High Water Marks in Hedge Fund Management Contracts

High Water Marks in Hedge Fund Management
Contracts
Zur Erlangung des akademischen Grades eines
Doktors der Wirtschaftswissenschaften
(Dr. rer. pol)
von der Fakultät für Wirtschaftswissenschaften
des Karlsruher Instituts für Technologie
genehmigte
DISSERTATION
von
Dipl.-Math. Margarita Sevostiyanova
Tag der mündlichen Prüfung: 14.12.2012
Referent:
Prof. Dr. Martin E. Ruckes
Korreferent:
Prof. Dr. Stefan Hirth
Karlsruhe, Dezember 2012
Contents
1 General Introduction
4
2 Time to Wind Down: Closing Decisions and High Water Marks in Hedge
Fund Management Contracts
12
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2
A Simple Model
2.3
The Fund Management Contract . . . . . . . . . . . . . . . . . . . . . . . 21
2.4
Implications
2.5
Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6
Robustness of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Hedge Fund Database Biases
38
4 Performance Smoothing of Hedge Funds
42
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2
The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3
Analysis: Period-Performance Fee . . . . . . . . . . . . . . . . . . . . . . . 56
4.4
Optimal Reporting Choice and Equilibria: Period Performance Fee . . . . 61
4.5
Optimal Reporting Choice and Equilibria: Performance Fee with High Water Mark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6
Comparing Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.7
Extended Model Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 General Conclusion
80
2
Appendix A
82
Appendix B
93
References
101
3
1
General Introduction
The fast-growing, extremely diverse and opaque hedge fund industry reached a record high
of US $2.13 trillion total assets under management in April 2012.1 According to rankings
in Absolute Return+Alpha 2 the 25 top-earning hedge fund managers earned altogether US
$14.4 billion in 2011. “Despite the industry’s overall recent poor performance, investors
haven’t shied away ... Since the financial crisis, investors have been drawn to hedge funds
because they have the ability to bet on all types of markets and don’t simply expect stocks
to move up.”3
Hedge funds are primarily private partnerships or investment funds open only to large
and sophisticated investors. The hedge fund industry is exempt from the mandatory registration with the US Securities and Exchange Comission (SEC) under the Investment
Company Act of 1940 which is intended to regulate investment companies such as mutual
funds.4 This regulatory exemption allows hedge funds to avoid the record-keeping requirements and substantial disclosure. Hedge funds use access to the capital markets via
private placement without any constraints to register their shares. Therefore, managers
have maximum flexibility in their portfolio choices and can employ an enormous variety of
strategies. In this context Ineichen (2003) states: “Hedge funds leverage the capital they
invest by buying securities on margin and engaging in collateralized borrowing. Betterknown funds can buy structured products without putting up capital initially, but must
make a succession of premium payments when the market in those securities trades up or
down. In addition, some hedge funds negotiate secured credit lines with their banks, and
some relative-value funds may even obtain unsecured credit lines.” Hedge fund strategies
vary enormously, but the main aim of most hedge funds is to raise and attempt to preserve
sufficient capital in order to generate positive absolute returns under all possible market
conditions.
Unfortunately, as an outsider or potential investor in a hedge fund, it is hard to receive
reliable data about the funds’ returns or to evaluate their performance. It is a well-known
1
“Hedge-Fund Assets Rise to Record Level”, Juliet Chung, The Wall Street Journal, June, 14th, 2012
2
Published in March, 29th, 2012
3
“Hedge Funds are Betting on Disaster”, Maureen Farrell, CNN Money, August, 23th, 2012
4
The act placed restrictions on the activities in which investment companies are allowed to engage.
For example, it forbade short selling. It required investment companies to file financial disclosure and set
limits on the fees they are allowed to charge; it also required that investment companies with more than
a certain number of investors register with the SEC. (Source: Morgan, Lewis & Bockius, LLP, 2005)
4
fact that hedge funds are reluctant to provide detailed information on their returns and
investment strategies. One possibility to obtain some information is to use the hedge fund
indices, that are constructed by using different available data and heterogenous selection
standards. However, problems can be caused by potential biases that can be included
in the data sources. According to Fung and Hsieh (2000) the organizational structure of
hedge funds, as private and often offshore vehicles, makes data collection a much more
onerous task, amplifying the impact of performance measurement biases. Indeed, hedge
fund indices show recent hedge fund performance within some degree of error, but at least
they help investors and hedge fund outsiders reasonably to characterize the directionality
of hedge fund performance.
Nevertheless, hedge fund performance data alone is not sufficiently informative for the
outsiders, can even prove misleading, especially if they cover only a short term. Performance over a period of a few years or even months could sometimes tell more about the
manager’s luck than skill. And past performance alone has never provided a reliable prediction of future success. All the more reason for the investors to identify a skillful hedge
fund manager. The investors expect better funds prospects and profitable allocation of
their investment if they believe to invest in a fund that is managed by an experienced
manager with required qualifications in her own special strategy.
It is generally thought that hedge fund returns are provided by the managers’ skills, their
depth knowledge of the fund’s strategy and their holdings: “It is possible to claim that
skillful managers outside their previous employers investment bank or fund management
operation prefer to manage portfolios that best show their skills and for which they get
best paid. An investment in a hedge fund is really an investment in a manager and the
specialized talent she possesses to capture profits from a unique strategy.”5 In addition,
the American economist and hedge fund manager Sanford J. Grossman argues that a
“fund’s return will be no better than its management and the economic environment in
which it produces its product” and that the performance of a hedge fund depends on the
underlying investment strategy and manager’s talent to implement this strategy. Gray
and Kern (2009) find overwhelming evidence that hedge fund managers have stockpicking
skills. The authors provide evidence that the hedge fund managers in their sample cannot
only identify outperforming stocks on average, but they are also able to distinguish the
best and the worst of the outperforming stocks.
For private persons the most common way to enter the world of hedge fund investing
5
“Hedge Funds Today: Talent Required”, Sanford J., Grossman, The Wall Street Journal, September,
29th, 2005
5
is to qualify as a private investor. “There are two basic categories of private investors:
accredited investors, who need a net worth of more than US $1 million [or an annual
salary of at least US $200,000]; and qualified purchasers, who need to have at least US $5
million in investment assets not including a primary residence or any property used for
a business.”6 Companies and institutional investors, that have at least US $25 million in
investment assets, have also the possibility to contribute their capital to a hedge fund by
generally qualifing as qualified purchasers. In order to overcome the registration in the
US Securities and Exchange Commission and Securities Exchange Act of 1934, the hedge
fund managers are legally allowed to accept either not more than 99 investors, where only
35 can be non-accredited, or not more than 1,999 qualified purchasers.
Hedge funds are popular investments, and investors are very competitive to contribute
their money to the fund. “They arrive every week, in ones and twos and groups of 10,
some of them coming straight from Sao Paulo’s Guarulhos International Airport. These
investors head for the dark-wood halls of Credit Suisse Hedging-Griffo as supplicants, asking to put their millions of dollars into one of the world’s top-performing hedge funds.”7
But once the fund is fully invested, the hedge fund investors have to accept severe contractual share restrictions and, thus, to deal with poor liquidity.
Hedge funds typically accept capital contributions at the beginning of each period and allow investors to withdraw capital at specified periodic intervals. To do this, the investors
must provide written notice to the hedge fund manager in advance of the permitted redemption date. Some funds impose in their contractual agreements a so called “lock-up”
which is a time-specified period, during which a new investor is restricted from redeeming
from the fund.8 If the investor decides to withdraw the capital after the lock-up-period,
in most cases she has to pay very high fees for doing so. Additionally, in order to retain
the balance of the investors’ capital, the hedge fund manager may have a contractually accorded authority to process only a portion of a redemption request by specifing a
limitation on what percentage of capital may flow out of a fund, known as a “gate”.9
6
”Hedge Fund Investing 101”, Lynn Sherman, Forbes, July, 15th, 2000
7
”Brazil Hedge Funds Beat U.S. Competitors by Investing in Bonds”, Bloomberg Markets Magazine,
June 7th, 2011
8
According to Belmont (2011) written notice has to be provided by the investors somtimes 90 or even
180 days prior to the redemption date. A typical lock-up period is one to two years.
9
Ang and Bollen (2009) compute that the cost of lock-up provisions and withdrawal suspensions can
be significant for investors. Lock-up provisions and gates vary with the liquidity of investments. For
example, in FrontPoint Partners’ FrontPoint-SJC Direct Lending Fund investors’ funds are locked up for
five years. (Source: Financial Times Online, “FrontPoint raises $1bn for new fund”, January 7, 2011.)
6
The contract that specifies contractual obligations for the hedge fund manager and her
investors sets not only the fund’s status with regulators and limits on capital withdrawals.
It also specifies the managerial involvement and fees that the fund’s manager charges and
how they are calculated. The first interesting contractual feature is that the managers of
most hedge funds invest large amounts of own money in their fund. Another feature is
that the contracts typically specify a guaranteed management fee between 0.5% and 2.5%
of funds’ assets under management and a fee based on a percentage of profits, known as
a performance fee, that can vary between 10% and 25% (compare Figure 1).
Brown, Goetzmann and Ibbotson (1999) suggest that the management fee is designed to
cover the hedge fund manager’s cost of the operating expenses and portfolio switching.
Fung (2011), on the contrary, argues that the management fee is one of the main components in hedge fund total compensation, when considering factors that incentivize the
management. The performance (or incentive) fee is calculated as a percentage of the differential between the funds new net profits earned in the last period and some hurdle rate
- in the majority of cases - the high water mark. A high water mark is the historic maximum of the fund net asset value’s previously seen at the end of one of the past periods.
The use of such kind of incentive contracts in the hedge fund industry foreshadows that
the manager’s reward depends not only on the recent performance level and on time, to
which the fee is paid, but also on changes of the fund’s past performance and the path10 ,
10
The manager’s payoff, depending at any time on the high-water mark, which is related to the maximum asset value achieved over time, is commonly understood as a path dependent payoff. Compare for
example Goetzmann, Ingesroll, Ross (2003).
7
on which the fund reached the current level. According to Agarwal, Daniel,
Naik (2011) performance based compensation contracts with high water mark provisions
provide managers with a call option. The manager receives the performance based fee
only if the fund value increases above a given maximum asset value achieved over time,
that is called by Panageas, Westerfield (2009) the “strike price”. When the hedge fund
loses value, the level on high water mark retains unchanged.
Implications of High Water Marks
Economists have provided suggestions for the role of the high water mark in hedge fund
management compensation contracts. But yet, it does not exist an unified theory on
whether the high water mark represents an optimal contractual solution between the
hedge fund manager and her investors. Many economists are thinking about the high
water mark as a form of insurance for investors: a hedge fund manager who first created
high returns on investment and afterwards loses a part of that capital cannot receive
performance fee payment until the loss has been made up. Further, the high water mark
prevent a manager from being paid twice for the same gains of the fund.
A broad field of research deals with increased risk-taking by the hedge fund manager
and tries to answer the question whether the high water mark provision in the hedge
fund manager’s compensation contracts leads to more risk-taking or not. Hodder and
Jackwerth (2007) and Chakraborty and Ray (2008) show theoretically that the manager’s
risk-taking crucially depends on a fund’s remaining life span. The authors of the first
paper model a situation in which the manager with the short-term perspective is willing
to take added risks only if the fund’s value is below the high water mark. In line with
this, Chakraborty and Ray (2008) develop a model which predicts hedge fund managers’
behavior by generally more risk-taking especially when the fund is below the high water
mark. To the contrary, Panageas and Westerfield (2009) find that even a risk-neutral
manager does not have additional incentives for higher risk-taking due to the high water
mark provision. Also empirical evidence is mixed regarding the question of whether high
water mark contracts boost managers’ risk-taking behavior. Ray (2009), for example,
finds that as soon a hedge fund falls below its high water mark the future expected
Sharpe ratio decreases. Additionally, these effects are strongest for funds that are closer
to the high water mark threshold. Brown, Goetzmann and Park (2001) and Aragon and
Nanda (2009), on the other hand, show that the changes in risk are not conditional upon
distance from the high water mark threshold.
Another implication of high water marks, discussed in the literature, provides prospects
8
for overcoming the problems associated with adverse selection. Theoretical results in a
multi-period model by Aragon and Qian (2010) suggest that a contract between managers
and investors which include high water mark provisions can be optimal and allow highly
skilled managers to signal their quality to the investors and, thus, to reduce the costs of
adverse selection. Aragon and Qian (2010) empirically support their findings and show
that high water mark provisions are more often used by smaller funds and funds with
shorter track records.
This thesis ties on and extends previous results from literature primarily examining incentives in management contracts with high water mark provision. Thereby, it is supposed
that the incentives set by contracts with high water marks are not limited to the existing
approaches covered by recent research. Two additional conceptions will be developed.
Firstly, a common observation in the hedge fund market is that well performing funds
shut down business for reasons not verifiable to outside observers. This leads to the question whether the contract with high water mark provision influences the managers’ funds
closing decisions. Secondly, the suggestion arises, especially due to the weak regulatory
oversight in the hedge fund industry, hedge fund managers might have little constraints
to misreport the funds recent performance, if doing so is beneficial to the manager. Especially, the contract that specifies high water mark provision may set different incentive
to managers compared to contracts without high water marks.
9
Research Questions and Structure
The objective of the thesis is to point out and analyze additional features of the high
water marks in hedge fund management compensation contracts which have not been
considered in the literature so far.
Empirical evidence reveals that investors in a hedge fund are not able to play a monitoring
role for the hedge fund manager. Thus, the question about the economic benefits and
effectiveness from the incentives set by hedge fund management compensation contract
remains legitimate. By assuming the fund manager to be better informed about the fund’s
investment strategy and fund’s prospects the manager is assumed to have a bargaining
power over the contract design and fund’s decision making.
Using approaches of asymmetric information we compare the incentive effects on hedge
fund managers depending on the structure of the fund manager’s compensation contract.
The basic question behind the both theoretical models presented in this thesis is whether
decisions caused by the incentive contract with the high water mark provision are beneficial or even optimal for the manager and for the fund’s investors if the hedge fund
manager is assumed to be better informed about the uncertain funds prospects.
In the following chapter we develop a rationale for the inclusion of the high water mark
provision to facilitate efficient closing of the hedge fund by their managers. Management
compensation contracts that include the high water marks specify lower expected fees
after periods of negative performance when fund closing may be warranted. Our approach
implies that by using the contract with the high water mark provision the fund’s manager
has incentives to close the fund more quickly upon periods of poor performance than if the
contract does not include a high water mark. If the fund with a high water mark provision
decides to continue after the periods of poor performance, the performance levels on an
after-fee basis in the following periods are expected to be superior to comparable funds
employing a period performance fee.
“Academic research on hedge fund performance readily admits to biases in commercially
available data ... These commercial databases have been the primary data source used
by academics and regulators to study hedge funds. Yet, the voluntarily nature of the
disclosure decision creates a host of biases that affect inferences on hedge fund performance
and risk.”11 In the third chapter we discuss the hedge fund data biases and the problems
that they cause in the field of empirical research methods.
11
Aiken, Clifford and Ellis (2010)
10
The fourth chapter studies hedge fund managers’ incentives for return smoothing that
can be caused by the contract with the high water mark provision. We show theoretically that managers of funds with high water mark provisions have stronger incentives to
underreport positive actual returns than managers of funds whose management contracts
specify period performance fees. Additionally, managers with high water marks have
strong incentives to overreport negative returns when doing so prohibits outflows. The
reason for this is that reporting, for example, a zero return rather than the actual negative
one does not affect the high water mark and therefore is inconsequential for future fees.
This pattern is not observed for managers whose fee income is determined by the fund’s
performance in each period.
11
2
Time to Wind Down: Closing Decisions and High
Water Marks in Hedge Fund Management Contracts
This section12 provides a rationale for the inclusion of high water mark provisions in hedge
fund management contracts. When hedge fund managers are better informed about future fund profitability than investors, contracts including high water marks provide the
fund managers with better incentives to efficiently close the fund than contracts with
linear performance fees. The model implies that funds with high water marks tend to
close more frequently upon periods of poor performance than their period performance
fee counterparts. If, however, such funds with high water mark arrangement decide to
continue, their after-fee performance is expected to be superior to comparable funds employing period performance fees. The model is also consistent with empirical evidence
that high water marks are more common in smaller funds and funds run by managers
without extensive track records.
2.1
Introduction
The literature on dynamic incentive provision typically proposes contracts that specify
compensation that is based on outcomes during the time period the manager can affect
these outcomes. For example, it is argued that a manager should be compensated for a
period’s activities exclusively based on that period’s outcomes rather than on previous
periods’ results as this may distort current incentives (see, for example, Holmström and
Milgrom, 1987). Hedge fund management contracts typically violate this property in that
the performance component of the management contract specifies that a performance
fee is based on the fund value at the end of a given period relative to the fund value’s
historic maximum rather than that at the beginning of the period.13 This fee structure
is frequently referred to as a performance fee with a high water mark provision. In
addition to performance fees, most hedge fund management contracts also specify a so
called management fee that is based on the value of assets under management.
12
The research presented in this section was performed in cooperation with Martin E. Ruckes; see
Ruckes and Sevostiyanova (2012a)
13
In the comprehensive sample of Agarwal et al. (2009), 80.1 percent of hedge funds display such a fee
structure.
12
The historic fund value and the fund value at the beginning of a period differ only when
periods of losses have occurred in the past. Then, a high water mark provision has two
effects compared to an otherwise identically structured fee based on period performance:
1) it reduces the expected fee amount paid to the hedge fund manager as a fee is applied
to a smaller base and 2) it introduces a convexity in the fee structure as a fee is only paid
on period performance above a strictly positive level. In a seminal contribution, Panageas
and Westerfield (2009) argue that these properties of a high water mark provision may
have desirable incentive effects in a dynamic context. They show that under a high water
mark contract, a risk neutral hedge fund manager displays risk averse behavior provided
that the fund’s horizon is sufficiently long. In case of a long fund horizon, a large share of
the fund manager’s expected income stems from future fees. As future fees are reduced
when the fund’s value is below its historic maximum, the manager tries to avoid reaching
such states. He will do so by limiting the risk of the fund’s holdings.
