates the given situation of a true price process Pt and two intermediate price estimates that deviate from the true prices. The deviating price estimates imply errors in the observed intermediate returns Qt. The figure is set up as
a simplifying situation for a private equity investment with initial investment P0 at
time zero and final liquidation value PT at time T.
pr i c e
PT
P̂ t
Pˆt ′
P0
t
t'
t i me
T
Fig. 2.1. True price process Pt and deviating price estimates at time t and t’ each being reported before final liquidation time T.
With respect to modelling the relationship between true returns Rt and the observed returns Qt’s a huge number of different specifications may be conceivable.
However, based on findings in the literature we believe that the following two different specifications for the time being are the most important ones.
Getmansky et al. (2003) assume that due to smoothing we observe period-t returns
which are weighted sums of the true period-t return and k lagged past true returns.
This means that, for i = 0, 1, ..., k, we have a process of the form
Managing Investment Risks of Institutional Private Equity Investors
265
Qt = ∑i =0 wi Rt −i ,
k
where the weights 0 ≤ wi ≤ 1 satisfy the restriction w0 + w1 + ... + wk = 1. Together
with the condition that true returns are iid this makes sure that expected returns
calculated on the basis of observed returns are an unbiased estimation for expected
true returns. However, as a result of the smoothing process, period-t variance will
be underestimated. In fact, according to our assumptions the following relationship will hold:
Var (Qt ) = Var ( Rt )∑i =0 wi2 ≤ Var ( Rt ).
k
Moreover, assume that we have two investments in two different private equity
funds, n and l. Let Corr(Rtn, Rtl) be the correlation coefficient between the unobserved true returns Rtn, Rtl, i.e. Corr(Rtn, Rtl) =E[(Rtn –E(Rtn))(Rtl –
E(Rtl))](Var(Rtn)Var(Rtl))1/2, and Corr(Qtn, Qtl) be the correlation coefficient between the observed returns Qtn, Qtl, i.e. Corr(Qtn, Qtl) =E[(Qtn –E(Qtn)) [(Qtl –
E(Qtl))](Var(Qtn)Var(Qtl))1/2, of the two funds under observation. Note that it
would make sense in this model to assume that Corr(Rtn, Rsl)=0 ∀ t ≠ s. In this
case the following relationship must hold under our assumptions:
Corr (Q , Q ) =
n
t
l
t
∑
k
i =0
win wil
∑ (w ) ∑ (w )
k
i =0
k
n 2
i
i =0
l 2
i
Corr (Rtn , Rtl ) ≤ Corr (Rtn , Rtl ).
Unless the condition win= wil ∀i applies, contemporaneous correlation between
observed returns of two different private equity funds is smaller than the correlation of true returns of the same funds. Hence, estimating multivariate distributions
on the basis of observed returns subject to smoothing leads to underestimation of
true return variance as well as true return correlation. Finally, as far as the correlation between observed returns and the return on the market portfolio is concerned,
a bias could arise as well. Assume that RtM is the true, observable market portfolio
return, for instance measured by an unbiased proxy like a market index. In this
case we are interested in estimating the correlation coefficient between the true
private equity return and the market index return, i.e. Corr(RtM, Rtl) =E[(RtM –
E(RtM))(Rtl –E(Rtl))](Var(RtM)Var(Rtl))1/2. Again, given that we can only observe
the smoothed proxy Qt, only the following correlation can be measured: Corr(RtM,
Qtl) =E[(RtM –E(RtM))(Qtl –E(Qtl))](Var(RtM)Var(Qtl))1/2. Note that it would make
sense in this model to assume that Corr(RtM, Rsl) = 0 ∀ t ≠ s. Then, on the basis of
our assumptions, we can show that due to smoothing correlation between returns
on private equity investments and returns on public equity investments will be underestimated. In fact, the following relation will hold:
Corr (RtM , Qtl ) =
w0l
∑ (w )
k
i =0
l 2
i
Corr (RtM , Rtl ) ≤ Corr (RtM , Rtl ).
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
To sum up, estimating private equity investment’s return distributions on the basis
of accounting-based appraisal values, like net asset values disclosed by general
partners, is subject to serious estimation biases. In fact, Emery (2003) presented
evidence in favour of so called stale pricing. In the context of a simple regression
analysis he showed that NAV based private equity returns adjust with a lag movement to public returns. Of course, on the basis of the analysis presented here it will
be possible to correct for this estimation bias, at least theoretically. Getmansky et
al. (2003) show how such a corrected estimation can be achieved by using the returns on a large sample of hedge funds. However, it should be noted that this correction method rests on the assumption that the smoothing process does not affect
the expected sample return and that the process, as described by the parameters 0
≤ wi ≤ 1, is stable over time. As smoothing may not only be due to informational
problems associated with illiquidity but also to deliberate actions set by general
partners, it is still an open question whether the methods discussed here will really
be sufficient for avoiding perceivable estimation biases. This is much more than a
statistical issue, because even small estimation errors can have a large impact on
asset allocation decisions. This issue will be treated in Section 3.
In addition to the problems already mentioned in the preceding section, we would
like to emphasize that the approach presented by Getmansky et al. (2003) also
rules out observational noise. In other words, it may well be that even if the general manager of a private equity fund would really like to disclose the true NAV he
is unable to do so because of the inability to infer the market value from available
information. More formally spoken, the above model assumes that all observations
Qt are purely based on true returns and, hence, that accounting-based valuation exactly matches market valuation apart from its slower reaction to news.
In evaluating return distributions of private equity investments, alternative
model specifications could be useful. Such models can allow for transitory deviations between the unobserved market value-based and the accounting-based returns. One possible specification would be to add observational noise to the above
introduced equation for Qt. In order to make our point in the simplest possible
way, we first rule out the existence of any smoothing at all. In that case Qt can be
modelled as a noisy observation of the true return
Qt = Rt + X t ,
where both returns are assumed to have identical unconditional expectations. The
noise terms Xt have zero unconditional expectation, E(Xt) = 0, are uncorrelated
with the true returns, Corr(Rt; Xt) = 0, and may exhibit linear dependence of the
first-order autoregressive type
X t = ρX t −1 + η t ,
with |ρ| < 1 and iid innovations ηt ∼ N(0, σ2). It then obviously follows
Managing Investment Risks of Institutional Private Equity Investors
267
Var (Qt ) = Var ( Rt ) + Var ( X t ) > Var ( Rt ).
Hence, although there is a possible effect of persistency in the observations Qt,
which is modelled through the dependent noise Xt, true return variance in this case
is over-, not underestimated. It can easily be seen that one cannot rule out this result even under the assumption of smoothed returns. Even though an estimation of
this model with observational noise may be performed within a state-space setting,
this is potential bad news as to what our ability of identifying the true return distribution of a private equity investment is concerned.
To sum up, specifying a reliable model for Qt might be a rather tricky task. And
for the time being we cannot even be sure whether observed NAV based returns
over- or underestimate true return variance.6
Instead of relying on smoothed and/or noisy asset value based returns one can try
to infer true investment returns from observable cash flow transactions between
private equity funds and their limited partners. Under this perspective the return
on a private equity investment could be measured by its internal rate of return
(IRR). However, one should be careful in putting the IRR simply in relation to asset value based returns observed on the public equity market. This is because the
IRR is a dollar-weighted return, while asset value based returns are time-weighted
returns. In other words, while an asset value based return over a period of length T
is simply the geometric mean of the single period realizations 1+Rt, the IRR is a
value-weighted average of these returns. Unless an investment consists of two
cash flows only, a single initial investment and a single final repayment, the IRR
would be different from the geometric mean of single period realizations 1+Rt.
The following simple example gives a flavour of the measurement relevance of
this difference. Assume a private equity investment where –for whatever reasons–
true market values are known and disclosed as NAV. The lifetime of the fund is assumed to be three years. Assume moreover that general partners define payouts to
limited partners in a way that the cash flows are generated according to Table 2.2.
Table 2.1. Unobservable true returns, true NAV as well as cash flows of a private equity
fund investment.
T
Rt
NAVt
CFt
6
0
-100
1
10%
110
0
2
20%
32
100
3
5%
33.6
33.6
As a corollary it should be noted that the model with observational noise generates an
unbiased estimation for the true return correlation of two different private equity funds as
well as for the return correlation of a private and public equity investment as long as the
additional assumption Cov(Xtn; Xtl) = 0 holds.
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
Based on NAV, the average true return over the fund’s lifetime is 11.5%,7 while
the IRR is 13.8%. Hence, observed IRR cannot be taken as an unbiased measure
for the true expected return Rt .8
According to this simple insight an inference from observed IRRs on unobserved Rt is not obtained straightforwardly. However, an inference may be possible on the basis of additional assumptions. For instance, the following solution
was proposed by Rouvinez (2003). In order to transform a multi-period cash flow
stream into a two period wealth comparison he defines a start date T0 and a terminal date T’. Any single private equity investment fund is raised a some point t ≥ T0
and liquidated at some point T ≤ T’, with t < T. Now, all the cash paid into the
fund is discounted back to period T0 , while all the cash paid out to investors is
compounded up to period T’. For this intertemporal transformation the risk free interest rate r is used; it is assumed to be constant. Hence, initial and terminal wealth
corresponding to cash flows generated by a single private equity fund can be calculated as follows:
∑
=∑
WT =
T
WT '
T
0
i =t
i =t
CFi in (1 + r ) − (i −T ) ,
0
CFi out (1 + r )T ' − i .
Here, CFiin resp. , CFiout is the cash paid into or paid out of the fund in period i.
The ratio of terminal wealth to initial wealth, basically, gives an information into
how many Euros a one Euro investment at time T0 will be transformed up to time
T’, given that the investment is exposed to private equity risk for a total period of
T-t. Now, the expected true return during the exposure to private equity risk, i.e.
E[WT/Wt], can be expressed as a function of time length of this exposure, i.e. T-t,
and the expected overall rate of return, i.e. E[WT’/WT0]. In fact, it can be shown
that the following relationship must hold:
 W
ln E  T
 W
  t
 W  

T' 

 + (T − t − T '+T0 ) ln(1 + r ),
  = ln E 

W

  T  
0


 Var  WT '  

 WT  
W 

 .
Var  T  = ln1 +
2
W




 t
W

E T'  


WT  

0
0
Using real life cash flow figures this expectation and variance can easily be calculated. In fact, E[WT/Wt] can be regarded as the expectation of a pooled rate of return of the whole asset class. According to the approach used by Rouvinez (2003)
7
8
It should be noted that the drop in the NAV from period 1 to period 2 is due to the payout
of 100. Hence, the return has to be measured on the basis of a payout corrected NAV.
For the IRR to be an unbiased estimator, Rt must be iid and only two cash flows are allowed to occur over the lifetime of the fund.
Managing Investment Risks of Institutional Private Equity Investors
269
it is supposed to be an estimation for the true return realised in the private equity
industry, as he assumes the relation RT-t =( WT/Wt)1/(T-t)−1 to hold. For the same
reason, therefore, the second equation yields the true return variance.
This approach has several drawbacks. First, we will get only an estimation for
an average rate of return over a longer period of time. Hence, no dependency with
market movements can be detected here. Second, this approach is not unbiased as
the return measured over the private equity exposure time depends on how the true
return generating process relates to risk free interest rates. One can see this problem very quickly by looking at the example presented above. Assume, for instance, that the risk free rate is 10%. In this case, applying the calculation proposed by Rouvinez would generate an average rate of return of 12.8%, which is
higher than the true average rate of return. On the other side, using a risk free rate
of 0% would lead to an average rate of return of 10.1%, which is again a biased
result. Third, this approach does not allow for estimating correlations within the
asset class.
Some of this criticism can be circumvented by using an approach introduced by
Chen et al. (2002), although this approach has some caveats as well. They start
from the presumption that for every single fund the relation WT/W0 = (1+IRR)1/T
holds, where for simplicity we assume t = 0. Now, this return is put into relation
with the returns on the public equity market over the same period. Therefore, the
following modified market model holding for every single fund i is specified:
(
)
Ti ln(1 + IRRi ) = β ln 1 + RTM−0 + ε i .
i
Here, RTi-0M denotes the public market return, as measured by a representative index over fund’s i lifetime, and β is the elasticity of fund returns to the market return common to all private equity funds. Moreover, it is assumed that ε ∼ N(αT,
σ2T) and Cov[εi,εj]=ρτij σ2 holds; here α is a fund specific return component per
unit of time common to all private equity funds, ρ is the per unit of time correlation of the non market driven part of all distinct pairs of funds’ returns, and τij is
the coexistence time of two funds i and j. Now, Chen et al. (2002) show that under
these assumptions a maximum-likelihood estimation for the asset class specific
parameters α, β, ρ and σ can be derived.
As mentioned, this approach has some drawbacks as well. The major one is the
fact that assuming all cash flows paid out by a private equity fund to be reinvested
at the IRR up to terminal date T is not correct. From the numerical example above
one can easily see that this would overestimate the true return of the fund. Of
course, in general it is not clear whether this approach yields an over- or an underestimation of true funds’ returns. In any case, however, the estimation would be
biased.
In order to circumvent these problems, we outline an idea for a third approach,
not yet discussed in the literature. Assume that from an ex-ante perspective the
true return process Rt can be explained by the CAPM. In that case the conditional
expectation of Rt can be written as
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
[
]
E Rt RtM = rt (1 − β ) + βRtM ,
where rt is the risk free interest rate governing in period t. Now, for every liquidated fund the following equation –using the conditional expectation of Rt– must
hold:
− CF0 = E
{∑
T
t =1
CFt (1 + IRR )
−t
}= E ∑
T
t =1
CFt
[∏
t
i =1
(1 + r (1 − β ) + βR )]
i
M
i
−t
.


It is a question to be left open here, whether this ex-ante equation can be transformed in an ex-post equation in a way that it will be possible to make an efficient
and unbiased estimation for the parameter β governing the return process of the
whole asset class. From this we can finally assess the distribution of the return
process, as the following must hold:
Var (Rt ) = β 2Var (RtM ) + Var (ε t ),
Corr (RtM , Rt ) = β
Var (RtM )
.
Var (Rt )
Obviously, an estimation of cross correlation of private equity investment returns
cannot be derived in the context of this approach, as we assume them to be from
one single distribution. The most important advantage of this approach is the fact
that we do not need any kind of reinvestment hypothesis with respect to a fund’s
cash flows. Therefore, no potential bias is induced due to the lack of any kind of
reinvestment assumption.
In the preceding section we showed how an assessment of distributional characteristics of true returns Rt could be derived from observed proxy returns Qt, or observed IRRs. Such inference included the first and second moment of the return
distribution F, i.e. the expected return on a private equity investment E(Rt) as well
as the return variance Var(Rt). Moreover, under certain circumstances we were
able to make an inference with respect to the correlation of private equity returns
with other asset classes Cov(RtM; Rt’) as well as with other investments in the same
asset class Corr(Rtn, Rtl).
Once distributional information is gathered from observed returns, most of the
commonly used risk management as well as asset allocation techniques can be applied to portfolios containing private equity investments. This, for instance, is especially true with respect to the Value-at-Risk (VaR) approach, where the task is
to infer some p-quantile, qp = F–1(p), of the distribution F of the returns. We point
out, however, that the particular features of private equity cast some doubt on
solely applying traditional risk management as well as asset allocation techniques.
Empirical results on private equity and alternative investment such as venture
Managing Investment Risks of Institutional Private Equity Investors
271
capital indicate large standard deviations of period returns as well as significant
skewness and excess kurtosis in the return distribution. This would have a particular impact on risk management, which we will discuss in the following section.
First, it should be noted that the asset value based estimation approaches presented
in Section 2.1 rely on the assumption that smoothing is kind of a stationary process. Especially, we ruled out that this process is driven by changing strategic goals
of reporting by general partners. In practice, however, one cannot rule out that the
degree of smoothing is related to incentives governing the behaviour of general
partners and, hence, will change over time depending on conditions not reflected
in the model. For instance, one might presume that adjustment to market prices,
given that they are privately known by general partners, is faster when they are increasing, especially when the whole market is in a positive mood, and slower
when they are decreasing. Therefore, the presumption that the estimated distribution F already integrates illiquidity effects in a sufficient manner should not be
taken as granted.9 The important consequence for risk management is that F is not
a stationary distribution. Unfortunately, no reliable empirical information is available in this regard as the techniques for integrating this kind of problem in the estimation of F have still to be developed.
Second, as long as we try to infer the true return distribution by looking at a
fund’s cash flow it is rather unclear as to what extent a liquidity discount is then
taken into account. One may presume that an investor forced to sell a stake in a
private equity fund prematurely faces an IRR which is considerably lower than the
IRR generated without premature liquidation. This is an important point, because
the expected returns derived under this methodology integrate an illiquidity driven
risk premium. This is an important aspect that has to be taken into account, especially for such groups of institutional investors that may face severe liquidity
shocks.
Third, it is well known from the risk management literature that asset return’s
distributions are not fully captured by the assumption of normality and independence. Special emphasis has been put in this context on the empirical regularity of
fat tails, i.e. the phenomenon that extreme realisations happen more frequently
than predicted under a normal distribution. As risk management is focused on extreme realisations, this is one of the most important theoretical and practical challenges. In fact, Ljungqvist and Richardson (2003) and Kaplan and Schoar (2003)
report that cash flow based private equity fund’s returns are heavily skewed in the
sense that there is a significant downside in the form of funds performing poorly
on a relative basis. However, Cochrane (2001) found a much less pronounced
9
Of course, also the traditional risk management literature dedicated some attention to illiquidity issues; cf. for example Jorion (2001, Chapter 14). However, as these authors
looked at traded assets, liquidity costs could be measured by bid-/ask-spreads, for instance. This would not be possible in the context of private equity.
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
skewness, if one switches from arithmetic to geometric returns of single venture
capital transactions. From a theoretical perspective one might expect returns to be
skewed because of the option-like payoff structure of risky claims. 10 This skewness should be more pronounced the higher the debt is relative to a firm’s market
value and the more the total firm value can be modelled as an option-like payoff
itself. The last point, at least, gives a strong indication that return skewness of
young and innovative business ventures should be more pronounced, as their investment projects often can be characterized as a real option. Therefore, the implementation of extreme value theory in risk management tools may be especially
important for investors exposed to private equity risks. The bad news are, however, that for the time being we do not have an empirically well founded understanding of extreme value behaviour in private equity investments.
Fourth, non-normality features of private equity portfolio returns may also relate
to what a particular institutional investor defines as a proper investing strategy.
The choice of such a strategy will depend on the institutional investor’s financial
goals as well as on his particular knowledge advantages. Two common generic
strategies are diversification and specialisation. While diversification lowers risk
as long as asset returns are not perfectly correlated and increases the degree of
normality from a financial risk management point of view, specialisation does not.
When following a diversification strategy investors seek constrained risk reduction for their overall asset portfolio. This strategy includes diversification not only
between companies and industries but also between financing stages especially
when venture capital investments are considered. When following a specialisation
strategy with private equity investments, investors increase overall risk relative to
diversification. The payoff from exposing a portfolio to diversifiable risk is that it
may offer rents from controlling activities. This is especially important for institutional investors acting as general partners, as there is an obvious principal-agent
relationship between the venture capital investor and its investee.11 Moreover, high
degrees of specialisation can be helpful in building up reputation and further in
gaining access to networks, to information flows as well as to deal flow from other
private equity investors. In a survey study of venture capital investors, Norton and
Tenenbaum (1993) give evidence that some considerable number of investors in
their sample follow such specialisation strategies. Of course, an empirical wellfounded understanding of the specific characteristics of private equity returns, especially as far as extreme value realisations are concerned, is even more important
for institutional investors following such a specialisation strategy.
10
It should be noted that financial theory models equity as a contingent claim –a call option– on firm assets. It predicts stronger nonlinearity of payoffs for out of the money call
options. For related empirical findings based on traded IPO aftermarket equity issues in
the German Neuer Markt see for example Wagner (2001).
11 Cf. in this regard Reid et al. (1997).
Managing Investment Risks of Institutional Private Equity Investors
273
Taking the aforementioned methodological problems in estimating the parameters
of the distribution F into account, one may not be surprised by the quite different
results obtained in the literature. As far as asset value based approaches are concerned, for the time being, there are two studies to be mentioned here. Emery
(2003), whose results should be interpreted cautiously as he did not explicitly correct for the smoothing problem, reports that returns calculated on the basis of biannual NAV US-funds data average to 15% for LBO-funds and 25% for venture
capital funds per year.12 The correlation with S&P500 returns is 56% resp. 64%.
The correlation between LBO- and venture capital funds’ returns is almost zero.
Kaplan and Schoar (2003) use a large data set provided by Venture Economics.
They try to overcome the smoothing problem by looking only at funds which have
already been closed or have been alive for at least five years. In fact, they can
show that the correlation between the rates of return calculated for this subsample
of funds is highly correlated to the IRR of the same subsample. Due to this restriction, however, the approach of Kaplan and Schoar (2003) is, in fact, not that different from a cash flow based approach. For a sample of more than 1’000 funds
they find an average IRR of 17%, while the median is 12%. The standard deviation
of the IRR is 32%. Moreover, they find evidence in favour of performance persistence.
As far as other cash flow based approaches are concerned, Ljungqvist and
Richardson (2003), for instance, find a median IRR of almost 20% for a US dominated sample of 73 funds with a cash flow history of at least nine years; the standard deviation of the IRR is 22%. Cochrane (2001) uses data from more than
16’000 single venture capital transactions and calculates an arithmetic return of
59% with a standard deviation of 100%. More interesting data, at least from an asset allocation perspective, has been reported by Chen et al. (2002). By using IRR
data of about 150 liquidated venture capital funds and by applying the maximum
likelihood estimation technique explained in Section 2.2, they calculate an average
venture capital fund return of 45%, with a standard deviation of 116% and a correlation with large capitalization stock’s returns of 4%. Finally, Rouvinez (2003) by
applying a cash flow based method already explained in Section 2.2 finds an expected return on a private equity investment of 14% and a standard deviation of
34%. His methodology does not allow for calculating correlation coefficients with
public market returns. Of course, these results may still be interpreted cautiously
as they have been derived despite of severe data restriction problems. It will therefore still take a couple of years until reliable empirical results will be available.
Nevertheless, it may become something like a stylized fact that venture capital returns are perceivably higher than returns on non-venture capital private equity investments. This seems to come along with a very much higher volatility as well as
12
The reason why Emery (2003) calculates the returns on a biannual basis relates to the
smoothing problem. In fact, he starts from the plausible presumption that smoothing effects vanish in the long run. However, whether a two year return period is already sufficient in order to overcome the smoothing bias is a question with not clear cut answer.
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
with a lower correlation to public equity returns. Given that future empirical research will corroborate this result, this will become an important issue in asset allocation decisions.
Tab. 3.2. Optimal portfolio weights of a private equity asset in a private and public equity
portfolio for different distributional parameters.
Var (Rt )
100%
100%
50%
50%
50%
30%
Corr(RtM, Rtl)
α
portfolio return
5%
10%
5%
10%
20%
30%
0,0%
3,0%
1,5%
14,2%
11,7%
6,2%
21,0%
8,0%
8,7%
8,3%
11,1%
10,6%
9,4%
12,6%
portfolio standard
deviation
15%
15%
15%
15%
15%
15%
15%
Finally, we would like to emphasize that the lack of clear cut empirical results
with respect to the conditional and unconditional distribution of private equity investment returns is a serious problem making any asset allocation decision a rather
tricky task. In fact, even a slight shift in the distributional parameters may have a
very large impact on portfolio allocation. Hence, even a slightly biased assessment
of these parameters could lead to dramatic errors in asset allocation. This becomes
clear from the next table, where we show how the optimal weight of a private equity investment α in an equity portfolio changes according to a change in the return variance as well as in the return correlation with public equity. In order to
calculate these weights we assumed that the expected return on a public investment is 8% with a standard deviation of 15%. Moreover, we assumed the expected
return on a private equity investment to be 30%. The weights for the private equity
asset were derived by maximizing the expected portfolio return under the constraint that portfolio variance is equal to the variance of a 100% public equity portfolio. Other asset classes, like bonds or real estate, are not taken into account here.
Private equity has become an increasingly important alternative asset class for institutional investors as it may offer return as well as diversification benefits relative to traditional stock and bond market investments. Despite the downturn of the
industry over the years 2001 to 2003, it is commonly believed that private equity
will become an even more important source of corporate financing over the years
to come. For this reason, understanding and managing risks associated with this
asset class is of crucial importance for institutional investors.
Managing Investment Risks of Institutional Private Equity Investors
275
This article aims at improving this understanding and, hence, to give a foundation for solving specific problems arising in the context of private equity risk
management and asset allocation decisions. As a starting point, we emphasized
that –in our view– the most specific characteristic of private equity is the lack of
an organised secondary market. Hence, investing comes along with low liquidity
or even illiquidity, i.e. with the impossibility to transact at a targeted point in time
and/or with the occurrence of substantial transaction costs.
There are two major consequences of the lack of an organised secondary market
that have been treated extensively in this paper. Firstly, illiquidity implies that it is
not possible to observe a continuous series of true investment returns over time. In
other words, illiquidity goes along with serious performance measurement problems. Section 2 was entirely devoted to this problem in that we tried to show to
what extent this sort of measurement problem can be resolved. Basically, there are
two ways for doing this. Either one tries to infer market values from reported asset
values, or one tries to infer true investment returns from realized cash flow based
investment returns. As we showed, both approaches are not free of serious restrictions. As a consequence, our empirical understanding of the risk-/return characteristics of this asset class is still incomplete and should be subjected to further research. Secondly, given that we would be able to overcome these measurement
problems we could apply several well-known risk management as well as asset allocation methods to the private equity asset class as well. However, in our view
there are some specific issues in this context that apply solely to portfolios exposed to private equity risk. We discussed these issues in Section 3.
As far as risk management is concerned, specific issues relate to the following
problems. Reliability of return distribution measurement is a rather serious problem, as one cannot rule out general partners to follow a strategic disclosure policy,
which would be very difficult to integrate in a statistical model of a return generating process. It is also unclear whether a cash flow based return inference model
would really capture the whole return impact of illiquidity. Moreover, there are
good theoretical reasons suggesting that private equity returns, especially when
they are related to venture capital investments, will be governed by a distribution
with much more pronounced fat tails than public equity returns. Finally, these issues may be faced in a different way depending on whether the investor follows a
diversification or specialization strategy. There are some preliminary empirical results indicating that specialization plays a much more important role in the private
equity industry than in the public equity industry. All these issues enrich risk management of portfolios exposed to private equity risk with rather specific problems.
Beyond these risk management issues, we also discussed issues related to asset
allocation decisions. Our major point here was to show that the empirical understanding of the risk-/return characteristics of the private equity class is, in fact, incomplete and, to a certain extent, contradictory. This is important as purposeful
asset allocation decisions can only be based on a well-founded empirical understanding of risk-/return characteristics. Moreover, we showed that even slight biases in the estimated distributional parameters can have a large impact on asset allocation decisions. This is one of the major reasons why we strongly emphasize
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Christoph Kaserer, Niklas Wagner, and Ann-Kristin Achleitner
the need for much more additional empirical and theoretical work on this asset
class.
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Kaplan S, Schoar A (2003) Private Equity Performance: Returns, Persistence and Capital
Flows. NBER Working Paper No. w9807
Ljungqvist A, Richardson M (2003) The Cash Flow, Return, and Risk Characteristics of
Private Equity. NBER Working Paper No. w9454
Lo A, MacKinlay AC (1990) An Econometric Analysis of Nonsynchronous-Trading. Journal of Econometrics 45: 181-212
Longstaff F (1995) How much can Marketability affect Security Values? Journal of Finance 50: 1767-1774
Norton E, Tenenbaum BH (1993) Specialization versus Diversification as a Venture Capital
Investment Strategy. Journal of Business Venturing 8: 431-442
Peng L (2001) A New Approach to Valuing Illiquid Asset Portfolios. Yale ICF Working
Paper
Reid GC, Terry NG, Smith JA (1997) Risk Management in Venture Capital InvestorInvestee Relations. European Journal of Finance 3: 27-47
Roll R (1981) A Possible Explanation of the Small Firm Effect. Journal of Finance 36: 879888
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Rouvinez C (2003) How Volatile is Private Equity? Private Equity International, June
2003: 22-24
Schilit WK (1993) A Comparative Analysis of the Performance of Venture Capital Funds,
Stocks and Bonds, and other Investment Opportunities. International Review of Strategic Management 4: 301-320
Scholes M, Williams J (1977) Estimating Betas from Nonsynchronous Data. Journal of Financial Economics 5: 309-328
Wagner N (2001) A Market Model with Time-Varying Moments and Results on Neuer
Markt Stock Returns. SSRN Working Paper
Carol Alexander1
1
Chair of Risk Management and Director of Research, ISMA Center, Business
School, University of Reading, UK
Operational risk has been defined by the Basel Committee as ‘the risk of financial
loss resulting from inadequate or failed internal processes, people and systems or
from external events’ (Basle Committee 2001). It includes legal risk, but not reputational risk (where decline in the firm's value is linked to a damaged reputation)
or strategic risk (where, for example, a loss results from a misguided business decision). The Basel Committee defines seven distinct types of operational risk: Internal Fraud; External Fraud; Employment Practices & Workplace Safety; Clients,
Products & Business Practices; Damage to Physical Assets; Business Disruption
& System Failures; Execution, Delivery & Process Management. Detailed definitions of each risk type are given in Annex 2 of the Basel working paper on operational risks.
Recent global trends in financial markets have increased many types of operational risks. De-regulation of financial firms has allowed the rapid growth of new
companies, some of which have dubious accountancy and other practices, and
subsequently we have witnessed a marked increase in company fraud. For example, the success of Enron arose from deregulation of energy market in US in early
1990’s. Information technology and systems risks have grown along with our increasing reliance on technology. Concentration of key financial services into a
single geographical location, such as custody services in the World Trade Center,
increases operational risks arising from damage to physical assets. Several operational risks arise from the increased complexity of financial instruments, where
banks now offer highly structured products having access to wide range of asset
classes across the world. With more complex instruments there is much less transparency in the trading, and an increase in several operational risks: systems risks
because of the reliance on new and complex systems; products and business practice risks because of the danger of mis-pricing and mis-selling these products; and
‘human’ risks in general because now only a few experienced people understand
the systems and the products.
With de-regulation of the industry, new risks have been acquired through expanded trading in capital markets. But capitalization, as a whole, decreased during the 70s and the early 80s and some individual banks, if not national banking
industries, became highly vulnerable. As a result, supervision and regulation of financial firms has extended capital adequacy requirements to cover more types of
risks. The first Basel Accord in 1988 covered only credit risks in the banking
book; the Basel 1 Amendment in 1996 extended this to market risks in the trading
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Carol Alexander
book; and now the new Basel 2 Accord that will be adopted by all G10 – and
many other – countries in 2007 will refine credit risk assessments so they become
more risk sensitive, and extend the calculation of risk capital to include operational risks. Also in Basel 2, minimum solvency ratios will now be applied to asset
management and brokerage subsidiaries, and well as to traditional banking operations.
The generic approach for risk sensitive operational risk capital assessments is
termed the ‘Advanced Measurement Approach’ (AMA). Why should financial
firms adopt an AMA? Notwithstanding regulatory pressures to use – or not to use1
– AMA for operational risk capital, it is not yet clear whether this more ‘risksensitive’ approach will in fact provide risk capital estimates that are less than
those based on simple rules. In fact, the simple ‘internal measurement approach’
(IMA) in which loss severity is not random, will certainly lead to much lower
capital charges than the ‘loss distribution approach’ (LDA).
But minimizing regulatory risk capital is, for the most part, just an academic
game. Minimum solvency ratios are not binding constraints for most large banks,
where regulatory risk capital estimation is simply a chore. More important for
most listed firms is their credit rating, and rating agencies certainly regard capitalization as one important element of the firm value. Also important is the reputation of the firm, and an AMA may be particularly important for banks that, under
the new Basel 2 rules for pillar 3, will need to disclose details of their quantitative
risk management methodologies.
With continuing de-regulation in financial markets, many new companies have
experienced rapid growth on the back of dubious accountancy practices. The consequent spate of large scale company frauds – Enron being one particularly important example – has led to new legislation in the US with the Sarbannes-Oxley Act
of 2002 now holding senior company executives personally liable for the accuracy
of corporate disclosures. Effective and reliable operational risk management has
now become the key to legal indemnity and, whilst some senior executives may be
satisfied with no more than qualitative control self-assessments, the more prudent
of these will invest in an operational risk assessment framework that is based on
the best industry practice.
Some AMA, aiming to provide maximum flexibility in model design, offer
many functional forms for frequency and severity distributions. Users are faced
with many modeling decisions, but may have little guidance on how the type and
source of data should influence their choice. The result can be a complex and expensive system that lacks transparency. Moreover, in order to deal with the diverse
data types and sources that are characteristic of an operational risk quantification,
these systems often resort to ad hoc procedures.
This chapter will argue that though understanding the main determinants of
AMA model risk and the proper methods for dealing with diverse sources and
types of data, the precise form of the AMA model can be specified. An AMA
methodology should be ‘data-oriented’, in the sense that the model should be de1
US regulators may require smaller banks to use of IRB credit risk assessment model as a
pre-requisite for AMA operational risk models.
Assessment of Operational Risk Capital
281
termined by the type and source of data used. These data can be in the form of risk
self-assessments or historical loss experiences (or ‘near misses’) and be obtained
from internal or external sources. Conversely, the design of risk self-assessments,
the internal loss experience database and the information required from external
data consortiums are each influenced by the design of the AMA model that they
will serve. It is therefore essential to understand the AMA model design at the earliest possible stage of designing and implementing an operational risk assessment
framework.
The outline of the chapter is as follows: the first section outlines the models
used for computing operational risk capital (ORC) as the Value-at-Risk metric of a
total operational loss distribution. For a given percentile and risk horizon, component ORC estimates are determined by the granularity of calculation chosen. Then
these are aggregated over all business lines and operational risk types to obtain the
total ORC estimate. The second section discusses issues surrounding operational
loss data and the third section focuses on aggregating different operational risks to
obtain the total ORC, and aggregating operational risks with market and credit
risks to obtain the total risk capital for the firm. The last section summarizes and
concludes.
Prior to implementing an AMA the firm must identify events that are linked to operational risks, and map these events to their operational risk ‘matrix’ such as that
shown in table 1 below. Each element in the matrix defines an operational risk
‘type’ by its business line and operational event category. In the AMA, operational
risk capital is first assessed for each risk type (for which the AMA is the designated approach – other methods may be used) and then aggregated to obtain the
total operational risk capital for the firm.
The AMA may be chosen only for the most important risk types, depending on
the nature of the business. For example, operational losses for clearing and settlements firms may be concentrated in processing risks and systems risks. The definition of the important event types and business lines will be specific to the firm’s
operations and should also take account of the granularity that is required for the
AMA calculations. Increasing levels of granularity are necessary to include the
impact of insurance cover, which may only be available for some event types in
certain lines of business. It is also desirable to isolate those elements of the matrix
that are likely to be dominant in the final aggregation, and should therefore be the
main priority for risk control.
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Carol Alexander
Table 1. Operational Risk Matrix
Internal
Fraud
External
Fraud
Employment
Practices
& Workplace
Safety
Clients,
Products
& Business
Practices
Damage
to Physical Assets
Business
Disruption &
System
Failures
Execution, Delivery &
Process
Management
Corporate
Finance
Trading &
Sales
Retail
Banking
Commercial Banking
Payment
& Settlement
Agency &
Custody
Asset
Management
Retail
Brokerage
The AMA may be chosen only for the most important risk types, depending on the
nature of the business. For example, operational losses for clearing and settlements firms may be concentrated in processing risks and systems risks. The definition of the important event types and business lines will be specific to the firm’s
operations and should also take account of the granularity that is required for the
AMA calculations. Increasing levels of granularity are necessary to include the
impact of insurance cover, which may only be available for some event types in
certain lines of business. It is also desirable to isolate those elements of the matrix
that are likely to be dominant in the final aggregation, and should therefore be the
main priority for risk control.
Unfortunately these can be precisely those risks for which the data are very subjective, consisting of expert opinions or risk self-assessments that have a large
element of uncertainty. Somehow, this uncertainty in the data must be included in
the risk model. Qualitative judgments must be translated to quantitative assessments of risk capital using appropriate statistical methodologies.
Operational risks may be categorized in terms of frequency (the number of loss
events during a certain time period) and severity (the impact of the event in terms
of financial loss). Very low frequency high severity risks, such as a massive fraud
or a terrorist attack, could jeopardize the whole future of the firm. These are the
risks associated with losses that will lie in the very upper tail of the total annual
Assessment of Operational Risk Capital
283
Expected Frequency
loss distribution. Risk capital is not designed to cover these risks. However, they
might be insurable. High frequency low severity risks, which include credit card
fraud and processing risks, can have high expected loss but will have relatively
low unexpected loss. Often these risks are mostly covered by the general provisions of the business. Otherwise they should be included in the risk capital but,
unless expected losses are very high, the risk capital will be far lower than that for
medium frequency, medium severity risks. These are the legal risks, the minor
frauds, the fines from improper practices, the large system failures and so forth. In
general, these should be the main focus of the AMA.
A probability-impact diagram, or “risk map”, such as that shown in figure 1, is a
plot of expected loss frequency vs expected severity (impact) for each risk type/
line of business. Often the variables are plotted on a logarithmic scale, because of
the diversity of frequency and impacts of different types of risk. This type of diagram is a useful visual aid to identifying which risks should be the main focus of
management control. These risks correspond to the black crosses in the dark
shaded region. The reduction of probability and/or impact, indicated by the arrows
in the diagram, may bring these into the acceptable region (with the white background) or the warning region (the light shaded region).
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
Expe cte d Se ve rity
Fig. 1. A Probability-Impact Diagram
The total operational loss refers to a fixed time period over which these events are
to be observed. This time period is called the risk horizon of the loss model to emphasize that it is a forward-looking time interval starting from today. For regulatory purposes, both operational and credit risk horizons are set at one year, but for
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Carol Alexander
internal purposes it is also common to use risk horizons of less than or more than
one year. For ease of exposition we shall here only refer to the one-year horizon –
hence the AMA aims to model the annual loss distribution.
Having defined a risk horizon, the probability of a loss event (which has no time
dimension) can be translated into the loss frequency, that is the number of loss
events occurring during the risk horizon. In particular, the expected loss frequency, denoted λ, is the product of the expected total number of events (including non loss events) during the risk horizon, N and the expected loss probability,
p:
λ = Np
(1.1)
Sometimes it is convenient to forecast λ directly – this is the case when we cannot
quantify the total number of events N and we only observe loss events – and in
other cases it is best to forecast N and p separately – for example, N could be the
target number of transactions over the next year, and in that case it is p not λ that
we should attempt to forecast using loss experience and/or risk self-assessment
data.
Loss frequency (the number of loss events occurring during the risk horizon) is
a discrete random variable: it can only take the values 0, 1, 2, …., N. A fundamental density for such a discrete random variable (which nevertheless is only appropriate under certain assumptions)2 is the well-known binomial density:
N 
h( n ) =   p n ( 1 − p ) N −n n = 0, 1, … , N
n
(1.2)
However, since p is normally small the binomial distribution can be well approximated by the Poisson distribution, which has the single parameter λ, the expected frequency, as in (1.1) above. Incidentally, λ is also equal to the variance of
the Poisson distribution. The Poisson distribution has the density function:
h( n ) =
λn exp( − λ )
n = 0, 1, 2, ……
n!
(1.3)
If the empirical frequency density is not well modeled by a Poisson distribution –
for example, one could equate the mean frequency observed empirically with
λ, but then find that the sample variance is significantly different from λ – an alternative, more flexible functional form is the negative binomial distribution
which has the density function:
2
We must assume that the probability of a loss event is the same for all the events in this
risk type, and therefore equal to p; and that operational events are independent of each
other.
Assessment of Operational Risk Capital
 α + n − 1 1 


h( n ) = 
 n  1 + β 
α
 β 


 1+ β 
285
n
n = 0, 1, 2,
(1.4)
The lognormal distribution for loss severity, L, has the density function:
g( l ) =
 1  ln l − µ  2 
exp  − 
  (l > 0)
 2 σ  
2π σ l


1
(1.5)
Thus the log of the loss severity, or log severity for short, ln L, is assumed to be
normally distributed with mean µ and variance σ 2. High frequency risks can have
severity distributions that are relatively lognormal, but for some severity distributions a better fit is provided by a two-parameter density. Often we use the gamma
density:
 l
l α −1 exp  − 
 β
g( l ) =
(l > 0)
β α Γ( α )
(1.6)
where Γ(.) denotes the gamma function or the two-parameter hyperbolic density:
g( l ) =
(
exp − α β 2 + l 2
2βB( αβ )
)
(l > 0)
(1.7)
where B(.) denotes the Bessell function.
For a given operational risk type we have a discrete probability density h(n) of
the number of loss events n per year, and a continuous probability density g(l) of
the loss severity, L. The annual loss distribution f(x) is then given by compounding
the two densities, under the assumption that loss frequency and loss severity are
independent.3 Figure 2 gives a diagrammatic representation of the relationship between the annual loss distribution and its underlying frequency and severity distributions. Marked on the annual loss distribution are the expected loss (the mean of
this distribution) and the unexpected loss at the 99.9 percentile (the unexpected
loss at the α percentile is the difference between the upper α percentile and the
mean of the annual loss distribution)
3
Under this assumption it is not necessary to specify conditional severity
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Carol Alexander
Annual Loss Distribution
Unexpected
Loss
th
99.9 Percentile
Annual Loss
Frequency Distribution
No. Loss Events Per Year
Severity Distribution
Loss Given Event
Fig. 2. The Loss Model
Often expected losses are already included in the normal cost of business – for example, the expected loss from credit card fraud my be included in the balance
sheet under ‘operating costs’. In that case operational risk capital (ORC) should
cover only the unexpected loss at some pre-defined percentile of the loss distribution. ORC is to cover all losses, except the highly exceptional losses, that are not
already covered by the normal cost of the business. The definition of what one
means by ‘highly exceptional’ translates into the definition of a percentile of the
loss distribution, where only losses exceeding this amount are ‘highly exceptional’
and these one should attempt to control using scenario analysis. The ORC is equal
to the unexpected loss (plus the expected loss, but only if expected losses are not
provisioned for elsewhere) at some percentile. The Basel Committee recommends
99.9% for the calculation of operational risk capital in banks; internally, for companies wishing to maintain a high credit rating, it is common to use even a higher
percentile.
Under certain assumptions (which are rather strong, but nevertheless appear to
be admissible by regulators) there are simple analytic formulae for the expected
loss and the unexpected loss in the annual loss distribution. This is based on the
‘internal measurement approach’ (IMA). The most general operational risk capital calculation is performed using the ‘loss distribution approach’ (LDA).
The IMA provides a useful benchmark, a lower bound for the operational risk
capital calculated using the full LDA. The basic formula given in the Basel 2 Accord is:
ORC = gamma × expected annual loss = γ × NpL
(1.8)
Assessment of Operational Risk Capital
287
where N is a volume indicator (a proxy for the number of operational events), p is
the probability of a loss event, L is the loss given event, and gamma (γ) is a multiplier. The γ factor is similar to a risk/return ratio and we shall see below that depends on the risk type but only through the loss frequency, it does not depend on
the size of the loss given event.
Note that NpL only corresponds to the expected annual loss when the loss frequency is binomially distributed and the loss severity is not regarded as a random
variable. Thus a very strong assumption of the IMA is that each time a loss is incurred, exactly the same amount is lost (within a given risk type).
The Loss Distribution Approach (LDA) requires a simulation algorithm to generate an annual loss distribution as follows:
1. Take a random draw from the frequency distribution: suppose this simulates n loss events per year;
2. Take n random draws from the severity distribution: denote these simulated losses by L1, L2, …Ln;
3. Sum the n simulated losses to obtain an annual loss X = L1 + L2 + …+
Ln;
4. Return to step 1, and repeat several thousand times: thus obtain X1, …..XM
where the number of simulations M is a very large number;
5. Form the histogram of X1, …..XM: this represents the simulated annual
loss distribution;
6. The ORC for this risk type is then the difference between the 99.9 th percentile and the mean of the simulated annual loss distribution, or just the
99.9th percentile, if expected losses are not provisioned for elsewhere.
1
1
Random draw
Frequency CDF
0
n random draws
Severit y CDF
0
n
L 3 …L 1 …L n
Fig. 3. Simulating the Annual Loss Distribution
Figure 3 illustrates the first two steps in the simulation algorithm. The use of empirical frequency and severity distributions is not advised, even if sufficient data
are available to generate these distributions empirically. There are two reasons for
this. Firstly, the simulated annual loss distribution will not be an accurate representation if the same frequencies and severities are repeatedly sampled. Secondly,
there will be no ability for scenario analysis in the model unless one specifies and
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Carol Alexander
fits the parameters of a functional form for the severity and frequency distributions.
The basic IMA ORC formula (1.8) can be extended to a set of analytic ORC
calculations, which depend on the choice of frequency density as follows:
Binomial ORC = γ LN p
(1.9)
Poisson ORC = γ Lλ
(1.10)
Negative Binomial ORC = γ Lα β
(1.11)
Notes:
1.
In each case γ is determined by the loss frequency distribution. Its value
may be taken from the appropriate tables (binomial, Poisson, or negative
binomial, and with or without expected loss in the ORC) [Alexander
(2003)];
2.
The ORC should increase like the square root of the expected frequency:
it will not be linearly related to the size of the banks operations;
3.
The ORC is linearly related to loss severity: high severity risks will therefore attract higher capital charges than low severity risks;
4.
The ORC depends on γ, which in turn depends on the dispersion in the
frequency distribution. High frequency risks will have much lower γ , and
therefore lower capital charges, than low frequency risks;
5.
Similar formulae apply to the calculation of operational risk capital at a
percentile other than 99.9%. In this case, the internal operational risk
economic capital is given by using a γ factor that is tabulated using the
required percentile.
6.
The inclusion of insurance implies a reduction in the ORC, whereby the
above formulae are multiplied by the factor (1 – r) where r is the recovery rate (Alexander 2002);
7.
The implicit assumption in all the analytic formulae given above is that
severity is not a random variable. That is, each time a loss occurs exactly
the same amount is lost, which is clearly very unrealistic. To include the
simple formulae above should be multiplied by the factor √(1 + (σl /µl)2)
where µl is the expected loss given event and σl is its standard deviation
(Alexander and Pezier 2003a). This ORC will be similar to the ORC calculated by the LDA (Alexander 2003a).
Note 7 leads to an important point: as loss severity becomes more uncertain, this
has a direct, almost linear impact on the capital charge.4 Thus for any given risk
4
A downwards adjustment in gamma is necessary when severity uncertainty is included,
but this is very small compared to the increase in the ORC resulting from the additional
factor √(1 + (σl /µl)2).
Assessment of Operational Risk Capital
289
type the ORC will always be much larger when calculated using the LDA methodology than it is under the IMA – or indeed when any other analytic formula
based on the assumption of non-random severity is used. This difference can be
particularly pronounced for operational risks where the loss amount is highly uncertain. For these risk types, the risk capital calculation based on the LDA can easily give a result that is 10 times larger than the IMA risk capital charge.
Internal and external loss experiences (or near misses) and risk and control selfassessment data each have a role to play in the quantitative assessment of operational risks. Another source of loss data is the public data on high severity operational losses that may be collected through tracking events that have been recorded
in the news.5 These data give some idea of the scale of very extreme losses that
could, in theory, be experienced. But they represent only the tail of the severity
distribution and not the more usual loss amounts in the center of the distribution.
Thus public data have limited use in a quantitative framework, although their
qualitative use can include the validation of self-assessments (Davidson 2003).
When different types and sources of data are to be used in the risk model, the
choice of functional forms to use in the loss model depends less on the goodness
of fit to historical loss experience data than the necessity to choose distributional
forms that
a.
b.
c.
d.
are appropriate for the type of data
allow the use of proper mathematical methods for filtering data,
admit the quantitative use of data from different sources
allow self-assessments to be validated and updated using external
data, and/or internal loss experience
There is no point in applying a statistical test to decide which frequency and severity distributions provides the closest fit to loss data: those with more parameters
will always fit better than those with less. However, this does not imply that one
should always choose the negative binomial frequency and the hyperbolic severity
distributions. In fact the choice of functional form really depends on both the type
of data – i.e. loss experience vs. self-assessments – and the source(s) of the data.
The binomial frequency is only used when a value for N can be specified. In that
case it has a direct application to risk self-assessment design, and Bayesian methods for combining internal loss experience with risk self-assessment data are easy
to derive (Alexander 2003b). The Poisson frequency is the most flexible of the
5
The OpVar loss database is a useful source of such information (see
www.opvantage.com).
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Carol Alexander
functional forms. It also has a direct application to risk self-assessments and it is
the standard distribution to apply to external data when information about the
“size” of contributing banks is restricted. By contrast, although the negative binomial distribution will always provide the closest fit to loss experience data, it is
very difficult to apply to risk self-assessment data. Moreover it can only be applied to external consortium data when the consortium data are scaled to a bank of
size unity.
For the severity distribution there are many good reasons to use a lognormal distribution (Alexander 2003b). Although this will not capture well the extremely
high impact losses these are, by definition, very rare indeed. They are the “news”
events that are recorded in public operational risk databases. They normally exceed the 99.99%-ile of the loss distribution and they do not require self-insurance
through risk capital. In fact, they may be associated with externally insurable
events. These are not the types of losses that regulatory capital should attempt to
guard against; nor are they relevant for the internal allocation of capital.
However high impact losses that lie in the upper tail of the severity density
should be included in the risk capital model. Some models advocate the use of
special distributions for fitting these ‘tail’ observations. The generalized Pareto
and other distributions from the class of ‘extreme value’ distributions have been
applied. However, in our view, the use of extreme value theory (EVT) is totally
inappropriate for modeling operational risks in banking. Whilst EVT may have
found useful applications to high frequency tic-by-tic financial market data, one
should not forget that it was introduced (almost 50 years ago) to model the distributions of extreme values in repetitive, independent identically distributed processes, such as those observed in the physical sciences (Gumbel 1958 and Embrechts 1991).To attempt to fit a generalized Pareto distribution, or any other
extreme value distribution, to the sparse and fragmented data that are available for
large operational losses is “a triumph of hope over reason”(Alexander 2003, p.
308). Moreover, once ‘fitted’ how can the distribution then be used with other
types and sources of data?
In many cases loss data should be filtered before fitting a distribution. Any extremely high impact losses, whilst (hopefully) being a major focus of risk control,
are not likely to give sensible risk capital estimates. If included in the loss data,
the risk capital estimate could be multiplied by a factor of 100.
Even internal loss data may require some degree of filtering. For example, loss
frequency may show a marked trend over time; either because the data collection
has become more efficient, in which case average frequency may have an upwards
trend, or because the risk culture within the institution is changing, and with the
increased awareness fewer operational losses are being made. These two trends
are in the opposite direction, so it can happen that no trend in frequency is apparent when both influences are having an effect. Given the scarcity of data it would
be very difficult to isolate the effect of each trend on annual data, but for medium
Assessment of Operational Risk Capital
291
to high frequency risks recording loss frequency on a monthly basis can allow
such trends to be identified, as in figure 4(a). If there is a distinct trend in loss frequency this needs to be identified, understood, and possibly filtered out of the data
(depending on its assumed cause). This filtering can reduce the skewness and leptokurtosis in the frequency density, as shown in figure 4(b).
Trend in Loss Frequency
160
Number of Loss Events
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Month
Fig. 4.a. Detecting a Trend in Frequency
De-Trended Frequency
6
Number of Months
5
4
3
2
1
0
1
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Number of Loss Events
Before de-trending, the (blue) frequency density is highly skewed and leptokurtic. After detrending the (purple) frequency density moves closer to normality.
Fig. 4.b. The Effect of De-Trending of Frequency
Of course, not all losses due to operational risks are recorded. The pilfering of office equipment is an act of internal fraud, but the loss amounts are normally too
small to be of importance to most banks. Therefore even internal databases set a
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Carol Alexander
"threshold" for the loss amount, so that only losses in excess of this threshold are
r ~ c o r d c dThe
. ~ effect of a threshold loss is to "truncate" the severity distribution:
the observations in a loss database are observations only on the truncated distribution, whose lower tail has been cut off. We need to "de-truncate" this severity distribution so that the parameters of the entire severity distribution, including the
lower tail, can be used in the loss model. Note that, when log severity is normal
there is a simple relationship between the truncated distribution and the untruncated distribution.
Many banks are now forming data consortiums that are aimed at the anonymous
sharing of loss experiences above a defined threshold a m o ~ n t If
. ~banks that are
engaged in similar operations are able to share their experiences over a long period of time, the consortium data have some very useful applications. But consortium data will need to be scaled so that it reflects the "size" of the contributing
banks' operations in the given line of business.
Let v ~denote
, ~ the "sizc" of the operations of bank k in this business unit during
time period t . A standard proxy for this "size" is the gross income in the given
business unit during that time interval. Knowledge of v ~would
, ~ allow a relatively
precise scaling of the data, but evcn if the consortium has been able to collect data
on v ~ it, is~very unlikely that the consortium will be able to make this very sensitive data available to its members. Possibly the consortium would reveal information on vk , the average size (e.g. avcragc gross income per business line) of each
contributing bank, averaged over a longer time period - c.g. during the period that
data has been reported. But it is possible that the consortium will decide that even
the information contained in vk is too sensitive to release to its members. Nevertheless knowledge of each vk - and preferably of cach ~ k , is~ ncccssary to scale
the external frequency and severity data so that it is suitablc for a given member of
the consortium. When sensitivity of the "size" data is an issue, the consortium has
the option to release data that has been pre-scaled to a bank of sizc unity.
Frequency requires scaling for lines of business where frequency is fairly homogeneous. For example, doubling the size of the business, more or less doubles
the number of losses. IIigh volume processes, such as those concerned with payment and settlements, agency and custody and retail banking, are in the lines of
business most likely to fall into this category. In some lines of business, such as
corporate finance, trading and sales and commercial banking, loss severity may bc
more or less homogeneous. This happens when increasing the sizc of the busincss
increases the size of loss rather than (or as well as) the number of loss events. In
that case external loss severity will need to bc scaled to reflect the size of thc contributing bank - and one way to do this is to assume that loss severity increases
proportionally with size.
Assuming frequency and severity are homogenous of degree one, the data scaled to a bank of size unity arc:
-
"Often,
the internal loss threshold will be less than the threshold set by an external data
consortium. For example, all losses over 5,000$ may be reported internally, but only
losses over 10,000$ may be reported to the consortium.
For example, see the Operational Riskdata Exchange Association (w~w.ol.x.com)
Assessment of Operational Risk Capital
293
•
•
Scaled frequency
= nk,t /vk,t
Scaled severity
= lk,t /vk,t
where nk,t denotes the total number of loss events during time period t and lk,t denotes the severity of a loss event occurring during time period t, for a given member bank k. Consortia not wishing to disclose all vk,t to participants can simply distribute these scaled frequencies and severities so that each internal bank –
knowing their own size vt – is able to scale them by the relevant amount.
!
Risk self-assessment can play an important role in the quantification some operational risk types. There may be no internal historical loss experience data for some
low frequency operational risks, such as certain types of fraud, legal and other
risks. And in any case where internal loss experience data are available, when a
firm has undergone a re-structuring of management and processes the historical
data may be of little relevance to future practices. Also, when operations have a
significant change in size, such as would be expected following a merger or acquisition, or a sale of assets, it may not be sufficient to simply re-scale the capital
charge by the size of its current operations. The internal systems, processes and
people are likely to have changed considerably and in this case the historical data
no longer have the same relevance for the future. Finally, even for risks where
there are plenty of internal loss experiences from processes that have not undergone a significant change during the period of data collection, the historical nature
of these loss events limits their use because data may have been collected under
conditions that will not continue into the future – for example, external market
conditions may be different during the next year – and so again, historical estimates will not be accurate forecasts of the future.
Risk self-assessments are designed to be forward looking – historical loss experiences are not. Internal risk self-assessments can therefore provide very useful
indications of expectations during the risk horizon of the risk model, provided
they are obtained using established procedures and are properly validated on an
ongoing basis.
The aim of a risk self-assessment is to obtain a forecast of the loss distribution,
in a given line of business and operational risk event type. However, taken alone,
these self-assessments are purely subjective. Therefore it is necessary to validate
them using some type of loss experience data. Internal loss experiences may only
be collected after some period of time, if at all. In that case self-assessments can
only be validated using external data, or the opinions of experts. When (scaled)
external loss experiences are used for the validation, there can be marked differences between the loss experience and self-assessment data that are easily explained by differences in operational risk management. Nevertheless, large differences arising during the validation process should certainly warrant detailed
checks of the self-assessment responses, and possibly also a modification in the
assessment design.
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Carol Alexander
Within any given event type and line of business, self-assessment questions are
often completed by a number of different individuals. For example, payments and
settlements encompass several different operational processes: execution, order
checking, transference, account management, and so forth. Within each of these
operational processes, an assessment of loss frequency and severity due to a single
event type – such as a systems failure – may be given by (at least) one individual.
If the bank aims to assess the ORC by lines of business that are less granular than
this – for example, according to the Basel recommendations – then the selfassessments from different processes within a given line of business need to be
combined into a single assessment of the frequency and severity covering the
whole of the payments and settlements business. Typically, this process will be facilitated by a “workshop” – perhaps overseen by an expert – where the individuals
discuss their assessments and make a collective decision about the overall operational loss profile of the business. This adds significantly to the subjectivity of the
self-assessment process, and the validation of such a “consensus” assessment becomes even more important.
It is also useful to combine self-assessment data with historical loss experience
data, so that both are reflected in the loss distribution forecast. Risk selfassessments can be a useful first step towards establishing an internal operational
risk database. However, if/when loss experience data do become available, the
firm should update the self-assessments to base their forecasts on both loss experience and their risk self-assessments.
" !!
By focusing on the proposed use of different types and sources of data, the implications of model choice can be fully explained. For example, suppose a firm employs a negative binomial frequency distribution and a gamma severity distribution for internal loss experiences. Then how can the severity data be de-truncated
and combined with data from an external consortium, which may of course use a
different truncation level? And if risk self-assessments are to play a role, how can
they be designed to be compatible with this choice of functional forms? How can
they be validated by risk assessments based on external data and updated using internal loss experiences? By addressing questions such as these and considering
which choices are crucial from the perspective of AMA model risk, the firm
should derive a ‘data-oriented’ approach to AMA model design with prescribed
model selection criteria that are entirely dependent on the way that multiple
sources of data are applied to the model.
The Basel Committee states: “The bank will be permitted to recognize empirical
correlations in operational risk losses across business lines and event types, pro-
Assessment of Operational Risk Capital
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vided that it can demonstrate that its systems for measuring correlations are sound
and implemented with integrity” (Basle Committee 2001). But correlation is not
an adequate measure of dependency for operational risks. Correlation only captures linear dependence and even in liquid financial markets correlations can be
very unstable over time. In operational risks is it more meaningful to consider
general dependencies rather than to restrict the relationships between losses to
simple correlation. In fact, the proper aggregation of operational risks should use a
copula function that allows for asymmetric dependencies that are larger in the tail
than the core of the annual loss distribution (Alexander 2003c).
Modeling dependencies between various operational risks is not a simple matter.
Yet modeling dependency between frequencies is an important issue when operational losses are grouped in time. In this case Frachot, Georges and Ronacalli
(2001) advocate the use of a multivariate extension of the Poisson distribution to
model correlated loss frequencies. Loss severities may also be dependent, as for
example when nominal losses are affected by the same macroeconomic variable
(e.g. an exchange rate or interest rate).
#
How should a bank specify the dependence structure between different operational
risks? The answer lies in linking the dependencies between operational risks to the
likely movements in common risk drivers.
Table 2. Key Risk Drivers and Key Risk Indicators for Different Operation Risks
Risk
Internal Fraud
External Fraud
Clients, Products and Business Practices
Employment Practices and
Workplace Safety
Damage to Physical Assets
Business Disruption and
Systems Failures
Execution, Delivery and
Process Management
KRD
Management & Supervision
Recruitment Policy (Vetting)
Pay Structure (Bonus Schemes)
Systems Quality
(Authentification Processes)
Product Complexity
Training of Sales Staff
Recruitment Policy (Discrimination)
Pay Structure
Safety Measures
Location of Buildings
Systems Quality
Back up Policies
Business Continuity Plans
Management & Supervision
Recruitment Policy (Qualification)
Volume of Transactions
Training of Back Office Staff
Pay Structure
KRI
Time Stamp Delays (Front Running)
Number of Unauthorized Credit
Card Transactions
Number of Client Complaints
Fines for Improper Practices
Number of Employee Complaints
Staff Turnover
Time off Work
Insurance Premiums
System Downtime
Number of Failed Trades
Settlement Delay
Errors in Transactions Processing
Key risk drivers (KRDs) are a fundamental tool for operational risk management.
These are the control variables that can be directly affected by management deci-
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Carol Alexander
sions. By contrast, management has no direct control over the key risk indicators
(KRIs), which are variables that are linked to operational losses and have some
causal dependency on one or more KRDs. Table 2 gives some examples of these
for various types of operational events.
It is impossible to specify a “correlation” matrix between all types of operational risks with any degree of uncertainty. However it is possible to model dependencies by examining the impact that changes in key risk drivers have upon
different categories of operational risks. This can be achieved through a scenario
analysis based on a statistical model where each KRI is explained by several KRD
factors, and where some factors are common to several KRIs. For example:
Scenario 1: Rationalization of the back office with many people being made redundant. Key risk drivers such as transactions volume, staff and skill levels, and
so forth are adversely affected. The consequent difficulties with terminations, employee relations and possible discriminatory actions would increase the “Employment Practices & Workplace Safety” risk. But other risks would also increase: the
reduction in personnel in the back office could lead to an increased risk of “Internal Fraud” and “External Fraud”, since fewer checks would be made on transactions, and there may be more errors in “Execution, Delivery & Process Management.” The other risk types are likely to be unaffected.
Scenario 2: Expansion of business in complex products, introducing a new
team of quantitative analysts. Model errors, product defects, aggressive selling and
other activities in the “Clients, Products & Business Practices” category may increase in both frequency and severity. But again, other risk may increase: Management and supervision may be adversely affected, and the new staff may be inadequately vetted during the recruitment procedure. Hence “Internal Fraud” could
become more likely and potentially more severe. “Business Disruption & System
Failures” will become more likely with the new and more complex systems. Finally there are many ways in which “Execution, Delivery & Process Management” risk would increase, including: less adequate documentation, more communication errors, collateral management failures and so forth.
$%
!
Consideration of scenarios such as this may lead to the conclusion that strong
positive dependencies do exist between different types of operational risks. However ‘correlation’ is a limited measure of dependency, which may be applicable to
linear dependencies between financial asset returns, but is likely to be inappropriate for capturing dependencies between operational loss distributions. Instead, one
could employ a ‘copula’, which generalizes the concept of correlation to nonelliptical distributions. If two densities f(x) and g(y) are independent then their
joint distribution is simply their product h(x,y) = f(x)g(y). If they are not independent then h(x,y) = c(x,y)f(x)g(y) where c(x,y) is the density function of the copula.
Very many copula functions are available to model dependencies that are more
general than correlation. For instance, ‘tail’ dependencies can be captured by the
Assessment of Operational Risk Capital
297
‘Gumbel’ copula. Only when the Gaussian copula is applied is correlation the dependency measure.
0. 15
0. 12
0. 09
0. 06
0. 03
0
0
5
10
15
20
Loss
25
Fig. 5. Two Annual Loss Densities
To understand the influence that such dependencies can have on the aggregate
ORC, consider a numerical example. The two annual loss distributions with density functions shown in figure 5 have been fitted by a mixture of two normal densities and a gamma density respectively.8 Different joint densities are now obtained using the Gaussian copula with ρ = 0.5, 0, −0.5 respectively; the Gumbel
δ copula with δ = 2 and the Gumbel α copula with α = 0.5. Note that
δ = 1, ρ = 0 and α = 0 all give the same copula, i.e. the independent copula. In
each case the aggregate ORC is obtained by convolution over the joint density
(Alexander 2003a).
Table 3. Aggregation of ORC Under Different Dependency Assumptions
Expected Loss
99.9th Percentile
ORC
ρ = −0.5
22.3909
41.7658
19.3749
ρ=0
22.3951
48.7665
26.3714
ρ = 0.5
22.3977
54.1660
31.7683
δ=2
22.3959
54.9715
32.5755
α = 0.5
22.3977
57.6023
35.2046
Table 3 summarizes the ORC in each case. Note that the mean (expected loss) is
hardly affected by the dependency assumption: it is approximately 22.4 in each
case. However the total ORC at the 99.9th percentile is very much affected by the
assumption one makes about dependency. It could be as small as 19.37 (assuming
correlation as the dependence measure, with the Gaussian copula and ρ = −0.5) or
8
The normal mixture has with probability 0.3 on the normal with mean 14 and standard
deviation 2.5 and has probability 0.7 on the normal with mean 6 and standard deviation
2. The gamma density has α = 7 and β = 2.
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Carol Alexander
as large as 35.2 (assuming asymmetric upper tail dependence with a Gumbel α
copula with α = 0.5).
!
&
The example above showed that small changes in the dependency assumption can