When the fund’s horizon is short, however, a high water mark provision’s implications
for managerial risk taking are much less clear. Specifically, a high water mark may lead
to excessive risk taking by the hedge fund manager caused by the convexity of the fee
structure (Hodder and Jackwerth, 2007, and Chakraborty and Ray, 2008). Indeed, the
average life span of a hedge fund is rather short. According to Malkiel and Saha (2005),
annual hedge fund attrition rates in the years 1994 to 2003 have been below 10 percent
only in one of the years.14
In this paper we argue that high water mark provisions have desirable incentive effects
especially for hedge funds with limited but uncertain horizons. They do so because high
water mark provisions facilitate the efficient closing of hedge funds by their managers.15
The profitability of a hedge fund’s strategy changes over time. At any point in time the
hedge fund’s manager is typically in a better position than investors to identify whether
the prospects of the fund’s strategy warrant the fund’s continuation. For example, while
investors may have to infer the quality of a fund’s strategy from recent performance its
manager possesses in depth knowledge of the fund’s strategy and holdings. Combined
with a close following of the markets relevant to the strategy this typically allows her to
14
Chan et al. (2005) report that for their sample of hedge fund liquidations “... half of all liquidated
funds never reached their fourth anniversary.” Note that fund liquidation does not necessarily mean
failure; see Liang and Park (2010).
15
The notion that fund closings are frequently instigated by fund management, is reflected, for example,
in the closing announcement of Atticus Global Fund: “This decision will come as a surprise to most of
you, especially given that we have received redemptions of less than 5% of capital ...”
13
better assess whether the recent performance tends to be temporary in nature or indicates
a permanent change of fund prospects. “The fund, which looked after $1.9 billion at its
peak, faced the prospect of spending the next few years trying to claw its way back to precrisis asset levels. Instead the founders decided to shut the fund and give investors their
money back ... For investors, it is generally a good thing if underperforming managers
are returning cash and not milking them for fees.”16
A fund manager’s incentives to close the fund are not necessarily aligned with those of
investors. Especially because negative fee payments are normally impossible to enforce,
even funds with poor prospects may generate significantly positive expected fees for the
manager. Thus, a management contract that leads to efficient fund closing needs to specify
low expected fees in circumstances in which fund closure may be efficient. Since this is
typically the case when recent performance has been poor, the high water mark’s effect
of reducing expected fees in these situations improves managerial incentives to close the
fund. Anticipating a more efficient fund closing decision increases investors’ willingness
to provide capital to the fund and in turn tends to increase expected fees for the fund
manager. The property that a management contract with high water mark generates
relatively low expected fees when fund performance has been poor, allows the manager to
set a relatively high performance fee rate. Doing so mitigates a second type of incentive
problem. Because the manager does not fully participate in the value gains of a fund,
she may close it even when the fund has performed well and fund prospects are intact. A
16
“Drowning in High Water Hell?”, The Economist, February 24th, 2012.
14
high performance fee rate implies high expected future fees when the fund value is at its
historic maximum, which leads to a low probability of fund closing.
We present a model that formally characterizes the above argument and show that linear
performance fee contracts with high water marks dominate those with period performance
fees when fee levels are set endogenously. Our approach implies that funds with high
water marks tend to close with a higher probability upon periods of poor performance
than their period performance fee counterparts. If, however, such funds with high water
mark arrangements decide to continue, their after-fee performance levels are expected to
be superior to comparable funds employing period performance fees.
In our model, performance fee levels are set to optimize managerial incentives whereas
management fees are typically used by the fund manager to extract rents. Optimal
management fees are lower if the performance fee structure contains a high water mark
than if it does not. In the former case, the non-negativity constraint of the management fee
may even be binding. Then, the performance fee serves also as the instrument for manager
to extract rents. When the probability of a deterioration of the fund’s prospects upon poor
performance is sufficiently low, a contract with period performance fee is even preferred
over that with high water mark. Given that small funds and funds run by managers
that lack extensive track records can be associated with relatively high probabilities of
deterioration of prospects such funds are expected to more frequently employ performance
fees with high water mark provisions.
Related Literature
Aragon and Qian (2010) provide a rationale for the inclusion of high water mark provisions
in hedge fund management contracts based on ex ante asymmetric information. Hedge
fund managers attempt to credibly signal their quality by offering a contract that pays
lower expected fees when performance is poor. As a contract containing a high water mark
tends to imply a lower fee for several periods, it is particularly well suited to be used as a
signaling device. Aragon and Qian (2010) show that high water marks can reduce excessive
closing caused by investor redemptions. In contrast, our approach focuses on the closing
decision by fund management and argues that high water marks not only reduces excessive
continuation by fund managers upon poor performance but also excessive termination
after strong returns.17 Also, in the model presented here an informational asymmetry
17
While the latter benefit of high water marks may appear of less economic consequence, cases of
fund closings by managers after strong results do exist. For example, Andrew Lahde, founder of Lahde
Capital, decided to close down his funds and to return money to investors after a return of 870 percent
the previous year [“Hedge fund returns money”, Financial Times Online, September 22nd, 2008].
15
arises after the contract is signed rather than beforehand. Deuskar, Wang, Wu and
Nguyen (2011) document in their empirical study that new unestablished hedge funds
tend to charge higher incentive and lower management fees. Such an initial fee structure
appears to be chosen by managers with superior skills and to predict better performance
of the fund with a higher survival rate. The empirical evidence suggests that the contract
in such form is adjusted to reflect beliefs updating about managerials skills on the funds
past performance. Deuskar, Wang, Wu and Nguyen (2011) find that the hedge funds that
face the choice between lowering the fee payment to the manager and fund closure after
the poor performance, tend more likely to close the fund down.
In our approach, the use of high water mark provisions plays a significant role in hedge
funds’ closing decisions. Empirical studies confirm the impact of high water marks on
closure rates of funds. Brown, Goetzmann and Park (2001), Aragon and Nanda (2009)
as well as Ray (2009) document that hedge funds whose value is further below their high
water marks close at higher rates. Anecdotal evidence also points towards the influence
of future expected fees on closing decisions: “Most funds close down because it does not
pay their managers to continue, not because their performance has been disastrous.”18
Particularly the high water mark contract component is thought to be responsible for
this behavior: “[The fact that they are still below their peak performance] has lead many
hedge funds to wind down rather than attempt to claw their way back to the point at
which they can earn performance fees.”19 Liang and Park (2010) find that hedge funds
with high water marks tend to close more quickly upon bad performance.
The remainder of the section is organized as follows. Subsection 2.2 presents the model.
Subsection 2.3 is concerned with the optimal contract design when performance fee levels are set to optimize managerial incentives and management fees are used by the fund
manager to extract rents. In subsection 2.4, additional empirical predictions related to
fund closing, after-fee fund performance and the level of management fees are derived.
Subsection 2.5 studies the parameter set in which the manager employs the performance
fee also to extract rents. In subsection 2.6 we provide some robustness analysis by allowing for intermittent capital redemptions and capital contributions by the fund manager.
Subsection 2.7 concludes this chapter.
18
“Hedge podge”, Economist, February 16th, 2008.
19
“Atticus closes flagship fund”, Financial Times Online, August 11th, 2009.
16
2.2
A Simple Model
We describe a stylized two-period model of hedge fund management contracting. During
the first period, information about the quality of the fund’s strategy is revealed which
may lead to a subsequent closing of the fund.
Fund Manager and Investor
Consider an investment manager who has an idea for an investment strategy with a time
horizon of two periods. The investment strategy is limited in scale: cash returns are linear
in initial investment, but any initial amount above V0 = 1 cannot be invested profitably.
The manager does not have financial wealth of her own and needs to raise capital from an
investor to implement her investment strategy. There exists an outside investor who has
one unit of capital to invest. Alternatively to operating a hedge fund in the second period,
the manager has a valuable opportunity to obtain an outside income of ω from working in
a different occupation, if the fund is inactive. This outside income ω is privately observed
by the manager at the beginning of the second period. Ex ante, ω is distributed according
to a cumulative distribution function F (ω) with density f (ω) on a support of [0, ωmax ].
The manager is assumed to have the entire bargaining power vis a vis the investor and
thus to make a take-it-or-leave-it offer to the investor. Both parties are risk neutral and
the risk free interest rate is zero.
Characteristics of Investment Strategy and Beliefs
Implementing the manager’s strategy implies that in each period, the invested amount
yields either a positive return RH > 0 or a negative return RL < 0. Consequently, fund
values after the first period are either V H ≡ 1 + RH or V L ≡ 1 + RL and after the
second period V HH ≡ (1 + RH )2 , V LL ≡ (1 + RL )2 , or V LH ≡ (1 + RL )(1 + RH ) ≡ V HL ,
where the first and second superscripts indicate the realized return in the first and second
periods, respectively. Returns are costlessly verifiable. At the time of contracting at date
0 the investor and the manager are symmetrically informed about the probabilities for
both positive and negative returns of the fund in the first period p ∈ (0, 1) and (1 − p),
respectively. It holds that
η := pRH + (1 − p)RL > 0,
(A1)
i.e. the investment strategy is profitable in the first period.
Investing in the first period generates information about expected second-period returns.
17
While a positive first-period return (state H) is uninformative for second-period prospects,
a negative first-period return tends to be associated with a deterioration of expected
returns.20 Concretely, given a negative first-period return, one of two return distributions
may materialize: one where the first-period return is a purely temporary phenomenon
and the probability of a positive second-period return remains at p (state L◦ ), and one
where the first-period return is indicative for second-period prospects and the probability
of a positive second-period return decreases to p − ε (state L− ), with ε ∈ (0, p). As we
are interested in whether fund closing takes place efficiently, we assume that the lowest
probability of a positive return in the second-period, p − ε implies a negative expected
surplus:
(p − ε)RH + (1 − p + ε)RL < 0.
(A2)
Assumption (A2) implies that it is efficient to close the fund if the investment strategy’s
prospects have deteriorated irrespective of the manager’s realisation of outside option. A
change in prospects occurs with probability 1 − θ in case of a negative first-period return
(see Figure 3).
We also assume that it is efficient to continue the fund after a positive first-period return:
1 + RH η ≥ ωmax .
(A3)
While assumption (A3) appears natural, it illustrates a central economic notion underlying
the model. When the fund has been doing well, investors want to see the fund continued.
In that case, the only possibly relevant distorting behavior by the manager is to close the
fund with positive probability.
Due to the intimate knowledge of her own investment strategy and close observation of
market development, the manager observes the true return distribution arising at date 1.
The investor is unable to observe the true return distribution and is only informed about
the first-period return of the fund. Upon observing a positive first-period return, the
investor’s probability for a positive second-period return remains at p and the probability
of a positive second-period return is p − ε(1 − θ) when he observes a negative first-period
return.
20
Assuming that a positive first-period return tends to be associated with an improvement of prospects
does not affect the results.
18
Compensation and Fund Closing
While ω characterizes the manager’s income during the second period if the fund is inactive,21 the manager is compensated in the form of ex ante specified fees during active
periods. The fees can be made contingent on the fund’s performance. Due to the manager’s limited wealth, fees cannot be negative in any period. This implies that after a fee
has been paid out, they are unaccessible to the investor in later periods.
We focus on two performance-based compensation arrangements:
• a period performance fee, where at the end of each period the manager receives a
constant fraction of the fund’s value gain during the period and nothing when the
fund loses value during the period,
• a performance fee with a high water mark provision, where at the end of each period
21
Introducing a positive outside income in the first period, does not affect the results as long as the
fund is still launched.
19
the manager receives a constant fraction of the fund’s value gain during the period
relative to the fund’s historic maximum value and nothing when the fund’s value is
below its historic maximum.
In addition to the performance fee, the manager charges a one-time management fee
k ≥ 0.22 In the following, we assume the non-negativity constraint to be not binding
in the optimal contract. Allowing for a fixed management fee in that way enables us to
separate the incentive effects of performance fees from the expected level of compensation.
We examine the case in which the inequality is binding in subsection 2.5.
As either of the two performance-sensitive arrangements specifies a payment of zero to the
manager in case of a negative return during the period, at maximum three payments have
to be specified. The contractually agreed performance-based payment upon a positive
first-period return is denoted by f H . Second-period performance-based payments are
dependent on first-period returns and are denoted by f HH and f LH , with the first and
second superscripts denoting the first-period and second-period returns, respectively (see
Figure 3). To simplify the analysis, it is assumed that the investor pays any fees separately
from the fund to the manager.23
Due to the inalienability of human capital, the manager cannot be forced to continue
the fund after period 1. Thus, the manager can close the fund at that date. It is not
possible to specify a fee that is contingent on the manager’s decision to close the fund. 24
Alternatively to the manager, the investor is able to effectively close the fund at date 1
if he is allowed to withdraw her capital from the fund. While hedge funds typically allow
investors to withdraw capital, many funds impose material restrictions on redemptions.
For example, “lock-up provisions” specify the time period that an investor has to at
least leave his capital in the fund for and “gates” limit the amount of funds that can
22
In practice, management fees are frequently paid periodically as a fraction of the assets under management. One purpose of the management fee is to cover a fund’s operational expenses. For example, the
investors in the funds Citadel Kensington Global Strategies and Citadel Wellington bear all the funds’
expenses directly in place of paying a management fee [see “Citadel Discusses Fees, Redemptions,” Wall
Street Journal Online, September 10th, 2010]. Findings by Deuskar et al. (2011), however, indicate that
a fund’s improvement in perceived quality tends to allow it to increase its management fee.
23
As long as the fund’s assets are sufficiently liquid, assuming that the cash to pay the fees are generated
by liquidating the corresponding part of the fund’s assets does not change the results.
24
Such a fee creates an additional moral hazard problem in that investors may withdraw funds strategically to preempt fund closing by the manager.
20
be withdrawn within a certain time span at the investor and/or the fund level.25 In the
following, we analyze a situation in which the investor is not permitted to withdraw capital
from the fund. In subection 2.6, we discuss if it can be optimal to allow the investor to
close the fund by redeeming his capital.
2.3
The Fund Management Contract
The sequence of events is as follows (see Figure 4). At date 0 the contract is signed and
investors provide financial capital. The fund manager invests this capital according to her
identified strategy. At date 1 the first-period return is observed by all parties and the fee
to the manager is paid as specified in the fund management contract. Then the manager
learns about her expected outside income ω and decides whether to continue the fund or
close it. If the fund is closed, all assets are liquidated at no cost and the proceeds are paid
to the investor. If the fund remains alive, assets are used according to the investment
strategy. At date 2, an alive fund’s return is observed, its assets are costlessly liquidated
and the proceeds distributed to the investor. The contractually agreed fee is paid to the
manager. If the fund is closed at date 1, the manager receives her outside income ω at
date 2.
25
Ang and Bollen (2009) compute that the cost of lockup provisions and withdrawal suspensions can be
significant for investors. Lock-up provisions and gates vary significantly with the liquidity of investments.
For example, in FrontPoint Partners’ FrontPoint-SJC Direct Lending Fund investors’ funds are locked
up for five years. [“FrontPoint raises $1bn for new fund”, Financial Times Online, January 7, 2011.]
21
Analysis
We first describe the fund manager’s optimization problem independent of the specific
structures of the performance fee discussed above. In this analysis, we represent the
performance fee structure in the fund management contract by A. Subsequently, we
compare the outcomes when using a period performance fee and a performance fee with
high water mark.
As the participation constraint of the investor can be satisfied by adjusting the fixed
performance fee, k, the performance fee structure serves two potential conflicts of interest
between investor and manager with respect to closing the fund. There is a potential
incentive for the manager to continue the fund even though doing so is not in the interest
of the investor, because she does not explicitly participate in losses the fund suffers.
The only way the performance fee arrangement is able to control this incentive is by
specifying relatively low expected future fees in the relevant states. The incentive for
excessive continuation is present in state L− and possibly in state L◦ , because negative
first-period returns tend to be the consequence of a worsening of fund prospects. There
is also a potential incentive to close the fund even though the investor would like to see
it continued. This is, because the manager participates only with a certain fraction in
the expected value gains of the fund. The contract can mitigate this incentive by offering
relatively high expected fees in the relevant states. The incentive for excessive closing is
present in state H and possibly in state L◦ .
To identify the optimal fund management contract, we first describe the manager’s closing
decision at date 1 and the investor’s participation constraint as well as the manager’s
objective function in general.
The manager decides whether or not to close the fund at date 1 based on the realization of
her outside income ω. She closes the fund whenever her outside income in period 2 equals
or exceeds her expected second-period fee income from operating the fund. For any given
fee structure and each of the three states at date 1, H, L◦ and L− , there is a level of ω
above which the manager closes the fund. Those cutoff levels depend on the performance
◦
−
fee arrangement are denoted by ω H (A), ω L (A) and ω L (A). Given our possible fee
◦
−
structures, it holds ω H (A) ≥ ω L (A) ≥ ω L (A). Then, the manager’s closing decision
can be characterized as follows: At date 1, the manager
22
never closes
for
closes iff prob(RH ) = p − ε
for
L
H
closes iff first period return is R and prob(R ) = p for
always closes
for
ω
ω
ω
ω
−
< ω L (A)
−
≥ ω L (A)
◦
≥ ω L (A)
≥ ω H (A).
The investor’s participation constraint depends on his anticipation of the manager’s closing decision. The investor’s participation constraint depends on the fund’s performance
and both the management fee, k, as well as the performance fee arrangement. The
performance fee arrangement affects the investors payoff not only through fee payments
to the manager but also via the manager’s closing choice at date 1. We drop the descriptor
(A) for brevity. The investor’s participation constraint is then given by
V0 ≤ −k +
H
H
HH
HH
HL
H
H
+p − f +F(ω )(p(V
−f )+(1−p)V )+(1−F (ω ))V
+
◦
◦
+(1−p)θ F (ω L ) p(V LH − f LH ) +(1 − p)V LL +(1 −F (ω L ))V L +
−
−
+(1−p)(1−θ) F (ω L )((p−ε)(V LH −f LH )+(1 −p+ε)V LL )+(1−F (ω L ))V L .