produce estimates of aggregate ORC that is doubled − or halved − even when aggregating only two operational risk types. Obviously the effect of dependency assumptions on the aggregation of many operational risks can be substantial. However, the most important dependency is not the dependency between one operational loss and another - it is between the costs and revenues of a particular activity.
Operational risks are mostly on the cost side, whereas the revenue side is associated with market and/or credit risks. In fact vertical dependencies, between a given
operational risk, and the market and/or credit risks associated with that activity,
are the most important dependencies to account for when estimating the total risk
of the bank. The effect of accounting for dependencies between different operational risks may be substantial, but this effect will be only marginal compared to
the effect of accounting for dependencies between operational, market and credit
risks in the total risk capital for the firm.
Table 4 illustrates this point by considering the economic capital (EC) data from a
sample bank (Alexander and Pezier 2003b). In table 4(a) the data have been scaled
so that the sum over all elements is 100, so that each element represents the percentage contribution of that risk to the total. The row totals give the aggregate EC
by business line and the column totals give the aggregate EC by risk type, under
the assumption that all risks are fully dependent. This is also shown in the first
column of table 4(b). The second column of table 4(b) gives the aggregate EC by
risk type and (in the last row) the total EC for the bank, under that assumption that
all risks are independent. The EC due to operational risk falls from 12.3 to 5.21,
and the total EC for the bank is only 1/3 of the total EC calculated under the assumption of full dependency.
Table 4.a. Economic Capital (EC) for a Sample Bank
Economic Capital
Corporate Finance
Trading and Sales
Retail Banking
Commercial Banking
Payment and Settlement
Agency and Custody
Asset Management
Retail Brokerage
Total
Market
7.2
13.3
3
2.1
0.3
0.7
6.5
0.9
34
Credit
4.8
7.3
10.3
24.1
1.6
1.3
3.4
0.9
53.7
Operational
2
3.8
2.2
1.3
1.2
0.9
0.6
0.3
12.3
Total
14
24.4
15.5
27.5
3.1
2.9
10.5
2.1
100
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299
Table 4.b. Aggregation of EC Under Different Dependency Assumptions
Market
Credit
Operational
Total
Full Dependency
Independent
34
53.7
12.3
100
16.88
27.91
5.21
33
Mixed Correlation
(a)
34
53.7
12.3
66
Mixed Correlation
(b)
16.88
27.91
5.21
50
The last two columns report the aggregate EC under the following assumptions:
(a) full dependence between business lines but independence between risk types
and (b) independence between business lines but full dependence between risk
types. In case (a) operational risk EC remains at 12.3, but the total EC is now only
2/3 of the total EC calculated under the assumption of full dependency. In case (b)
total EC is now 50% of the original total, and the relative contribution of operational risks has fallen from 12.3% to 10.42% of the total. In general, assuming less
than full dependence between risk types has the effect of making the smaller risks
(like operational risk) appear even smaller.
There are many reasons why firms should adopt an AMA. In addition to a certain
pressure on banks by some regulators, rating agencies may require financial firms
to assess their operational risks using the best practice of the industry. This may be
particularly so for firms whose capitalization has been low, for example because
their brokerage, agency and custody services or their specialization in transactions
processing has attracted little or no capital charge for market or credit risks. In
general, the proper assessment of operational risks is a key element of the operational risk management framework and is therefore an important determinant of
the value of the firm. Finally, following the Sarbannes-Oxley Act in the US in July
2002, the accurate assessment of operational risks can be key to the legal indemnity of its senior executives.
With institutions already mapping events to operational loss categories and
building warehouses of operational risk data, the pivotal issue is increasingly the
analytical methodologies, the so-called ‘AMA’. Even though the data collection is
still at a relatively early stage, the AMA model design will influence the data collected, so users already need to know the modeling methodology, even if it is not
fully implemented until a later stage.
Operational losses are firm specific and contextual, so the appropriate analytical
approach varies from user to user, depending on the type and quality of data available. The standard packaged application of the ‘loss model’ approach has traditionally offered the end-user complete flexibility to define the analytical method-
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Carol Alexander
ology, an approach that has led to complex and expensive AMA systems. But
blind use of descriptive modeling can produce perverse results. The data need to
be analyzed within a coherent and mathematically consistent framework.
There are three sources of model risk in the AMA risk models. In increasing order of importance these are: inappropriate choice of functional forms for the frequency and severity distributions; incomplete data and inadequate data handling;
and inappropriate assumptions about dependencies between operational risks and
between operational, market and credit risks. Much of the recent research on operational risk assessment has focused almost exclusively on the first of these
sources of error. But far greater risk model errors stem from the use of selfassessment and/or external data, and from the consequent mis-handling of these
data as they are filtered, scaled and combined. The design of a self-assessment
should take into account the need to validate self-assessments using other types or
sources of data, and the distinct possibility that the self-assessment data will be
combined with loss experience data when generating a forecast of the loss distribution.
By far the most important type of AMA model risk is that associated with risk
aggregation. Indeed, this is also true for market and credit risks. To invest time
and expense in developing an accurate assessment of a particular type of operational risk within a given business line seems rather asinine when the end result is
to be aggregated using extremely crude assumptions, such as full dependencies between all risks.
In conclusion, there is a clear need for a prescriptive and data-oriented modeling
methodology that is specifically designed to handle multiple types of data, replacing the complexities associated with flexibility and description with the relative
simplicity of a prescribed solution. In addition this solution must provide a link
between the drivers that are the focus of risk control and the risk capital assessment so that the proper aggregation of operational risks is determined through the
identification of a common set of key risk drivers.
Alexander C (2002) Rules and Models: Understanding the Internal Measurement Approach
to Operational Risk Capital RISK 15: 1 pp S2-S5
Alexander C (2003a) Statistical Models of Operational Loss. In: Alexander C (ed.) Operational Risk: Regulation, Analysis and Management FT – Prentice Hall, Pearson Education, London
Alexander C (2003b) Technical Document for OpRisk2021 Module 1
www.2021solutions.com
Alexander C (2003c) Operational Risk Aggregation. In: Operational Risk, 4:4 pp
Alexander C, Pezier J (2003a) Binomial Gammas, Operational Risk 2:4 pp
Assessment of Operational Risk Capital
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Alexander C, Pezier J (2003b) Assessment and Aggregation of Banking Risks 9th Annual
Round Table Report of the International Financial Risk Institute (IFCI). www.ifci.ch
Basle Committee (2001) Working Paper on the Regulatory Treatment of Operational Risk,
Consultative Paper 2.5, September 2001, available from www.bis.org
Davidson C (2003) Banks use Op Loss Databases for Qualitative Analysis. Operational
Risk 4:11 p18.
Embrechts P, Kluppelberg C, Mikosch T (1991) Modeling Extremal Events. Springer Verlag, Berlin
Frachot AP G, Roncalli T (2001) Loss Distribution Approach for Operational Risk. Credit
Lyonnais, Paris http://gro.creditlyonnais.fr/content/rd/home_copulas.htm
Gumbel EJ (1958) Statistics of Extremes. Columbia University Press, New York
Wilhelm Kross1
1
Director & Principal Consultant, Value & Risk AG, PO Box 1705, Bad Homburg, 61287, Germany, Tel. +49(0)170–461 8923, Wilhelm.Kross@Valuerisk.com
Abstract: ‘Operational Risk’ (OpRisk), interpreted in the context of Basel II as
‘the risk of loss resulting from inadequate or failed internal processes, people and
systems or from external events’, introduces a challenge to financial services providers and institutions who have to date predominantly controlled their organization’s exposure to market and credit risk. It is no surprise that approaches chosen
and methods employed are often dominated by the needs of corporate controlling,
accounting and regulatory reporting functions. This observation raises no immediate concerns, however, a lack of proactive cooperation from risk management
counterparts can render controlling-driven OpRisk initiatives inefficient and ineffective over time. Moreover, to portray a worst-case evolution, attempting to address Basel II’s OpRisk by means of introducing another set of independent software tools into an already complex IT infrastructure; generating yet another pool
of practically irreconcilable data originating from a mix of objective measurements and individual subjective assessments of uncertain quality; and employing
practically incompatible analysis methods; may actually raise an organization’s
exposure to OpRisk, hence defeating the objective.
Beyond reactively addressing stakeholders’ and managers’ requirements, appropriately implemented OpRisk management infrastructure, process and people
investments can be business enablers – turning risk management into an opportunity and addressing the requirements of Advanced Measurement Approaches
(AMA) as a desirable side-effect. Early-mover financial services providers and a
larger number of industrial organizations have in these efforts demonstrably generated net value. This article attempts to outline a generic roadmap for organizations who have in the past placed little or no emphasis on the management aspects
of OpRisk – allowing the early introduction of those evolutionary steps known to
generate considerable value quickly; highlighting some appropriate choices at
policy level; and building on practical lessons learnt in managing typical hurdles
throughout the implementation of suitable OpRisk management systems and processes.
Keywords: Operational Risk, Risk Management Strategy, Cost Effectiveness, Value Generation, Advanced Measurement Approach
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Wilhelm Kross
‘Operational Risk’ (OpRisk), interpreted in the context of the Bank for International Settlements’ various documents on Basel II as ‘the risk of loss resulting
from inadequate or failed internal processes, people and systems or from external
events’, introduces a challenge to financial services providers (FSPs) and institutions who have to date predominantly controlled their organization’s exposure to
market- & credit risk. While a fully automated implementation of minimum capital requirement calculations by business line, certain computer models and some
newly designed regulatory reports may be a challenge, the overall topic ‘OpRisk’
is not exactly new. A variety of internal and regulatory policy driven approaches
to managing OpRisk aspects can be found in all organizations affected by Basel II.
What really is new for FSPs will be the level of transparency and integration required to address so-called ‘Advanced Measurement Approaches’ (AMA), and the
intensity and the process oriented focus to be expected in future reviews and audits performed by regulating authorities.
Of course, considering how most FSPs have structured their business operations, many affected parties will attempt to develop and implement IT solutions allowing them to run their OpRisk controlling and reporting as a more or less fully
automated additional reporting feature. Attempting to solve the overall problem by
means of designing yet another ‘one-size-fits-all’ IT solution may not yield the desired results, though. Introducing another set of independent software tools into an
already complex IT infrastructure; generating yet another pool of practically irreconcilable data originating from a mix of objective measurements and individual
subjective assessments of uncertain quality; and employing practically incompatible analysis methods from a portfolio vs. a system and process oriented focus;
may actually raise an organization’s exposure to OpRisk, hence in some regards
defeating the objective and raising costs. Furthermore, a variety of operational
and cultural hurdles must be taken in order to create an environment compatible
with the new levels of transparency – which in turn imply hazards to the overall
success of the exercise and again raise the organization’s exposure to OpRisk.
This article attempts to demonstrate that these typical approaches are not the
best choice an organization can make, even over the short term. Instead, this article outlines a generic roadmap for organizations who have in the past placed little
or no emphasis on the management aspects of OpRisk – allowing the early introduction of those evolutionary steps known to generate considerable value quickly;
highlighting some appropriate choices at policy level; and building on practical
lessons learnt in managing typical hurdles throughout the implementation of suitable OpRisk management systems and processes. To set the stage this article initially highlights common approaches and likely pitfalls in what organizations will
likely need to address to become ‘AMA compliant’, followed by a discussion of
desirable side-effects in OpRisk management, and then followed by a framework
to analyze value generation potentials and set strategic or tactical priorities.
Operational Risk: The Management Perspective
305
Before engaging in a discussion on potential pitfalls on the road to AMA compliance a brief discussion of commonly practiced techniques and approaches is appropriate. The number of related books, articles, and brochures published is too
large to be quoted here; a select number and some recent workshop presentations
can be traced via Internet search engines. In addition, a variety of related documents was published by the Bank for International Settlements (see www.bis.org).
As was discussed elsewhere1 an in-depth analysis of approaches and methodologies chosen demonstrates that most organizations who have engaged in an explicit analysis and controlling of OpRisk, have started collecting internal loss data
and have developed an appreciation of future loss scenarios in moderated and
checklist-aided self-assessments. Approaches chosen seem to have been supported
by IT applications, however, the make-or-buy decision and the coverage of and
inherent assumptions made in commercially available and self-developed IT applications are heterogeneous. Fewer organizations are tracking key risk indicators
in addition to other key performance indicators commonly analyzed by their accounting, controlling and reporting functions. The use of generic risk maps and
external loss databases appears to be relatively common. Advanced formalized
hazard assessments seem to be employed by a select number of organizations perceived to be first-movers. The degree of cross-industry information exchange on
OpRisk aspects has perceivably increased as the increasingly large number of related conferences and workshops indicates; it may hence be argued that it is a
question of time until common knowledge converges within FSPs and to a lesser
extent with stakeholder groups and auditors & regulatory authorities.
These developments raise no immediate concerns. Most of the approaches chosen seem to address the new requirements formulated in the context of Basel II, at
least those aspects which have been stipulated as being likely to be implemented
in national law and financial supervision regulations. In summary, to be on the
safe side, FSPs would be well advised at this stage to consider at least the following aspects should they choose to implement a minimalist approach to OpRisk:
• Develop an operational framework and a policy for OpRisk, including operational guidance on consistent measurement and evaluation approaches
• Capture operational risk factors, collect and analyze OpRisk indicators
• Develop a loss database, prepare the integration of external loss databases
• Develop interfaces to other operational and risk control systems
• Integrate results into capital allocation models (risk/return management)
• Integrate results into financial & regulatory reporting systems
• Participate in data and information exchange relating to OpRisk
• Perform regular reviews of data collection and evaluation activities, and the
organizational OpRisk management effectiveness and related costs
1
see Kross and Schulz, 2004; Kross and Dobiey, 2003; Kross, 2000, 1999a, 1999b, 1999c.
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Wilhelm Kross
Engaging in these rather ‘minimalist’ first steps, may level the ground for AMA
and reduce the number of undesirable surprises for OpRisk reporting under the
Standard- or Basic Indicator Approaches as it evolves in the next few years.
The discussion has introduced several aspects, which organizations do address, or
should address, in order to become and remain compliant with regulatory requirements that may realistically be expected in the foreseeable future. Since the Standard- and Basic Indicator Approaches ignore risk reduction efforts and due to the
additional reporting and regulatory capital provision efforts lead to increased cost
profiles, it is a question yet to be answered whether they do in fact generate value
for anybody. Those who may instinctively be convinced that they do, would be
well advised to research the extensive body of literature on real option valuation
techniques, and in particular take into consideration that poorly implemented and
poorly integrated controlling activities may reduce managerial flexibility 2. Furthermore, as was argued earlier in this article, poorly implemented IT systems may
similarly raise exposures to OpRisk3. To quantify all pro’s and con’s would certainly extend beyond the scope and reach of this article; it is therefore simply
submitted here that a business case in favor of addressing OpRisk with the Standard- or Basic Indicator Approaches will be difficult. On the other hand, as is outlined in the next sections below, AMA clearly does present choices to convert
OpRisk management into an opportunity potentially generating net value.
Based on this discussion the Standard- or Basic Indicator Approaches seem to
make sense only in exceptional circumstances or as a short-term compromise. The
balance of this article therefore focuses predominantly on cost-effective means
and ways to address challenges for AMA compliance. While regulating agencies
have to date been reluctant to stipulate exactly which minimum requirements are
to be met, it is submitted as a working hypothesis that the following topic areas
become a focus of regulators’ and auditors’ scrutiny in years to come:
• The various ‘minimum’ requirements mentioned in the previous section
• The degree of independence between OpRisk controlling and management
• The completeness and comprehensiveness of self-assessment scenarios, calibrated to losses collected in a loss database
• The range of influence of loss scenario valuation, possibly tracing consequence chains with formalized hazard assessment techniques
• Computer-assisted evaluation and analysis models, possibly to be reviewed
independently from applications in specific organizations
2
3
For further information on real options techniques see Hommel et al. (2003) and Hommel
et al. (2001).
see also discussions published elsewhere (Kross, 2002a, 2002b)
Operational Risk: The Management Perspective
307
• Valuation techniques, in particular those valuations derived from the one or
other form of subjective assessments, and those used for the calculation of
overall loss distribution functions and their extreme ranges
• The degree to which OpRisk issues are addressed explicitly, using analysis
techniques appropriate for underlying residual risk & inherent correlations
As Morgan and Henrion (1990) submitted more than a decade ago “Professional
standards for model building are non-existent. The documentation of model and
source data is in an unbelievably primitive state.” “But despite, or perhaps because
of, the vast uncertainties inherent in most ... models, it is still not standard practice
to treat uncertainties in an explicit probabilistic fashion, outside the relatively
small fraternity of decision analysts.” These statements still bear some validity today, and are not limited to industry-specific applications – which introduces the
identification of potential obstacles and pitfalls.
Potential obstacles, implementation hurdles and sub-optimal results in an attempt to address AMA can conceivably be subject to various degrees of severity
and impact. The worst possible scenario would likely be the one of a total failure,
which may express itself in the form of a bankruptcy or the loss of the banking license. Partial failures, the darker range of the so-called ‘shades of grey’, may express themselves in the form of major compliance issues identified by regulators,
financial auditors, stakeholders and financial analysts. Somewhat less detrimental
would be the wide range of inefficiencies with their various indirect impacts;
spending far too much in OpRisk controlling and management, short-sightened IT
solutions, and approaches leading to major future repair and upgrading are worth
mentioning in this context. Reaching from the net cost effect into the net value
generation range, the other extreme can be classified as one of developing sustainable competitive advantage through OpRisk initiatives. This latter concept is discussed in subsequent sections on desirable side-effects further below, while the
following paragraphs focus on the various forms of OpRisk initiatives resulting in
net costs due to inappropriately managed implementation hurdles, inefficiencies
and conceptually sub-optimal approaches.
Instead of discussions at great depth, the following figures (see also Kross and
Schulz, 2004; Kross and Dobiéy, 2003) provide a summary of typical issues and
pitfalls experienced in individual or group subjective assessments – in particular
the variables affecting validity and defensibility …
308
Wilhelm Kross
Typical
Typical Issues
Issues relating
relating to
to the
the assessment
assessment of
of current
current and
and future
future performance
performance of
of OpRisk
OpRisk initiatives
initiatives
4Incomplete
4Incomplete analyses
analyses of
of hazards
hazards without
without appropriately
appropriately detailed
detailed cause/consequence
cause/consequence chains,
chains,
inappropriate
inappropriate levels
levels of
of detail
detail or
or aggregation,
aggregation, availability
availability of
of the
the right
right data
data at
at the
the right
right time
time
4Definition
4Definition of
of ‚acceptable
‚acceptable risk‘
risk‘ varying
varying and
and layered
layered for
for level
level of
of management
management and
and responsibility
responsibility ??
4Future
4Future strategy
strategy scenarios
scenarios and
and framework
framework for
for calibrating/weighting
calibrating/weighting of
of decision
decision criteria
criteria &
& objectives
objectives
4Development
4Development of
of short/long
short/long term
term alternatives
alternatives for
for desired
desired outcomes
outcomes &
& management
management measures
measures
Variables
Variables affecting
affecting cost,
cost, validity,
validity, defensibility
defensibility
4Comprehensiveness (several aspects ... whole universe)
4Detail (very few ... very many)
4Accuracy and precision (large potential errors ... high precision)
4Formalism (informal ... very formal)
4Iteration (single analysis ... multiple alternatives)
4Uncertainty (reflecting contingencies … probabilistic approach)
4Participants (internal work ... full justification to all stakeholders)
4Timing (ad-hoc = brainstorming ... inductive = from goals backwards)
4Communication (no explanation...continuous dissemination of all information to stakeholders)
Fig. 1. Issues and variables affecting validity and defensibility
… and potential traps in decision and probability assessments:
Traps
Traps -- decision
decision analysis
analysis
4Optimal alternatives not considered, poor comprehension, inappropriate constraints
4Inappropriate, incomplete or overlapping objectives, non-linear and/or dependent tradeoffs,
unspecified assumptions
4“Motivational” biases (management, expert, conflict, conservative, impressions)
4“Cognitive” biases (anchoring, availability, base rate, representative, coherence, conjunctive
distortions, overconfidence)
4Disagreements among multiple stakeholders (if partnering)
4Risk communication (explicit & open process, disclosure of weaknesses)
Traps
Traps -- analysis
analysis of
of quantities
quantities and
and schedules
schedules (deterministic,
(deterministic, probabilistic)
probabilistic)
4Poor conceptual model (misunderstood, missing scenarios, uncertain attributes & consequences)
4Inappropriate analytical model (too simple or too complex)
4Insufficient information re.: input parameters (data & judgment biases; variability & uncertainty;
random events/processes; contingencies)
4Data errors, analytical simplifications, non-representative samples
4Input & output correlations and inter-dependencies poorly incorporated
4Logistics, interpretation of trends, and changes during life cycle
Fig. 2. Typical traps in decision and uncertainty analysis
In addition, OpRisk analysts should develop an awareness of the impact of subjective assessments, both in future scenarios and combined losses (i.e., model assumptions) relating to failure events. Quantitative estimates may originate from a
range of individual subjective assessment techniques, as is outlined below …
Operational Risk: The Management Perspective
Problem
Problem Areas
Areas of
of Data
Data Quantification
Quantification
Uncertainty in Data
Inappropriate
Inappropriate quantification
quantification
of
of uncertainties
uncertainties
Insufficient
Insufficient problem
problem definition
definition
Unspecific
Unspecific assumptions
assumptions
Prejudice
Prejudice
Capacity
Capacity for
for analytical
analytical thinking
thinking
Interviewee
Interviewee does
does not
not reflect
reflect
uncertainties
uncertainties consistently
consistently //
correctly
correctly
Inaccurate
Inaccurate definitions
definitions of
of base
base
parameters
parameters and
and environment
environment
leads
leads to
to incorrect
incorrect results
results
Assumptions,
Assumptions, used
used for
for
evaluation
evaluation are
are not
not obvious
obvious
and
clear
and clear
Evaluation
Evaluation does
does not
not mirror
mirror
the
the actual
actual knowledge
knowledge of
of the
the
interviewee
interviewee
Limited
Limited understanding
understanding of
of
complexities
complexities and
and data
data
variability
variability
309
Techniques
Techniques of
of
data
data quantification
quantification
Self evaluation
Self evaluation with
Check lists / hints (e.g.
multiple choice)
Self evaluation based
on statistically available
data base
Expert witness
testimony
Aggregated individual
evaluations
Time series of
individual evaluations
Plausibility-checked &
calibrated evaluations
Probability encoding
Choice
Choice of
of evaluation
evaluation techniques
techniques must
must be
be adjusted
adjusted to
to the
the overall
overall framework:
framework: the
the simpler
simpler the
the better.
better.
Backward
Backward planning:
planning: determine
determine ultimate
ultimate defensibility
defensibility &
& credibility
credibility requirements
requirements for
for the
the required
required data.
data.
Forward
action:
pragmatic
commercial
framework
determines
approach,
subsequently
depending
Forward action: pragmatic commercial framework determines approach, subsequently depending on
on the
the
circumstances
circumstances further
further alignment
alignment &
& adjustment
adjustment takes
takes place
place until
until data
data quality-requirements
quality-requirements have
have been
been met.
met.
Fig. 3. Managing individual subjective assessments
... or from the far more comprehensive range of group subjective assessments,
each coinciding with a different set of challenges4.
Problem
Problem Areas
Areas and
and Possible
Possible Results
Results of
of Group
Group Assessment
Assessment
(complexities
(complexities of
of individual
individual subjective
subjective assessments
assessments also
also apply
apply here)
here)
Uncertainty
in Data
Datenunsicherheit
Disagreement
Disagreement w.r.t.
w.r.t. definitions
definitions
and
and underlying
underlying assumptions
assumptions
Convergence
Convergence
Inability
Inability to
to eliminate
eliminate systematic
systematic
errors
errors and
and evaluation
evaluation errors
errors
Judgements/evaluations
Judgements/evaluations are
are based
based on
on
different
different information
information sources
sources
Consensus
Consensus
Disagreement
Disagreement w.r.t.
w.r.t. the
the interpretation
interpretation
of
of the
the available
available base
base data
data
Disagreement
Disagreement
Differing
Differing opinions
opinions w.r.t.
w.r.t. the
the magnitude
magnitude
of
of the
the level
level of
of concern
concern
Skewed
Skewed results
results due
due to
to battles
battles in
in the
the
trenches,
trenches, political
political correctness,
correctness, etc.
etc.
Forms
Forms of
of Opinion
Opinion
&
& Decision
Decision Making
Making
Mechanical
Mechanical
Aggregation
Aggregation
Dominant individual value
Calibrated individual value
Average value calculation
Calibrated average value
Group statistics
Weighting methods
Psychological
Psychological Methods
Methods
Plausibility of the individual
evaluations
Open forum
Delphi panel
Group probability encoding
Formal group evaluation
Selection of evaluation techniques to be adapted to suit the overall circumstances: the simpler the better.
Group evaluation is not always the sum or the average of the individual evaluations.
Process of evaluation is a critical factor, also for demand of additional evaluation & sensitivity analysis.
Depending on circumstances further adjustment until data quality requirements have been met.
Fig. 4. Managing group subjective assessments
These graphical representations are considered self-explanatory; outlining a comprehensive set of challenges potentially experienced by decision makers in search
4
For further information on challenges in subjective probability assessments see Shapira
(1995), Morgan & Henrion (1990), Keeney (1992), Goldratt (1992), Howson & Urbach
(1989), Kahnemann et al. (1982), Puppe (1990), and von Winterfeldt & Edwards (1986).
310
Wilhelm Kross
of approaches to OpRisk which do not imply partial failures. It is submitted that
an application of appropriate OpRisk techniques on the residual OpRisk leads to
continuous improvement in this regard.
Even less detrimental than partial failures but significantly more difficult to demonstrate in terms of net value destruction, is the wide range of inefficiencies with
their various indirect impacts. For example, overspending in OpRisk controlling
and management, creating bottlenecks through short-sightened IT solutions, and
approaches leading to major future repair and upgrading, have already been mentioned in this context. Less obvious are ‘softer’ issues like the net impacts of addressing OpRisk with an unbalanced and stifling controlling vs. management approach, or simply addressing OpRisk with those resources in the risk controlling
department which are not suitable for higher challenges.
Analytically, inefficiencies are best highlighted in a process context. The competitive benchmark would be the construct of a lean process which leads to the interim or ultimate goals efficiently and effectively. This introduces several contextual issues though. First, different organizations may choose different strategies
and different ultimate goals. Hence what constitutes a value-added activity vs. an
inefficiency may differ widely. Second, regulatory thresholds and the respective
focuses of future regulatory audits are still somewhat uncertain. Hence ‘how safe
is safe’ remains to be debated. Third, OpRisk initiatives cannot be seen entirely
removed from other initiatives in an organization. Hence what adds net value to an
organization because it addresses an organizational issue or introduces desirable
side-effects, differs widely from one organization to another. Fourth, as was demonstrated elsewhere5, valuation techniques are still in a rather primitive state, at
least those commonly practiced in real-life organizations and by financial analysts.
Hence what is or can be measured to identify a net value generating initiative vs.
an inefficiency is highly debatable.
Yet another consideration often integrated poorly, is the risk communication
strategy. The reader may imagine the potential impact should an ex-employee of a
large bank, or a saboteur for that matter, divulge in-depth information on experienced losses to the media; many board members may not have their communication strategies prepared for such events, which coincidentally is a surprisingly
clear indication of commonly practiced management paradigms in banks. The degree of inefficiency or failure is of course uncertain and depends on the specific
circumstances; poor risk communication strategies may well fall into the category
of partial or ultimately total failures. The reader is advised that a large number of
books and articles have been published to share experience gained and some theoretical models relating to risk communication. Professional associations like the
Society for Risk Analysis (www.sra.org) offer a platform for information ex5
For further information on a range of traditional approaches to valuation see Khoury
(2004), Copeland et al. (2000) and Jorion (2001).
Operational Risk: The Management Perspective
311
change, which insight decision makers may consider useful. So while some relevant valuable experiences can be drawn from, the general level of awareness is
still somewhat removed from operational realities.
So besides a few experts’ opinions and a flurry of consultants promising lean
approaches with guaranteed success, little is known to identify and properly
evaluate inefficiencies. The only realistic compromise an organization may choose
is to develop its own competitive benchmarks, which as a first step requires the
development of a policy framework and an overall process against which progress
can be rated. A generic framework which organizations may modify to develop
and evaluate their own strategy, is presented towards the end of this article.
! "#$%&
The discussion has so far highlighted a variety of aspects which can go completely
wrong, as well as partial failures and inefficiencies. Of course, while partial failures may be considered fundamentally unplanned and undesired, some inefficiencies are intentional given the set of objectives an organization may need to balance
over the short vs. medium vs. long term. This introduces the question whether beyond reactively addressing short-term bottlenecks, FSPs may choose to manage
OpRisk-initiatives somewhat below conceivable optimum benchmarks given
complementary choices to eliminate risks or cost drivers, or to open opportunities
for value drivers that will ideally pay for themselves sooner or later.
It has been argued that OpRisk initiatives have the potential to generate value.
To date this argument has remained qualitative, probably due to the fact that net
benefits are not traced back to original decisions to implement the one or other
OpRisk initiative. It is also not in the interest of real-life organizations to state
how inefficient they had been before a miraculous solution was found. However,
at a more aggregate level, it is known that OpRisk coincides with good housekeeping. As was demonstrated (Süssmeier-Kross, 2003) a strong correlation exists between those FSPs who have transferred OpRisk management considerations into
day-to-day managerial decision making and cost reduction or continuous improvement frameworks, and those FSPs who have been able to reach comparatively exceptional performance in terms of cost-effectiveness and flexibility. In
other words, given a certain level of maturity in OpRisk management, organizations will have learnt to operate significantly more efficient and effective than
their competitors do. So desirable side effects are a topic worth further research.
Regrettably the current state of transparency in this regard is rather poor. Best
practices are widely unknown, underlying success factors in achieving best practices are even more intransparent, and successful applications of multi-attribute
analysis and evaluation frameworks to assess the performance of OpRisk initia-
312
Wilhelm Kross
Screening
Objectives
3.
ALTERNATIVES
4.
PERFORMANCE
ASSESSMENTS
- E.g., time, financial
- Social / reputational
Desirabilities
Preferences
5. DECISION
ANALYSIS
-Scenario comparison
-Alternatives ranking
-Preferred alternative
Achieved Score
2. DECISION
CRITERIA
GOAL
Identify Preferred
Alternative(s)
1
Prob. Rank #1
1.
Cumulative Probability
tives seem not to have been published to date. This is surprising since the wealth
of literature on decision and performance analysis techniques is tremendous6.
The following figure (Kross and Schulz, 2004; Kross, 1999c) shows a generic
decision and uncertainty analysis framework potentially useful for OpRisk initiatives and related sensitivity analyses. As it outlines, desirable side effects must be
identified and measured against the primary objectives of an organization. These
objectives are probably best identified by means of influence diagram techniques,
using which decision makers identify what really is important for their organization (i.e., end objectives) vs. what is important because it influences these fundamental aspects (i.e., means objectives). It is submitted here that FSPs should care
about the time and money dimensions, and a third dimension capturing reputational aspects, in their attempts to evaluate the performance of their OpRisk initiatives. Several authors (Clemen and Reilly, 2001; Kross, 2000, 1999a, 1999b) offer
further insight into the use of influence diagrams and their translation into decision
trees. The recent book ‘Strategy Maps’ (Kaplan and Norton, 2004) offers further
insight in this regard and matches the findings of decision analysis techniques with
balanced scorecard design and implementation.
1
1
2
3
4
5
6
7
8
?
0
0
200
NPV of Cost ($1,000,000s)
Summary of Achievement Scores
1 +/- 0% +25 % +50 % +100 % +150 %
0
1…
8 1 … 81 … 8 1 … 81 …
Scenario Reference
Summary of Probability Rankings
+/- 0% +25 %
0
Feedback
8
+50 % +100 % +150 %
Scenario Reference
Fig. 5. Generic decision analysis framework for OpRisk initiatives
So techniques, by which the multiple dimensions of performance of OpRisk initiatives can be valued, are known, as is a framework in which these can be applied in
an integrated manner. Furthermore, a minimum set of objectives against which
performance is to be rated, has been identified for OpRisk initiatives. Hence the
6
for further information on uncertain multi-attribute decision analysis techniques see
Keeney & Raiffa (1993), Keeney (1992), Morgan & Henrion (1990) and Clemen &
Reilly (2001).
Operational Risk: The Management Perspective
313
remaining challenge relates to the successful implementation and the use in reallife situations. Of course real-life organizations may use these techniques following different goals than academics would, and refrain from refining analyses to the
n-th degree in favor of order-of-magnitude sensitivity analyses. Using a probabilistic framework in which subjective assessments are managed with all their inherent strengths and weaknesses, renders this choice legitimate.
It is considered likely that OpRisk initiatives will be demonstrated to imply a
multitude of desirable side effects – basically a variety of factors which enhance
performance in terms of reducing overall risk, reducing overall cost and time, enhancing managerial flexibility and improving reputational aspects. Furthermore,
positive correlations between these dimensions will likely be identified. Organizations would hence be well advised to make choices which help them to maximize
overall utility or desirability, as a weighted performance rating in terms of achieving the abovementioned dimensions. Organizations may not need to engage in
elaborate research to identify relevant factors since some are already known (e.g.,
positive correlation between risk & cost), or foreseeable (e.g., positive correlation
between good house keeping, demonstrated low OpRisk and an outstanding valuation of qualitative aspects in rating agencies’ recommendations).
' ()*+
Reaching from a net cost effect and possibly an overall risk increase into the net
value generation range, and possibly towards the other extreme classified as the
one of developing sustainable competitive advantage through OpRisk initiatives,
are strategic choices an organization will need to make. There are various pitfalls
along the road, failure potentials and inefficiencies which in addition to resource
and time constraints need to be considered. Taking into consideration the various
findings introduced earlier in this article, the following figure attempts to highlight
some of the most important success factors and priorities for OpRisk initiatives,
with a specific focus on the integration of data capturing techniques:
314
Wilhelm Kross
Data capture and
plausibility checks
by means of
appropriately designed
and implemented
questionnaires ???
Early focus on
enterprise-wide
objectives in risk management,
efficient integration into
other initiatives
in the organisation
Conceptual models &
pragmatic approach
to managing
data quality issues
and data gaps
Consideration of
uncertainties and
variability in cost& risk evolution
and in management
Appropriate
approach for the
management of respective
risk types/factors, short-/
long-term adjusted to
acceptable residual risk as
Optimised management
per corporate strategy
of enterprise performance:
acceptable short- vs. long-term,
transparent, cost-efficient,
regulatory compliant & part
of organisational culture
Appropriate methods
to analyse & evaluate
and to compare all
risk types/factors with
causes & consequences
with a sensible level of
detail & accuracy
Fig. 6. Relevance of questionnaires and moderated data capturing techniques for success of
OpRisk initiative
Mapping these success factors against the minimalist approaches for the Standardand Basic Indicator Approaches presented earlier, and against the commonly practiced techniques and approaches, it is apparent that several issues are not addressed appropriately – leading to inefficiencies and possibly to partial failures.
For the purpose of the rather superficial discussion in this article the following
corrective measures can be formulated as a list of priorities:
• Irrespective of the approach chosen to address OpRisk under Basel II the
first steps are a corporate policy and an operational framework.
• Even minimalist approaches should not be underestimated, as the variety of
complexities highlighted in this article suggests. Some rather senior resources should be allocated to implement the corporate OpRisk strategy.
• A central aspect often neglected is how OpRisk information will be communicated. An implementation strategy should ideally identify at the onset who
should receive which form of information and when. Ultimately it will not
be possible and not be sensible to keep everything confidential.
• While questions relating to methodology may be best allocated to an independent controlling department, the appropriate separation of OpRisk controlling and management functions is critical and needs to be implemented.
The measurement of performance necessitates a multi-dimensional analysis
framework. Skills required to perform such analyses may not be available in
a typical controlling function in a medium size FSP. Furthermore, it is essential that information flow be improved such that (de-)centralized OpRisk
managers have access to a wealth of information in a timely manner. For
•
Operational Risk: The Management Perspective
315
some FSPs this may imply a cultural shift, as will the recognition that
OpRisk management implies an elevated degree of transparency in causes
and consequences related to sub-optimal performance.
• OpRisk analysis and evaluation will need to be complemented with a few aspects not commonly addressed in controlling departments in FSPs. Particularly, due to the process related view reflected in OpRisk analyses, some
fundamental data gaps may prevail over the short to medium term, which
can only be addressed with subjective assessments. Measures to enhance
data validity and defensibility and to filter for inherent management biases
(Shapira, 1995) will become a central issue. Regression analyses may not
always be possible. A probabilistic framework will need to be established,
which implies a management paradigm shift for those who believe that everything is under control. Data may need to be aggregated bottom-up, which
is not necessarily compatible with portfolio models commonly used for market and credit risk in FSPs. And foremost, managing OpRisk with key indicators like a ‘Value-at-Risk’ may not be possible over the short term since
probability assessments based on subjective judgment data display significant inaccuracies and weaknesses in the extreme ranges (e.g., 99,95 %).
• Any OpRisk initiative should be designed to allow a relatively easy and efficient, staged upgrade to AMA. Ultimately, as is argued in this article, AMA
is the only permitted approach to addressing OpRisk under Basel II which
has the potential to generate net value.
These priorities are at least to some extent reflected in the generic roadmap presented in the following section.
, -.&/
The discussion has so far introduced most aspects an organization will need to
consider one way or other, in order to address OpRisk efficiently and effectively.
The generic roadmap presented herein reflects several fundamental components :
• OpRisk starts with a policy and with an appreciation of what is available.
• OpRisk needs to establish an appreciation process-related interdependencies
and causes and consequences leading to net value destruction.
• Semi-quantitative techniques help to establish the OpRisk framework, however, these are gradually replaced with quantitative data.
• Risk mitigation efforts and ‘residual risk’ must be reflected explicitly.
• OpRisk is ultimately integrated into the organizational culture through a continuous improvement framework.
• Although quick-wins are always possible, most desirable side effects may
need some time to become apparent.
The following figure includes a graphical representation of a generic roadmap:
316
1
Wilhelm Kross
Corporate Policy :
policy – operational risk;
definition of unacceptable risk,
available budgets,
risk management strategy
2
3
Op. Risk Reporting :
loss database (internal, external);
syst. selection software op. VAR,
key indicators (op. VAR system)
key indicators (management)
Description of work flow,
system landscape, human
interaction; possible events,
gap/hazard analysis, consequ.
chains, cause/effect relationships
4
Description of approaches to
proactive/reactive management;
for each, expected results;
performance assessment models,
responsibilities, training
5
Often Available (in some form):
- Policies & procedures handbooks
- 4-eyes-principle, internal revision
- System- / user handbooks
- Graphics on interfaces / data flow
- Performance- / cost measurement
- Process modelling / documentation
- Work procedures, role descriptions
- Quality assurance, approval/review
- Experiences from earlier mistakes
- Monitoring of project portfolio
- Loss database (internal / external)
- Experience OpRisk Management
- Emergency procedures, Y2K, etc.
Possibly data quality, defensibility
and completeness unsatisfactory ...
9
10
Insurance scenarios :
for insurable risk/consequence
factors, cost comparison;
avoidance/outsourcing :
comparison of cost effectiveness
6
Expected results :
decision support, communication
tool, comparison of scenarios,
description of processes/residual
risk per scenario
Subsequent iterations :
extension – expert system,
holistic internal model,
reduced capital provision
for operational risk
Gradually quantitative approach:
conditional probabilities,
consequence chains,
measured at least in time/money
7
Derive conclusions for scenarios :
critical aspects, cost effectiveness,
available budgets over time,
(un)acceptable residual risk,
provisions for expected cost
8
Planning and implementing :
scenario analysis w. residual risk,
prioritization;, planned reduction
and/or insurance and/or
outsourcing of risk factors
Fig. 7. Generic approach to implementing OpRisk
A discussion which decision makers may introduce in their respective organizations, is in how far their current approaches address the various aspects highlighted in this roadmap and in the above discussion, and which steps should be introduced in order to become AMA compliant and open the door for opportunities
through effective OpRisk management.
0 +
This article demonstrates that current OpRisk approaches in many FSPs can be
classified as somewhat sub-optimal. Considerable pitfalls exist even in minimalist
approaches, which can however be managed. Attempting to simply address all issues with a fully automated IT solution will likely introduce more problems than
are solved.
AMA approaches are, as is argued in this article, the only choice a FSP has to
address OpRisk in a way which does not simply introduce additional costs and
possibly even raise the organization’s exposure to OpRisk. To manage inherent
challenges on the road to AMA compliance an organization will need to shift
away from several established management paradigms, develop additional skills,
possibly re-design the way in which controlling and management cooperate, accept pragmatic compromises, and treat subjective assessments in an explicit probabilistic fashion.
Operational Risk: The Management Perspective
317
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23rd/24th, 2003, Hannover, Germany (proceedings to be published in 2004)
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21st – 24th, 2002, Humboldt-Universität Berlin, Germany
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Kross W. (1999a) Risk Assessment and Methods for Conducting Risk Assessment, seminar
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von Winterfeldt D, Edwards W (1986) Decision Analysis and Behavioral Research. Cambridge University Press, Cambridge
PART 2
Insurance Risk Management
Martin Nell1 and Andreas Richter2
1
University of Hamburg, Institute for Risk and Insurance, Email:
martin.nell@rrz.uni-hamburg.de
2
Illinois State University, Department of Finance, Insurance and Law, Email:
arichter@ilstu.edu
Abstract: Dramatic events in the recent past have drawn attention to catastrophe
risk management problems. The devastating terrorist attacks of September 11,
2001 incurred the highest insured losses to date. Furthermore, a trend of increasing losses from natural catastrophes appears to be observable since the late
1980s. The increase in catastrophe losses triggered intensive discussion about the
management of catastrophic risk, focusing on three issues. First, considering the
loss potential of certain catastrophic events, the insurance markets’ capacity does
not seem to be sufficient. An approach to address this capacity issue can be seen
in passing certain catastrophic risks to investors via securitization. Second, after
the events of September 11, 2001, the government’s role as a bearer of risk became an increasingly important issue. Finally, as has been recently demonstrated
by the floods in Europe of August 2002, problems of protecting against catastrophic threats do not only exist on the supply side but also on the demand side. Thus
policymakers are considering the establishment of mandatory insurance for fundamental risks such as flood and windstorm. This paper addresses aspects of these
three issues. In particular, we are concerned with the extent to which state or government involvement in the management of catastrophic risk is reasonable.
JEL classification: G1, H4, L5
Keywords: Catastrophic Risk, Risk Management, Public-Private Partnership
*
The authors would like to thank Brandie Williams for very helpful research assistance.
Project support from the Katie School of Insurance and Financial Services at Illinois
State University is gratefully acknowledged.
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Martin Nell and Andreas Richter
Dramatic events in the recent past have drawn attention to catastrophe risk management problems. The devastating terrorist attacks of September 11th, 2001 incurred the highest insured losses to date. According to current estimates, property
and business interruption insurance losses alone amount to 19 billion USD. Estimates of the total insured losses (including, in particular, life and liability insurance) range from 30 to 77 billion USD.1
The extent of these consequences leads to a reassessment of a risk category that
had been, until that point, either ignored or underestimated. An intensive discussion was triggered among insurance practitioners and economists about ways to
reorganize the financing of terrorism-related risks.2 This discussion, on the one
hand, indicates open questions with respect to the optimal design of risk management tools for a given individual catastrophe risk situation. On the other hand,
from a more fundamental point of view, it highlights the problem of how a society
should in principle deal with such risks, particularly how man-made disaster risks
should be allocated. Aspects of both, individual management of catastrophic risk
as well as societal decisions regarding the allocation of man-made catastrophe
risk, will be tackled here.
Prior to September 11th, the highest insured losses had, by far, been incurred by
natural catastrophes. In particular, one has to mention the accumulation of major
natural disasters at the beginning of the 1990s, including Hurricane Andrew in
1992 and the Northridge earthquake in 1994.3 Furthermore, while the (yearly)
man-made disaster losses, before 2001, seemed rather flat, a trend of increasing
losses from natural catastrophes appears to be observable since the late 1980s (see
Fig. 1.1.).
1
2
3
See Zanetti et al. (2002).
See, e.g., Nell (2001), Rees (2001).
The total insured consequences (excl. liability) of hurricane Andrew, according to current
estimates, amount to 20.2 billion USD, and losses resulting from the Northridge earthquake to 16.7 billion USD. The most dramatic (in terms of insured losses) man-made catastrophe before the World Trade Center and Pentagon terrorist attacks was the explosion
on platform Piper Alpha in 1988 (3 billion USD), the thirteenth-biggest event in the list
of insured events from 1970 through 2001. See Zanetti et al. (2002), p. 23.
Catastrophic Events as Threats to Society
323
40
35
[USD bn indexed to 2002]
30
25
20
15
10
5
Man-made disasters (excl. liability)
2000
1990
1980
1970
0
Natural castrophes
Fig. 1.1. Natural and Man-made Catastrophes 1970 – 2002
(Data: Swiss Re, Economic Research & Consulting; Zanetti et al. 2003)
One could detect an increasing frequency of catastrophic events as well as an increase in the average amount of loss per event. In particular, the increase in size
can largely be attributed to the growing density of population and the geographic
concentration of values in catastrophe-prone areas.4 With respect to the apparent
increase in number of such disasters, one might refer to implications of climate
change as well as to stochastic factors. Certainly, a part of the recent accumulation
of natural catastrophes was due mainly to random influence or coincidence. This
becomes obvious for the case of earthquakes in 1999: Although the number of severe earthquakes was not unusual, these events were perceived as a very singular
accumulation, since in a short time span several densely populated areas were hit. 5
The just-mentioned increase in catastrophe losses triggered intensive discussion
about risk management of catastrophic risk, focusing on the following three issues:
1. Considering the loss potential of certain catastrophic events, the insurance markets’ capacity does not seem to be sufficient. One example is the
series of insurer bankruptcies following Hurricane Andrew. Hurricane
Andrew, of course, was a major natural disaster. Still, it incurred losses
much smaller than the amounts today’s estimates assign to certain scenarios: Catastrophic events resulting in insured losses of 100 billion USD or
more, which might lead to a partial collapse of insurance markets, are
considered possible. An approach to address this capacity issue can be
seen in the so-called alternative risk transfer (ART) transactions, which
4
5
See, e.g., Zanetti et al. (2001) or Berz (1999).
See Nell and Richter (2001), pp. 237-238.
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Martin Nell and Andreas Richter
emerged after the late eighties’ and early nineties’ severe natural disasters. These transactions would directly pass certain catastrophic risks to
investors via securitization. Therefore, a significant share of the earlier
work on the management of catastrophic risk concentrates on ART and
its potential to cover catastrophic risk.
2. After the events of September 11, 2001, the government’s role as a bearer
of risk became an increasingly important issue. This development was
mainly triggered by the fact that insurance companies around the world
cancelled contracts with airlines and airports. The terrorist attacks of
New York and Washington had induced a major reassessment of air traffic-related liability exposure, such that insurance companies were only
willing to offer coverage at significantly increased rates. As aircrafts
without sufficient liability coverage would not be given permission to
take off, air traffic was in danger to cease more or less entirely. Facing
this scenario, many governments provided state guarantees for airlines
based in their countries. Additionally, many states participate in the different types of risk-sharing arrangements recently introduced to cover
terrorism risk in various countries. This again triggered a political debate
on the advantages and disadvantages of such state intervention with respect to the insurance of terrorism risk.
3. Finally, as has been recently demonstrated by the floods in Europe of
August 2002, problems of protecting against catastrophic threats do not
only exist on the supply side but also on the demand side. In Germany,
for instance, the proportion of insured victims was quite small, although
coverage would have been easily available in almost every affected region. As a result, policymakers are now considering the establishment of
mandatory insurance for fundamental risks such as flood and windstorm.
This work will address aspects of these three issues. In particular, we are concerned with the extent to which state or government involvement in the management of catastrophic risk is reasonable. As, in principle, we are in favor of publicprivate partnerships, one goal of this paper is to identify key elements of a meaningful division of labor between public and private institutions.
We will proceed as follows: Section 2 addresses recent approaches to financing
catastrophic risk via the capital markets. Problems of covering terrorism risk are
discussed in section 3. In section 4 we deal with potential inefficiencies in the demand for catastrophe coverage and their implications, and section 5 concludes
with a brief assessment of our findings.
The extreme losses from natural catastrophes in the early 1990s lead to a temporary shortage of catastrophe reinsurance, as reinsurers became more cautious and
therefore limited the supply, withdrew from the catastrophe risk market or (espe-
Catastrophic Events as Threats to Society
325
cially after Hurricane Andrew) even went bankrupt.6 In addition to this, one could
easily imagine natural disaster scenarios producing even much higher losses. For
instance, the estimated insured loss potential is about 60 billion USD for a severe
hurricane hitting the U.S. east coast and 100 billion USD for a major California
earthquake.7 Especially, when reference is also made to the enormous potential of
economic losses – 100 billion USD for the former, 300 billion USD for the latter
event – these scenarios seem to show the capacity limits of traditional insurance
markets.
Furthermore, it has to be assumed that an event of this size would again cause a
series of insolvencies in the reinsurance market. Therefore, a significant part of the
currently provided capacity might not be available when needed.8
The seemingly existent reinsurance capacity gap, combined with an increase in
catastrophe coverage prices that followed hurricane Andrew,9 set off a search for
ART solutions. The focus was primarily on tools that would enable the direct
transfer of risk using the financial markets, via so-called insurance-linked securities. Contributions with respect to an extension of capacity could be expected if,
for example, the issuance of marketable securities was able to attract additional
capacity from investors who are not otherwise related to the insurance industry.10
Capital market insurance solutions could be observed since 1992. The following
provides a brief overview of some basic forms of insurance-linked securities. 11
At the end of 1992 the Chicago Board of Trade (CBOT) started trading futures
on catastrophe loss indexes and related options.12 These instruments were based
upon underlying indexes representing the development of losses for certain regionally defined markets.13 An index was computed using actual loss data from a
subset of insurance companies having business in the respective area. The deriva-
6
Holzheu and Lechner (1998), p. 11.
See Durbin (2001), pp. 298-299.
8
For an approach to measuring the (re)insurance markets’ capacity for catastrophe risk,
see Cummins et al. (2002).
9 See, in particular, Froot (2001), p. 540.
10 To motivate the interest in financial market solutions for the transfer of insurance risk,
authors often refer to the size of the financial markets or their daily fluctuations in comparison to the size of a major natural catastrophe (see, e.g., Durbin, 2001, p. 305, Laster
and Raturi, 2001, p. 13, or Durrer, 1996, pp. 4-5). For example, a 250 billion USD event
would only represent less than 0.5% of the total market value of publicly traded stocks
and bonds of 60 trillion USD (Laster and Raturi 2001, p. 13).
11 For a more comprehensive discussion of insurance risk securitization design possibilities
as well as for data concerning transactions in this field see, e.g., Durrer (1996), Baur and
Schanz (1999), Belonsky et al. (1999), and Laster and Raturi (2001).
12 See Durrer (1996), pp. 9-11.
13 Contracts based upon catastrophe losses in the entire U.S. were available as well as contracts based upon loss data collected for smaller regions, in particular the states characterized by extremely high natural catastrophe risk (see Durrer 1996, p. 9).
7
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Martin Nell and Andreas Richter
tive tools were primarily aimed at insurance companies as a means to hedge their
individual catastrophe losses. 14
The CBOT options turned out not to be very successful.15 Over the past few
years, transfer of insurance risk via the financial markets has mainly been carried
out using over-the-counter securities, such as, for example, catastrophe bonds (cat
bonds) or contingent capital, instruments that enable a direct transfer of risk to investors.
A cat bond is a bond in which the interest and/or – depending on the specific design – the principal is (partially) forgiven when a pre-defined catastrophic event
occurs. The typical structure of a cat bond issue is as follows: 16 A special purpose
vehicle (SPV) is set up, usually as an offshore reinsurer, which is located, for example, in the Caymans, its purpose solely being the handling of that specific securitization. The SPV reinsures the primary and backs up this contract through the
issuance of the cat bond. The principal invested is held in trust. If no loss occurs,
principal and interest are paid back to the investors, whereas in case of a loss this
amount is reduced by the reinsurance coverage that goes to the primary.
Contingent capital could, for instance, be provided through equity put options: 17
A primary issues put options on its own equity, i.e., it purchases the right to sell