(1)
While the first line of (1) contains the fixed management fee to be paid to the manager,
lines 2 to 4 describe investor’s payoffs in the three states weighted with the probabilities
with which the states occur. Each payoff depends on the fee structure both directly and
indirectly via the fee structure’s impact on the manager’s closing decision. In equilibrium,
the manager will set the management fee, k, to its maximum value provided that the investor is willing to provide capital. Therefore, the investor just breaks even in equilibrium
and (1) is fulfilled with equality.
Because the fund manager is able to appropriate the entire rent, she maximizes the expected surplus generated by the fund’s investments. The surplus also takes into account
the manager’s income outside the fund. The expected surplus varies with the manager’s
◦
−
closing decisions at date 1, represented by ω H (A), ω L (A) and ω L (A). By assumption
(A1), η denotes the fund’s expected return if the success probability is p, the expected
◦
−
surplus, S(ω H , ω L , ω L ), can be written as
◦
−
S(ω H , ω L , ω L ) = −1 − E(ω) +
H
H
H
H
H
+p F(ω )(1+R )(η+1)+(1−F (ω ))(1+R +E(ω|ω ≥ ω )) +
(2)
◦
◦
◦
+(1−p)θ F (ω L )(1 +RL )(η+1)+(1−F (ω L ))(1 +RL +E(ω|ω ≥ ω L )) +
L−
L
H
L
L−
L
L−
+(1−p)(1−θ) F (ω )(1+R )(η+1−ε(R −R ))+(1−F (ω ))(1+R +E(ω|ω ≥ ω )) .
23
Optimal Contracting without High Water Mark
First we consider the case in which the manager selects a period performance fee such that
the performance-based fee in each period amounts to a constant additional fraction a ≥ 0
of the gain in fund value during the period and nothing in case of a decrease in fund value.
If the first-period fund return is positive, the manager receives a performance fee of f H =
a(V H − V0 ) = aRH at the end of the first period. If the second-period return is positive,
the fee depends on the fund’s first period return. In case of a positive first-period return,
the manager’s second-period fee is f HH = a(V HH − V H ) = a(1 + RH )RH upon a positive
second-period return. This implies that for the level on manager’s expected outside income
equal to apRH (1 + RH ) := ω H (a) the manager is indifferent between managing the fund
in the second period and launching his expected outside income. In case of a negative
first-period return, the second-period fee is f LH = a(V LH − V L ) = a(1 + RL )RH .26 Thus,
◦
in state L◦ the manager closes the fund for ω larger than apRH (1 + RL ) := ω L (a) and in
−
state L− for ω above a(p − ε)RH (1 + RL ) =: ω L (a).
Optimal Contracting with High Water Mark
Consider now a fee structure that specifies a linear performance fee, e
a ≥ 0, with a high
water mark provision. A high water mark specifies that a performance fee is based on the
difference between the fund’s value at the end of the period and the historic maximum
of fund values provided that this difference is positive. Because the fund value at date
0, V0 = 1 is (trivially) the historic maximum of fund values, the fee level f H = e
aRH
is the same as under a period performance fee. The same applies to V H and therefore
f HH = e
a(1 + RH )RH . If the first-period return is negative, the historic maximum of
fund values remains its initial value V0 = 1. This implies that with a high water mark
f LH is structurally different from its period performance fee counterpart. It is given
by f LH = max{0, e
a(V LH − 1)}, which we assume to be strictly positive. Thus, f LH =
e
a((1 + RL )(1 + RH ) − 1) = e
a(RH (1 + RL ) + RL ) > 0. For a given value of e
a > 0, f LH
in case of a performance fee with high water mark is strictly smaller than that in the
absence of a high water mark.
The corresponding closing thresholds for ω are defined as follows:
ω H (e
a) := e
apRH (1 + RH ),
◦
ω L (e
a) := e
ap(RL + RH (1 + RL )),
−
ω L (e
a) := e
a(p − ε)(RL + RH (1 + RL )).
26
Note that there is a convexity in the fee structure despite the seemingly linear contract.
24
Due to its smaller base, the fee percentage with high water mark e
a can be larger than
its period performance counterpart a without inducing the manager to continue the fund
in states L◦ and L− . For a given percentage fee, the manager’s optimal closing policy in
state H is identical in both performance fee regimes, as the structure of the relevant fee,
f HH is not affected by a high water mark.
Comparing the manager’s incentive constraints and the resulting expected total surplus
levels of the manager between the contracts with and without a high water mark yields a
central result:
Proposition 1 The optimal contract with a high water mark provision yields at least as
high a payoff to the fund manager as the optimal contract with a period performance fee.
Proof: See Appendix A.1.
A contract with a high water mark (weakly) dominates a contract with a period performance fee. The formal argument for this is as follows: By selecting an appropriate fee
level, a high water mark contract is able to generate an identical closing policy in the
downward states L◦ and L− as any given contract with period performance. The fee
percentage of the high water mark contract is higher than that of its period performance
counterpart. This typically reduces the manager’s incentive to close upon a positive firstperiod return. Only if the optimal period performance contract implies the continuation
of the fund with probability one in state H, is it possible that the two types of contracts
yield the same payoff to the manager.
Uniform Distribution of Outside Income
To make the benefits of a performance fee with high water mark more transparent, we
now assume that the fund manager’s outside income is uniformly distributed, i.e. that
ω
the manager’s outside income ω has cumulative distribution function F (ω) = ωmax
with
1
density f (ω) = ωmax on [0, ωmax ]. To allow for a relatively wide spectrum of outside income
levels and to simplify the analysis, we also assume that ωmax = 1 + RH η (compare
(A3)).
The following proposition presents the optimal fee choice dependent on the investor’s
participation constraint (1) and the manager’s closing decision:
25
Proposition 2 When the manager’s outside income in period 2 is uniformly distributed
H
on
0, 1 + R η , the optimal contract with a high water mark provision is given by
e
a∗ , e
k ∗ with
p2RH(1+RH)2 η+(1−p)(1 +RL)(RL +RH(1+RL)) pθη+(p−ε)(1−θ)(η−ε(RH −RL))
e
a∗ =
p3 (RH(1+RH))2 +(1−p)(RL +RH(1+RL))2 p2 θ+(p−ε)2 (1−θ)
and
e
k∗ = η − e
a∗ pRH .
The optimal contract with a high water mark provision specifies a higher performance fee
parameter and a lower management fee than the optimal contract with a period performance fee, (a∗ , k ∗ ).
From an ex ante perspective, the optimal contract with a high water mark leads to a higher
probability of closing in state L− , to a lower probability of closing in state H, and to a
strictly larger payoff to the hedge fund manager than the optimal contract with a period
performance fee.
Proof: See Appendix A.2.
Compared to the optimal contract with a period performance fee, the optimal contract
with a high water mark reduces excessive continuation by generating a lower expected
fee from continuing the fund when closing is efficient. The fund manager is compensated
for this reduction in expected fees by larger fee payment state H. Because in that state
closing is not efficient, the contract with high water mark further improves efficiency buy
curbing the manager’s incentive to close the fund too often. The manager indeed strictly
prefers a contract with high water mark to one without whenever it affects her closing
decision. A high water mark arrangement is more efficiently able to utilize the superior
information of the manager about fund prospects. Thus, the model identifies a rationale
for including high water mark provisions in hedge fund management contracts based on
closing considerations.
For a numerical example, Figure 5 displays the expected surplus of a hedge fund under
both performance fee structures as a function of the performance fee parameter. It illustrates that for any fixed level of the performance fee parameter, the expected surplus with
high water mark is larger than with period performance fee. The figure also shows that
26
the optimal performance fee parameter is higher with high water mark than with period
performance fee.
Adjusting the fixed management fee allows the manager to design the performance fee
structure in a way to optimize incentives. Because the structure of the performance fee
with high water mark is better suited to align incentives between the investor and the fund
manager, compensation via a management fee is lower than when a period performance
fee is used. 27
The numerical examples in Table 1 show that the optimal contract with period performance fee includes a significantly lower performance fee parameter than the optimal
contract with high water mark. The latter is better able to control both the incentive
to excessively close in the upward state and the incentive to excessively continue upon
a deterioration of the fund’s prospects. The contract with high water mark generates,
however, excessive closing in the downward state when fund prospects remain intact.
27
Theoretical papers on the use of high water marks tend to ignore management fees. One notable
exception is Lan, Wang and Yang (2011) who find that the management fee discourages risk taking by
the fund manager.
27
This holds both relative to the efficient closing policy and relative to the policy generated
by the contract with period performance fee. As in this state in the middle of the spectrum the difference in surplus from continuation compared to closing are relatively small,
the misaligned incentives between investor and manager are less consequential than the
benefits of a high water mark structure in the extreme states.
28
2.4
Implications
The model provides a number of implications on closing behavior, performance and contract design.
a) Funds with high water marks tend to close more frequently upon negative performance.
Corollary 1 When the manager’s outside income in period 2 is uniformly distributed on
0, 1 + RH η , the the closing probability upon a negative first-period return is higher for
the optimal contract with a high water mark provision than for the optimal contract with
a period performance fee.
Proof: See Appendix A.3.1.
The use of high water mark provisions improves hedge funds’ closing decisions. After
periods of poor performance, high water marks reduce excessive fund continuation.28
Empirical studies confirm the impact of high water marks on closure rates of funds.
Liang and Park (2010) find that hedge funds with high water marks tend to close more
quickly upon bad performance.29
b) Funds with high water marks tend to outperform funds without high water marks after
periods of poor performance.
Corollary 2 Suppose the manager’s outside income in period 2 is uniformly distributed
on 0, 1 + RH η . Then, conditional on fund continuation after a negative first-period
return, the expected second-period after-fee return of a fund with a high water mark provision exceeds that of a fund with a period performance fee.
Proof: See Appendix A.3.2.
Contracts with high water marks provide improved incentives for closing by specifying
lower expected fees when the fund is under water. Thus, hedge funds with high water
marks tend to have better after-fee performance when returns have (recently) been poor
relative to otherwise comparable funds with period performance fees.
28
This implication contrasts with the prediction in Aragon and Qian (2010) that high water marks
reduce the probability of fund closing upon negative returns, because higher expected after-fee fund
returns reduce investors’ incentives to withdraw capital.
29
The authors, however, don’t explicitly test for the statistical difference between the two parameter
estimates.
29
c) In funds with high water mark contracts a higher expected performance tends to be
associated with higher management fees.
Corollary 3 Suppose the manager’s outside income in period 2 is uniformly distributed
on 0, 1 + RH η and the contract contains a high water mark provision. Then, an
increase in the level of the positive return, RH , leads to an increase in the management
fee, e
k∗.
Proof: See Appendix A.3.3.
The level of the management fee is fund managers’ instrument to extract the surplus the
fund generates. As the magnitude of a positive return, RH , increases, the management fee
increases as well.30 Deuskar et al. (2011) find that successful funds tend to increase their
management fees suggesting that these increases of the management fees reflect higher
return expectations.
2.5
Extension
Non-negativity of the management fee and probability of performance deterioration
In the model described above, the performance fee is set to align closing incentives of the
fund manager with those of the investor whereas the adjustment of the management fee
allows the fund manager to extract the expected surplus. This is possible, because we focus
on the parameter space for which the optimal contract specifies a positive management
fee. If, however, the requirement that the management fee be non-negative becomes
binding, the specified performance fee affects the expected level of cash returns to the
investor. Concretely, only a suboptimally low performance fee from an incentive point
of view satisfies the investor’s participation constraint. In the following, we examine
this case maintaining the assumption of the uniform distribution of outside income on
0, 1 + RH η and show that it is present when the likelihood of the deterioration of the
fund’s prospects is relatively small.
30
A change in parameters not only affects the magnitude of the fund’s expected return but also the
optimal performance fee rate and therefore the expected fee income appropriated by the performance fee
component alone. Thus, although other surplus-increasing parameter changes, such as increases in RL or
p tend to be associated with higher management fees, there are parameter combinations such that this
is not the case.
30
Proposition 2 and Appendix A.2 reveal that the restriction on the management fee, k ≥ 0,
is binding if and only if the optimal performance fee rate derived in subsection 2.3 exceeds
η 31
. It turns out that the optimal performance fee parameter of a contract with period
pRH
performance fee never exceeds that value (see Appendix A.4). Therefore, we focus in
the following on the derivation of the high water mark performance fee parameter, e
a∗ .
Due to the convexity of the optimization problem, the new optimal contract has the form
(e
a∗ , k = 0).
The new value of the period performance fee rate with a high water mark provision, e
a∗k=0 ,
can be derived from the investor’s participation constraint (1) and solves the following
equation:32
(1−p) L
(R +RH(1+RL))2 (p2 θ+(p−ε)2 (1−θ))+
ωmax
(1−p)
+e
a∗k=0
(1+RL)(RL+RH(1+RL)) pθη+(p−ε)(1−θ)(η−ε(RH−RL)) +
ωmax
+(1+p(1+RH))(η−e
a∗k=0 pRH ) = 0.
− (e
a∗k=0 )2
(3)
The performance fee rate e
a∗k=0 is smaller than the expected surplus maximizing performance fee rate e
a∗ derived in Proposition 2.
To gain a better understanding of the possible states in which the performance fee with
high water mark provision e
a∗ increases beyond the threshold pRηH and its consequences, we
relate it to the model parameter θ, which characterizes the probability of a deterioration of
the fund’s prospects. Recall that in case of a negative first-period return the deterioration
of prospects occurs with probability 1−θ.
Analyzing the new corresponding optimal contract and expected fund surplus leads to
the following result:
Proposition 3 Suppose the manager’s outside income in period 2 is uniformly distributed
on 0, 1 + RH η . The surplus maximizing performance fee rate e
a∗ is decreasing in θ.
There exists a critical value for θ, θ◦ ∈ (0, 1), below which the management fee restriction,
k ≥ 0, is binding.
Note that if a performance fee parameter larger than pRηH is chosen, the probability of fund continuation in state H is equal to one. Thus, a further increase in the performance fee parameter does not
affect the closing probability in that state.
31
32
The lengthy closed form solution of (3) is presented in Appendix A.4.
31
There is a critical value of θ, θb ∈ (θ◦ , 1), below which the optimal contract with period
performance fee leads to a strictly higher expected surplus than the optimal contract with
a high water mark provision.
Proof: See Appendix A4.
For a given performance fee rate, a high water mark provision specifies a larger difference
in fee income for the manager across states. While this is always beneficial from an
incentive standpoint, the additional restriction of a non-negative management fee may
significantly impair the manager’s desirability of a high water mark provision but not
that of a period performance fee. There exist circumstances in which a contract with
period performance fee leads to a larger expected surplus and is therefore preferred to a
contract with a high water mark. Hence, the choice of a period performance fee may not
only be determined by the absence of signaling considerations as shown in Aragon and
Qian (2010); it may also be a consequence of the more severe effects of limited liability
restrictions on contracts with high water marks.
32
Contracts with period performance fees are preferred only if the probabilities of the deterioration of funds’ prospects are relatively low. It appears reasonable to assume that
investors assign significant probabilities of downward adjustments of fund prospects to
small funds or those run by managers that lack extensive track records. Thus, the derived
result is consistent with findings by Aragon and Qian (2010) that those types of funds
more commonly employ high water mark provisions.
2.6
Robustness of the Results
Intermittent Redemption by the Investor
So far, we have abstracted from allowing the investor to withdraw funds after period 1.
Given the linear investment technology, the investor either wants to redeem all or none
of his funds. Thus, allowing for the intermittent redemption of funds, the investor has
the opportunity to effectively close the fund. In the following, we discuss some of the
main aspects of including the investor’s option to redeem funds intermittently in the fund
management contract. We do this maintaining the assumption of a uniform distribution
of outside income as given in subsection 2.3.
First, note that it is never superior to allow the investor to redeem capital intermittently
when the fee structure is designed in a way that the option is never exercised. Doing so
only introduces additional restrictions.
If there are circumstances in which the investor closes the fund, he does so only upon
a negative first-period return and under inferior information than the manager. His
information is inferior in two ways: the investor cannot distinguish between states L◦ and
L− , and also is not informed about the realization of ω.
The investor leaves his capital in the fund if doing so increases his expected cash flows.
In case of negative first-period return, fund withdrawal yields the investor a cash flow of
the date-1 value of the fund, which is given by the fund’s gross value, V L .
Consider the case that the investor withdraws his capital from the fund with certainty
after a negative first-period return. Note that in this case only the fees f H and f HH are
paid with positive probability. Because these fee payments are independent of whether
the contract specifies a period performance fee or contains a high water mark provision, no
discrimination between these two performance fee structures is necessary. The investor’s
break even constraint in this case is given by:
33
V0 ≤ −k +
+p − f H +F(ω H )(p(V HH −f HH )+(1−p)V HL)+(1−F (ω H ))V H +
+(1−p)V L .
Based on this constraint, the maximal level of performance fee parameter a that the
investor is willing to accept is obtained if k is set to zero and is equal to pRηH .33
The manager anticipates intermittent redemption and consequently the fund’s closing by
the investor after a negative first-period return. Then, fund’s expected surplus is:
S rd (ω H ) = −1 − E(ω) +
+p F(ω H )(1+RH)(η+1)+(1−F (ω H ))(1+RH +E(ω|ω ≥ ω H )) +
+(1−p)V L
⇔ S rd (ω H ) = η − E(ω) + p F(ω H )(1+RH)η+(1−F (ω H ))E(ω|ω ≥ ω H ) .
The manager maximizes the expected fund surplus with respect to her optimal closing
policy in state H, which is a function of the performance parameter a. Given assumption
(A3), the expected surplus in state H is at least as high as the maximal level of her
outside opportunity ωmax . Thus, the manager chooses the maximum possible continuation
probability F(ω H ) equal to 1. Then, the optimal performance fee parameter is equal to
a∗ = pRηH .
Now we are in a position to compare the expected surplus with intermittent redemption
by the investor to the one generated by the contract derived in Proposition 2.
Proposition 4 When the manager’s outside income in period 2 is uniformly distributed
on 0, 1 + RH η , the optimal contract with intermittent redemption by the investor leads
to a strictly lower expected surplus than the optimal contract without intermittent redemption.
Proof: See Appendix A5.