shares to a counterparty at a pre-specified price in the case of a certain event, such
as the individual catastrophe losses exceeding a threshold. This put option would
enable the primary to recapitalize, at conditions negotiated ex ante, after major
losses, which might be crucial since a catastrophe would typically reduce the surplus of many insurers in the affected region. It may, therefore, also create a shortage of capacity, implying that access to capital would be particularly attractive in
that situation.
Cat bonds have had the biggest market share among recent insurance risk
securitization transactions.18 These bonds are mainly used by primary insurers and
reinsurers to substitute or supplement traditional reinsurance or retrocession19.
14
The typical insurance derivatives transaction at the CBOT would be so-called “call
spreads” which enable a primary insurer to duplicate the structure of a typical nonproportional reinsurance contract, but based upon the underlying index. In a nonproportional reinsurance contract the primary bears losses up to a certain amount – called retention –
and is compensated by the reinsurer for the exceeding part of the losses. Additionally, the
reinsurer’s share of the risk is usually limited by an upper bound.
15 See Müller (2000), p. 216, and Laster and Raturi (2001), p. 5. However, it seems likely
that, medium-term, derivative instruments can play an important role for catastrophe risk
transfer (see Laster and Raturi, 2001, p. 17). At the moment, similarly structured instruments receive attention in a related field: The hedging of weather risk. In areas of business for which success heavily depends on weather conditions, companies, as for example energy providers, try to hedge these risks through weather derivatives (see Müller,
2000, pp. 217-221).
16 See, e.g., Belonsky et al. (1999), p. 5.
17 See, e.g., Doherty (2000), pp. 615-616.
18 See Laster and Raturi (2001), p. 19.
19 Retrocession is the reinsurance purchased by a reinsurance company.
Catastrophic Events as Threats to Society
327
It has to be emphasized, however, that such instruments can, of course, also be
attractive risk management tools for companies from other branches. As an example, reference can be made to the cat bond hedging earthquake risk that was issued
by Tokyo Disneyland in 1997.20
As in traditional (re)insurance, the trigger mechanism for a cat bond can be the
actual individual losses from certain catastrophic events. For instance, a transaction can be designed in such a way that no or only reduced interest is paid to the
investors, implying that coverage is available for the hedging party, if the latter’s
actual catastrophe losses exceed a pre-negotiated threshold. Naturally, a contract
could define more than one threshold, triggering different amounts of coverage.
Another possibility would be to tie the contingent payment from a cat bond to a
market index, as in the above-mentioned CBOT options. Obviously, a market index can be useful as the underlying for a cat bond or other kinds of insurancelinked securities, if the individual portfolio structure is a sufficiently good representation of the entire market. The main advantage of an index, besides its contribution to alleviate standardization, is the fact that, compared with reinsurance, it is
largely out of the primary’s control.21
If, finally, a risk securitization transaction is based upon technical parameters
describing the intensity of a catastrophic event (parametric trigger), manipulation
can be completely excluded. Examples for this kind of an underlying are the Richter scale reading of an earthquake or the strength of a hurricane, observed in a certain region over a certain period specified by the contract. The usefulness of such
parameters arises from their correlation with an event’s insured consequences. A
parametric trigger has the additional advantage that the relevant numbers are usually available very quickly. Contrasting this, a market index typically needs a long
time until it is fully developed, in particular due to time-consuming problems of
loss-settling.22
A special case of the parametric trigger is a modeled trigger, for which the procedure would be as follows: In the case a relevant event happens, a simulation is
run, based upon certain observed parameters, that generates an estimate for the
losses from the primary’s actual portfolio. The simulation result then determines
the amount to be paid to the primary. If the model is completely specified ex ante,
this underlying can also not be influenced ex post by the primary. A modeled trigger can be helpful, e.g., for situations where the number of combinations of potential parameter realizations and outcomes does not allow for every single case to be
explicitly mentioned in the contract.
20
See Müller (2000), pp. 215-216.
See the next section for a more detailed discussion of this point.
22 Parametric triggers are also typically used in securitizing weather risk (see footnote 15).
In this context, in particular the temperature and the amount of rain are useful as underlying random variables. As an example, one can refer to a transaction recently carried out
by a German energy provider hedging against excessive rainfall. A large number of this
energy provider’s customers are farmers and therefore need greater amounts of energy
for their watering systems when rainfall is not sufficient.
21
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Martin Nell and Andreas Richter
As was mentioned above, the demand for ART solutions is usually explained
via the supposedly limited supply of traditional hedging tools. According to this
rationale, the total capacity of the world’s (re)insurance markets would not be sufficient for covering certain catastrophe risks.23 This explanation, however, seems
to be of only limited validity. Additional risk financing capacity could also be
generated through extending capital funds held by the insurance industry or
through market entries in the insurance markets. The latter, in fact, could be observed during the 1990s following hurricane Andrew: Immediately after this event
reinsurers were very reluctant and in particular the Lloyd’s reinsurance market
went through a heavy crisis, leading to a decline in the supply of catastrophe coverage.24 Nevertheless, the available reinsurance capacity definitely increased over
the next few years as more capital flowed into the industry. In particular, reinsurers located in the Bermudas were a major source for additional capacity provided
during this period. Companies specialized in natural catastrophe reinsurance were
set up and the Bermudas quickly became a very important market.25
According to these considerations, the attractiveness of insurance risk
securitization cannot be convincingly explained via capacity shortages in the
reinsurance industry. Thus, for further insight one needs to turn to the specific
economic advantages these tools might have, compared with insurance.
Risk transfer through the financial markets can be carried out in many different
ways. Naturally, the economic assessment of such instruments depends to a great
extent on the specific design chosen, and in particular on certain institutional characteristics. The following, however, will not be concerned with the very details of
institutional design, but rather with certain basic features defining important criteria for an economic comparison of risk securitization and traditional (re)insurance.26
A first interesting economic explanation for risk securitization is the fact that,
depending on the underlying random variable, certain kinds of these tools offer an
instrument to address moral hazard. A typical insurance or reinsurance contract is
an indemnity contract, i.e., it is designed in such a way that contingent payments
are connected directly to the insured’s, or respectively the primary’s, stochastic
actual losses. Therefore, (re)insurance coverage can be perfectly correlated with
the losses – at least so far as a monetary equivalent of the actual consequences can
be determined. This, however, also implies that moral hazard is a major problem
23
See, e.g., Durrer (1996), Cholnoky et al. (1998), Bantwal and Kunreuther (2000), Cummins et al. (2004).
24 See Holzheu and Lechner (1998), p. 14.
25 For example, the global market share of the Bermuda reinsurance market developed from
0% to 5% between 1992 and 1997. Being specialized in natural catastrophe risk, it benefited from increased premiums in this segment and from relatively lower natural catastrophe losses between 1995 and 1997 (Holzheu and Lechner, 1998, pp. 12-21).
26 For an introduction to the economic comparison of risk securitization and insurance see,
in particular, Doherty (1997), Froot (1997), Croson and Kunreuther (2000).
Catastrophic Events as Threats to Society
329
of insurance markets:27 In most cases insured risks can be influenced by the insureds who would also usually have a significant unobservable discretion with respect to their actions. Thus, insurance coverage induces changes in the insureds’
behavior. This phenomenon can be observed in primary insurance, but also in the
relationship between a primary insurer and its reinsurer. A primary is in charge of
risk selection and monitoring as well as settling losses with its customers. Considering the fact that it would normally be impossible or prohibitively expensive for
the reinsurer to monitor these activities, reinsurance relationships will usually be
characterized by asymmetric information. As a consequence, a primary’s carefulness can be expected to decrease in the amount of its reinsurance coverage.
As was mentioned above, the coverage from many risk securitization transactions does not directly depend on the actual losses but on some other random variable, which is correlated with the losses. If the trigger is a market loss index,
moral hazard is limited to the primary’s contribution to the index. By making use
of a parametric trigger the moral hazard problem can even be avoided. However,
the reduction or elimination of moral hazard incurs a certain cost. Typically, the
less the underlying random variable can be influenced by the primary, the less useful is the contingent coverage as a hedging vehicle. The resulting mismatch between the loss and the coverage is called basis risk.28 For instance, an earthquake
might not trigger the payment from a cat bond, since its strength is too low, even
though substantial damages are caused in the primary’s portfolio. On the other
hand, a realization of basis risk could be that coverage from the cat bond is actually paid to the hedging primary although no significant individual losses are observed from that particular event.
Another aspect important for the comparison of risk securitization and reinsurance are the transaction costs incurred by the respective instruments. A product
that ties its payments to an exogenous index reduces or avoids administrative costs
such as costs from loss handling or monitoring. One advantage for the case of a
parametric trigger can be the above-mentioned fact that determination of due
payments is fast and less problematic. But also acquisition costs might be partially
spared by making use of the financial markets.
As is often argued, insurance risk is not or is only weakly correlated with market
risk, implying that the price for insurance risk securitization should include just
small risk premiums. This results in a potential advantage over reinsurance for the
following reasons: By purchasing reinsurance shares an investor participates in the
company’s entire risk portfolio, including, e.g., its investment performance or the
risk of mismanagement. As opposed to this, risk securitization enables investors to
assume a pure position in the very specific catastrophe risk category and in that
sense expands their opportunity set.29 Furthermore, as empirical evidence indi27
For a discussion of moral hazard problems in insurance markets, see, among many others, Holmström (1979), Shavell (1979), or Nell (1993).
28 For an analysis of the trade-off between moral hazard and basis risk in a combination of
an index-linked securitization product and an insurance product that covers a part of the
basis risk, see Doherty and Richter (2002).
29 See, e.g., Froot (1999).
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Martin Nell and Andreas Richter
cates,30 an important cause for high cost of reinsurance is risk-averse decisionbehavior of reinsurers, particularly when it comes to dealing with catastrophic
risk.31
A further potential advantage of certain types of catastrophe risk securitization
is the fact that with these tools default risk can be more or less completely
avoided.32 This is important since, in particular, certain natural disaster hazards
impose a significant insolvency risk for reinsurance companies active in that business, implying that their contracts are subject to default risk. This is due to the potential of a regional accumulation of losses as it is typically incurred by catastrophic events. The threat of loss accumulation leads to high correlation between the
different local primaries’ portfolios and, therefore, between claims from different
contracts in a reinsurer’s portfolio. For the single primary insurer, this leads to an
increased default risk or credit risk with respect to catastrophe reinsurance. 33 In
contrast, risk securitization can be carried out in such a way that it is free of or
subject to only very little default risk: The funds invested in a cat bond, for instance, are collected ex ante which implies that the credit risk for the primary insurer is reduced to the default risk connected with the investments made by the
trustee.34
So far, insurance-linked securities have not been as successful in the market as
was first expected. For instance, the total volume of transactions carried out until
2001 amounts to about 13 billion USD.35 Compared to the size of the reinsurance
market, this is not very significant. The catastrophe excess of loss coverage pur-
30
See Froot (2001), who looks at catastrophe reinsurance data. He finds, for example, that
the average ratio of premiums over expected losses between 1989 and 1998 was higher
than 4.
31 In a perfect market the risk premiums included in the price of a risk securitization product would not differ from the risk premium for the same risk held by a reinsurer. For a
discussion of the various imperfections explaining additional cost of risk originating from
a (reinsurance) company’s risk averse decision-making, see among others Greenwald and
Stiglitz (1990), Dionne and Doherty (1993) or Nell and Richter (2003).
32 See, e.g., Croson and Kunreuther (2000), pp. 30-31, Laster and Raturi (2001), p. 14.
33 As was mentioned before, an illustrative example for the realization of default risk was
hurricane Andrew, which led to a number of insolvencies in the reinsurance market. The
following years were also characterized by a massive drop of the number of reinsurance
companies due to a series of mergers and acquisitions (see Holzheu and Lechner, 1998).
Considering that major factors determining a reinsurer’s risk of insolvency are its worldwide spread and financial strength, this tendency of consolidation might – among other
issues – also be a consequence of a growing awareness of default risk. See also Laster
and Raturi (2001), p. 14: That default risk is an issue in reinsurance contracting is also reflected by market shares. In 1999, for example, among the world’s 100 biggest reinsurance companies, only 20% of premiums were written by companies rated (by Standard &
Poor’s) below AA.
34 The use of catastrophe options also avoids default risk to a great extent, as usually obligations are guaranteed by the exchange (see, e.g., Laster and Raturi 2001, p. 18).
35 Munich Re ART Solutions (2001), p. 11.
Catastrophic Events as Threats to Society
331
chased in the worldwide reinsurance market in the year 2000, e.g., amounted to
107 billion CHF.36
The at first rapid increase in the use of new financial risk transfer instruments
halted in the late 1990s after a decrease in reinsurance prices. 37 Consistent with
our discussion, insurance-linked securities do not seem to play a major role as a
means to expand the available catastrophe risk financing capacity. However, these
products have introduced new tools to address problems of default risk and in particular moral hazard, and in that sense can indirectly help to expand the limits of
insurability. Although recent transactions favor index triggers, the resulting basis
risk of such securitizations seems to be the primary explanation for the reluctance
of many risk-managers in the use of alternative risk transfer products.
Nevertheless, the impact of future major disasters on reinsurance capacity and
pricing might cause the growth of the market for insurance-linked securities gain
speed again – as private risk management tools, but also as a component of a public risk management strategy. Terrorism risk is one example of a risk category
where coverage generated through cat bonds can be an interesting alternative or
addition to traditional insurance solutions. Kunreuther (2002), for instance, suggests incorporating federal cat bonds as an element of a public-private approach to
covering terrorism risk.38 As will be discussed in the next section, government involvement became an issue after the events of September 11, since the terrorist attacks caused another capital and capacity shock for the insurance and, in particular, the reinsurance industry.39 Generally, the resulting increase in catastrophe
reinsurance prices again provided a framework for a medium-term gain of the insurance-linked securities’ market share, as insurers might reconsider the structure
of their risk management portfolio.
After the attacks of 9/11, an intensive discussion set off about the role of the state
in managing terrorism risk. On the one hand a topic was the temporary issuance of
state guarantees concerning the risk of “war and terrorism” for airlines and airports, on the other hand a general involvement of the state in covering terrorism
was discussed. In this context, two questions deserve particular attention: First, we
need to determine when and for which type of risk there would not be sufficient
insurance coverage available. Second, it should be analyzed under which condi-
36
Durbin (2001), p. 301.
See Laster and Raturi (2001), p. 18.
38 Sovereign cat bonds have also been discussed in a different problem context: For instance, Croson and Richter (2003) discuss the usefulness of sovereign cat bonds issued
by developing countries for the primary purpose of generating conditional funds for infrastructure emergency repairs after catastrophic events.
39 See, e.g., Doherty et al. (2003).
37
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Martin Nell and Andreas Richter
tions (if at all) the state should assume risk. These questions will be discussed in
the following.
The development of modern industrialized societies is inseparably connected
with the emergence of insurance markets and the supply of coverage for new risks.
Since the majority of people are risk-averse, the opportunity of transferring risk to
risk-taking institutions such as insurance companies enables them to engage in
risky activities they otherwise would avoid. In a world of risk-averse individuals
undertaking risky activities is productive, such that insurance supply enhances
welfare.40
Such positive influence of insurance, however, can only occur under the condition that insurance companies posses sufficient information to be able to at least
approximately price risks based upon their expected losses. If this information is
not available or if it has to be ignored, for instance due to a political decision that
prohibits certain premium discrimination, the following inefficiencies are unavoidable: Individuals’ decision-making will display an insufficient level of care,
since higher effort would not be reflected in a lower insurance premium. Closely
related to this phenomenon is the problem that production technologies would be
chosen that are too risky from a welfare economics point of view – technologies
which yield rather high returns if no loss occurs but which are suboptimal due to
their high risk.41 These unfavorable implications of non-risk based pricing constitute major moral hazard problems (see section 2). As was emphasized before,
moral hazard cannot be completely resolved in insurance markets. However, competition in these markets forces insurance companies to include any available relevant information in their pricing of insured risks. Where insureds still possess unobservable discretion in choosing their level of care, incentives can be set via
instruments such as, for instance, coinsurance or deductibles.
For most risks coverage is provided through private insurance markets. Yet, it
can be observed that insurance companies do not supply protection at all for certain types of risk or that they are reluctant to offer the desired level of coverage.
Two main reasons explain why in such situations no or only insufficient coverage
would be available: First, for certain categories of risks moral hazard problems
can become too severe because, e.g., the insureds’ influence on the risk is very
significant. Obviously, state intervention cannot be considered a useful tool to
solve this issue: Typically the state would not have the better information concerning insureds than the insurance companies, and therefore would not be better in
dealing with moral hazard.
Second, problems with the supply of insurance can also arise where the loss potential of a single event is so enormous that the entire industry cannot provide sufficient capacity to cope with it. This can be the case for some types of catastrophic
risk which are characterized by high correlation and therefore a tendency to incur
cumulative losses (see section 2). Examples are, in particular, natural disasters
such as floods, windstorms and earthquakes, but also war and terror-related risks.
40
See Sinn (1986) for an explanation of the importance of risk as a production factor and
the welfare-increasing effect of insurance coverage.
41 See, e.g., Nell (1990).
Catastrophic Events as Threats to Society
333
Furthermore, areas exist in which even single losses – if entirely covered – could
exhaust the insurance markets’ capacity, as, e.g., the liability risk connected with
nuclear power plants. Typically, insurers would offer only rather low amounts insured and thus very limited coverage for these hazards.
Where the private markets’ supply of insurance is insufficient due to capacity
restrictions of the entire industry (including alternative sources, as discussed in
section 2), state supplied protection may be considered as an option. The state
would usually be able to provide much more capacity than the private sector.
For terrorism risk obviously this kind of scenario materialized. Right after September 11, the coverage offered in the insurance markets did not meet the demand
for protection. The motivation for state guarantees granted to airlines and airports
was the fact that insurance companies quite uniformly cancelled existing policies.
Thus, private markets were only offering limits that would, for many routes, not
even be sufficient to fulfill the minimum requirements. Serious trouble for the airline industry was imminent. Negotiations between airlines and airports on the one
side and insurers on the other side were particularly difficult: The insurance industry, in the aftermath of the terrorist attacks, completely reassessed their liability
exposures in the context of air traffic. Therefore, liability coverage in this area
was, if at all, only offered at drastically increased premiums. While up to that
point the collision of two passenger aircrafts had been considered the worst case
scenario, now the focus was shifted to the possibility of even much more dramatic
events. Additionally, in the political context following September 11, a significantly increased likelihood was commonly assigned to attacks on aircrafts and
skyjackings.
In this situation, in which contracts needed to be fundamentally adjusted, a temporary issuance of state guarantees was adequate, as they helped to keep air traffic
going and to give the involved parties enough time for negotiations. Nevertheless,
state liability can only be useful if expected costs of terrorism risk are still internalized by the air traffic industry. Under no circumstances these guarantees should
be utilized as a means of subsidizing an industry whose structural problems have
not been initialized by September 11 but existed before. Costless guarantees, as
have been provided by the British government, are certainly the wrong way to address the problem. Contrasting their approach, the price of such state protection
for the industry should rather be much higher than insurance premiums paid before September 11, considering the dramatically changed risk situation.
As mentioned above, the state guarantees were meant to provide the negotiating
parties with some time for adjusting their contracts. A more fundamental issue,
however, is the question whether a general state guarantee for terrorism risk is
necessary. Due to the quite high and difficult to estimate loss potential, the insurance industry was only willing to supply rather limited coverage. This would have
led to losses from terrorist attacks remaining uninsured to a great extent. This, in
turn, would have caused efficiency losses: Socially beneficial activities threatened
to not be carried out, since investors would not be willing to take the risk associated with these activities. Thus, it can be reasonable for a state to provide additional protection. However, pricing must be based upon the actual risk: Otherwise,
terrorism risk would not sufficiently be taken into account in decision-making,
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Martin Nell and Andreas Richter
which would lead to underinvestment in security technologies and excessively
risky construction and production investments.
The German approach to solving the problem of terrorism risk coverage is one
way of involving the state in catastrophe risk financing. In 2002, an insurance
company by the name of Extremus was formed whose task it is to supply protection for large-scale losses from terrorist attacks. Extremus insures exposure units
with amounts insured exceeding 25 million EUR. It offers coverage for losses up
to 13 billion EUR per year, which, according to current estimates, should be sufficient. Private insurers and the state provide the capacity jointly. The German insurance industry covers the first 1.5 billion EUR. International reinsurers are responsible for the second layer of 1.5 billion EUR. In excess of this, the state is
liable for 10 billion EUR.
A major drawback in the Extremus structure is the fact, that, at least up until today, it does not apply any premium discrimination based upon major risk factors
such as location and type of a building. So, given its current design, this solution
subsidizes highly exposed buildings by charging insufficient premiums while
owners of low risk properties would be overcharged. However, considering the
amount of thought that has been put into assessing terrorism risk since September
11, this shortcoming of the current solution certainly does not need to be permanent. Changes in the rating schedule could make Extremus an example of a socially beneficial state guarantee approach.
Many examples can be found of obviously not very helpful state intervention in
the insurance markets. In some instances, the just-mentioned subsidization of high
risks even is an expressed goal of such an instrument. In these cases, premiums for
certain high-risk exposures are basically considered “too high”. An interesting example can be seen in homeowners insurance in the state of Florida. Here, one
could recently observe a strong increase in construction on the coast, although these areas are characterized by a particular severe windstorm risk exposure. One
reason for this development seems to be that, because of state regulation, premiums for homeowners insurance in this area are far below the adequate risk-based
level. As a consequence, the risk of windstorm damage is not or not sufficiently
internalized when settlement decisions are made.42 Furthermore, homeowners
typically underinvest in protection measures against windstorm. The deficit resulting from inadequately low insurance premiums for buildings in the coastal area is
compensated through premiums charged in other regions or for other insurance
products. So, on top of the above-mentioned problems, insurance regulation here
also subsidizes the above-average income group of people who tend to settle in the
coastal area.
Moreover, there are forms of state intervention that imply systematic exploitation of the state through the private sector. In France, for instance, fundamental
risks such as flood and earthquake are insured on a mandatory basis at uniform
premiums. French primary insurers have the right to pass these risks to a state-run
reinsurer that is furnished with an unlimited state guarantee. Obviously, this cre42
On state regulation of homeowners insurance and its consequences see, e.g., Klein and
Kleindorfer (1999). See also Russell (1999), pp. 227-244.
Catastrophic Events as Threats to Society
335
ates a severe adverse selection problem, as it provides strong incentives for insurance companies to pass on only the bad risks. Not surprisingly, the reinsurer experienced significant losses, although in total insurance of fundamental homeowners risks was highly profitable.
These considerations allow for the following conclusion: State intervention in
the context of catastrophe risk financing can be socially desirable, if insurance capacity provided by the private sector is not sufficient. Recent experience, however,
teaches that regulation in this area is often based upon the wrong motivation or designed poorly. Therefore, economic advantages and disadvantages need to be analyzed thoroughly for every specific case of state intervention in insurance markets.
!
" The floods of August 2002 in Central Europe demonstrated that problems in catastrophe risk management do not only exist on the supply side but also on the demand side for catastrophe coverage. It became obvious that only a small proportion of victims had purchased insurance against these losses and that, for instance,
in Germany the density of insurance against these hazards was rather low. In
Germany, flood risks can be covered through a fundamental risk („Elementarschaden“-) extension of homeowners and contents policies as well as certain
commercial coverages. However, only about 3.5% of German homeowners and
roughly 9% of contents policies include this extension.43 Even taking into account
that many buildings are located in areas with insignificant flood risk, it needs to be
asked what might be the reasons for this low market penetration.
Looking at the supply side, we find that insurers would only in extremely floodprone areas be reluctant to offer this additional coverage. But even there, this insurance is available. Typically, the policy would just include increased deductibles
and certain obligations regarding loss prevention measures, and rates would be
higher. Still, the fundamental risk extension is in general not very costly: For instance, the premium for an amount insured of 300,000 EUR in a region with medium flood risk (likelihood of a flood of 2%-10%) would be about 50-60 EUR.44
Thus, the low insurance density cannot be attributed to insufficient supply, its
reasons must be found on the demand side. Two different explanations should be
considered in this context:
The first explanation is that people seem to underestimate their exposure to natural disaster risk. For instance, the results of a study carried out in areas with high
flood risk suggest that individuals systematically underestimate the likelihood of
natural catastrophes.45 Also, one can often observe that after a major catastrophic
event the demand for protection against such hazards significantly expands and
during a period without any event decreases, as individuals update their beliefs.
43
See Schwarze and Wagner (2002), p. 596.
See Schwarze and Wagner (2002), p. 596.
45 See Kunreuther (1976).
44
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Martin Nell and Andreas Richter
their beliefs. However, justifying government intervention in the form of mandatory insurance on the grounds of this rationale requires a quite paternalistic view
of the role of the state.
More important, from our point of view, is the following second explanation:
The 2002 floods as well as other catastrophic events have shown that victims to a
great extent receive assistance from the state and from private sources. Since this
emergency aid is usually based upon the actual loss of a victim, insurance and
other sources of compensation are direct substitutes. The low demand for fundamental risk coverage and insufficient loss prevention can therefore be explained
by the potential victims’ anticipation of (costless) non-insurance assistance.
It can be assumed that state assistance and private help for people who suffer
major losses from a natural disaster, are politically and socially unavoidable.46
However, we need to be aware that this considerably reduces incentives to invest
in loss prevention on an individual as well as on a collective basis (such as, in the
case of flood risk, risk adjusted development decisions, the creation of flooding
areas or the moving of oil tanks to upper floors in a building).
The anticipation of emergency aid and the resulting insufficient loss prevention
make a strong case for a regulatory intervention in the form of mandatory insurance against flood risks. This rationale is also quite widely accepted in other contexts: For instance, mandatory savings for the purpose of funding retirement,
which exist in various forms in most countries, are usually justified via the potential anticipation of state help in case of old age poverty.
Compulsory insurance, thus, is a useful component of catastrophe risk management.
# Over the last two decades, the frequency as well as the size of natural catastrophes
has increased considerably. Simultaneously, predictions about possible losses
from certain catastrophe scenarios have been adjusted to significantly larger
amounts. Additionally, the events of September 11 demonstrated the enormous
dimension of terrorism risk. These developments initialized a lively discussion
among economists as well as politicians on ways of improving catastrophe risk
management. One particular concern was the question of how a more comprehensive coverage of consequences of natural disasters could be achieved. Furthermore, approaches were discussed that aimed at limiting the size of losses by
means of increased loss prevention.
The discussion on improving financial protection against catastrophic risk at
first focused on the supply of insurance coverage. This was driven by the concern
that losses incurred by certain disasters could exhaust insurers’ capacity and cause
46
Therefore, the suggestion sometimes expressed in the literature, to not provide any emergency help for individuals in catastrophe-prone areas (see, e.g., Epstein 1996), is politically not feasible.
Catastrophic Events as Threats to Society
337
the insurance markets to collapse. Anticipating this, catastrophe risk underwriting
policies became more and more restrictive.
A reduction of this problem seemed to be possible through instruments that directly transfer insurance risks via the financial markets. Such transactions could be
observed since the early nineties. Initially, high expectations were placed on the
so-called alternative risk transfer, based upon the argument that the assumed low
correlation between market risk and catastrophe risk and the resulting diversification opportunities would attract substantial capacity. Up until today, however,
these expectations have not been fulfilled: The trade in certain types of alternative
risk transfer tools (in particular the CBOT catastrophe index options) has ceased,
other instruments, such as cat bonds, are being used in the markets, but the number of these transactions is still rather low. So far, a significant increase of catastrophe coverage through the securitization of insurance risk has not been achieved,
and nothing indicates that this will change much in the near future.
September 11 forced discussions to address the role of the state as a potential
risk bearer. This was triggered by the fact that many governments decided to provide state guarantees for airlines and airports to keep air traffic up, since insurance
companies denied to offer protection at former conditions and/or rates. Additionally, insurers generally were unable to supply sufficient capacity to cover terrorism risk. In several countries, therefore, the state is now strongly involved in the
financing of terrorist risk. This kind of state intervention can be beneficial, provided that its sole function is to extend capacity for catastrophe coverage. If, on
the other hand, it leads to a renunciation of premium discrimination, state intervention can even be harmful, as it implies the reduction of loss prevention.
Furthermore, catastrophe risk management problems also exist on the demand
side. It can be observed that even if sufficient coverage would be available, the
demand for certain types of catastrophe coverage is low. This is problematic since
the rationale behind the low demand is an anticipation of governmental or private
emergency aid that would be granted after the occurrence of a disaster. The Oder
flood as well as the August 2002 floods demonstrated that emergency aid can be
sufficient to compensate for the entire loss incurred by such natural catastrophes.
Since the individual amount of emergency aid is usually based upon the actual
loss of a victim, it can be viewed as a direct substitute for insurance. In this
framework, it is rational not to purchase insurance but to rely on catastrophe
emergency aid. From the perspective of societal management of catastrophic risk,
however, this is highly unsatisfactory, as it destroys any incentive for loss prevention. To solve this problem, policymakers should consider mandatory insurance
approaches, which, of course, would also need to utilize risk-based premiums in
order to avoid the just-mentioned incentive issues.
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Ulrich Hommel and Mischa Ritter1
1
European Business School – International University, Schloss Reichartshausen,
D-65375 Oestrich-Winkel, Germany
Abstract: Insurance and financial markets are converging as (re-)insurers are
searching for new ways of expanding their underwriting capacities and managing
their risk exposures. Catastrophe-linked instruments have already established
themselves as a new asset class which offers unique profit and diversification opportunities for the investor community. This chapter analyzes the principal forces
behind the securitization of catastrophic insurance risk and thereby highlights key
factors which determine to what extent and with what means other forms of insurance risks (in particular other types of property & casualty, longevity, health and
weather risks) can be transferred to financial markets in the future.
JEL Classification: G150, G220, G130
Keywords: Catastrophe Risk, Securitization, Derivatives, Reinsurance, Risk Management
Alternative Risk Transfer (ART) is one of the most rapidly growing segments of
the reinsurance business today. Its objective is to develop alternative distribution
channels (e.g. insurance coverage for captive demanders) and to identify ways of
expanding the capital base to satisfy the ever-growing appetite for reinsurance,
most importantly via the securitization of insurance risks. ART is spearheading the
process which works towards bridging the gap between insurance and financial
markets and increasingly exposes reinsurers to competitive pressures from investment banks. This chapter focuses on the one area where the transfer of insur*
We thank Petra Riemer-Hommel and Robert Vollrath for helpful comments and Gudrun
Fehler for editorial assistance. The usual disclaimer applies.
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Ulrich Hommel and Mischa Ritter
ance risk to financial markets has already been carried out successfully, catastrophe (or CAT) risk. The analysis will highlight key aspects which will gain equal
prominence when organized and over-the-counter (OTC) markets begin to absorb
other forms of insurance risks, most imminently other types of property & casualty risks as well as longevity and health risks (see also Riemer-Hommel/Trauth in
this volume). Not all forms of financial innovation related to securitizing CAT risk
have been successful. All exchange-traded instruments have basically ceased to
exist in recent years but are nevertheless included in this chapter since they have
provided unique forms of managing CAT-like exposures. Some of the key reasons
for these failures will be further detailed below as well.
We have witnessed an extraordinary increase in the frequency and severity of
natural catastrophes1 in recent decades which can largely be explained by an overall increase in population density, the appreciation of property values in industrialized countries, and an increase of insured property values in high-risk regions.2
Particularly problematic are so-called massive-loss events with insured damages
of USD 1 bill. or more since they may threaten the solvency of the insurance industry as a whole. Property-casualty insurers have faced a total of 36 events between 1970 and 2003 each with an insured loss of over USD 1.5 bill., 28 since
1990 alone.3 The most prominent examples are Hurricane Andrew (USD 20.9
bill., 1992), the Northridge Earthquake (USD 17.3 bill., 1994), Taifun Mireille
(USD 7.6 bill., 1991), and Hurricanes Daria and Lothar (both USD 6.4 bill.,
1990/1999). With the rise of global terrorism, man-made disasters have also become a central issue for managing CAT-like exposures. The “September 11” attack (2001) has for instance caused total insured losses of USD 21.1 bill. (in 2003
USD).
Traditional insurance markets lack the ability to supply sufficient coverage for
existing CAT risk exposures. The U.S. insurance industry for instance controls an
1
2
3
Natural catastrophes are defined as non-man-made events leading to insured property
losses of at least USD 36.4 mill. (in 2003 USD). Examples are flood, storms, earthquakes, droughts and avalanches. The definition of the lower bound follows the annual
statistics on natural and man-made catastrophes published in Swiss Re: sigma (available
electronically at http://www.swissre.com). Based on insured losses, other lower bounds
for natural and man-made disasters are: shipping (USD 14.6 mill.), aviation (USD 29.3
mill.). The lower bound for total economic loss has been set equal to USD 72.7 mill. See
also Zanetti (2004), p. 41.
See Borden/Sarkar (1996), pp. 1-2. The U.S. insurance industry has for instance recorded
damage claims of USD 75 bill. between 1989 and 1995 compared to just USD 51 bill.
between 1950 and 1988 (measured in 1995 USD). Population density has increased by
75% along the U.S. Southeast Atlantic Coast between 1970 and 1990 compared to
approx. 20% for the U.S. as a whole; insured coastal property grew by 69% between
1988 and 1993 alone (see Lewis/Murdock in Froot 1999, p. 52). Whittaker (2002) argues
that economic damages resulting from natural disasters will increase by approximately
100% every 10 years in the coming decades.
See Swiss Re, sigma 1/2004, p. 38. Figures are expressed in 2003 USD. See also Munich
Re, topics geo 2004, p 14-15.
New Approaches to Managing Catastrophic Insurance Risk
343
equity capital base of USD 350 bill.4 A single earthquake with an epicenter near
Orange County (CA) or a hurricane coming to shore in the vicinity of Miami (FL)
may already impose damage claims of up to USD 50-100 bill. on U.S. property &
casualty insurers, an amount large enough to probably put a number of insurance
providers in a state of financial distress.5 Primary insurers have responded by reducing the availability of insurance coverage (e.g. homeowner insurance in high
risk areas) and by raising insurance premiums and deductibles. State legislators
have in some cases reacted by setting up public insurance schemes (e.g. California
Earthquake Authority, 1996), by establishing guaranty funds6 (e.g. Florida Hurricane Catastrophe Fund) and by passing legislative moratoriums which prevent
primary insurers from stopping to supply CAT risk coverage.7 In this context, it is
also important to note that the gap between total economic losses and losses actually insured widens significantly as we move from the developed world to transition economies and underdeveloped countries.8
The primary reason for the property & casualty insurers’ continued overexposure to CAT risk has been the limited availability of traditional reinsurance coverage which is, however, not surprising given the reinsurance industry’s narrow
capital and surplus base of USD 42 bill. (1997).9 The rising demand for reinsurance has pushed CAT reinsurance premiums up by 72% between 1990 and 2002
alone while the primary insurers’ attachment points (equivalent to deductibles)
have for instance risen by 73% between 1985 and 1994.10 The average coverage
for massive-loss events exceeding USD 5 bill. had risen markedly in the U.S.
since 1970 but still failed to reach 30% in 1994.11 Hence, the insurance industry’s
4
See Cummins/Doherty/Lo (2002), p. 558.
See Thomas (1997), p. 15 and Sandor in: Himick/Bouriaux (1998), p. 5. Scenario analysis has shown that the primary insurers’ ability to cover disaster claims may be as low as
50% (see Durrer 1996). Total damages caused by the 1995 earthquake in Kobe (Japan)
caused reconstruction expenses of approx. USD 100 bill., only USD 1 bill. of those were
insured (see Kleindorfer/Kunreuther (Challenges Facing the Insurance Industry) in Froot
1999, p. 151).
6
Guaranty funds follow the principle of deposit insurance to protect policy holders from
insurer default. Guaranty fund statutes applying to property & liability insurance have
been in force in 52 U.S. jurisdictions (including Puerto Rico) since 1981 (see Lee/Smith
1999, p. 1438).
7 The state of Florida has for instance introduced regulatory constraints which prevent insurers from raising the rates and decreasing their exposures to homeowner insurance
risks in areas with above-average disaster probabilities (see Klein/Kleindorfer (1999a),
pp. 6).
8 See for instance Munich Re, topics geo 2004, p 14-15, also Ritter (2004).
9 Froot/O’Connell (The Pricing of U.S. Catastrophe Insurance) in Froot (1999), pp. 195225) demonstrate empirically the presence of an aggregate reinsurance gap.
10 See Dunleavy et. al. (1997), pp. 59. As discussed by Froot (1999), the growth of reinsurance premiums exceeded the growth in expected losses by factor 3 in 1992 and 1993.
CAT reinsurance premiums are also subject to strong cyclicality with historical peaks in
1994 and 2003 (based on data provided by SwissRe).
11 See Froot (Introduction) in Froot (1999), p. 2-4.
5
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Ulrich Hommel and Mischa Ritter
key challenge since the early 1990s had been the search for alternative means of
reinsuring CAT risks.
The world’s equity and fixed income markets with a capitalization of more than
USD 30 trillion coupled with the organized and over-the-counter (OTC) markets
for derivative instruments provide the capability and aspiration to absorb some of
these exposures. As outlined in section 2, CAT-linked securities offer new portfolio diversification opportunities for investors given their statistical independence
from systematic market risk. In addition, traditional CAT reinsurance returns still
exceed the risk-free rate which represents the adequate minimum required return
in the absence of arbitrage and capital constraints. The discussion in section 3 explains that contract design may either aim at replicating the payoff structure of
traditional reinsurance contracts or generate payoff patterns tailored around the insurer’s specific risk management need. As discussed in section 4, the use of CAT
derivatives further enables insurers to obtain a more favorable risk profile, above
all by reducing their overall CAT risk exposure, by obtaining a superior geographical diversification of their contract portfolio and by eliminating so-called
moral hazard risk. Section 6 provides an overview of the various hedging strategies to be pursued with CAT derivatives. A key issue related to the usage of CAT
derivatives is the valuation of these contracts which is discussed in section 6. Finally, the concluding remarks (section 7) analyze the implications of securitization
and disintermediation for the future development of the reinsurance industry, in
particular its role as an agent for the pooling and bearing of risk. 12
CAT-linked securities enable insurers to reduce and restructure their existing
risk exposures and thereby enhance their risk-bearing capacities. It therefore
seems appropriate to begin the discussion by reviewing some of the economic rationales why insurers should engage in active risk management. On a generic
level, corporate risk management can be justified on the basis of market imperfections which imply an invalidation of Modigliani-Miller’s (1958) irrelevance theorem. Specifically, corporate (as opposed to investor-based) hedging activities may
raise shareholder value by13
(a) reducing the agency costs of equity financing (risk preference problem, performance signaling) and debt financing (underinvestment and asset substitution problem) associated with asymmetric information;
(b) reducing the corporate tax burden in an environment with convex tax schedules (e.g. statutory progressivity, loss forward and tax credit provisions);
(c) reducing the transaction costs of hedging associated with scale effects, asymmetric information (i.e., inability of shareholders to determine the firm’s true
exposure) or structural access barriers;
12
See also the contribution of Riemer-Hommel/Trauth on the securitization of longevity
risk in this volume.
13 Hommel (2000) and Pritsch/Hommel (1997) provide an extensive discussion of the theoretical foundations of the various risk management motives as well as their empirical
support. See also Dufey/Srinivasulu (1983) as well as Hommel and Kaen (both in this
volume).
New Approaches to Managing Catastrophic Insurance Risk
345
(d) reducing the expected costs of financial distress (i.e., reducing the probability
of encountering financial distress and the direct as well as indirect costs associated with financial distress);
(e) improving the availability of internally generated cash flows in an environment with higher absolute costs as well as rising marginal costs of external financing;
(f) helping optimize the firm’s risk portfolio (i.e., by helping to eliminate noncompensated risk exposures).
While all of these motives are more or less applicable to the insurance business
as well14, it must be recognized that there are important differences. Insurers do
not consider risk an undesirable side effect of doing business; it is their business.
They pool, repackage, cede as well as retrocede, diversify, and bear all forms of
risk exposures. Maintaining the ability to deliver on their contractual obligations
lies at the heart of their risk management activities (d).15 In addition, consolidation
pressures resulting from the internationalization of the insurance business force
market participants to pursue market expansion strategies which imply the need to
expand underwriting capacity with internal or external means (e). Financial markets offer unique opportunities for contract-based replication of CAT risk exposures which enables insurers to cede exposures directly to (primarily institutional)
investors.
Insurers are typically specialized in certain product segments and regions, if
only for historical reasons, and may therefore be prevented from exploiting their
full market potential in order to avoid the overexposure to certain risk factors.
Traditional reinsurance offers opportunities to cede some of these risks but is often
said to require special premiums to compensate for agency risk given the reinsurers’ limited ability to evaluate the risk profile of the primary insurer’s contract
portfolio.16 Financial markets provide additional opportunities for the optimization
of an insurer’s risk portfolio (f), for instance via CAT swap markets, and help to
reduce the exposure to agency risk as well as its price (a).
The market success of CAT-linked securities appears to be assured given that they
can be considered zero-beta assets, i.e., CAT insurance losses are not systemati14
See in this context also Cummins et. al. (2001) and the sources cited therein.
Empirical support has for instance been provided by Gron (Insurer Demand for Catastrophe Reinsurance) in Froot (1999), pp. 23-44. The relevance of the financial distress motive can be demonstrated by the fact that U.S. property & casualty industry has paid out
more than 100% of premiums for non-catastrophe losses in every year between 1987 and
1996; the portion of premiums paid out for catastrophe losses have fluctuated between
1% and 10.5% over the same time period (see Cummins/Doherty/Lo 2002). In order to
avoid financial distress, any pay-out above 100% must be covered by investment income
and equity.
16 See for instance Froot (2001) for a discussion of this aspect.
15
346
Ulrich Hommel and Mischa Ritter
cally correlated with the returns of any other asset classes or of the market as a
whole.17 According to the Capital Asset Pricing Model, the expected return
(minimum required return) must therefore equal the risk-free rate ( R f ).
(
E [RCAT ] = R f + β CAT , Market ⋅ E [R Market ] − R f
)
β CAT =0
=
Rf
(1)
It also follows that CAT derivatives constitute good investment propositions if the
CAT-linked asset’s Sharpe Ratio (SR) exceeds the one of the initial portfolio position (P) multiplied with the correlation coefficient between portfolio and CAT security ( ρ ). This condition is satisfied for any return above the risk-free rate. 18
SRCAT =
E [RCAT ] − R f
σ CAT
≥ ρ⋅
E [R P ] − R f
σP
⇒ E [RCAT ] ≥ R f
(2)
In order for the CAT-linked security to raise the SR, its portfolio weight ( w ) must
satisfy the Black/Litterman condition for ρ = 0 :19
E [RCAT ] ≥
(1 − w)⋅ β CAT , Pˆ ⋅ E [R Pˆ ]
1 − w ⋅ β CAT , Pˆ
(3)
where P̂ represents the new portfolio position including the CAT-linked security.
As shown by Froot et. al. (1995), investments in fully collateralized CAT insurance contracts easily met this standard when securitization started to get under
way in the early 1990s. Average returns between 1970 and 1994 exceeded the
TBill rate by 224 basis points (= 2.24%). Excess returns for national exposures
exceeded the ones for regional exposures and were higher for contracts covering
lower layers which are more frequently impacted by loss claims.
In addition to offering a favorable risk-return pattern on a stand-alone basis, the
introduction of CAT-linked securities has also generated unique portfolio diversification opportunities leading to a shift-in of the efficient frontier. 20 As demonstrated by Froot et. al. (1995), adding increments of CAT exposure to a diversified
portfolio may imply reductions in return but also triggers substantial increases in
the Sharpe Ratio by lowering portfolio volatility. While Froot et al. recommend a
portfolio weight of 25% for CAT-linked investments based on historical return
data, Litzenberger et al. (1996) and Durrer (1996) derive a more modest recommendation of 1-2%.
17
Froot et. al. (1995) estimate correlations with other major asset classes (such as bonds
and stock) and show that correlations vary between -0.13 and 0.21 but do not differ from
zero in a statistically significant fashion.
18 See also Cole/Sandor (1996), pp. 798-800.
19 See Litzenberger et. al. (1996), p. 83.
20 See for instance Durrer (1996) and Kielholz/Durrer (1997).
New Approaches to Managing Catastrophic Insurance Risk
347
CAT reinsurance can be obtained with the objective to either transfer risk to another party or to provide financing in the occurrence of a disastrous event. The existing transaction structures can basically be grouped into the following four categories:21
• Traditional Reinsurance: Transaction which involves the transfer of insurance
risk from a party who has assumed such risk from a third party by means of an
insurance contract (ceding insurer) to another party who is specialized in assuming such risk (reinsurer). This transaction type emphasizes the transfer of
CAT risk.
• Financial Reinsurance: Transactions which combine traditional reinsurance
products with financing components for purposes of generating surplus relief,
catastrophe coverage, tax optimization, and income smoothing. The emphasis
is typically placed on risk financing rather than on the transfer of risk. Contracting parties are primary insurers and reinsurers. Finite risk reinsurance as
one example of financial reinsurance may be used to complement specific coverage gaps (e.g. unhedged loss layers, settlement risk) rather than the full underlying risk.
• Securitization: Transfer of (insurance) risk from one party who has assumed or
is directly subject to such risk to another party via the issuance of securities
with a risk-linked payoff structure (e.g. CAT bonds). The transactions represent a mixture of risk transfer and risk financing with an emphasis typically
placed on the latter. Of particular relevance in this context are also contingent
financing facilities which provide financial support at the discretion of the
covered party or upon the occurrence of a catastrophic event (e.g. Contingent
Surplus Notes).
• Derivatives: Transfer of (insurance) risk by means of issuing derivative instruments with a risk-specific underlying (e.g. loss index).22 If the underlying is
sufficiently standardized to allow exchange-trading, then we also speak of insurance commoditization. Depending on the specific contract design, this
transaction type may emphasize the transfer of risk (e.g. CAT swaps) or risk
financing (e.g. CAT equity puts).
According to our definition, the distinguishing feature of securitization relative to
derivatives transactions is the presence of an initial risk exposure which is transferred to investors by the virtue of being packaged into a financial instrument.
However, wewill apply this definition rather loosely and include under this heading liability-backed securities as well as structures with embedded CAT derivatives (e.g. bond issues with coupons linked to a CAT loss index).23
21
See also Goldman/Pinsel/Rosenberg in: Himick (1998), pp. 80-82.
See Dufey (1995) for a general discussion of the principle of bundling/unbundling of
risk.
23 See Shimpi (1997) for a discussion of this distinction.
22
348
Ulrich Hommel and Mischa Ritter
Traditional CAT reinsurance contracts are typically treaty-based rather than facultative and are specified as excess-of-loss (non-proportional) policies, i.e., the reinsurer covers all claims which exceed a certain minimum level (stop-loss cover),
also called attachment point. In essence, the primary insurer exchanges exposure
to peak losses resulting from so-called correlated event risk24 against the payment
of a constant premium stream. The attachment points function as deductibles and
help to reduce the reinsurer’s moral hazard risk. The same objective may, however, be achieved with a coinsurance arrangement (also called pro rata coverage)
where insurer and reinsurer share the losses in constant proportions or, alternatively, with protective covenants which place behavioral constraints on the insurer’s behavior (e.g. with respect to the acquisition of new business in disasterprone areas). It is not uncommon for the reinsurer to cap his obligations by defining an upper attachment point (limit) beyond which he will no longer offer compensation for insurance losses, i.e., he merely offers coverage for a loss layer. As a
result, the primary insurer retains the peak of the loss exposure but, in return, reduces his exposure to counterparty (credit) risk, i.e., the risk that the reinsurer becomes insolvent as a result of the catastrophic event.
The reinsurer may alternatively choose to offer so-called state-contingent reinsurance which is based on the objective severity of a catastrophic event (e.g. total
damages, disaster severity index value such as the Richter scale) rather than on the
insurer’s actual losses.25 By doing so, the reinsurer protects himself against moral
hazard risk but exposes the primary insurer to additional basis risk. This is also the
direction taken by ART-based CAT risk coverage.
The main difficulty of insuring CAT risks is to resolve the apparent mismatch
between a stable premium flow and fluctuating insurance claims given that insurance providers lack the incentive to build up CAT-contingent surplus funds. 26 For
one, the investment of ex-ante capital tends to generate sub-par returns and therefore depresses the insurer’s overall profitability. Second, due to the lack of a binding commitment mechanism, some insurers would find it individually rational to
convert these funds into underwriting capacity and compete for business by driving down insurance premiums, thereby depressing industry profitability even
more.27 Third, insurers set up as corporations may invite raiders to undertake hostile takeover with the build-up of ex-ante capital while mutual insurers may be
pushed into a roll-back of rates by their membership or consumer advocates. In
addition, insurers face a number of regulatory constraints such as accounting re24
CAT risk represents correlated event risk since if one insurance claim is in the money,
then so will be all other claims belonging to the same risk class. See also Croson/Kunreuther (2000), pp. 25-26 for an illustration.
25 See Croson/Kunreuther (2000), p. 27.
26 See also Harrington et. al. (1995). Colquitt et. al. (1999) however demonstrate that cash
holdings of property-liability insurers vary, as expected, negatively with cash flow variance, capital market access and greater short-term demand for cash. Garven/LammTennant (1996) identify leverage (+), size (-) and the length of the payout tail (+) as explanatory factors for the level of reinsurance demand.
27 See Smith et. al. (1997), p. 28.
New Approaches to Managing Catastrophic Insurance Risk
349
strictions on the build-up of contingency surplus (e.g. under U.S. FASB No. 5)
and tax provisions which treat additions to ex-ante capital and interest on these
funds as taxable income.28 As documented by Jaffee/Russell (1997, p. 215) for the
case of the U.S., insurers also seem to lack a dynamically consistent premium
strategy which takes CAT-related loss peaks into account and which ensures a certain smoothness in the premium development. Following Hurricane Andrew, average insurance rates increased by 65% between 1992 and 1995. The Northrigde
earthquake triggered requests for rate increases from 101 insurance companies.
State Farm for instance applied for a 97.2% increase in CAT-linked policy premiums and received regulatory approval for a 65% hike.
The traditional CAT reinsurance gap has led to the emergence of a number of
ART-based solutions, including several financial-market-based transaction structures. Figure 1 gives an overview. The specific contract features and implied hedging strategies are easily understood given the previous discussion:29
• Exposure Matching: Contract design may aim at replicating the payoff patterns
of traditional reinsurance. Plain-vanilla call options display the same payoff
structure as traditional excess-of-loss insurance contracts as long as they are
written on the same underlying. Call (bull) spreads (1 long call with strike
price X ′ plus 1 short call option with strike price X ′′ > X ′ ) replicate an excess-of-loss contract with a lower as well as an upper attachment point.
• Proportional Coinsurance: Insurers may instead simply choose to part with
some of their contract portfolio and exchange the proceeds against an eventindependent cash flow stream by employing so-called portfolio transfers. They
have the flavor of an arbitrage opportunity if the premium differential between
traditional and ART-based reinsurance is positive and, in essence, replicate
proportional (pro rata) reinsurance contracts which are however less common
for CAT-linked coverage.
• Ex-Post Capital Provision: Given that insurers lack the proper incentives to
accumulate surplus funds ex ante, they are in need of a mechanism which generates an automatic infusion of capital following a catastrophic event. CAT
equity puts and intermediated debt on the basis of standby lines of credit satisfy these objectives. CAT bonds may also serve the same purpose if the insurer can delay or avoid the repayment of some of the principal (so-called ‘Act
of God’ bonds).
• Portfolio Diversification: Insurers with an undiversified contract portfolio are
exposed to correlated event risk. CAT swaps enable insurers to diversify
across product lines and geographical regions, for instance by exchanging cash
flow streams associated with home owner insurance policies in Florida against
those of automobile insurance policies in California.
• Funding Cost Reduction: Bonds and surplus notes with a CAT-linked coupon
payment enable insurers to reduce their cost of debt following a disastrous
event. While this alternative is clearly less attractive than some of the other alternatives, it was the solution most readily accepted by the investor community
28
29
See Jaffee/Russell (1997), pp. 210.
A comparatively more limiting classification has been proposed by Doherty (1997).
350
Ulrich Hommel and Mischa Ritter
with a total volume of more than USD 3.5 bill. by the end of 2003 (based on
natural disasters only).
Some of these hedging strategies will be discussed in more detail in section 5.
CAT Futures
CAT Options
CAT Bonds /
CATCAT-Linked Notes
Portfolio Transfers
CAT Equity Puts
Property Catastrophe
Swaps
Standby
Line of Credit
• Futures contract based on the ISO (Insurance Services Office) index, traded on the
CBOT between 1992 and 1993, replaced by ISO call option spreads in 1993 and
PCS call option spreads in 1995.
• Options based on insurance loss indices (typically call options). Traded on the
CBOT as call spreads on the basis of nine PCS (Property Claims Services) loss
indices since 1995 until 2000 and on the Bermuda Commodities Exchange on the
basis of Guy Carpenter loss indices since 1997 until 1999.
• Special-purpose financing vehicle (SPV) issues interest-bearing debt certificates
with CAT-linked repayment provisions and/or coupon payments. SPV writes a
CAT reinsurance contract and collects the accruing premium. Issue proceeds and
premiums are collected by a trustee who keeps the funds in a collateralized
account and repays principal and interest contingent on the occurrence of a CAT
event. Contingent surplus notes require that in the occurrence of a CAT event, the
collateralized investments are replaced by insurer-issued surplus notes. OTCinstruments first introduced in 1995 (Surplus Notes) / 1996 (Bonds).
• Insurers cede parts of its contract portfolio to a special purpose vehicle which
issues equity and debt securities to investors (other insurers, mutual funds, hedge
funds, etc.). First introduced in 1996.
• Insurers acquire put options as part of the traditional reinsurance which entitle
them to issue new equity to reinsurers at a contractually fixed price when a
catastrophic event occurs. First introduced in 1996.
• Exchange of catastrophe risk exposures between primary insurers, reinsurers,
brokers and corporations on the basis of swap contracts traded on the Catastrophe
Risk Exchange, New York (CATEX, active since 1996) or as OTC instrument
(e.g. risk-referenced total return swaps).
• Financial intermediaries give loan guarantees in case a catastrophic event occurs.
First introduced in 1995.
Source: D’Arcy/Grace (1993); Dunleavy et. al. (1997); Durrer (1996a), S. 11-15; Shepherd (1997); Smith et. al. 1997, S. 33-4.
Fig. 1. Standard Instruments for Alternative CAT Risk Transfer
Financial markets offer a number of advantages relative to intermediated reinsurance, above all flexibility, i.e., the ability to adjust reinsurance coverage on short
notice, for instance when a catastrophic event (hurricane) approaches. In addition,
exchange-traded instruments enhance the transparency of CAT risk pricing and
help to bring down the mark-up resulting from the market power of traditional reinsurers. They have also proven to open the path to multi-year and multiple-peril
policies which reinsurers have so far been reluctant to offer.
!
The discussion of the previous section has already highlighted that ART-based
CAT reinsurance not only involves an elimination of existing risk exposures but
also an alteration of the insurer’s risk profile, specifically a tradeoff between dif-
New Approaches to Managing Catastrophic Insurance Risk
351
ferent types of risk which are summarized in Figure 2.30 When choosing between
different coverage alternatives, insurers must therefore balance reductions and increases in risk exposures with explicit as well as implicit risk premiums paid and
received. It is further necessary that insurers take a dynamic stance and incorporate expectations regarding mitigation efforts31, regulatory changes and related aspects into their hedging decisions.
CA T Risk
Credit Risk
Ex A nte
Basis Risk
• Risk exposure to catastrophe events, initially embedded in property & casualty
insurance contracts and possibly ceded to reinsurers or transferred to financial
markets via securitization. It consists of two components, underwriting (magnitude
of loss claims) and timing risk (when claims are submitted).
• Risk that the counterparty (e.g. reinsurer) is not able to fulfill its contractual
obligations and cover the losses resulting from a CAT event.
• Risk exposure resulting from a mismatch between the underlying for the
reinsurance payoff (e.g. loss index) and the insurer‘s contract portfolio.
Intertemporal
Basis Risk
• Risk exposure resulting from changes in the insurer‘s risk profile if reinsurance
payoffs are based on the contract book at initiation of reinsurance coverage.
Model
Basis Risk
• Risk exposure resulting from a mismatch between the actual catastrophe and the
actuarial models used for calculating insurance payoffs and premiums.
Settlement Risk
• Risk exposure resulting from a mismatch between the ease of determining total
losses associated with the insurer‘s own portfolio vs. determining the loss figure
relevant for fixing the reinsurance payoff.
A gency Risk
(Moral Hazard Risk)
Risk)
• Arises in the presence of asymmetric information with regard to the risk profile of
the insurer‘s contract portfolio and behavior. Insurers may behave opportunistically
after acquiring reinsurance by increasing the loss potential (by writing additional
high-risk contracts which fall under reinsurance coverage) of their contract
portfolio and by failing to undertake actions which limit the loss potential for the
reinsurer (by shirking on the claims adjustment and payment process).
Source: Croson/Kunreuther (1999), Doherty (1997).
Fig. 2. Key Risk Components of CAT Insurance and Reinsurance Contracts
Traditional reinsurance enables insurers to unload some of their CAT exposures
but, in return, they expose themselves to credit risk, i.e., the risk that the reinsurer
becomes insolvent. Credit risk represents CAT risk in disguise since insolvency
will be highly correlated with the occurrence of CAT-type events. In addition, the
reinsurer is exposed to moral hazard risk if the primary insurer has the ability and
30
This overview focuses on the risk components immediately relevant for the design of
ART-based reinsurance. Other risk factors are adverse selection as a form of agency risk,
contract risk (uncertainty regarding when and to what extent the insured party exercises
the rights granted by the primary insurance contract), premium risk, expense risk and investment risk.
31 See for instance Kleindorfer/Kunreuther (1999b) and Kunreuther (1997).
352
Ulrich Hommel and Mischa Ritter
incentive to engage in opportunistic behavior.32 The agency cost resulting from
this contracting deficiency will however be borne by the party seeking coverage in
the form of higher reinsurance premiums. Traditional reinsurance does typically
not involve any ex-ante basis, model basis or settlement risk given that the insurer’s actual losses serve as the underlying of the payoff function. Intertemporal
basis risk can easily be dealt with by adjusting the coverage, i.e., by ceding more
exposures or by for instance acting as a retrocessionaire.
Exchange-traded CAT derivatives (options, futures, swaps) permit insurers to
trade off the costs of moral hazard risk against accepting some basis risk by linking the payoff from reinsurance protection to a generally accepted loss index
rather than the user’s actual contract position. They offer the added benefit that
exposure to credit risk can be avoided via the definition of position limits and
margin requirements. Settlement risk, however, arises due to the fact that the aggregate damage appraisal process does typically not follow the same principles as
the in-house claim assessment procedures.
Finally, over-the-counter instruments (e.g. bonds, surplus notes, portfolio transfers, equity puts) represent the intermediate case where the insurer can choose to
what extent coverage should be custom-tailored. Winterthur’s Hail Bond issue33
for instance links the coupon payment to the total number of hail damage claims
filed by holders of Winterthur car insurance policies for a single-day event. Moral
hazard risk is partially controlled by raising the knock-in barrier (initially 6000
damage claims) with the aggregate number of Swiss car insurance policies but
nevertheless is still relevant given that the issuer can alter the geographical distribution of his claim portfolio and thereby increase exposures in hail-prone areas.
" #$%&
$
This section highlights the use of CAT-linked securities as risk management tools
using two examples, the replication of traditional reinsurance with PCS call
spreads and the provision of ex-post capital as well as the reduction of funding
costs with CAT bonds.
32
See Bohn/Hall (The Moral Hazard of Insuring Insurers) in Froot (1999, pp. 363-384) for
a discussion of the moral hazard problem in the context of supplying reinsurance coverage by means of a guaranty fund. See in this context also Han et. al. (1997).
33 See Hess/Jaggi (1997) for a description of the bond issue and Schmock (1999) for a discussion of the role of model basis risk in this particular instance.
New Approaches to Managing Catastrophic Insurance Risk
353
!
Among the CAT-linked instruments issued in the OTC market, CAT bonds have
proven to be the most successful contract innovation.34 From the investor’s point
of view, this instrument represents a straight bond plus a short option position
which entitles the issuer upon exercise to lower the coupon payment and/or repay
only a fraction of the original principal. In the latter case, the literature also refers
to this type of instrument as an „Act of God” bond. The implicit writing of an option at the time of issue is typically compensated with a higher coupon payment
but may for instance also require the issuer to repay the principal above par.
Table 1. Risk Capital of Catastrophic Bond Issues35
Under
$50 MM
1997
1998
1999
2000
2001
2002
2003
Total
2
3
3
1
0
1
0
10
$50 MM
and over,
under
$100 MM
1
3
1
2
0
1
0
8
$100 MM
and over,
$200 MM
$200 MM
and over
Avg. Deal
($MM)
Median
Deal
($MM)
1
1
5
4
7
2
4
24
1
1
1
2
0
3
4
12
126.6
105.8
98.4
126.2
138.1
174.1
216.7
139.2
90.0
63.0
100.0
135.0
150.0
162.5
180.0
122.5
A total of approx. 60 CAT bond issues have been brought to the market since
1997, only a minority of those issued by non-financial institutions (e.g. DisneyLand Tokio, Universal Film Studios, Pylon-Transaction). Total issue volume since
1994 has been approx. USD 8.4 bill (see Table 1). Each bond issue typically consists of several tranches, i.e., while all bonds are based on the same CAT risk exposure, the issue itself is divided into several risk classes with an individualized
credit rating the AAA portion for instance being fully collateralized while the
lower-rated remainder is only partially secured (see also Table 2).36 By doing so,
insurers are able to target different investor groups, in particular institutional investors (e.g. pension funds, life insurers) which are prohibited from investing in
low-grade issues. SwissRe has been the only issuer to set up so-called CAT bond
programs allowing the flexible issuance of a strip of bonds based on a pre-
34
For a listing of OTC issues, see for instance Bernero (Second Generation OTC Derivatives and Structured Products) in Himick (1998), pp. 53-58).
35 Source: Guy Carpenter (2004), p. 4.
36 See for instance Froot/Seasholes (1997).
354
Ulrich Hommel and Mischa Ritter
specified set of CAT risk exposures (e.g. Pioneer 2002 Ltd.37). In the United
States, CAT bonds are normally issued according to SEC Rule 144a implying that
they can only be sold to qualified institutional investors and that the investors’ information rights are distinctly below exchange-trading standards.
Table 2. Ratings of Catastrophe Bond Tranches Based on Issue Number and Volume in
USD Mill. (1997-2003)38
1997
1998
1999
2000
2001
2002
2003
Total
#
1
1
1
1
0
0
1
5
B
USD
15.00
21.00
20.00
100.00
0.00
0.00
163.85
319.85
#
3
5
9
7
9
9
12
54
BB
USD
453.00
657.60
877.90
815.50
896.80
695.15
624.94
5020.89
#
2
0
1
3
0
2
6
14
BBB
USD
37.00
0.00
50.00
141.00
0.00
261.25
814.50
1303.75
#
0
0
0
0
1
0
1
2
A
USD
0.00
0.00
0.00
0.00
50.00
0.00
26.50
76.50
#
0
0
0
0
0
0
0
0
AA
USD
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
#
1
3
1
0
0
0
0
5
AAA
USD
82.00
22.50
1.40
0.00
0.00
0.00
0.00
105.90
CAT bonds come with an average maturity of approx. 3 years (based on total volume outstanding in 2003) reflecting the investors’ sentiment against the provision
of truly long-term risk coverage.39 The key feature of any CAT bond issue is the
specification of the insurance trigger (i.e., index value for industry-wide losses,
company-specific claims or parametric event specifications above which CAT
protection starts to set in) as well as the specification of the payoff function for the
in-the-money region (see Table 3).
Table 3. Maturity and Triggers of CAT Bond Issues (1997-2003)40
1997
1998
1999
2000
2001
2002
2003
Total
1
2
2
7
5
3
2
0
0
19
1
0
0
1
1
1
2
6
Maturity in Years
3
4
1
0
3
4
3
4
4
19
0
0
0
0
1
2
1
4
5
10
0
1
2
1
0
0
2
6
1
0
0
0
0
0
0
1
Indemnity
Trigger
3
5
4
2
1
1
2
18
Other
Triggers
1
0
2
3
6
7
5
24
The majority of CAT bond issues have been based on U.S. exposures (East/Gulf
Coast hurricanes with USD 3.6 bill., California Earthquake with USD 2.9 bill.) be37
Pioneer included six tranches based on five parametric indices tied to natural perils. See
SwissRe Capital Markets (2003), p. 3-4.
38
Source: Guy Carpenter (2004), p. 6. S&P Ratings supplemented by Fitch where requiered.
39 See Guy Carpenter (2004) and table 3.
40 Source: Guy Carpenter (2004), p. 5.
New Approaches to Managing Catastrophic Insurance Risk
355
tween 1997 and 2003 while European- and Japanese-based issues only represent a
total of USD 4.2 bill. In comparison, the relevance of other geographical regions is
negligible in comparison (USD 1.4 bill.).41
CAT bond issues are typically structured in a fashion that allows the insurer to
obtain a form of risk coverage which receives the same accounting treatment as
traditional reinsurance. For this purpose, the bond is actually issued by a special
purpose vehicle (SPV) which transfers the proceeds to a (partially/fully) collateralized trustee account42 and writes a regular reinsurance contract to the insurer. Principal and interest are returned to the investor the same way unless a contractually
specified catastrophic event occurs in which case some of the funds are diverted to
satisfy the insurer’s loss claims.43 Figure 3 depicts the typical transaction structure.
Conditional
Loss Coverage
Premium
= Rate on Line
Reinsurance
Contract
(Re-)Insurer
(Re-)Insurer
CAT-linked Principal + Coupon Payment
(Ex-Post Capital Provision, „Act of God Bonds“)
Coupon
Payment
Proceeds
Premium
Issue Proceeds
Special Purpose
Vehicle
Issue Proceeds
(Investment)
Investors
Principal + CAT-linked Coupon Payment
(Funding Cost Reduction)
Transactions/Flow of Funds at Initiation
Trust Company
(Collaterization)
Collaterization)
Transactions/Flow of Funds after Initiation
Fig. 3. Transaction Structure for CAT Bonds
Several parties have to be involved to bring a CAT bond issue to market. Placement of the issue is typically handled by one or more investment banks while contract structuring is done by the lead investment bank or the issuer itself. Risk modelling and pricing are generally outsourced to a specialized modelling agency.
Legal structuring are handled by specialized legal firms. 44 Standard rating agencies evaluate the quality of the issue. Other cost-generating activities involve
41
See Guy Carpenter (2004), p. 7.
Collateralization can be obtained by investing the proceeds in government securities.
43 See Tilley (1995) for a more technical introduction.
44 In recent years, approx. 70% of all transactions have been managed by Skadden, Arps,
Slate, Meagher & Flom LLP.
42
356
Ulrich Hommel and Mischa Ritter
printing of the prospectus, creation of the SPV, road show and the ongoing management of the issue.45
Liquidity for secondary trading is established by a small number of specialized
market makers (among them SwissRe, Goldman Sachs, Lehman Brothers and
Cochran-Caronia). While, not surprisingly, Bouzouita/Young (1998) argue that
secondary market liquidity is the primary determinant for the demand of CATbased instruments, liquidity is rather low compared to other instruments belonging
to the same risk class.46 Following the discussion in section 2, CAT bonds represent an attractive investment opportunity on the basis of pure return prospects as
well as of their portfolio diversification potential. Using a Sharpe-rate-based
analysis, Canabarro et al. (1998) show the CAT bonds offer significantly higher
returns than other fixed-income securities with an analogous default probability.
Excess-of-loss Contract Replication with PCS Call Option Spreads
In order to replace the rather unsuccessful catastrophe futures and options contracts which were based on the comparatively much coarser ISO index, the Chicago Board of Trade had initiated the trading of catastrophe option contracts based
on the loss indices of the Property Claims Services (PCS) in 1995. The main objective had been the creation of a contract which allows the replication of traditional excess-of-loss reinsurance, in this case via the purchase of a call spread
(combination of a short and a long call position). Trading was discontinued in
2000 due to lacking investor demand. Table 4 summarizes the historical contract
features and Figure 4 gives an overview of the evolution of trading volume. At its
peak, the market provided an additional reinsurance capacity of USD 89 mill. 47
45
Ritter (2004) reports average transaction figures based on Canter (1999) and survey evidence (2003): economic strucutring (USD 1.5 mill., USD 1.5 mill.), legal structuring
(USD 300-350,000, USD 220,000), modelling (USD 150-200,000, USD 150,000), rating
(USD 150-250,000, USD 150,000), other expenses (n/a, USD 150,000). Total expenses
for 2003 have been in absolute figures approx. USD 2.2 mill. for an average transaction
size of USD 150 mill. Running expenses amount to approx. USD 180,000 per year. Actual transaction costs will of course vary with specific bond characteristics, for instance
deal complexity. See also Froot (1999c), p. 23 and d’Agostina (2003), p. 21.
46 See also Lane (2001), p. 2.
47 See Laster/Raturi (2001), p. 21.
New Approaches to Managing Catastrophic Insurance Risk
Number of
Contrac ts (,000s)
357
Open
Interest (,000s)
4.0
8
3.5
7
3.0
6
2.5
5
2.0
4
1.5
3
1.0
2
0.5
1
0
Sep 95
Sep 96
Sep 97
Sep 98
Sep 99
0
Fig. 4. Trading Volume of PCS Call Option Spreads (1995-2000)48
Table 4. Characteristics of PCS Options
Option Type
Loss Cap
Unit of Measurement
Quotation Units
Strike Prices
Loss Period (Specification)
PCS Indices
Max. Daily Price Change
Position Limits
Development Period
Settlement Date
Settlement
Trading Time
European calls and puts on aggregate property & casualty loss claims
small caps: industry losses from USD 0 to USD 20 bill.
large caps: industry losses from USD 20 bill. to USD 50 bill.
1 index point = USD 200 value = USD 100 mill. damages
1/10 of an index point = value of USD 20
defined as multiples of 5 (small caps: 0 to 195 for calls, 5 to 200 for
puts; large caps: 200 to 495 for calls, 205 to 500 for puts)
1st quarter (March), 2nd quarter (June), 3rd quarter (September), 4th
quarter (December), calendar year (Annual)
national (U.S.), regional (East, Northeast, Southeast, Midwest, West),
states (California, Florida, Texas) – contracts on the Californian and
Western index have annual loss period, national contracts are offered
on a quarterly as well as annual basis, all other contracts have a quarterly loss period
small caps: 10 index points, large caps: 20 index points
aggregate position of 10,000 options, reporting requirement for positions of 25 contracts or more
12 months after the loss period has ended (time allotted to the recording of insurance claims with reference to the loss period)
last business day of the development period (also last trading day), inthe-money options are automatically exercised at 6 p.m.
cash settled; small caps: min{USD 200 times index settlement value,
USD 40,000}; large caps: max{min{USD 200 times settlement value
of the index, USD 100,000}, USD 40,000}
8:30 a.m. – 12:30 p.m. (local time)
Source: Chicago Board of Trade (1995) http://www.cbot.com/
48
Source: CBOT data based on Laster/Raturi (2001), p. 22.
358
Ulrich Hommel and Mischa Ritter
Determining the optimal hedge portfolio for the coverage of a particular loss layer
requires the application of the following procedure. Assume that the insurer
wishes to acquire coverage for a layer of magnitude y above firm-specific losses
Y . The hedge calculation requires the insurer to identify the corresponding PCS
spread contract and the number of contracts needed to generate the desired coverage. PCS call spreads have a generalized payoff structure π with
π = max{L − X 1 , 0} − max{L − X 2 , 0} .
(4)
L represents total loss claims resulting from the catastrophic event. X 1 and
X 2 > X 1 are the strike prices of the implicit long and short option position. The
units of measurement are index points, each representing USD 100 million (= 10 8)
in insured damages. The insurer is assumed to have a market share of α ⋅100% .
Firm losses are estimated to deviate from industry-wide losses by a factor λ .
The specification of the call spread uses the end points for the desired cover as a
starting point and adjusts for market share and exposure deviation. The strike
prices for the implicit option positions are given by
X1 =
1
Y 1 
⋅ ⋅ 
10  α λ 
8
X2 =
Y + y 1 
⋅
⋅ .
λ
10  α
1
8
(5a/b)
The number of call spread contracts required to cover the range between Y and
Y + y is given by
N=
y
( X 2 − X 1 )⋅V
(6)
where V represents the USD value of an index point. Following the general reinsurance practice, the cost of spread reinsurance is expressed as the rate on line
which is defined as the premium paid divided by the gross amount of risk transferred. Figure 5 summarizes the results graphically. Numerical examples can be
found in Canter et. al. (1996) and Hommel (1998).
The insurer is still exposed to ex-ante basis risk. First of all, the relationship between firm and industry losses will be non-constant or may even be a non-linear
function. Second, the relationship between firm and industry losses will always
have a stochastic element given that the relative market position is never uniform
across hazard-prone regions.49 As a consequence, it is not feasible to fully predict
how many contracts will be needed ex post to ensure full coverage.
49
This has been the main reason for the introduction for CAT options on the Bermuda
Commodities Exchange which were based on zip-code-based loss indices published by
Guy Carpenter (see Mullarkey/Froot et. al. 1998 for a detailed description and a comparison with other index alternatives). Trading of these contracts was, however, also discontinued in 1999 for the same reasons. See also Major (Index Hedge Fund Performance) in
Froot (1999, pp. 391-426) for an analysis of the role of the index choice for hedging performance. See in this context also Harrington/Niehaus (1999).
New Approaches to Managing Catastrophic Insurance Risk
Payoff
359
Firm Losses
Slope = 108 ⋅ α ⋅ λ
Y+y
X1
X2
Strike
Price
Y
X1
Call Spread Payoff Profile
X2
Loss
Index
Loss Layer Transformation
Fig. 5. Hedging with PCS Call Spreads
Besides representing a substitute for traditional reinsurance50, PCS call options
have been used51
• to improve the geographical diversification of the insurer’s contract portfolio
by engaging in buying as well as selling of PCS call spreads (based on different PCS indices),
• to obtain coverage for loss layers not included in traditional reinsurance (typically layers with larger attachment points),
• to swap exposures by buying and selling PCS call spreads for different layers
of the same PCS index instrument (e.g. butterfly spreads),
• to increase reinsurance coverage in the wake of a catastrophic event, 52
• to replace traditional single-peril reinsurance immediately after a catastrophic
event.
The valuation of CAT securities requires above all a probability assessment of the
insured losses within a specified geographical region which can be generated with
one of the following two methods:53
50
See O’Brien (1997) for a comparative (albeit somewhat preliminary) performance analysis of alternative hedging strategies (buy-and-hold, periodic readjustment, threshold adjustment) with PCS options.
51 See also Hommel (1998), p. 214.
52 Trading activity tends to increase significantly immediately before and during major
catastrophic events (see also Cantor et. al. (1996), p. 100).
53 See Litzenberger et. al. (1996), pp. 78-80.
360
Ulrich Hommel and Mischa Ritter
•
Estimation with Historical Time Series Data: Forecasts can be obtained by using the historical time series of loss ratios, e.g. the aggregate losses measured
by the PCS loss index divided by total premiums earned in lines of insurance
business with CAT exposure. In order to obtain a meaningful predictor, the
forecast procedure must include adjustments for population growth, growth in
insured property values, changes in weather patterns and changes of other explanatory variables.
• Forward-Looking Simulations: „Ground-up” assessment of loss probabilities
using seismological, meteorological and economic data on the basis of simulations with randomly generated disaster scenarios. Modeling is based on data
from historically recorded catastrophes as well as on subjective disaster assessments.54
Actuarial pricing of insurance contracts applies these procedures to determine the
expected value of covered losses and applies a loading margin to control the probability of ruin and thereby ensure the underwriter’s solvency. 55
Basic financial pricing is founded on the no-arbitrage principle, i.e., the value of
financial securities must equal the risk-adjusted present value of future net cash
flows. The pricing of derivatives typically relies on the principle of risk-neutral
valuation which applies an equivalent martingale measure as a risk-adjusted expectation operator and employs the risk-free rate of interest as the relevant discount factor. 56 In contrast, the valuation models for CAT securities suggested by
the literature fall into one of the following categories:
• Standard Arbitrage Approach: Cummins/Geman (1995) have proposed an arbitrage-based methodology using an instantaneous stochastic process to describe the evolution of a loss claims index consisting of a geometric Brownian
motion to capture the stochastic timing of claims and a Poisson jump process
to represent the catastrophic event itself. Since the underlying represents a sum
of claim payments, it is treated as an Asian option and formally priced on the
basis of a Monte Carlo simulation. Cummins/Geman, however, fail to take into
account that jump sizes are by their very nature random which necessarily implies that markets are incomplete. Hence, there does not exist a unique martingale measure to price these assets.57 In addition, it can also be subject to discussion whether the uncertainty regarding claim reporting should be an
essential feature of a CAT pricing model. Aase/Ødegaard (1996) for instance
study a marked point process which characterizes the average time between
events as a Poisson process and the severity of the event as a Gamma process.
Geman/Yor (1997) model the aggregate claims as a jump diffusion process (ignoring claim reporting uncertainty) and derive „quasi-analytical” solutions on
the basis of Laplace transformations.
54
For a detailed how-to introduction, see for instance Hutton (The Role of Computer Modeling in Insurance Risk Securitization) in Himick (1998), pp. 153-168.
55 See for instance Dong et. al. (1996).
56 See Neftci (1996), pp. 101-143.
57 See Embrechts et. al. (1997), p. 509.
New Approaches to Managing Catastrophic Insurance Risk
361
• Markets Approach: Chang et al. (1996) use a so-called „randomized time approach” to derive implied option values from observed futures prices. It lacks
any usefulness given that CAT futures trading has been suspended with the introduction of PCS call options in 1995.
• Preference-Based Approach: Another strand of the literature attempts to circumvent market incompleteness by developing a valuation model with microfoundations, i.e., by employing a utility maximization framework which captures the agents’ attitudes towards risk given that a perfect hedge is not feasible.58 While this approach allows us to derive unique prices for CAT instruments, we sacrifice an essential property of standard security pricing, the
independence from investor preferences. The results are of limited use since
they are driven by the specification of the utility functions, even more so if we
need to assume that preferences are uniform across all agents (representative
agent technique59).
Overall, it must unfortunately be concluded that there does not exist an exact
model for the pricing of CAT securities given market incompleteness. Future contributions will only be able to alleviate the problem to a certain degree.
In closing, it needs to be emphasized that the introduction of CAT-linked securities has not been an unambiguous success so far. Exchange-traded instruments
have basically ceased to exist but the bond premium puzzle is slowly losing its
relevance, i.e., CAT premiums appear to slowly close in on those of comparable
high-yield bonds as the market has seen a record year in 2003 with a total volume
of USD 1.73 bill.60 Bantwal/Kunreuther (2000) had identified a number of potential explanations for this premium puzzle, among them excessive risk aversion
(especially with respect to basis risk)61, myopic loss aversion, ambiguity aversion,
the lacking ability to understand the risks involved and the costs of acquiring the
58
See Aase (1995) for an application of this concept. See also Embrechts (1996) and
Meister/Embrechts (1995) for a general discussion of this problem, including the use of
the Esscher principle of probability transformation for obtaining a unique martingale
measure.
59 See also Cox/Pedersen (1998) for an application of this principle.
60 A detailed analysis of the 1997 USAA bond issue has been provided by Froot (2001).
Evidence on rising CAT bond demand has been reported by d’Agostino (2003, p. 28) and
Guy Carpenter (2004), p. 4).
61 Moore (1999), p. 35, derives an implied relative risk aversion of magnitude 30 which is
consistent with the value range necessary to explain the equity premium puzzle. A decision-maker with these preferences would for instance assign a certainty equivalent value
of USD 51,209 to a 50:50 gamble between receiving USD 50,000 and USD 100,000 (see
Bantwal/Kunreuther 2000, p. 80).
362
Ulrich Hommel and Mischa Ritter
necessary know-how.62 Thus, at this point, financial markets have merely the ability to complement traditional and financial reinsurance. The continued inability to
obtain reinsurance coverage for higher loss layers has led to proposals favoring the
introduction of government-backed (but overall self-supporting) excess-of-loss
policies for aggregate claims between USD 25 bill. and USD 50 bill., mainly because government agencies have the ability to easily manage any intertemporal
mismatches between premium inflows and loss claim outflows.63
Financial markets are already in the process of penetrating other lines of insurance, the current emphasis being placed on the securitization of life insurance contracts.64 Persisting consolidation pressures force life insurers to obtain acquisition
currency. In addition, they face the added problem that most of the costs associated with life insurance contracts are to be paid in the early stages which raises the
short-term financing burden of horizontal acquisitions. The securitization of the
contract portfolio’s future cash flows could help to resolve this problem to some
degree and, as an additional bonus, converts the economic value of a contract portfolio into regulatory capital. However,it needs to be emphasized that securitization
merely serves as a financing vehicle in this instance rather than as a hedging tool.