33
For the proof see Appendix A.5.
34
Given that the manager has private information, granting intermittent redemption rights
to the investor is not optimal. Thus, the model implies that intermittent redemption
rights are typically used for reasons other than increasing the efficiency of the fund’s
closing policy. They may, for example, be included because of liquidity needs by hedge
fund investors.
Capital Contribution by the Manager
So far, it has been assumed that the manager does not invest own financial wealth in the
fund. Actually, hedge fund managers typically do invest their own capital in the fund.34
The following arguments introduce the case in which the manager possesses financial
wealth of A > 0 and contributes it to the fund. The initial investment amount that can
be invested profitably is V0 = A + Y ≡ 1, where A is the part indicates the manager’s
contribution and Y = 1 − A investor’s, respectively.
If the invested amount yields a positive return RH > 0, the fund’s value increases after
the first period to V H . We can distinguish between the manager’s A(1 + RH ) and the
investor’s (1 − A)(1 + RH ) shares, respectively. Analogous is the wealth development
after a negative first-period return RL < 0 with decreasing value V L and the manager’s
A(1 + RL ) and the investor’s (1 − A)(1 + RL ) shares, respectively. This allocation of the
share proportions between both parties is also kept constant in the second period. As
described in the basic model the investor still has the same participation constraint (1).
The modification is that he is now interested in changes in the portion of (1 − A)V0 .
The manager’s decision whether to operate the fund after the first period or to close it
depends on the realization of her outside income ω. In the new model setting the cutoff
levels of ω depend not only on the period performance fee arrangement but also on the
fund’s fraction that was generated by the manager’s investment A. Taking into account
◦
−
the adjusted values of ω H , ω , ω L , the manager’s closing decision and the expected surplus
◦
−
S(ω H , ω , ω L ) remain unaffected, as described (2) in the basic model.
Recall that the optimal contract in case A = 0, that was described in proposition 2, is
given by (e
a∗ , e
k ∗ ). After comparing the expected total fund surplus levels, with respect
to the manager’s incentive constraint between the contracts with the period performance
34
Agarwal et al. (2009), report that “Discussions with industry practitioners suggest that often the
manager reinvests all of the incentive fees earned back into the fund.” Thus, they calculate the manager’s
coinvestment as the cumulative value of the incentive fee reinvested together with the returns earned on
it.
35
fee and the period performance fee with a high water mark provision, we can state the
following result:
Proposition 5 When the manager contributes financial wealth A > 0 to the fund invest ment and her outside income in period 2 is uniformly distributed on 0, 1 + RH η she
chooses the optimal contract with the high water mark provision (e
a∗ , e
kA∗ ). The optimal
performance fee rate e
a∗ does not depend on A. The optimal management fee e
kA∗ and the
◦
−
expected total fund surplus S(ω H , ω L , ω L ) increase with increasing A.
From an ex ante perspective, the increase in financial wealth A > 0 leads to a lower
probability of closing in state H, to a lower closing probability in state L◦ and to higher
closing probability in state L− .
Proof: See Appendix A6.
Thus the manager investing her own capital in the fund brings about further alignment
of interests. It is obvious to see that the higher the financial contribution the higher the
expected loss in the case of low return realization. In order to prevent expected loss the
manager’s incentive to close the fund as efficiently as possible in each of the states increases
with the amount of her investment in the fund. Findings by Agarwal et al. (2009), show
that higher levels of managerial ownership, in the funds which use incentive contracts
with inclusion of high-water mark provision, are associated with superior performance.
The numerical examples in Table 3 show the changes of closing probabilities in different
states as the managers capital contribution increases.
Because the optimal performance fee rate e
a∗ is independent of whether the manager
contributes financial wealth or not, the main results of Proposition 2 do not change.
2.7
Conclusion
In this section we studied the choice between two different types of performance fee
structures in hedge fund management contracts: fees based solely on the performance
during the preceding period and fees based on the performance relative to the historical
fund value maximum. It provides a rationale for the inclusion of the latter, so-called high
water mark provisions, based on the argument that such structures facilitate efficient fund
closing. Significant levels of expected fees in states that potentially warrant fund closing
provide incentives for fund managers to continue the fund even when doing so is inefficient.
Management contracts with high water mark provisions specify lower expected fees after
36
periods of negative performance when fund closing may be warranted. In equilibrium,
managers receive higher fee rates and thus higher compensation in case of a continuously
positive value development of the fund.
Our approach implies that funds with high water marks tend to close more quickly upon
periods of poor performance than their period performance fee counterparts. If, however,
such funds with high water mark arrangements decide to continue, their performance
levels on an after-fee basis are expected to be superior to comparable funds employing
period performance fees. The model is also consistent with empirical evidence that high
water marks are more common in smaller funds and funds run by managers without
extensive track records.
37
3
Hedge Fund Database Biases
For investors in a hedge fund or fund of hedge funds it can be very hard or even impossible
to obtain a “true” or even “fair” information about the hedge fund’s performance. There
are not many ways of collecting information about levels of the hedge fund performance
as an outsider (and even as an accepted investor in a hedge fund). One possibility is
to use the hedge fund indices, that are constructed by using different available data and
heterogen selection standards. The reason for this inhomogeneity is the absence of a fully
representative hedge fund data base, that could cover the data of all active hedge funds.
The same problem is faced by researchers in empirical research. For empirical research on
hedge funds the availability of high-quality data is the determining factor. The empirical
literature on the hedge fund reported returns shows a large variety of different results due
to different data samples.
Due to lack of the regulatory oversight hedge funds are not required to report their
returns or any other information, such as size of managed assets, the investment strategy,
pursued by the manager, or the portfolio composition. A unique, comprehensive and
publicly accessible database containing the track records of all, especially active hedge
funds simply does not exist. The publicly available hedge fund data contain voluntarily
disclosured monthly investment performance of some hedge funds. Many empirical studies
use as a main data source for their analysis the largest hedge fund data providers TASS,
CTA, HFR or CISDM. The first problem the researcher has to deal with is the need of
combining the data from various data sources in order to collect sufficient data sample.
Through the wide usage of these databases by researchers and practitionals the hedge
funds’ can exploit this status to convey some specific messages or signals to the audience.
Thus, the empirical estimates of hedge funds performance are sometimes overstated and
come from biased data sources. Brown, Goetzmann, Ibbotson, and Ross(1992), Fung and
Hsieh (2000), Liang (2000), Jorion and Schwarz (2010) and Edwards and Caglayan (2001)
cover these well-known data biases extensively in the hedge fund literature.
Survivorship bias
The most common and easily fixable data bias in a hedge fund study is the survivorship bias. Survivorship bias occurs when the database does not include the returns of
hedge funds that have stopped reporting their performance during the observable period.
The academic literature estimates that survivorship bias increases returns from 0.16% to
6.67%, p.a. depending on the observation period, but there exists wide disagreement in
explaning why hedge funds stop reporting their returns to the data gathering services.
38
On one hand, many academic studies argue that hedge funds which perform poorly during the observed periods relative to the other funds stop reporting in order to hide the
actual losses on their investments and to avoid harming their reputation (see, for example, Malkiel and Saha, 2005) or capital outflows. Thus, when analysing track records of
hedge funds it can happen that the sample of current funds will include only those that
have been successful in the past, while some hedge funds were closed because of a poor
performance. In this case, survivorship bias causes reported hedge fund performance in
the database to appear higher (and in the most cases better) than the true actual average hedge fund performance. On the other hand, hedge funds may stop reporting their
performance because they have already collected sufficient capital contribution for their
investment strategy and therefore are not interested in attracting more investors.
Instant History or Backfill Bias
Another possible problem associated with the hedge fund data is named instant history
bias. Instant history or backfill returns occur when a hedge fund is added to a database but
has been operating for many periods of time before making first report to this database.
The academic literature has produced several estimates of the instant history bias on
performance, which range between 0.05% and 4.35% p.a. Many hedge fund strategies,
for example, may be running and generating returns for a while before they are offered
to potential investors, to see whether the strategy is successful. Then, the manager can
decide to report the data to a commercial database. In this case, the database includes
the historical data and the past hedge fund’s performance from when the hedge fund
was not part of the database. Hedge funds that are unable to generate high performance
and good track records with their strategy are unwilling to disclose their history to the
potential investors. The backfill bias occurs because often only the managers with good
hedge fund past performance are the ones who want to be included in a database.
Self-Selection Bias
Self-selection bias occurs if only funds with good performance report their performance to
a database. This effect can create an upward bias, which can be limited due to the fact,
that hedge funds with continuously good performance stop sometimes their reportings as
they have reached an optimal size of assets under management and, thus, do not need to
attract more investors.
According to Aiken, Clifford and Ellis (2010) there exist quite a number of other biases
that could affect databases. As they point out: “Funds have some discretion as to the
timing of their reports to the databases. ...In most cases, the fund has up to 3 months to
39
file its monthly return. ... A fund with poor performance in a given month may have the
incentive to delay reporting, increase the funds risk, and hope for a better outcome in the
next month. If the strategy works, both monthly returns are listed. If it does not work,
the fund never reports either return to the database. A similar version of this bias occurs
when a previously delisted fund is allowed to rejoin the live funds file. If a fund is willing
to fill all gaps in its time-series of returns, it is allowed to rejoin the database. As it is
likely that only funds that performed well during their delisting period will re-list, these
features of the commercial data will impart a further upward bias on the return data.”
The academic literature proposes several suggestions for overcoming problems associated
with biases in hedge fund indices. For instance, using fund of hedge fund indices to
estimate the performance of the hedge fund market leads to the results that are less
likely to be affected by issues such as survivorship bias or backfilling bias (see Fung
and Hsieh (2000)). The track records of funds of hedge funds seems to be almost free
of the many biases contained in databases of individual funds. Another idea suggests
to use the database that contains information on when hedge funds actually joined the
database (for example Hedge Fund Research (HFR) database contains this information).
Deleting all the the backfill observations in the selected data set can help to reduce or
even to eliminate the backfill bias. Ben-David, Franzoni, Landier and Moussawi (2011)
combine in their empirical study a list of hedge funds (by Thomson-Reuters), mandatory
institutional quarterly portfolio holdings reports (13F), and information about hedge fund
characteristics and performance (TASS) in the conviction that the 13F filings are not
affected by the self-selection and survivorship bias. Agarwal, Daniel and Naik (2011)
exclude the first two years’ data for each fund from their empirical analysis to tackle
backfilling bias.
Nevertheless, the above described problems caused by database biases should be considered while trying to measure the hedge fund’s performance. Fung and Hsieh (2004) mean
that “existing hedge fund indices, while helpful in providing investors with an idea on
the current progress of the industry on average, offer little clues to ... questions [of asset
allocation and performance measurement].”
In the following chapter we argue that additionally to the problems associated with the
biased hedge fund data the investors in a hedge fund can additionally make a mistake
while trying to measure and evaluate the hedge fund’s performance using available data.
Namely, by disregarding the hedge fund managers contractual provisions. We develop a
general theoretical framework to describe the managerial incentives that can be caused by
two different contracts: that include performance fee with the high water mark provision or
40
period peformance fee. We can show that the hedge fund voluntary reported performance
is strictly dependent on the specification of the contractual agreement between the fund
manager and the fund investors. In this context, we are paying close attention to the role
of high water mark provisions in the hedge fund management compensation contract.
41
4
Performance Smoothing of Hedge Funds
This section35 shows that performance fees with high water marks cause hedge fund managers to smooth performance. High water marks provide both the incentive to underreport
returns in good times as well as to overreport returns during difficult periods. In good
times, reporting a high return increases the high water mark and diminishes fees from
future fund flows. During difficult periods, when the fund is under water, overreporting
has little consequences on future fees income, because doing so does not effect the basis
for future fees.
4.1
Introduction
Hedge fund returns appear to be smoother than the returns of mutual funds or common
stocks. Compared, for example, with the S&P 500’s monthly returns in the period from
January 2009 to November 2011 (compare Figure 6), monthly hedge fund returns are much
less volatile. For quite some time, both investors and researchers have been suspecting
that at least part of this return smoothness is due to the reporting practices of many hedge
funds. For example, according to De Souza and Gokcan (2004): “...there is a high degree
of serial correlation in most hedge fund strategy monthly returns, which causes excess
smoothness in their return series. This excess smoothness typically leads investors to
understate both the true volatility of these strategies and their correlation with traditional
asset classes and will significantly overstate the true Sharpe ratios.” One way hedge fund
managers indeed appear to be able to massage their results is by using their discretion in
valuing illiquid assets. Many hedge fund strategies include investments in illiquid assets
in their portfolios and in the most of the cases it is not easy to determine the net value
of the assets under management. For less liquid or tradable assets there are frequently
no market prices available. The manager has the opportunity to linearly extrapolate
the approximate price between two observable prices or to use smoothed broker (dealer)
quotes. Empirically, Green (2010) documents “patterns in hedge fund returns that suggest
that reporting manipulation is significant and pervasive for hedge funds with discretion
in valuing their portfolios of illiquid assets.” One other way of affecting the reported value
of assets under management is by deliberately reporting false values of assets in their
portfolios as suggested by the results of Cici, Kempf and Puetz (2011). This may be
facilitated by a lack of regulatory oversight compared to other investment management
35
The research presented in this section was performed in cooperation with Martin E. Ruckes; see
Ruckes and Sevostiyanova (2012b)
42
vehicles such as mutual funds.
The smoothing of earnings is a pervasive phenomenon in many economies not only among
financial but also non-financial firms. One goal of earnings smoothing is to reduce income
stream fluctuations in order to conceal relevant information that can be used to measure
corporate performance, to assess managerial ability or to predict the future earnings. For
example, De Fond and Park’s (1996) empirical analysis supports the notion that issues
of job security provide powerful incentives for managers to smoothen reported income
levels. Managers appear to be “borrowing” earnings from the future for current use, if
actual earnings are poor but expected earnings are high. Conversely, managers “save”
current earnings for future use, if the current performance is high but expected future
performance is poor. Fudenberg and Tirole (1995) show that even managers whose firms
have substantial earnings today and expect high earnings in the future may have an incentive to smooth earnings. This is the case when future earnings are still uncertain and
the negative consequences of reporting low future earnings is significant for the manager.
We build on Fudenberg and Tirole’s (1995) insight and show that standard features in
hedge fund management contracts can have a profound impact on income smoothing, because they affect the manager’s benefits of reporting income streams with low volatilities.
Specifically, we compare contracts that pay a performance fee to the fund manager whenever the reported value of the fund’s assets increases relative to that at the beginning of a
period (period performance fee) versus a performance fee that is paid only if the reported
43
asset value of the fund exceeds its historic maximum (performance fee with high water
mark).
When reporting the value of their funds’ assets, hedge fund managers that maximize
expected fee payments take into account both the performance fee generated by the report
and the expected fund flow resulting from the report. The expected fund flow affects future
fees from managing the fund. When a negative return in the future leads to significant
outflows of capital, a manager with a positive current return may report a lower one
in order to avoid having to report a negative return in the future. This is the case
even when reporting a positive return today increases the managers perceived ability to
investors provided that ability depreciates sufficiently quickly.
While the structure of the performance fee of a fund is inconsequential for fund flows
themselves it affects managers financial consequences from these flows. Consider, for example, a fund with a performance fee with high water mark where the manager observes a
positive return that, if reported truthfully, raises the high water mark for future periods.
In case of future inflows the manager does not financially benefit from these unless the
fund’s asset value exceeds the new high water mark. This is different if the manager
decides to report a lower than the actual return. Then she benefits more from future
inflows, because of a lower high water mark. This is different for a manager who operates
an otherwise identical fund with a period performance fee. Given that the historically
maximal fund value is irrelevant for fund fees, the described consideration does not affect
the manager’s reporting decision. As a consequence, managers of funds with performance
fees with high water marks have stronger incentives to underreport positive actual returns
than managers of funds whose management contracts specify period performance fees.
Similarly, managers with high water marks have strong incentives to overreport negative
returns when doing so prohibits outflows. The reason is that reporting, say, a zero return
rather than the actual negative one does not affect the high water mark and therefore
is inconsequential for future fees. Again, this is different for managers whose fee income
is determined by the performance in each period. Reporting a zero return rather that
the actual negative one affects future returns negatively as the coming period’s return is
based on this period’s reported asset value. In sum, fund managers whose performance is
measured against their funds’ high water marks have stronger incentives to report muted
returns than those of funds specifying period performance fees. Our result is therefore
consistent with empirical evidence in Green (2010), who finds more significant smoothing
by funds whose management contracts contain high water mark provisions. One implication of this finding is that researchers are ill-advised to compare the performance of
funds with high water mark fee structures to those with period performance fees based on
44
Sharpe ratios. For example, Aggarwal, Daniel and Naik (2009) find significantly higher
Sharpe ratios of funds with high water marks. At least part of this difference may be due
to a higher level of strategically motivated return smoothing by managers of funds with
high water marks. Another implication of our model is that learning about the manager’s
quality and a fund’s future prospects occurs more rapidly in funds that do not include
high water marks in their management contracts.
45
Related Literature
Theory.
As mentioned above, our model uses the general framework of Fudenberg and Tirole
(1995) as a basis. An important foundation in their, and our, model is that managers
cannot permanently misreport their returns. If they provide a biased performance report
today, future reports will have to be biased in the opposite direction. The authors study a
manger who is concerned with job security and ignore any aspects of explicit performancebased contracting. Our model incorporates explicit performance-sensitive contracts that
are commonly used in the hedge fund industry. We argue that option-like compensation
contracts such as those that include high water marks are likely to induce their manager’s
incentives for smoothing their returns. In addition, our approach incorporates the effects
of fund flows on managers’ decisions to smoothen income.36
Acharia and Lambrecht (2012) present an alternative approach to income smoothing.
They argue that managers fear a ratchet effect when reporting high earnings: investors
increase their expectations of future earnings if the firm presents high current earnings.
In order to avoid disappointments, managers are hesitant to report high earnings levels
truthfully. This setting assumes relatively limited levels of information by investors and
appears to be better suited for firms with individual investors rather than sophisticated
institutional investors as is the case for hedge funds.
Jylhä (2011) studies the hedge fund manager’s motives to misreport the funds returns.
The author can show three new extensions for the empirical literature on hedge fund
return misreporting, namely: misreporting is more prevalent in funds with capital flows
that are strongly dependent on past performance, it is more prevalent in young funds,
and more prevalent in times of capital outflows. The results presented by Jylhä (2011)
suggest also the idea, that the hedge fund manager tries to represent the fund to appear
as more attractive to the investors than it is in reality by misreporting her returns. On the
contrary to our approach Jylhä (2011) assumes in his model the true hedge fund returns to
be randomly drawn from a normal distribution and completely independent of any action
taken by the manager. Thus, the fund manager in this approach can only misreport
the true returns to the outsiders. The true fund returns in our model are assumed to
be dependent on manager’s skills and the quality of her investment strategy. The main
difference in our model compared to Jylhä (2011) is that we describe smoothing of the
36
For the hedge fund manager, losing her job is equivalent to the withdrawal of the entire capital by
investors. Insofar, our model derives the manager’s cost of losing her job endogenously as the expected
value of forgone fees.
46
reported returns to be costly for the manager. Henceforth, the manager has to bear the
costs of smoothing in the subsequent period. Another important difference is that the
hedge fund manager in the model of Jylhä (2011) is compensated only via management
fee, which sets completely different incentives compared to the performance fee chosen in
our model.
To our knowledge, Dutta and Fan (2012) is the only paper that also incorporates compensation contracting into their approach to income smoothing. The authors derive an
optimal contract when earnings manipulation is possible. Given that their paper is not
specifically designed to study the behavior of hedge fund managers, they do not look at
typically used contract clauses and changes in firm size generated by inflows and outflows
of capital.
Empirics.
Many studies document a positive serial correlation in the self-reported hedge fund returns
as a result of deliberate misrreporting. Bollen and Pool (2008) argue in their empirical
analysis that the structure of hedge fund incentive contracts and the competitive nature
of the industry gives hedge fund managers stronger incentives to overreport losses than
underreport gains. Bollen and Pool (2008) call this behavior “conditional smoothing”.
The authors construct and test a statistical model to screen fraudulent smoothing in the
hedge fund returns and can show that a hedge fund manager tend smooth losses more
than gains, which results in higher serial correlation when funds perform poorly. Bollen
and Pool (2008) make it possible with their model to detect deliberate cheating in hedge
fund returns. The authors argue that for some funds the most likely reason for return
smoothing is simply fraud, by which the hedge fund managers misreport the true returns
in order to reduce the fund’s return volatility and thus to achieve higher Sharpe ratios.
Also, in their second empirical analysis, Bollen and Pool (2009) find a discontinuity in
the distributions of monthly hedge fund returns, pooled across funds and over time, due
to prevalence of misreporting or more precisely, due to temporarily overstated returns.
In order to make their hedge funds more attractive for existing or potential investors
or in order to prevent capital outflows the incentive for manipulating returns can be
enormous for the manager. Ben-David, Franzoni, Landier and Moussawi (2011) present
evidence that hedge funds engage in manipulation of stock prices for reporting purposes.
The authors show that these manipulations are more attractive for funds that are better
diversified, funds that have experienced poor performance in the last period or funds with
a very high recent performance with the goal to attract investors’ attention.
47
Getmansky, Lo, and Makarov (2004) document also a very high serial correlation of reported hedge fund returns. The authors offer many possible explanations for the presence
of this serial correlation. They suggests the serial correlation may be a proxy of illiquidity
in hedge fund investments. Hedge funds often use different share restrictions in their
contracts such as lock-up periods or redemption periods for fund withdrawal. Such share
restrictions avoid short-term capital outflows and make it possible for managers to invest
in illiquid assets more easily. In order to evaluate the illiquid assets in their portfolios
hedge fund managers can for example obtain value estimates from brokers who simply
extrapolate past market values. Getmansky, Lo, and Makarov (2004) suggest as a second
possible explanation for the presence of high serial correlation the purposeful managerial
smoothing of contemporaneous and lagged returns in hedge funds. Lack of transparency
and regulation and also the special fee structure allow the hedge fund managers to misreport their returns in order to charge higher fees.
Agarwal, Daniel, and Naik (2011) find a positive relation between contractual provisions
(high water marks and long restriction periods on investor redemptions) and performance.
They can show that returns are significantly higher in December compared to the rest of
the year. This findings suggest that the hedge fund manager revise their returns upwards
in order to earn higher fees.
Green (2010) analyzes the discontinuity in the distribution of the hedge fund reported
returns. He uses two proxies for the variation in managers’ incentives to manipulate
reported returns: whether a fund has a high water mark provision and whether excess
market returns are negative. Green (2010) is able to show that high water marks and
restriction periods are positively associated with fund returns for quantiles of the return
distribution with losses and small-to-moderate gains. In contrast to Agarwal, Daniel, and
Naik (2011), the results from Green (2010) support the idea that contractual provisions are
negatively associated with returns for quantiles with moderate-to-large gains. The most
interesting finding suggests that hedge funds with high water mark provisions and funds
in their first two years of reporting to a public database show the greatest discontinuity
in the distribution of reported returns.
This section proceeds as follows. In the next subsection we introduce the economic environment of the model and set up the optimal reporting problem. In subsection 4.3 we
provide an theoretical discussion of the structure of the compensation contract, which can
be choosen between the contract with the period peformance fee and the contract with
the high water mark provision. Subsections 4.4 and 4.5 contain our analysis of the optimal reporting choice and derivation of the equilibria, if the contract cpecifies the period
48
performance fee and if the contract specifies performance fee with the high water mark
provision, respectivelly. Subsection 4.6 compares findings of the both previous sections.
In subssection 4.7 we present an extended model version, and a final subsection of this
chapter contains concluding remarks.
4.2
The Basic Model
We consider a risk-neutral investment manager who has an idea for an investment strategy
with a time horizon of three periods. The manager does not have financial wealth of
her own, and must therefore borrow an initial amount V0 = 1 from outside investors to
implement her investment strategy. There exists a pool of outside investors with sufficient
wealth to invest. Outside investors are risk-neutral.37 The interest rate in the economy
is normalized to zero.
Characteristics of the Investment Strategy
In the model there are four points of time t = i, i ∈ {0, 1, 2, 3}. At time t = 0 the
manager and the investors are symmetrically informed about the fund’s prospects and
investors decide whether to make an initial investment of V0 = 1. If investment took place,
implementing the manager’s investment strategy implies that per period the invested
amount generates either a positive (gross) return u > 1 or a negative (gross) return
d = 1/u < 1 (see Figure 7). If the first-period return is u, we will call the manager being
of “u-type”. If the first-period return is d, we will call the manager being of “d-type”. We
assume in our model that at each point in time only the latest performance of the fund
is informative for the fund’s prospects in the following period. More precisely,
• if the fund’s current performance is positive, its return is equal to u − 1 > 0, the
manager’s probability for a positive return in the following period is equal to p ∈
(0, 1),
• if the fund’s current performance is negative, its return is equal to d − 1 < 0, the
manager’s probability for a positive return in the following period is equal to pε,
, 1). 38
with ε ∈ ( 1−p
p
37
We will exclude the possibility of income smoothing that may result from investors’ risk aversion,
because risk avers investors prefer a less volatile return pattern.
38
By such definition of ε we first assume, that the manager’s success probability in the following period
decreases once she experienced losses, thus ε < 1. Second, we assume the decreased success probability
49
Given the assumption that the future expected profit depends only on the current return
and not on the return levels in the past we explicitly assume the process governing return
changes to be first order Markov. Consequently, the term about the manager’s type refers
only to her first-period return u or d and not to her success probability in the following
periods. To see the economic reasoning for this assumption consider the possibility of the
manager’s ability for portfolio management and her professional skills that may change
over time. Additionally, the market can adopt the idea of manager’s investment strategy,
making it less profitable over time. Poor performance can be considered as a “loss by
accident”without any information for the future performance, but can also be indicative
for the future performance.
The hedge fund’s returns generated are observed only by the manager but not by the
investors. The investors are assumed, however, to be able to observe the actual value of
of pε to be still larger than the smallest possible failure probability, pε > 1 − p, in order to create positive
probability for the manager to be able to recover past losses in the future periods.
50
assets under management in t = 2 and t = 3, but not after the first period.39 The value
of assets under management in t = 1 can be V1 ∈ {u, d}. At the end of the second period,
the value is V2 ∈ {u2 , 1, d} and at the end of the last period, in t = 3, the fund’s value of
assets under management can be V3 ∈ {u3 , u, d, d3 }.
At the beginning of the first period there is no current return and, thus, neither the
manager nor investors know exactly the fund’s prospects in the first period.40 Ex-ante,
there exists a mass υ ∈ (0, 1) of managers with a first-period probability for a positive
return of p. Analogously, with the mass 1−υ managers have a first-period probability for
a positive return of pε. υ is common knowledge. It maintains that the manager’s and the
investors’ prior belief of the manager’s success probability in the first period to be equal
p is given by:
µ := υp + (1 − υ)pε.
(A1)
To ensure that it is optimal to launch the fund we assume additionally that the investment
strategy is profitable in the first period:
µu + (1 − µ)d ≥ 1.
(A2)
We also assume that the investors are willing to invest in the fund.
Manager’s Report in t = 1
The only value of assets under management unobservable to the investors is that at date
1. Thus, the new value V1 in t = 1 is the manager’s private information and specifies her
type. As stated above, the second and the third period values, V2 and V3 , are perfectly
observable for both parties. After observing her current performance and, thus, her type
in t = 1 the manager reports to the investors.
By privately observing the first-period asset value the manager has the opportunity to
misreport it in t = 1. v1 denotes the manager’s reported asset value. When realizing
V1 = u the manager can either report the true value v1 = u to the investors or report
the lower value v1 = 1, thus, underreporting the first-period return. Analogously, when
realizing V1 = d the manager can either report the true value v1 = d to the investors or
39
We adopt the assumption first presented by Fudenberg/Tirole (1995), in order to simplify the future
calculation. The more general assumption about the unobservability of the second-period return on the
investors’ side does not change the results.
40
Note, there is no adverse selection problem.
51
report the higher value of v1 = 1, thus, overreporting the first-period return (see Figure
8). Thus, we assume, that the manager is unable to report v1 = d if the true value of
assets equals V1 = u and the manager is unable to report v1 = u if the true value of assets
equals V1 = d. By limiting the level of misreporting to a value that falls between the two
possible values of V1 ∈ {u, d}, we implicitly assume that there are significant falsification
costs for the manager to deviate too much from the true asset value. For example, beyond
a certain degree of misreporting it requires the manager to hold a suboptimal portfolio,
possibly of illiquid securities, that allows her to report the desired asset value.
At time t = 2 the manager observes the second-period realization i ∈ {u, d}. We assume
that the manager is unable to manipulate her reported return for the second time. Thus,
the investors observe the true new value of assets under management V2 ∈ {u2 , 1, d2 }.
This assumption has the important implication that overreporting (underreporting) the
52
true value V1 affects the reported return in the second period negatively (positively).
Possible Changes in Net Asset Value
The initial net asset value of the hedge fund is the investors’ capital contribution of one
unit in t = 1. The value of assets under management changes over time by the way of
realized investment returns. Additionally, the investors who invest in the hedge fund can
contribute or withdraw money from the fund depending on the reported asset value at
date 1 and subsequently the observed value of assets under management. In t = 1 we
assume the investors’ to withdraw capital, if they assume the expected fund’s performance
to be insufficient. In t = 2 we assume the investors’ either contribute or withdraw capital,
based on their assumption of the expected performance of the fund.
Manager’s Compensation
There are two ways hedge fund managers typically profit from her investment strategy.
Charging a fee that is proportion to the value of the assets under management (management fee) and, additionally, receiving a percentage of the fund’s increase in asset value
(performance fee). Hedge funds commonly use one of two types of performance fee: the
manager either participates in any of the fund’s value gains or, alternatively, the losses
experienced by the fund in prior periods must first be recouped by compensating gains
before further performance fees are paid. Due to the manager’s limited liability, fees usually cannot be negative in any period. This implies that after a fee has been paid out,
they are unaccessible to the investors in later periods.
We take as exogenous the two following different compensation contracts:
• a period performance fee (f ≥ 0) that is paid out to the manager at the end of each
period as a constant fraction of the fund’s value gain during the period and not
when the fund loses value during the period,
• a performance fee with a high water mark provision (h ≥ 0) that is paid out to the
manager at the end of each period as a constant fraction of the fund’s value gain
during the period relative to the fund’s historic maximum value and not when the
fund’s value is below its historic maximum.
In both compensation contracts the underlying principle is that the manager is rewarded
for her performance, which is calculated as the increase in net asset value of the fund
(in the second contract - there is a high water mark above which these performance fees
apply). Thus, the value of assets under management is the valuation basis for the level of
53
the manager’s fee. Consequently, capital withdrawals and capital inflows may play a key
role in managerial incentives while deciding on which report is optimal for her in t = 1.
To simplify the analysis, we first normalize the management fee to zero. It is also assumed
that the investor pays any fees separately from the fund to the manager.41
Investors’ Beliefs and Responses
The investors’ information in t = 1 is the manager’s reported value of assets under management at the end of the first period v1 ∈ {u, 1, d}. Observing a report of v1 ∈ {u, d}, the
investors know the value realization in the first period with certainty and, therefore, know
also the manager’s type. The only one state that can lead to an asymmetry of information
is the one in which the manager’s report in t = 1 is v1 = 1. The investors know that the
true fund value is either u or d, but are unable to verify the true value. Based on the
information of the reported first-period return, investors update their prior belief µ about
the manager’s type according to the Bayes’ rule and, thus, her probability of a positive
return in the following period. Contingent on their observation and updated belief, the
investors decide whether to withdraw the entire capital from the fund or not. For a sufficiently low belief probability of investors to face a u-type manager the capital outflow
occurs with probability λ1 ∈ (0, 1). We denote the critical belief below which withdrawals
can occur as µ1 ∈ (0, µ). This implies that if the investors assess the probability of facing
a u-type manager at least as high as at the outset, they do not withdraw capital. If,
however, this probability is sufficiently small, a withdrawal occurs with probability λ1 .
At time t = 2 investors observe the true second-period value of assets under management
V2 ∈ {u2 , 1, d2 }. Based on the observed value and the new belief about the manager’s
type and, thus, her probability of a positive return in the following period, the investors
respond either with an additional capital contribution of ∆ = 1 with probability η ∈ [0, 1]
or the withdrawal of the entire capital with probability λ2 . Recall that the descriptor
u-type refers to the fund’s return in period 1. Therefore V2 = 1 implies that the secondperiod return of a u-type manager is negative. Thus, at date 2 a high probability of
facing a u-type implies a relatively low probability of a positive return in period 3. Thus,
there exists a critical investor belief about the probability of facing a u-type above which
a withdrawal of funds can occur. This critical belief is denoted by µ2 ∈ (µ, 1). Consistent
with our assumption on µ1 , µ2 > µ implies that in the absence of learning about the
manager’s type investors do not withdraw their capital. We assume that λ2 is signifi-
41
As long as the fund’s assets are sufficiently liquid, assuming that the cash to pay the fees are generated
by liquidating the corresponding part of the fund’s assets does not change the results.
54
cantly larger than the probability of withdrawal at the preceding date, λ1 . This implies
that investors sufficiently strongly penalize negative performance after periods of good
performance. This is necessary in models of this type to create a sufficient importance
of second-period performance for first-period reporting (for a discussion of this issue see
Fudenberg/Tirole,1995).
Sequence of Events
The sequence of events is summarized in Figure 9. At date 0 the contract is signed and
investors provide financial capital of V0 = 1 to the fund. The fund manager invests that
capital amount according to her investment strategy. At date 1 the first-period return
i ∈ {u, d} is observed only by the manager. Based on the observed first-period return
the manager learns about her type and decides whether to report her true realization of
i, i ∈ {u, d}, or to misreport the return, by announcing v1 = 1. The fee is paid to the
manager based on the reported value of assets under management v1 ∈ {i, 1}, i ∈ {u, d},
as specified in the fund’s management compensation contract. Investors observe the
manager’s reported value of assets under management v1 ∈ {i, 1}, i ∈ {u, d}. If investors
observe or with sufficiently large probability believe that the first-period return is negative
and, thus, the manager’s type is d, they decide to withdraw their capital with probability
λ1 . In the remaining cases there is no capital outflow after the first period. If the fund is
closed, all assets are liquidated at no cost and the proceeds are paid to the investors.
If the fund remains alive, assets V1 ∈ {u, d} are used according to the investment strategy
in the following period. At date 2, the fund’s new true value of assets under management
V2 ∈ {u2 , 1, d2 } is observed by all parties. The fee is paid to the manager based on the
value V2 as specified in the fund management compensation contract. For the second time,
the investors decide whether to leave the fund or not. If investors observe or at least with
55
sufficiently large probability believe to experience a decrease in the value of assets under
management compared to the previous period V2 < V1 , they withdraw the investment
with the probability λ2 . In the remaining cases they invest an additional capital amount
of 1 in the fund with probability η. If the fund is closed, all assets are liquidated at no
cost and the proceeds are paid to the investors.
At date 3, if the fund is still alive, the third fund’s return is observed by all parties, its
assets are costlessly liquidated and the proceeds are distributed to the investors. The
contractually agreed fee is paid to the manager.
4.3
Analysis: Period-Performance Fee
First we study the manager’s reporting and investors’ investment decisions when the fund
management contract specifies a period performance fee. The analysis uses the concept
of backward induction, whereby the manager and the investors reason backward from the
end of the third period to the beginning of the first period in order to determine which
choices are rational at each stage.
Investors Beliefs and Responses in the Third Period
In t = 2 the are three possible values of assets under management, V2 ∈ {u2 , 1, d2 }. V2
is observed by both parties . There exist a total of 7 different reported fund “histories”,
that can describe the development of the reported fund asset values up to date 2. By
history we denote the tuple of first- and second-period reported asset values: {v1 , V2 }.