The transfer of alternative risk to the market also forced an other ongoing development. Based on the understanding that more than 20% of the US economy is directly weatherdependent,65 capital market participants are trying to develop a
product which allows to shift this weather risk to the capital market. Initiated
through the deregulation of the energy market in the US, the weather derivatives
market seems to get more and more importance. The weather derivatives market
seems to overhelm the cat bond market with a capacity of over USD 7.5 bill. since
1997.66 Starting in the US, now worldwide seems the weather trading to be the
most growing area in ART. Currently most traded contracts are based on temperature (cooling or heating degree days are used as basis), but also rain and snowfall,
humidity and sunshine etc. are possible. The contract structure can be like a vanilla option, a swap, option collars, or exotic options.
Longevity risk can also be expected to become an important issue as medical research is coming closer to discovering the genetic code which drives the human
aging process. Given that these advances will take place in discrete jumps and
may lead to rapid shifts of the life expectancy of policyholders, life insurers face
potential exposures of tremendous proportions. Following the CAT example, the
resulting reinsurance gap will trigger a search for alternative risk coverage, for in62
Rode/Fischhoff/Fischbeck (2000) take a behavioral finance approach and support the
presence of psychological motives for the existence of the premium puzzle.
63 See in particular Lewis/Murdock (1996) and Lewis/Murdock (Alternative Means of Redistributing Catastrophic Risk in a National Risk-Management System) in Froot (1999),
pp. 51-85). Cummins et. al. (Pricing Excess-of-Loss Reinsurance Contracts against
Catastrophic Loss) in Froot (1999), pp. 93-141) incorporate the specific features of the
proposal into a general valuation model.
64 e.g. Vita Capital Ltd. capturing a SwissRe issue linked to mortality.
65 See Bossley (1999a), p. 42-43.
66 See Swiss Re, sigma 1/2003, p. 39.
New Approaches to Managing Catastrophic Insurance Risk
363
stance in the form of mortality (or survivor) bonds whose performance depends on
shifts of the mortality tables.
The discussion of this chapter has demonstrated that the successful transfer of
insurance risk to financial markets requires a reinsurance or financing gap, investor interest (i.e., additional diversification opportunities), palatable hedging opportunities for the insurance industry and the ability for both sides to agree on a
methodology for pricing these risks. These factors explain why CAT securities
have been introduced in the first place, why market volume has been rather modest so far and what other segments of insurance business will follow the path of
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Shimpi P (1997) The Context for Trading Insurance Risk. The Geneva Papers on Risk and
Insurance 22(82) January : 17-25
Smith RE, Canelo EA, Di Dio AM (1997) Reinventing Reinsurance Using the Capital
Markets. The Geneva Papers on Risk and Insurance 22(82) January : 26-37
SwissRe Capital Markets (2003) Insurance Linked Securities Quarterly. SwissRe, New
York
Tilley JA (1995) The Latest in Financial Engineering: Structuring Catastrophe Reinsurance
as a High-Yield Bond. Morgan Stanley & Co. Inc., New York
Whittaker M (2002) Climate Risk to Global Economy. United Nations Environment
Program Finance Intiatives (UNEP-FI).
Zanetti A, Enz R, Heck P, Green JJ, Suter S (2004) Natural Catastrophes and Man-Made
Disasters in 2003. Sigma 1
Zanetti A, Enz R, Menzinger I, Mehlhorn J, Suter S (2003) Natural Catastrophes and ManMade Disasters in 2002. Sigma 2
Christopher L. Culp1
1
Kirchgasse 4, CH-3812 Wilderswil (BE), Switzerland or 540 North State Street
# 3711, Chicago, IL 60610, U.S.A.
Abstract: Alternative risk transfer (ART) refers to the products and solutions that
represent the convergence or integration of capital markets and traditional insurance. The increasingly diverse set of offerings in the ART world has broadened the
range of solutions available to corporate risk managers for controlling undesired
risks, increased competition amongst providers of risk transfer products and services, and heightened awareness by corporate treasurers about the fundamental
relations between corporation finance and risk management. This chapter summarizes the dominant products and solutions that comprise the ART world today.
Alternative Risk Transfer (ART) includes those contracts, structures, and solutions
that enable firms either to finance or transfer some of the risks to which they are
exposed in a non-traditional way.1 ART is all about “convergence” – the convergence of capital markets and insurance, the convergence of corporate finance and
risk management, the convergence of swap dealers with (re-)insurance companies,
and so on.2
The increasingly diverse set of offerings in the ART world has broadened the
range of solutions available to corporate risk managers for controlling undesired
risks, increased competition amongst providers of risk transfer products and services, and heightened awareness by corporate treasurers about the fundamental relations between corporation finance and risk management.
This chapter presents a descriptive overview of the major products and solutions in the ART market today. After reviewing in Section 2 the origins of the
term “ART” in the form of “captives,” Section 3 then describes “finite risk” pro1
2
Not all risk, of course, is “undesired.” Some risk is necessary for the profitable operation
of a business. How to distinguish between core risks that a company is in business to
bear and other risks is the subject of lengthy discussions in Culp (2001, 2002a, 2004).
The convergence of corporate finance and risk management that the proliferation of ART
products has encouraged is not emphasized in this article, purely for reasons of length.
For that analysis, interested readers are requested to consult Culp (2002a, 2002b, 2002c,
2002d).
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grams (including some recent controversy that has surrounded these innovative
structures). The products discussed in Sections 2 and 3 tend to be used primarily
for risk finance (i.e., raising funds at a fixed price to smooth the cash impact of a
loss) rather than risk transfer (i.e., actually shifting the adverse impact of a risk to
another firm). Sections 4 and 5 review two ART forms used for true risk transfer –
multi-line, and multi-trigger programs. Section 6 then explains how the structured
finance world is increasingly becoming a part of the ART universe, both for risk
finance and risk transfer applications. Section 7 summarizes a relatively new ART
form known as contingent capital, and Section 8 concludes.3
ART first gained widespread acceptance as an industry term in the 1970s to describe organized self-insurance programs. As insurance markets hardened and
produced rising premiums and declining capacity, corporations wanted to emphasize to insurers that they could often seek the protection they needed through alternative means, the most obvious of which was self-insurance. Even today, more
and more corporate treasurers are reminded that their weighted-average cost of
capital – i.e., the cost of raising capital to self-insure – should be approximately
the limit they are willing to pay for external capital provided through an insurance
program.
The decision to self-insure is called a retention decision. A planned retention is
a risk to which a firm is naturally subject and that the firm’s security holders are
prepared to bear on an ongoing basis. A planned retention may occur either because the alternative – risk transfer – is too expensive, or because the risk is considered integral to the firm’s core business activities and operating profits. (Culp
2001, 2004)
A funded planned retention is a retained risk for which a firm sets aside funds to
smooth the cash flow impact of future losses. A firm may wish to obtain funds
now to cover a subsequent loss, for example, if post-loss funding costs are expected to rise dramatically in response to the announcement of the loss. The practice of pre-funding retained risks is broadly known as risk finance.
Self-insurance is a form of risk finance. Not all self-insurance, however, is
ART. Many self-insurance schemes are either indirect or informal, such as earmarked reserves. Funds allocated to such loss reserves neither get the favorable
tax and accounting treatment afforded for true insurance or qualified selfinsurance, nor are such funds immune from the temptation managers may have to
use the funds in some other way.
The major ART forms designed to facilitate the pre-loss funding of planned retentions are significantly more “formalized” than simply earmarking balance sheet
reserves. These ART forms are discussed in the sections below. Although many
3
Although the material here is original, portions of this chapter draw heavily from Culp
(2002a).
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371
originally were and still are primarily intended only as a source of risk finance,
some of the more recent structures discussed below include some degree of risk
transfer, as well.
To set aside funds as self-insurance against future losses in a way that does not
arouse investor suspicions that the money could be spent in some other way, the
alternative risk financing structure called a captive emerged in the 1970s. In its
most basic form – called a single-parent captive – the sponsoring firm sets up a
wholly owned subsidiary that is also a licensed (re-)insurance company and then
purchases insurance from itself by way of the new captive subsidiary. The equity
capital of the captive is usually minimal – just enough for the captive to obtain an
insurance or reinsurance license in a captive-friendly domicile. 4 Having obtained
this license, the captive then writes explicit insurance contracts to the ceding sponsor to cover the risks the sponsor wishes to pre-fund in exchange for explicit premium payments. In certain circumstances, the premium paid is tax-deductible.
Captives are commonly used by firms to insure high-frequency, low-severity
loss events for which the ceding sponsor has a relative stable historical loss experience. The expected losses in such cases are usually relatively easy to estimate,
and the premium collected by the captive for providing insurance is set equal to
those expected losses in present value terms over some risk horizon– usually a
year. If the present value of actual losses exceeds the present value of expected
losses charged by the captive to the sponsor as a premium, the risk borne by the
captive – and ultimately the ceding sponsor – is called underwriting risk.
Because loss claims do not necessarily arrive in the same time period (e.g., year)
that premium is collected, however, the captive also faces both timing and investment risk. Timing risk is the risk that the assets acquired by the captive using premium income to fund future claims grow at a rate that may be perfectly correct in
a present value sense – i.e., after a year the assets may be worth exactly the total
claims paid – but that may be too low to finance the unexpected arrival of a lot of
large claims early in the insurance cycle. In other words, timing risk is the risk that
the captive’s assets are inadequate at any discrete point in time to fund its current
liabilities. Investment risk is the related risk that market risk on the assets acquired
by the captive to fund its claims results in an unexpected shortfall of assets below
insurance liabilities.
Like any other insurance company, a captive manages its underwriting risk by
attempting to price its insurance to cover expected losses but manages its timing
and investment risk through its technical reserves. Technical reserves represent the
future claims expected on the insurance contracts the captive has written to its
ceding sponsor corporation and come in two types – unearned premium reserves,
and loss reserves.
4
Captive-friendly domiciles exist both on-shore (e.g., the State of Vermont) and off-shore
(e.g., Bermuda, Singapore, and the Channel Islands).
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Christopher L. Culp
In most insurance lines (e.g., liability and property), policy coverage lasts one
year and premium is payable at the beginning of the policy year. Although premium is collected in advance, it is earned only as time passes if a claim has not
occurred. Unearned premium is premium that has been collected and may still
need to be used to cover an as-yet-un-submitted claim. The unearned premium reserve is thus the proportion of premium that must be set aside to honor future expected claims.
The technical reserves an insurance company maintains to honor any future
claims – known or unknown – above the unearned premium is called the loss reserve. Loss reserves may be set aside for losses that have been reported and adjusted, reported but not adjusted, incurred but not reported (IBNR), or for loss adjustment expenses.
Like more traditional insurance companies, captives engage in one of two types
of reserve management methods for financing the claims arising from their liabilities (Outreville 1998). Under the first method – the capitalization method – the
captive invests the premium collected from the ceding sponsor in assets and then
uses those assets plus the return on those assets to finance subsequent insurance
claims. Captives using the capitalization method usually attempt to keep assets
funded by premium collections linked to the technical reserves of the liabilities for
which premium was collected. Technical reserves at captives using the capitalization method tend to be medium- or long-term, as are the assets invested to back
the corresponding liabilities.
The compensation method, by contrast, is a “pay-as-you-go” system in which all
premiums collected over the course of a year are used to pay any claims that year
arising from any insurance coverage the captive has provided to its sponsor. Under
this method, no real attempt is made to connect assets with technical reserves. All
premium collected is used to fund mainly short-term assets, and those assets collectively back all technical reserves for all insurance lines.
One important implication of the differences in reserve management styles is the
captive’s potential demand for reinsurance. If the captive becomes concerned that
the funded retention should actually have been transferred rather than retained, the
captive structure makes it easy for the firm to acquire selective reinsurance for
risks about which the ceding sponsor and the captive may have been especially
worried.
A major attraction of the captive structure – like pure self-insurance – is the retention of underwriting profits and investment income on assets held to back unearned premium and loss reserves. If the actual losses underwritten by the captive
are lower than expected, the sponsor can repatriate those underwriting profits –
plus any investment income – in the form of dividends paid by the captive to its
sole equity holder, the sponsor.
Local laws, regulations, or tax requirements often require firms to obtain local
insurance coverage. In this case, firms may opt for a captive structure in which the
captive is incorporated and chartered as a reinsurance company rather than an insurance company. In this case, the ceding sponsor then buys its coverage from a
locally recognized insurer (called a fronting insurer), which then reinsures 100%
of the exposure with the captive. In some cases, multiple fronting insurers are re-
Alternative Risk Transfer
373
quired to provide recognized cover to different operating subsidiaries of multinationals, as illustrated in Figure 1.
Sponsor/Self-Insurer
(Parent Corporation)
Premiums
Claims
Payments
Equity
Dividends
Premiums
Fronting Insurer
(Locally Licensed Insurance Company)
Claims
Payments
Captive
(Licensed Reinsurance Company)
Fig. 1. Single-parent captive with fronting insurer
The costs of setting up a captive are often surprisingly low. Most of the costs go to
the fronting insurers (if required) and to the captive manager (i.e., the firm retained to run the captive, process claims, and the like). Despite the relatively affordable nature of a captive, not all firms wishing to self-insure find single-parent
captives to be the ideal solution. Some firms, for example, self-insure cyclically
based on how high external premiums are. For such firms, the costs of constantly
setting up and dismantling captives (or allowing one to remain open but idle) can
get prohibitive quickly.
Firms that opt not to establish single-parent captives have several alternatives
available. The difference between most of these alternatives is the degree to which
the structures facilitate pure risk financing versus actual risk transfer. This distinction will become clearer as the alternatives are discussed below.
One way to enjoy the benefits of a captive without setting up a single-parent captive is to share captive ownership through a multi-parent or group captive – i.e., a
captive insurer whose equity ownership is held by several firms rather than just
one. Such structures are also similar in design and operation to mutual insurance
companies, risk retention groups, and other cooperative-style insurance companies.
A group captive, for example, is a captive reinsurance company that collects
premium from multiple sponsors and in turn agrees to underwrite certain risks of
those sponsors. The premium, investment income, expenses, and underwriting
risks are all pooled. The loss-sharing regime, in turn, may be proportional to the
premiums paid into the captive or fully mutualized. In either case, group and
multi-parent captives involve some degree of risk transfer through the pooling of
claims by the multiple sponsors.
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Christopher L. Culp
Group captives are often set up by industry trade associations on behalf of their
members. Energy Insurance and Mutual Limited, for example, is the group captive
representing numerous U.S. electricity and gas utilities. The benefits of pooling
premiums and risks allow the group captive to achieve a smoother time profile of
loss payouts than would be possible in any individual participant’s situation.
If a firm does not wish to establish a single-parent captive but also does not wish
to engage in risk transfer through the pooling arrangements typically associated
with mutuals, two alternative structures are still available in the self-insurance
realm. The first is called a rent-a-captive. A rent-a-captive is essentially similar to
a multi-parent group captive except that the participating corporations relying on
the captive for insurance do not actually own any part of the rent-a-captive and do
not pool their risks with one another.
Rent-a-captives are set up, maintained/managed, and owned by market participants like (re-)insurance companies or insurance brokers for the benefit and use of
corporate customers. The customers in turn remit premium payments to a fronting
insurer that then cedes the premium to the rent-a-captive through facultative reinsurance to give the customers coverage for losses on the risks they wish to retain.
The rent-a-captive itself typically maintains “customer accounts” for participants
in which premiums are credited and claims booked. In addition, investment and
underwriting income are tracked and may be returned to the participants, usually
when the rent-a-captive contract is terminated. Unlike a multi-parent captive, the
individual customer accounts in a rent-a-captive are segregated.
From the perspective of a self-insuring participant, the rent-a-captive works
much like a single-parent captive except that ownership rights and dividends now
accrue to a third party.5 Figure 2 illustrates.
Self-Insurer
Premiums
Rent-a-Captive Owner
Claims
Payments
Equity
Dividends
Premiums
Fronting Insurer
(Locally Licensed Insurance Company)
Claims
Payments
Rent-a-Captive
Fig. 2. Rent-a-Captive
Some have expressed concerns, however, that rent-a-captives do not achieve true
customer account segregation – specifically, that the customer accounts are on paper only, but that the actual funds are commingled. Participants then worry that
5
The rent-a-captive may also require collateral from participants in excess of premium
paid to pre-fund later losses.
Alternative Risk Transfer
375
the commingled assets of the captive may be mis-invested, yielding reserve losses,
or that the loss of one firm could be ex post mutualized in the event of the captive’s insolvency.
As a result of some of these concerns about rent-a-captives, captive management organizations have been offering a second alternative for corporations wishing to self-insure in a non-mutualized captive structure without setting up a singleparent captive. Called protected cell companies (PCCs), these entities are set up
essentially like a rent-a-captive except that customers have ring-fenced and genuinely segregated, bankruptcy remote accounts.
!"
Once known as financial reinsurance and unique to the insurance industry, finite
reinsurance or finite risk products now provide insurance companies and nonfinancial corporates alike with an important source of risk finance. Like captives,
finite risk products are primarily intended to help firms pre-fund a retained risk
that the firm wishes to self-insure. Increasingly, however, finite risk programs also
contain some degree of risk transfer, as well, thus offering corporate risk managers
a way to pre-fund certain losses and transfer others.
A major distinction between the types of finite risk ART forms available in the
market today is whether or not the liability whose timing risk is being managed
with a finite risk product has or has not already been incurred. Retrospective finite
risk products are intended to help firms manage the timing risks of existing liabilities of the firm, whereas prospective finite risk solutions cover contingent liabilities that have not yet been formally assumed by the firm. In the case of an insurance company seeking financial reinsurance through finite risk products,
retrospective finite risk products cover past underwriting years and prospective
products cover current or future underwriting years. For a corporation, the distinction is essentially the same except that the liabilities being managed are not acquired through an underwriting process but are instead the result of some business
decision(s) made by the firm that alter its natural risk profile.
A Loss Portfolio Transfer (LPT) is the cession by a firm of all remaining unclaimed losses associated with a previously incurred liability to a (re-)insurer. In
addition to paying an arrangement fee, the cedant also typically pays a premium
equal to the net present value of reserves it has set aside for the transferred liability plus a risk premium to compensate the (re-)insurer for the timing risks of the
assumption. A LPT thus enables a firm to exchange an uncertain liability in the
form of a stream of unrealized losses over time for a certain liability whose present value is equal to the expected NPV of the unrealized losses plus a risk premium and a fee.
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Christopher L. Culp
The principal risk that the cedant transfers to the (re-)insurer through a LPT is
the timing risk that losses or claims arrive at a much faster rate than expected. In
that case, the investment income on the reserves – and perhaps the reserves themselves – may be inadequate to fund the losses. A time series of losses that occur
more slowly than expected, by contrast, will represent an opportunity for a net
gain that the (re-)insurer would typically share with the cedant. LPTs thus are risk
financing mechanisms through which firms can address the timing risk of a liability.
LPTs usually include aggregate loss limits, as well as exclusions for certain
types of risks not arising directly from the ceded liabilities. Per-loss deductibles
are sometimes also included in LPTs by (re-)insurers. Because the timing of
losses ceded in an LPT can sometimes be extremely long-term, the cedant may
also demand some kind of surety from the assuming (re-)insurer. Letters of credit,
collateral, or bank guarantees may be requested by a cedant to prove the financial
integrity if the (re-)insurer has questionable credit quality.
LPTs can be attractive sources of risk finance for various reasons. LPTs can also
benefit non-insurance, corporate customers seeking to swap an uncertain liability
stream for a fixed payment today. LPTs can help corporations with captives, for
example, wind up certain self-insurance lines if the firm alters its retention decision for certain risks. LPTs are also useful to non-financial corporations in securing risk financing for run-off solutions, especially in the area of environmental
claims and clean-up cost allocation.
In order to qualify as legitimate insurance transactions for tax and accounting
purposes, however, finite risk structures must involve some degree of underwriting
risk for the (re-)insurer. Accordingly, LPTs are often coupled with finite reinsurance contracts known as adverse development covers (ADCs) – essentially just
excess-of-loss coverage above a certain minimum attachment point and up to a
limit. The lower attachment point of an ADC is usually close to the cedant’s current reserves.
Consider, for example, a firm with a workers comp exposure to asbestos claims
from its personnel over the next five years. The firm estimates the terminal value
of its five-year liability at €10 million five years hence and has set aside the present value of that amount in reserves – say, €9 million. The firm remains concerned, however, about two risks: that a large claim will occur earlier than
planned, and that total losses will exceed estimated losses. A typical finite transaction would involve a LPT and ADC in which the firm cedes the €9 million to a reinsurance company. In return, the re-insurer agrees to cover €12.5 million in
losses. The insurer is exposed to timing risk on the first €10 million in claims and
underwriting risk on the remaining €2.5 million.
Adding an ADC to a LPT in a finite structure is not merely a question of tax and
accounting. The ADC can also play an important role for the cedant by protecting
the firm against the risk that realized losses on an existing liability are higher than
reported and forecast. ADCs are commonly used, for example, to cap old liabilities that are of concern in a merger or acquisition. When the acquiring firm or
merger partner is concerned that a liability could be much greater than the target
firm has planned for in its reserve holdings, the cession of risk through an ADC
Alternative Risk Transfer
377
can provide the target firm with a good remedy to such concerns on the part of its
suitor.
!
The potential benefits to corporates of finite risk programs are significant. As this
range of benefits has become better understood, interest in these products has significantly increased in the past several years. In particular, finite gives corporates
an intermediate solution for situations in which the retention level is uncomfortably high but a pure risk transfer solution is unavailable or too expensive. Finite
products also provide an alternative to risk transfer solutions that directly impact a
firm’s working capital layer. By partially funding retentions outside the working
capital layer, firms can increase their debt capacity.
Finite risk programs can also have other potentially significant benefits for corporates. Finite can help firms stabilize an insurance budget and, when properly
constructed and accounted for, reduce earnings and/or cash flow volatility. Finite
also allows firms to create off-balance-sheet provisions for unusual risks (e.g., extreme “tail” risk events, exotic risks, operational risks, etc.).
In addition, finite risk programs are widely regarded as important devices for
combating adverse selection problems through positive signaling. A firm that enters a charge-off against its earnings for a liability that has not been fully realized,
for example, may be suspected of possessing superior information about the liability that leads to under-reporting. A firm wishing to counter such fears by investors
can take out an ADC to lock in its liability at the charge-off amount and thus signal its confidence that the charge-off was indeed correct (Shimpi 2001).
Turner & Newall, a United Kingdom motor components manufacturer, utilized
an ADC for signaling purposes – i.e., to combat a concern amongst investors and
analysts that it had inadequately reserved against a major liability. 6 The liability
for Turner & Newall was a series of asbestos claims associated with some of its
discontinued operations.
Turner & Newall self-insured its asbestos claims by establishing a captive and
then reinsured some of that underwriting risk with an ADC for $815m XS
$1,125mn. The ADC had a 15-year tenor and, like other finite risk products, contained an agreement for a partial premium rebate if actual loss developments were
favorable relative to its reserve holdings after the 15 years.
In a more general case, the multinational firm Hanson PLC was concerned when
it acquired building materials company Beazer PLC that Beazer’s discontinued
U.S. operations would create an impediment to growth for the new conglomerate.
Hanson self-insured the liabilities of Beazer’s U.S. operations through a captive,
and the captive, in turn, acquired $80mn. XS $100mn.7 in an ADC in perpetuity.
6
7
The details of this example are discussed in GGFP (2000).
The notation $A XS $B refers to an excess-of-loss reinsurance treaty with a lower attachment point of $B and a coverage level of $A. In the Hanson/Beazer example, the
ADC thus reimbursed the firm for any losses above $100 million up to $180 million.
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Christopher L. Culp
In this manner, Hanson ring-fenced the liabilities of Beazer’s discontinued U.S.
operations using an ADC.
Although all of the above examples involve retrospective coverage for liabilities
already incurred, finite reinsurance can also be applied prospectively to liabilities
that have not yet been incurred – e.g., a policy line that an insurance company intends to offer but has not yet, or an environmental liability that a firm will incur
when a factory goes into future production.
"#$% &
'
The origin of finite risk traces back to the “time and distance” policies once commonly used in Lloyd’s by insurance syndicates to smooth the volatility of their
earnings and premium income. In a time and distance policy, one insurer makes a
premium payment to another insurer in exchange for insurance coverage that exactly equals the terminal value of the premium income stream. This allows the cedant to stabilize the volatility associated with any claims by swapping a cash flow
stream with uncertain timing for a certain cash flow stream.
Time and distance policies are no longer considered legitimate insurance outside
of Lloyd’s. In the United States, for example, an insurance contract must include
some element of underwriting risk and true risk transfer to be distinguished from a
financing or depository arrangement. Prior to the failure of Enron, most considered a “10:10” rule reasonable – viz., if there is at least a 10% chance that at least
10% of the policy risk is borne by the insurer, the contract is “insurance.” Today, a
“20:20” rule is generally applied. On a €10 million finite policy, for example, the
insurance provider must essentially face at least a 20% chance of incurring a €2
million underwriting loss. Otherwise, the deal is considered a financing arrangement.
A recent enforcement action by the U.S. Securities and Exchange Commission
(SEC) reminds us of the importance of these requirements. The SEC announced in
September 2003 that it had reached a settlement with American International
Group (AIG) for payment by AIG of a civil penalty in the amount of $10 million.
The penalty was based on AIG’s role in helping a firm called Brightpoint perpetrate an accounting fraud using a finite product.8
In October 1998, Brightpoint – a distribution and outsourcing firm – announced
that it expected to take a one-time charge-off associated with the closure of its
U.K. division in the range of $13 to $18 million. By December of 1998, the estimate of the loss had grown to $29 million. But rather than restate, Brightpoint and
AIG entered into a transaction that ostensibly allowed Brightpoint to report actual
losses in the estimated range.
The AIG/Brightpoint agreement was a finite program with both retrospective
and pro