Observing the value of assets under management of V2 = u2 the investors can face either
a reported fund history of {v1 = u, V2 = u2 } or {v1 = 1, V2 = u2 }. Based on one of
the two possible reported fund histories the investors’ update their beliefs about the
manager’s type and her probability of a positive return in the next period. Investors’
beliefs are defined by a conditional probability: α(u|i, u2 ), i ∈ {u, 1}, the probability
that, conditional on observing the reported fund history of {v1 ∈ {u, 1}, V2 = u2 } the
manager is of u-type and, thus, have realized a positive return of u in the first period.
If investors are certain to face a u-type manager the investors expect a probability of p
for a positive return in the following period. However, the investors know that the asset
value in the first period cannot be 1. As a consequence, the value of V2 = u2 is sufficiently
informative for the investors, so, that their belief is identical upon observing the history
of {v1 = u, V2 = u2 } or {v1 = 1, V2 = u2 } in this state:
56
P (V2 = u2 |V1 = u)P (V1 = u)
P (V2 = u2 |V1 = u)P (V1 = u) + P (V2 = u2 |V1 = d)P (V1 = d)
pµ
=
= 1,
pµ + 0(1 − µ)
⇔ α(u|1, u2 ) ≡ 1.
α(u|u, u2 ) =
By observing in t = 2 an increase in the fund’s asset value over the last period the investors
respond with an additional capital contribution of 1 with probability η. Conditional on
the investors’ belief of α(u|u, u2 ) = 1 or α(u|1, u2 ) = 1, the u-type manager’s expected
period performance fee payment in the third period is given by:
E u (f |u2 , 1) = f p(u − 1)(u2 + η).
For a value of assets under management equal to V2 = d2 in t = 2, the investors can observe
a reported fund histories of {v1 = d, V2 = d2 } or {v1 = 1, V2 = d2 }. Like in the previous
case, the negative return in the last-period and the knowledge of impossibility to obtain
a zero return in the first period are sufficiently informative for the investors to learn the
manager’s type. For the reported fund’s history of {V1 ∈ {d, 1}, V2 = d2 }, the investors
believe with probability α(u|i, d2 ), i ∈ {d, 1}, that the manager’s type is u is:
P (V2 = d2 |V1 = u)P (V1 = u)
P (V2 = d2 |V1 = u)P (V1 = u)+P (V2 = d2 |V1 = d)P (V1 = d)
0µ
=
= 0.
0µ+(1−pε)(1−µ)
⇔ α(u|d, d2 ) ≡ 0.
α(u|1, d2 ) =
Therefore, the investors are sure, that the fund manager is a d-type manager and has
performed poorly in both periods. The investors assess the next-period probability of
a positive return to be pε and respond with capital withdrawals with probability λ2 .
Anticipating the possible capital outflow in t = 2 the d-type manager’s expected period performance fee payment in the third period, conditional on the investors’ belief of
α(u|i, d2 ) = 0, i ∈ {d, 1}, is given by:
E d (f |d2 , 0) = f (1−λ2 )pε(d−d2 ).
57
The third possible value of assets under management that the investors can observe in
t = 2 is V2 = 1. There are three potential reported fund’s histories, that can lead to this
outcome.
• The investors may observe a fund history of {v1 = u, V2 = 1}. Then the investors
know that the fund performed well in the first period but experienced a loss in
the second period. Based on this information the investors believe with probability
α(u|u, 1) ≡ 1 that they face a u-type manager. Because of last period’s loss, the
probability of a positive return in the following period is pε. As a response, the
investors withdraw their capital with probability λ2 . Contingent on investors’ belief
of α(u|u, 1) ≡ 1 and the possible capital outflow, the u-type manager’s expected
period performance fee payment equals:
E u (f |1, 1)) = f (1−λ2 )pε(u−1).
• The investors’ may observe a reported fund history of {v1 = 1, V2 = 1}. In this
state the investors are unable to distinguish whether the true first-period return
was positive, equal to u, or negative, equal to d. Generally, the investors place a
probability α(u|1, 1) ∈ [0, 1] on the fund’s manager being u typed.
For sufficiently low probabilities to face a u-type manager, α(u|1, 1) < µ2 , investors
assign a sufficiently high probability for a positive second-period return – and therefore also the third-period return – that they are willing to contribute additional
capital of 1 with probability η to the fund. For α(u|1, 1) ≥ µ2 , investors respond
with the capital outflow with probability λ2 .
If the fund indeed performed well in the second period, the d-type manager’s probability of a positive return is equal to p and her expected fee payment, contingent
on the investors belief of α(u|1, 1), equals:
E d f |1, α(u|u, 1) =
(
f p(u−1)(1 + η) f or
f (1−λ2 )p(u−1) f or
α(u|u, 1) < µ2
α(u|u, 1) ≥ µ2 .
If the fund performed poorly in the second period the u-type manager’s probability
of a positive return is equal to pε and her expected fee payment, contingent on the
investors belief of α(1|1, 1), equals:
E u f |1, α(u|1, 1) =
(
f pε(u−1)(1 + η) f or
f (1−λ2 )pε(u−1) f or
58
α(u|1, 1) < µ2
α(u|1, 1) ≥ µ2 .
• The investors may observe a reported fund history of {v1 = d, V2 = 1}. Then the
investors know that they face a d-type manager. Additionally, the investors know
with certainty, that the fund performed well in the second period. Based on their
belief of α(u|d, 1) = 0 the investors assign a probability of p for a positive return
in the following period and respond with capital contribution of 1 with probability
η. The d-type manager anticipates a possible capital contribution in this state and
expects in the third period a value of period performance fee equal to:
E d (f |1, 0)) = f p(u−1)(1 + η).
Investors Beliefs and Responses in the Second Period
In t = 1 the are two possible fund returns: a positive return of V1 = u and a negative
return of V1 = d, but there are three possible reports the manager can make, and, thus,
the investors can observe: v1 ∈ {u, 1, d}. Recall, that by the realization of u the manager
is able to report v1 ∈ {u, 1} and by the realization of d the manager is able to report
v1 ∈ {1, d}.
By observing the reported return of v1 = u the investors receive a credible signal about
the true fund performance and the fund’s prospects for the following period. Namely,
they assess the probability of a positive fund return in the second period to be p.
Given a first-period reported return of v1 = u and investors beliefs as described above,
the u-type manager’s expected performance fee in t = 1 is given by:
E u (f |u, 1) = f (u−1)+f p(u2 −u)+p2 (u−1)(u2 +η)+(1−p)(1−λ2 )pε(u−1) .
(4)
By observing a reported return of v1 = d the investors also receive a credible signal about
the true fund performance and the fund’s prospects for the following. Specifically, the
investors are certain that V1 = d and that the probability for a positive return in the
second period is pε. As a response in t = 1 the investors withdraw their capital with
probability λ1 . The fund is still alive in the second period with probability (1−λ1 ).
Then, the d-type manager’s expected period performance fee at time t = 1 is given by:
E d (f |d, 0) = f (1−λ1 ) pε(1−d)+p2ε(u−1)(1+η)+(1−pε)(1−λ2 )pε(d−d2 ) .
59
(5)
If investors observe a first-period reported value of assets under management equal to
v1 = 1 they know that the true return is either u or d, but are unable to distinguish
whether the message originates from the u- or d-type manager.
Depending on investors’ beliefs α(u|1) ∈ [0, 1] about the manager’s type and, therefore,
her probability of gererating a positive return in the second period, the investors can
respond either with a withdrawal of capital, which occur with probability λ1 , or not.
• If in t = 1 investors’ with sufficiently high probability α(u|1) ≥ µ1 believe, that
the true first-period return is positive, they will stay with the fund and there is no
capital outflow. In this situation the investors believe that it is relatively likely that
they face a u-type manager who underreports her true return by announcing a lower
value of assets under management of 1. If the manager in fact realized a return of
u in the first period, her probability of a positive second period return is still p and
her expected fee payment at time t = 1 by reporting v1 = 1 is given by:
E u f |1, α(u|1) ≥ µ1 = f p(u2−1)+p2 (u−1)(u2 +η)+(1−p)(1−λ2 )pε(u−1) . (6)
If the manager in fact experienced a low last-period performance of d but overreported the value of assets under management, by announcing v1 = 1, her probability
of achieving a positive second-period return decreases to pε. But given investors’
belief of α(u|1) ≥ µ1 the d-type manager does not expect a capital outflow and her
expected fee payment in t = 1 by reporting v1 = 1 is given by:
E d f |1, α(u|1) ≥ µ1 = f p2 ε(1−λ2 )(u−1)+(1−pε)(1−λ2 )pε(d −d2 ) .
(7)
• Observing the reported return of v1 = 1 investors may have a sufficiently low belief
α(u|1) < µ1 that the manager is a u-type manager. As a response, the investors
withdraw their capital with probability λ1 .
If the manager in fact is u-type and realized a positive return of u, she has a high
probability of a positive return of p in the second period. But by announcing v1 = 1
and given investors’ belief of α(u|1) < µ1 , she is still affected by a possible capital
outflow in t = 1. In this state the u-type manager’s expected amount of period
performance fee is equal to
60
E u (f |1, α(u|1) < µ1 ) =
(8)
= f (1−λ1 ) p(u2−1)+p2 (u−1)(u2 +η)+(1−p)pε(u−1)(1+η) .
If the manager in fact experienced a low first-period performance of d, she has a
lower probability for a positive return in the second period of pε. By announcing
v1 = 1, the d-type manager’s expected period performance fee payment in t = 1,
contingent on investors’ belief of α(u|1) < µ1 , is given by:
E d f |1, α(u|1) < µ1 = f (1−λ1 ) p2 ε(u−1)(1+η)+(1−pε)(1−λ2 )pε(d −d2 ) . (9)
4.4
Optimal Reporting Choice and Equilibria: Period Performance Fee
The findings discussed in the previous subsection describe an incomplete information
signaling game. The informed player, the manager, has the choice between truthful reporting and misreporting of her first-period return and, thus, between revealing her true
type or not. The manager chooses first her optimal reporting. Then her uninformed
opponents, the investors, choose their optimal response based on updated beliefs about
the true fund return, the manager’s type and, thus, theprospects for the following period.
Now we describe a systematic procedure to search for a Perfect Bayesian Equilibria in
pure strategies.
Definition: In the model, the Perfect Bayesian Equilibrium (PBE) is a set of strategies and beliefs that satisfy each of the following three conditions:
1. The strategies of the manager and the investors are sequentially rational.
2. Based on the observed reported return investors update their prior beliefs about
the manager’s success probability in the following period according to Bayes’ rule.
3. At the out-of-equilibrium information sets, beliefs are derived, using Bayes’ rule,
from the beliefs at the information sets that precede the information set in question
and players’ continuation strategies as implied by their equilibrium strategies, if
possible.
In the equilibrium, the manager’s expectations about the investors strategy is consistent
with the investors’ expectations about the manager’s strategy. Each party chooses a
61
best response to what it believes the other party will choose to do. Taking into account
the manager’s decision and the corresponding responses of the investors in each of the
decisions nodes, we can analyze if the following four candidate strategy profiles constitute
a Perfect Bayesian Equilibrium:
• pooling(1,1): is a pooling strategy profile for the informed player, in which the
manager independent of her first-period return announces v1 = 1 in t = 1. Investors’
optimal responses are contingent on their updated beliefs.
• separating(u,d): is a separating strategy profile for the informed player, in which the
u-type manager truthfully reports her first-period return by announcing v1 = u, and
the d-type manager truthfully reports her first-period return by announcing v1 = d,
in t = 1. Investors’ optimal responses are contingent on their updated beliefs.
• separating(u,1): is a separating strategy profile for the informed player, in which
the u-type manager truthfully reports her positive first-period return by announcing
v1 = u and the d-type manager overreports her negative first period return by
announcing v1 = 1. Investors’ optimal responses are contingent on their updated
beliefs.
• separating(1,d): is a separating strategy profile for the informed player, in which
the u-type manager underreports her positive first-period return of u by announcing
v1 = 1, and the d-type manager truthfully reports her negative first period return
of d by announcing v1 = d. Investors’ optimal responses are contingent on their
updated beliefs.
For each strategy profile we have now to calculate the investors’ beliefs. Contingent on
investors’ beliefs we have to consider their optimal responses in each state.
We begin with the pooling strategy profile pooling(1,1).
In a pooling equilibrium the manager reports v1 = 1 independent of her type and, thus,
whether her true first-period return was positive, equal to u, or negative, equal to d.
Bayesian updating in t = 1, by observing a misreported value of v1 = 1, implies that
investors do not learn anything about the manager’s type, her success probability and
fund’s prospects. Hence, the investors posterior belief α(u|1) coincides with their prior
belief, µ. Given that investors were willing to invest capital into the fund in t = 0, we
assume that in the absence of learning in a pooling equilibrium they will stay with the
fund for at least one period longer. Thus, in pooling(1,1), there is no capital outflow in
t = 1.
62
In t = 2, if the investors observe V2 = u2 they are certain to face a u-type manager. By
observing V2 = d2 they are certain to face a d-type manager. Only by observing V2 = 1
the informational asymmetry is still unsolved. By observing V2 = 1 investors are able to
form beliefs that are more precise than those in t = 1. Because V2 = 1 is a true value it
is informative for investors. More precisely, after observing {v1 = 1, V2 = 1}, the investors’
posterior belief about the u-type manager in t = 2 is equal to
P (V2 = 1|V1 = u)P (V1 = u)
P (V2 = 1|V1 = u)P (V1 = u)+P (V2 = 1|V1 = d)P (V1 = d)
(1−p)µ
< µ.
=
(1−p)µ + pε(1− µ)
α(u|1, 1) =
Investors’ posterior belief of facing a u-type, α(u|1, 1), is lower than their prior belief, µ.
An increased probability of facing a d-type manager is good news for investors, because
it is the d-type manager who realized a positive second-period return. Thus, investors
expect a probability of p for a positive return in the third period and contribute additional
capital of 1 with probability η to the fund.
Given the investors’ optimal response, we can now determine the manager’s optimal
reporting choice, especially, whether the manager has an incentive to deviate from the
previously described pooling strategy profile or not. The announcement of v1 6= 1 occurs
off-the-equilibrium path according to the strategy profile pooling(1,1), implying that the
investors’ off-the-equilibrium beliefs can be arbitrarily.
• If the true fund history is {V1 = u, V2 = 1} (implying that the manager is a u-type),
her expected fee payment, given investors’ beliefs in pooling(1,1) strategy profile,
α(u|1) = µ in t = 1 and α(u|1, 1) < µ2 in t = 2, is described by:
(10)
E u f |1, µ, α(u|1, 1) < µ2 =
= f p u2 −1+p(u−1)(u2 +η)+(1−p)ε(u−1)(1+η) .
If the u-type manager deviates towards the truthful reporting of her first-period
return, by announcing v1 = u, she sends a credibly signal about her type to the
investors. Her expected fee payment will be as described in (4). Comparing the
u-type manager’s expected fee payments given truthful reporting of v1 = u (4)
63
with the expected fees payment given underreported value of v1 = 1 (10) leads to
the following result:
E u (f |u, 1) ≥ E u f |1, µ, α(u|1, 1) < µ2 ⇔
η≤
1 − pελ2
:= η u
pε
⇔
λ2 ≤
1 − pεη
:= λu2 .
pε
The value η u defines the probability-weighted threshold amount of expected additional capital contribution below which the strong-type manager reports truthful
her first-period return and above whom she has an incentive to underreport. The
option to avoid the capital outflow and additionally to receive an expected amount
of η contributed to the fund in t = 2, in the case of low second-period performance,
is valuable to the u-type manager. The incentive to underreport the true first-period
return and, thus, her type is higher for increasing levels for η.
Analogously, λu2 denotes an equivalent threshold value for the withdrawal probability
in t = 2. More precisely, with increasing levels of λ2 the u-type manager’s incentive
to pool with the d-type, by announcing v1 = 1, increases. In order to prevent capital
outflow in t = 2 in the case of poor second-period performance the u-type manager
forgoes part of her period performance fee in t = 1.
Remark : As soon as the numerical values of λu2 or η u are larger than 1, it means
that the u-type manager reports the true first-period return of u with probability 1.
By analyzing the values λu2 and η u we can make an interesting observation about the
success probability pε. In order to incentivize the u-type manager to underreport
her first-period return, the threshold probability λu2 (or η u ) has to be below 1, more
precisely, due to η ∈ [0, 1], it holds:
1 ≥ ηu =
⇔ pε ≥
1 − pελ2
pε
1
.
1 + λ2
In summary, the remark implies that the u-type manager has an incentive to underreport her first-period return for η > η u (λ2 > λu2 ) only if the probability of a positive
1
return for the d-type manager, pε, is sufficiently high pε ∈ [ 1+λ
, 1).
2
• If the true fund history is {V1 = d, V2 = 1} (implying that the manager is a d-type),
her expected fee payment, given investors’ beliefs in pooling(1,1) strategy profile,
α(u|1) = µ in t = 1 and α(u|1, 1) < µ2 in t = 2, is described by:
64
E d f |1, µ, α(u|1, 1) < µ2 =
= f p2 ε(u−1)(1+η)+(1−pε)(1−λ2 )pε(d−d2 ) .
(11)
Consider now, that the d-type manager announces her true first-period return, v1 =
d. By doing so, the d-type manager credibly signals her type to the investors. Her
expected fee payment by truthfully reporting her first-period return generates a
belief investors of 0 to face a u-type manager. In this case the d-type manager’s
expected fee is described as in (5). Comparing the d-type manager’s expected fee
payment in both regimes leads to the following consideration:
E d (f |d, 0) ≥ E d f |1, µ, α(u|1, 1) < µ2 ⇔
(12)
1 − λ1
p(u−1)(1+η) + (1−pε)(1−λ2 )(d−d2 )
≥
λ1
1−d
Given investors’ beliefs in strategy profile pooling(1,1), the d-type manager faces
a trade-off between accepting capital outflow already after the first period, but
receiving an expected amount of (1−λ1 )pε(1−d) in t = 2 when truthfully reporting
v1 = d, and avoiding any capital outflow in t = 1, but also missing the expected
fee amount by reporting v1 = 1 in t = 2. For very small values of λ1 the d-type
manager always reports her true first-period return v1 = d, especially for λ1 = 0 we
have the following result:
λ1 = 0 ⇒
pε(1 − d) > 0 ⇒
E d (f |d, 0) > E d f |1, µ, α(u|1, 1) < µ2 .
In terms of λ2 and η we can solve the inequality (12) with the following results:
η≤
(1−λ1 )(1−d) − λ1 (1−pε)(1−λ2 )(d−d2 )
− 1 := η d
λ1 p(u−1)
(1−λ1 )(1−d) − λ1 p(u−1)(1+η)
d
⇔ λ2 ≥ 1 −
:=
λ
2 .
λ1 (1−pε)(d−d2 )
65
After comparing the thresholds η u and η d (or, analogously λu2 and λd2 ) we can find a critical
value
λ∗1 :=
ε(1−d)
u(1+pε(1−λ2 ))−pε2 (1−λ2 )(d−d2 )+ε(1−d2 −p(1−λ2 )−λ2 (d−d2 ))−1
so that it holds: η d ≥ η u for all λ1 ≤ λ∗1 . Summarizing the considerations about the
pooling strategy profile (1, 1) by given investors’ beliefs of µ in t = 1 and α(u|1, 1) < µ2
about the u-type manager and corresponding responses in t = 1 and t = 2 leads to the
following result:
Proposition 6 Suppose the hedge fund management contract specifies a period performance fee, f . The pooling strategy profile pooling(1,1) can be supported as a PBE if and
1
and
only if pε ≥ 1+λ
2
(
η u f or λ1 > λ∗1
η>
η d f or λ1 ≤ λ∗1 .
We now proceed with the separating strategy profile separating(u,d).
In the strategy profile separating(u,d) the u-type manager reports her true first-period
return of u by signaling her type to the investors. The d-type manager reports also her true
first-period return of d. Observing the reported first-period returns of either u or d, the
investors can perfectly distinguish between both manager types, thus, the informational
asymmetry is resolved for all following periods. The u-type manager’s expected period
performance fee in this case is given by (4). The d-type manager’s expected period
performance fee in this situation is given by (5).
Now we analyze whether the u- or the d-type manager has an incentive to deviate from
the truthful reporting in t = 1. If investors observe a reported value of assets under
management v1 = 1, the investors must still update their belief of α(u|1). However, such
reported value occurs off-the-equilibrium path according to the given strategy profile
separating(u,d). In consequence, the investors’ belief can be arbitrarily specified, i.e.,
α(u|1) ∈ [0, 1]. In Appendix B.1 all possible expected rewards for the u- and the dtype manager are calculated, contingent on all possible beliefs in this case. We show the
following result:
Proposition 7 Suppose the hedge fund management contract specifies period performance
fee, f . The separating strategy profile separating(u,d) can be supported as a PBE for
(
η u f or λ1 > λ∗1
η≤
η d f or λ1 ≤ λ∗1 ,
66
if pε ≥
1
,
1+λ2
and for all η ∈ [0, 1], if pε <
1
.
1+λ2
Proof: See Appendix B.1.
Figure 10 illustrates the results of Proposition 7: for high probabilities for capital outflow
in t = 1, λ1 > λ∗1 , the u-type manager has first an incentive to deviate towards misreporting for relatively high probabilities for additional capital contribution η > η u . In this
state the d-type manager will always mimic the u-type, as soon, as u-type misreports her
first-period return. Thus, the strategy profile pooling(1,1) is a PBE as soon as the u-type
manager misreports her first-period return (see the upper part of Figure 10).
With decreasing values of λ1 ≤ λ∗1 the d-type manager’s incentive for overreporting of her
first-period return decreases. Thus, not until η > η d the d-type manager has an incentive
to hide her true type by overreporting her first-period return. In this state for η ≤ η d the
u-type manager has also to reveal her type (see the lower part of Figure 10).
The separating strategy profiles separating(u,1) and separating(1,d) can not be supported
as a PBE.
In the strategy profile separating (u, 1) the u-type manager reports her true first-period
return of u by signaling her type to the investors. Even though the d-type manager
overreports her true first-period return, the investors nevertheless learn her type. The
same happens in the strategy profile separating (1, d). As soon as the d-type manager
credibly signals her type to the investors perfectly distinguish between both manager
types. Appendix B.2 shows that neither the separating(u,1) nor the separating(1,d) can
be supported as a PBE.
67
If the hedge fund manager contract specifies a period performance fee, the manager’s
opportunity for misreporting is valuable only in certain parameter settings. A pooling
equilibrium requires a sufficiently high probability of a positive return of the d-type manager as well as specific parameter levels for λ1 , λ2 and η (see Figure 10).
4.5
Optimal Reporting Choice and Equilibria: Performance Fee
with High Water Mark
Consider now a situation where the hedge fund management contract specifies a performance fee with high water mark. The fund’s initial value V0 = 1 is the first high water
mark. The nodes V1 = u, V2 = u2 and V3 = u3 in Figure 11 characterize further possible
high water marks. In case of a loss in the first period, the only possible fund value that
sets a new high water mark is V3 = u (see Figure 11).
68
u-Type Manager’s Decision
The first positive fund’s realisation V1 = u specifies the second high water mark and the
manager’s performance fee with the high water mark provision by truthful reporting of
v1 = u equals to h(u − 1). By observing the new high water mark the investors identify
the manager as a u-type manager and the informational asymmetry is resolved for the
following periods. If investors observe the third high water mark V2 = u2 in t = 2 they
contribute additional capital of η to the fund. If the second-period return is poor and
the new value of assets under management drops to V2 = 1, investors by observing the
history {v1 = u, V2 = 1} withdraw all of their capital with probability λ2 . The u-type
manager’s expected performance fee with the high water mark provision at time t = 1,
given reported return of v1 = u and investors’ belief of 1 on u-type manager, equals to:
2
2
E (h|u, 1) = h(u−1)+hp u −u+p(u−1)(u +η) .
u
(13)
By anticipating a poor performance in the second period the u-type manager has an
option to underreport the first-period return in order to prevent capital outflow in t = 2.
The underreported fund’s value of v1 = 1 is uninformative for investors in t = 1. For a
given investors belief α(u|1, 1) < µ2 on the u-type the u-type manager is able not only
to prevent capital outflow in t = 2, but additionally to get capital contribution of 1 with
probability η. On the one hand, underreporting is costly for the u-type manager in this
state, because he has to surrender the fees in value of h(u − 1). On the other hand, by
performing well in the third period, the u-type manager is able to set a new high water
mark and to receive a fee amount of (1+η)(u−1). To deceide whether underreporting is
beneficial in t = 1, the u-type manager compares her expected fee payment by thruthful
reporting of v1 = u and by announcing v1 = 1.
d-Type Manager’s Decision
Observing a poor first-period realisation of V1 = d the d-type manager can thruthfully
report v1 = d. In this case capital outflow occures in t = 1 with probability λ1 . If
investors observe the true manager’s type in t = 1, there exist only one state in which the
d-type manager is able to receive a positive fee amount, namely by realising V3 = u in the
third period. By offering her true type to the investors the d-type manager’ expected fee
payment in t = 1 is given by:
E d (h|d, 0) = h(1−λ1 )p2 ε(1+η)(u−1).
69
(14)
By using the oppotunity of overreporting the d-type manager can take an advantage if
investor’s belief on the u-type is α(u|1, 1) < µ2 in t = 2. Thus, by observing V2 = 1 in
t = 2 investors’ are willing to contrubute additional capital of 1 with probability η to the
fund, if they with sufficiently high probability beliefe to observe an increase in the value
of assets under management during the second period.
Equilibria
First we investigate whether the strategy profile pooling(1,1) can be supported as PBE.
As discussed in the basic model in the strategy profile pooling(1,1) both manager types
are reporting v1 = 1 at the end of the first period. Investors have a belief of µ in t = 1
and a belief of α(u|1, 1) < µ2 in t = 2 that the manager’s type is u. Thus, the investors do
not withdraw any capital at the end of the first and the second periods, but additionally
contribute capital of 1 with probability η in expectation in t = 2. Contingent on the
investors’ beliefs and their responses in the given strategy profile, the u-type manager’s
expected fee payment in pooling(1,1) is given by:
E u h|1, µ, α(u|1, 1) < µ2 = hp u2 −1+p(u2 +η)(u−1)+(1−p)ε(1+η)(u−1) .
(15)
Comparing the u-type manager’s expected fees with the high water mark provision given
truthful reporting (13) and underreporting (15) in the strategy profile pooling(1,1) leads
to the following threshold:
2
2
E (h|u, 1) = h(u−1)+hp u −u+p(u−1)(u +η)
≥ E u h|1, µ, α(u|1, 1) < µ2 ⇔
u
η≤
1−pε
:= ηeu .
pε
Note, that the u-type manager’s decision does not depend on the withdrawal probability
λ2 . Additionally, in order to obtain correct numerical values for ηeu ∈ [0, 1] it has to hold
pε ∈ [ 12 , 1].
Analogously, contingent on the investors’ beliefs and their responses in the strategy profile
pooling(1,1), the d-type manager’s expected fee payment is given by:
E d (h|1, µ, α(u|1, 1) < µ2 ) = hp2 ε(1+η)(u−1).
70
(16)
Comparing the d-type manager’s expected fees with the high water mark provision given
truthful reporting (14) and underreporting (16) in the strategy profile pooling(1,1) shows,
that the d-type manager has never an incentive to deviate from her strategy in strategy
profile pooling(1,1):
E d (h|d, 0) = h(1−λ1 )p2 ε(1+η)(u−1)
≥ E d h|1, µ, α(u|1, 1) < µ ⇔
(1−λ1 ) ≥ 1 −→ ∃ only f or
λ1 ≡ 0.
In summary, the strategy profile pooling(1,1) can be supported as PBE for η ∈ [e
η u , 1] if
pε ∈ [ 12 , 1].
In Appendix B.3 we prove whether remaining candidate strategy profiles can be supported
as PBE. The following proposition summarizes the findings.
Proposition 8 Suppose the hedge fund management contract specifies a period performance fee with high water mark provision, h. For all λ1 , λ2 ∈ (0, 1) only the pooling
strategy profile pooling(1,1) can be supported as a PBE for
η ∈ (e
η u , 1),
if pε > 0.5. If pε ≤ 0.5 only the separating strategy profiles separating(u,d) and separating(u,1)
can be supported as a PBE for all η ∈ [0, 1].
Proof: See Appendix B.3.
The threshold ηeu that defines whether the separating or the pooling equilibrium occurs
in the market depends only on the u-type manager’s decision and not on the d-type’s.
The d-type manager prefers to overreport her first-period return as soon as the investors
beliefs on the u-type in t = 2 by observing the reported value of 1 to the second time is
smaller than their prior belief of µ. In this situation it is irrelevant for the d-type manager
whether there is capital outflow in the first period or not. The only one valuable option
for her is the investors capital contribution η in t = 2.
71
4.6
Comparing Equilibria
Corollary 4 The hedge fund manager, independent of her type and the type of the contract, has a greater incentive for misreporting of her first-period return with increasing
values of pε.
If the contract specifies the period performance fee, the manager’ incentive for misreporting
of her first-period return increases with increasing values of λ1 and λ2 .
Proof: See Appendix B.4.1.
The first part of Corollary 4 is intuitive, since the expected management compensation
level as a percetage of expected future profits should be sufficiently large to incentivise
the manager for misreporting. The u-type manager is willing to hide her type only, if her
expected fee payment by achieving V3 = u after the poor second-period performance is
high enough to recover her due fees. The d-type manager’s achiving of the state V2 = 1 is
also more likely, if her success peobability pε increases. The second part of Corollary 4 says
that in order to overcome capital outflow in t = 1 and t = 2 the manager misreports her
first-period return for high levels of λ1 and λ2 . More interesting fact is that if the contract
specifies performance fee with the high water mark provision, the withdrawal probability
λ2 do not play such a great role in the manager’s decision making. Additionally, as soon
as the probability λ1 is positive, the d-type manager is always preffering to pool with the
u-type.
Comparing Propositions 6, 7 and 8 yields the central result of our model.
Proposition 9 There exist parameters where return smoothing occurs when the hedge
fund management contract specifies a performance fee with high water mark, but return
smoothing is absent for a management contract with period performance fee. The opposite
does not occur.
Proof: See Appendix B.4.2.
Comparing the cutoff’s for η, it is easy to see that if the management compensation
contract specifies the period performance fee the threshold value η u is larger that the
counterpart ηeu if the contract specifies the performance fee with the high water mark
provision:
1 − pε
1−λ2 pε
= η u ≥ ηeu =
pε
pε
⇔ 1 ≥ λ2 .
72
In other words, the difference between the both parameters, η u and ηeu , decreases with
increasing λ2 , especially for λ2 ≡ 1 we have: η u = ηeu . Furthermore, the managerial
optimal reporting choice in t = 1 does not depend on the withdrawal probabilities λ1
and λ2 if the manager’s compensation contract specifies performance fee with the high
water mark provision. To make the futher discussion more intuitive, we use a numerical
example presented in Table 4.
If the contract specifies the period performance fee, in the numerical example (see Table
1
4) the pooling equilibrium occurs for pε ≥ 1+λ
= 0.77 (shown in Proposition 6). If
2
the contract specifies performance fee with the high water mark provision, the pooling
equilibrium occurs already for smaller values pε ≥ 0.5 (shown in Proposition 8). With
increasing values on pε the needful expected amount η to incentivise the manager for
misreporting decreases.
Proposition 9 shows that management contracts with high water marks are more vulnerable to managerial return smoothing than those with period performance fees. The reason
for this is that both types of managers have stronger incentives to report v1 = 1 when
they receive fees based on value gains relative to the historic fund maximum. Note that
the incentive to smooth returns does not depend on the magnitude of the fees, f and h,
Thus, even if h were to be higher than f – because period performance fees are paid in
more instances that performance fees with high water mark – the incentives to smooth
73
returns do not change.
To see the economic reasoning for Proposition 9 consider first a u-type manager whose
management contract specifies a performance fee with high water mark. If she reports V1
truthfully, this raises the high water mark for future periods. In case of future inflows the
manager does not financially benefit from these unless the fund’s asset value exceeds the
new high water mark. This is different if the manager decides to report a lower than the
actual return. Then she benefits more from future inflows, because of a lower high water
mark. This is different for a manager who operates an otherwise identical fund with a
period performance fee. Given that the historically maximal fund value is irrelevant for
fund fees, the described consideration does not affect the manager’s reporting decision.
As a consequence, managers of funds with performance fees with high water marks have
stronger incentives to underreport positive actual returns than managers of funds whose
management contracts specify period performance fees.
Managers with high water marks also have stronger incentives to overreport negative
returns when doing so prohibits subsequent outflows. The reason is that reporting a
zero return rather than the actual negative one does not affect the high water mark and
therefore is inconsequential for future fees. Again, this is different for managers whose fee
income is determined by the performance in each period. Reporting a zero return rather
that the actual negative one affects future returns negatively as the coming period’s return
is based on this period’s reported asset value. In sum, fund managers whose performance
is measured against their funds’ high water marks have a stronger incentives to report
smuted returns than those of funds specifying period performance fees.
Corollary 5 If the management compensation contract specifies performance fee with the
high water mark provision h and the withdrawal probability λ2 is smaller than 1, smoothing
of the hedge fund reported returns occurs already for smaller success probabilities pε as if
the contract specifies the period performance fee f .
Proof: See Appendix B.4.3.
74
4.7
Extended Model Version
The basic model considers not many posibilities to achieve a high water mark, especially,
if the first-period return was low equal to d. If the contract specifies performance fee
with the high water mark provision, the d-type manager is able to receive a positive fee
amount only in the third period by achieving a value of assets under management equal
to V3 = u. Also the u-type manager does not have possibility to receive a positive fee
amount once the second-period return was low.
Consider an extension of the basic model in the following way. Denote by RH > 0 the
fund’s positive net return, so the positive gross return equals to u := 1+RH . Analoguously,
denote by RL < 0 the negative fund’s return, so the negative gross return equals to
d := 1+RL . (Note, that the assumption u = 1/d can be still satisfied for some parameters
RH and RL but is not longer general.)
Now there are several possibilities to achieve new high water marks in additional states
compared to the basic model (see Figure 12). At first consider the u-type manager: after
the truthful reporting of v1 = u the new high water mark is set. If the u-type manager is
additionally successfull in the second and also in the third periods, she recives twice the
performance fee with the high water mark provision, exactly as in the basic model. Once
the u-type manager underperforms in the second period there is now a new possibility to
set a high water mark, if the manager succeeds in the third period. In this state the new
value of assets under management equals to V3 = u2 d = (1 + RH )2 (1 + RL ). The new
high water mark is set if V3 > V1 , this condition is satisfied for
RL + RH (1 + RL ) > 0.
(A3)
The second-period return of V2 = ud = (1 + RH )(1 + RL ) is also a new high water mark
relative to V0 = 1, if (A3) is still satisfied. Consider now the d-type manager: after
the truthful reporting V1 = d or overeporting V1 = 1 the new high water mark is set if
V2 = ud > 1 and, thus, (A3) is satisfied. The third high water mark is possible, if the
d-type manager is also successfull in the third period by generating V3 = u2 d, which value
is larger than the last high water mark V2 = ud.
Investors’ beliefs and managerial incentives for misreporting of her first-period return
do not change compared to the basic model. If the management compensation contract
specifies the period performance fee f , the modified thresholds42 , that are crucial for
42
For proofs see Appendix B.5.
75
determining the Perfect Bayesian Equilibria, are
u
ηnew
=
1 − λ2 pεud
pε
⇔
λu2 new =
1 − pεη
pεud
and pε ≥
1
.
1 + λ2 ud
(17)
The new threshold for λ1 is given by
λ∗1 new :=
εd
.
1+εd(1+d)+pεd(1−λ2 )(u−dε)+εd2 λ2
If the management compensation contract specifies the period performance fee with a
high water mark provision h, the Perfect Bayesian Equilibria remain also alike in the
basic model with the following thresholds:
u
ηenew
=
u(1−d)(1−pε)−λ2 pεu(ud−1)
pε(u−1)
76
⇔
eu new = u(1−d)(1−pε)−pεη(u−1)
λ
2
pεu(ud−1)
and pε ≥
u(1−d)
.
u(1−d)+λ2 u(ud−1)+u−1
We now can better analyse how the change in expected returns affect managerial optimal
reporting policy. Recall, that if the contract cpecifies period performance fee, the strategy
u
, 1]. Analogously, if the contract specifies
profile pooling (1,1) is a PBE for η ∈ [ηnew
performance fee with high water mark provision, the strategy profile pooling (1,1) is a
u
, 1].
PBE for η ∈ [e
ηnew
u
We begin with numerical examples that provide some intuition about the changes in ηnew
u
for varying values on pε, λ2 , RH and RL .
and ηenew
u
u
as functions of λ2 and pε.
and ηenew
a) Numerical Example: ηnew
Suppose the parameters to be set to the following values: RH = 5%, which means that the
fund’s positive gross return is set to the value u = 1.05, and RL = −3%, which means that
the fund’s negative gross return is set to the value d = 0.97. The withdrawal probability
in the second period λ2 can take all values between 0 and 1. Use the result from above
(17) to see that the values for pε are not permitted to be to small, more precise, it holds:
pε ∈ [0.45, 1].
u
u
ηnew
and ηenew
as functions of λ2 and pε are presented in Figure 13. On the vertical axis
is shown the analytical difference between the probability weighted values of additional
77
u
capital contribution if the contract specifies period performance fee ηnew
and performance
u
fee with the high water mark provision ηenew . If we additionaly consider that η ≤ 1, it
is obviously in our example that PBE pooling(1,1) exists for all values pε ∈ [0.45, 1],
u
) if the contract specifies performance fee with the
λ2 ∈ [0, 1] (above the graph of ηenew
high water mark provision. If the contract specifies period performance fee (compare the
u
), the PBE pooling(1,1) does not exist for high values on pε and λ2 . In
graph of ηnew
general, the example shows the same result like the basic model (compare Proposition 9),
that managerial incentive for misreporting increases with increasing values on pε and λ2 .
43
u
u
as functions of RH and RL .
and ηenew
b) Numerical Example: ηnew
u
u
Consider now ηnew
and ηenew
as functions of RH and RL . The remaining parameters are set
to the following values: the withdrawal probability λ2 in t = 2 equals 30%, the decreased
success probability pε equals 60%. We assume additionally the positive net return, RH ,
varying between 0 and 30% (u ∈ [1.0, 1.3]), the negative net return, RL , between −30%
and 0 (d ∈ [0.7, 1.0]). Figure 14 demonstrates the numerical example. Remarkable is the
u
sensitvity of the function ηenew
(RH , RL ).
43
For general proof see Appendix B.4
78
4.8
Concluding Remarks
In this chapter we compared contracts that pay a performance fee to the hedge fund
manager whenever the reported value of the fund’s assets increases relative to that at the
beginning of a period (period performance fee) versus a performance fee that is paid only
if the reported asset value of the fund exceeds its historic maximum (performance fee
with high water mark). We can show that standard features in hedge fund management
contracts can have a profound impact on income smoothing, because they affect the
manager’s benefits of reporting income streams with low volatilities. When reporting the
value of their funds’ assets, hedge fund managers that maximize expected fee payments
take into account both the performance fee generated by the report and the expected
fund flow resulting from the report.
The main result shows that managers of funds with performance fees with high water
marks have stronger incentives for both, underreporting positive actual returns and overreporting negative actual returns when doing so prohibits outflows.
The smoothing of hedge fund returns requires that not only funds with poor performance
overreport the value of their portfolio but also that funds with solid performance are
willing to pool with them. The latter is the case when the expected increase in fee
income of reporting solid performance todayis outweighed by the negative effects on fee
levels of potentially having to report poor performance in the future. We show that this
relationship holds more commonly for fund managers whose performance fees are paid
only if the reported value of the fund’s holdings exceeds its historic maximum.
The smoothness of the reported fund returns typically leads investors to understate both
the true volatility of these strategies and their correlation with traditional asset classes.
As a concequence, the investors are unable to detect suitable methods for performance
and risk measurement.
79
5
General Conclusion
This thesis compares two different types of performance fee structures in hedge fund
management compensation contracts under information asymmetry. We differ between
two incentive structures stipulated between the hedge fund manager and the investors:
firstly, fees based solely on the performance during the preceding period and, secondly,
fees based on the performance relative to the historical fund value maximum - the high
water mark. We find that the type of the contract fundamentally influences both, the
manager’s and the investors’ optimal incentive structures.
In the first theoretical model we assume the hedge fund manager to be better informed
about the future fund profitability than the investors. We assume, additionally, that the
fund manager’s incentives to close the hedge fund are not necessarily aligned with those
of the investors. The hedge fund management contract that leads to the efficient closing
should specify low expected fees in circumstances in which fund closure may be efficient.
This is typically the case when the recent performance has been poor. We develop the
argument about the efficient fund closing and can further show that the compensation
arrangement between the hedge fund manager and the investors’ optimally includes the
high water mark provision opposed to the contract which employs period performance
fee.
The crucial benefit of the high water mark provision contract is that it reduces the manager’s expected performance fees in a state of poor realized returns. This enhances the
manager’s incentives to efficiently close the fund before realizing potential losses. But
also for the investors a compensation contract that specifies a high water mark provision
is beneficial, since they anticipate the manager’s more efficient fund closing. Given this
anticipation we can show that the investors have an increased willingness to provide capital to the hedge fund. However, in periods of poor fund performance the investors may
refuse to withdraw their investment from the fund.
Finally, in the context of the optimal closing policy, the efficient incentive effects for high
water mark contracts have their limits. Since the manager does not fully participate in
value gains, under some conditions she may close the fund in states where it has still
intact prospects.
The second main chapter of this thesis models a three period theoretical framework where
managers report about their earnings, which can result in return smoothing. There are
two different types of managers in the market who either realize positive or negative firstperiod returns. Depending on the realized first-period return the managers learn about
80
their type. We assume this to be hidden information, observable only to the manager.
This scenario sets the manager into the position to either truly report or misreport her
actual earnings.
We explicitly model managerial incentive problems contingent on the choice of the compensation contract and recognize that the high water mark provision contract is more
vulnerable to managerial return smoothing, compared to the contract with period performance fee, and therefore can imply inefficiencies on the investors’ side. The reason for
this is that both types of managers have stronger incentives to misreport their first-period
return when they receive fees based on value gains relative to the historic fund maximum.
In line with current empirical literature we also find that managers under high water mark
contracts have stronger incentives to overreport negative returns, when doing so prohibits
subsequent outflows. Consequently, in case the manager reports smoothed reported fund
returns, the investors have to accept the inefficiency, which results in slow learning about
the manager’s quality and therefore the fund’s prospects.
An interesting implication of this result is that the smoothed reported fund returns in turn
influence the investors’ withdrawal policy. Since the investors are unable to learn about
the manager’s quality and the fund’s prospects, they cannot efficiently decide whether to
withdraw their capital or to make an additional investment.
The effects of high water marks in hedge fund management contract design may not only
be restricted to the factors covered in this thesis, such as efficient fund closing or return
smoothing. Further research may also take into account other influencing factors. For
instance high water marks may also have material implications on the manager’s incentives
concerning portfolio choice or redemption policy.
81
Appendix A
We use the following notation for all proofs:
The indicator
( function of a subset A of a some set X is a function IA : X → {0, 1} defined
1 if x∈A
as IA (x) :=
0 if x∈A.
/
φI := p2RH(1 + RH)2 η+(1−p)(1 +RL)(RL I+RH(1+RL)) pθη+(p−ε)(1−θ)(η−ε(RH −RL))
2
3
H
H 2
L
H
L 2 2
ϕI := p (R (1+R )) +(1−p)(R I+R (1+R )) p θ+(p−ε) (1−θ)
ξ := p(1 + RH )2 + (1 − p)(1 + RL )2
H
H
For parameters RH > 0, RL ∈ ( −pR
, −(p−ε)R
),
1−p
1−p+ε
is straightforward that φI, ϕI, ξ > 0.
44
p ∈ (0, 1), ε ∈ (0, p) and θ ∈ (0, 1) it
A.1 Proof of Proposition 1. Consider a contract with the period performance fee rate
a > 0 and a contract with a high water mark provision and performance fee rate e
a > 0,
so that the following condition is satisfied:
RH (1 + RL )
e
a=a L
.
(R + RH (1 + RL ))
|
{z
}
>1,
sinceRL <0
Note, that the performance fee rate with a high water mark provision e
a is strictly larger
than the corresponding period performance fee rate a in this setting. The manger’s closing
−
−
threshholds in state L− are equal in both regimes: ω L (a) ≡ ω L (e
a), and also in state L◦ :
◦
ω L (e
a) = a
RH (1 + RL )
L
H
L
H
L
L◦
p(R
+
R
(1
+
R
))
=
apR
(1
+
R
)
=
ω
(a).
(RL + RH (1 + RL ))
Calculating the corresponding closing thresholds in state H yields:
RH (1 + RL )
> ω H (a)
ω (e
a) = apR (1 + R ) L
|
{z
} (R + RH (1 + RL ))
|
{z
}
=ω H (a)
H
H
H
>1
44
Follows from the assumptions (A1) and (A2).
82
Using (1) and comparing the fund’s expected surplus under the contract with the period
performance fee a and the contract with the performance fee with a high water mark
provision e
a yields:
◦
−
◦
−
S(ω H , ω L , ω L ) (e
a) ≥ S(ω H , ω L , ω L )(a)
H
H
H
⇔ (1+R )η F (ω (e
a))−F (ω (a)) +(1−F (ω H (e
a)))E(ω|ω ≥ ω H (e
a))−(1−F (ω H (a)))E(ω|ω ≥ ω H (a)) ≥ 0
Z ωmax
Z ωmax
H
H
H
ωf (ω)dω ≥ 0
ωf (ω)dω −
⇔ (1+R )η Pr(ω (a) ≤ ω < ω (e
a)) +
{z
}
|
ω H (a)
ω H (e
a)
>0,
ω H (a)<ω H (e
a)
since
Z
ω H (e
a)
⇔
(1+RH)η − ω f (ω)dω > 0
ω H (a)
The last integral is positive for all ω ∈ [0, ωmax ] with ωmax = (1+RH)η.
A.2 Proof of Proposition 2. Rewrite the equality (2)
◦
H
L◦
S(ω , ω , ω
L−
−
ωH
ωL
ωL
) = η+p
(1+RH)η+(1−p)θ
(1+RL)η+(1−p)(1−θ)
(1+RL)(η−ε(RH−RL))
ωmax
ωmax
ωmax
◦
−
Using definitions for ω H , ω L and ω L we receive:
◦
−
a2I
IφI
+ 2ωmax
.
S(ω H (a), ω L (a), ω L (a)) = η + ωamax
The expected surplus-function, as a parabola, is twice continuously differentiable at aI.
The first derivative of the function equals to zero at the extrema:
(
e
a∗ if I ≡ 1
φI
aI =
:=
ϕI
a∗ if I ≡ 0
The second order condition yields
◦
−
d2 S(ω H , ω L , ω L )
−1 3 H
H 2
L
H
L 2 2
2
p (R (1+R )) +(1−p)(R I+R (1+R )) (p θ+(p−ε) (1−θ))
=
daI
ωmax
and is constant and negative for all parameters in the domain of the definition. This
shows that the expected surplus has a unique global maximum at aI.
Because the investor must break even, a maximal possible amount that can be collected
H
in kI with respect to the investor’s participation constraint (1) is equal to: kI =
η − aIpR .
a∗ , e
k ∗ and
Thus, the optimal contract with a high water mark provision is given by e
optimal contract with period performance fee is given by (a∗ , k ∗ ) .
83
Lemma 1 The performance fee rate a∗ in the optimal contract with period performance
fee is smaller than the performance fee rate e
a∗ in the optimal contract with high water
mark provision e
a∗ . The management fee k ∗ is larger than e
k∗.
Proof: We show that e
a∗ =
φ1
ϕ1
>
φ0
ϕ0
= a∗ . Recall that ϕI > 0, so we have
0 < φ1 ϕ0 − φ0 ϕ1
ε
ε H
ε 2
H
L
L
⇔ (R +2R (1+R )) θ+(1− ) (1−θ)) > R (1+R ) θ+(1− )(1−θ)(1 − (R −R ))
|
{z
}
p
p
η
L
H
L
>RH(1+RL)
ε
ε
ε
⇔ θ+(1− )2 (1−θ) >θ+(1− )(1−θ)(1 − (RH−RL))
p
p
η
ε
ε H
⇔ (1− ) > (1 − (R −RL))
p
η
L
⇔ 0 R → T rue
With e
a∗ > a∗ we can argue: e
a∗ pRH > a∗ pRH ⇔ η − e
a∗ pRH < η − a∗ pRH ⇔ e
k∗ < k∗.
For kI ≡ 0, we have 0 = η − aIpRH ⇔ aI = pRηH , thus the restriction on management fee,
k ≥ 0, is binding iff the optimal performance fee rate exceeds the value pRηH . In the next step we show that from the ex ante perspective, the optimal contract with a
high water mark provision leads to strictly lower closing probability in state H. Pr(state
H) is equal to1 − F (ω H (a)), hence it is sufficient to show that:
H (a∗ )
H (e
a∗ )
F (ω H (a∗ )) = ωωmax
< ωωmax
= F (ω H (e
a∗ )) ⇔ ω H (a∗ ) < ω H (e
a∗ ):
ω H (e
a∗ ) ≡ ω H (a∗ ω H (e
a∗ ) ≡ e
a∗ pRH (1 + RH ) > a∗ pRH (1 + RH ) ≡ ω H (a∗ ).
We show now that from ex ante perspective, the optimal contract with a high water mark
provision leads to higher closing probabilities in states L◦ and L− . Proof for state L◦ :
◦
◦
ω L (a∗ ) > ω L (e
a∗ )
⇔ a∗ pRH (1 + RL ) > e
a∗ p(RL + RH (1 + RL ))
⇔ φ0 ϕ1 RH (1 + RL ) > φ1 ϕ0 (RL + RH (1 + RL ))
RL
⇔ RH (1 + RL ) (φ0 ϕ1 − φ1 ϕ0 ) > φ0 ϕ1 |{z}
| {z }
{z
}|
{z
}
|
>0
>0
>0
<0
Analogously for state L− :
−
−
ω L (a∗ ) > ω L (e
a∗ )
a∗ (p − ε)RH (1 + RL ) > e
a∗ (p − ε)(RL + RH (1 + RL ))
84
use the same proof as in state L◦ . A.3 Proofs of Corollaries 1, 2 and 3.
◦
◦
A.3.1 Proof of Corollary 1: With proof of Proposition 2 we have: ω L (a∗ ) > ω L (e
a∗ ).
Thus, the closing probability upon a negative first-period return in state L◦ for the optimal
contract with high water mark provision is larger than the corresponding probability for
◦
◦
a∗ )) > 1 − F (ω L (a∗ )).
the optimal contract with period performance fee: 1 − F (ω L (e
−
−
Analogously we have 1 − F (ω L (e
a∗ )) > 1 − F (ω L (a∗ )) in state L− . Thus, the weighted
closing probability upon a negative first-period return for the optimal contract with high
water mark provision is higher than the corresponing weighted closing probability for the
optimal contract with period performance fee:
◦
−
◦
−
θ(1 − F (ω L (e
a∗ ))) + (1−θ)(1 − F (ω L (e
a∗ ))) > θ(1 − F (ω L (a∗ ))) + (1−θ)(1 − F (ω L (a∗ )))
◦
−
◦
−
⇔ θ F (ω L (a∗ )) − F (ω L (e
a∗ )) + (1−θ) F (ω L (a∗ ) − F (ω L (e
a∗ )) > 0
{z
}
{z
}
|
|
>0
>0
A.3.2 Proof of Corollary 2: In the first step we use Proposition 2 and Corollary 1 to
see: f LH (a∗ ) = a∗ RH (1 + RL ) > e
a∗ (RL + RH(1 + RL)) = f LH (e
a∗ ). Conditional on fund
continuation after a negative first-period return, the investor has a posterior belief to face
◦
◦
−
state L◦ equal to θF (ω L )/(θF (ω L ) + (1 − θ)F (ω L )), that is independent of whether the
contract specifies a high water mark provision or a period performance fee, because
◦
◦
θω L (a∗ )
θω L (e
a∗ )
=
θω L◦ (e
a∗ )+(1−θ)ω L− (e
a∗ )
θω L◦ (a∗ )+(1−θ)ω L− (a∗ )
◦
◦
−
⇔ ω L (e
a∗ )ω L− (a∗ ) = ω L (a∗ )ω L (e
a∗ )
p
p
=
.
⇔
2p − ε
2p − ε
In the second step we compare the expected fund’s second-period after-fee return in both
regimes:
◦
◦
θF (ω L (e
a∗ ))p(V LH − e
a∗ (RL +RH(1+RL)))
θF (ω L (a∗ ))p(V LH − a∗ RH (1+RL ))
>
θF (ω L◦ (e
a∗ )) + (1 − θ)F (ω L− (e
a∗ ))
θF (ω L◦ (a∗ ))+(1−θ)F (ω L− (a∗ ))
⇔ V LH − e
a∗ (RL +RH(1+RL)) > V LH −a∗ RH (1+RL )
⇔ a∗ RH (1+RL ) > e
a∗ (RL +RH(1+RL)).
85
A.3.3 Proof of Corollary 3: The manager’s outside income in period 2 is uniformly
distributed on 0, 1 + RH η . Thus the
contract with the
manager chooses the optimal
φ
high water mark provision given by e
a∗ = ϕ11 , e
k∗ = η − e
a∗ pRH . The management fee
can be rewritten as e
k ∗ = pRH + (1 − p)RL − e
a∗ pRH = pRH (1 − e
a∗ ) + (1 − p)RL . Recall
that e
a∗ depends on RH , so the first derivative of e
k ∗ equals:
de
k ∗ (RH )
d
d
RH ∗ dϕ1
dφ1 H
∗
L
∗
(e
a
=
pR (1−e
a )+ H (1−p)R = p (1 − e
a )+
−
) >0
dRH
dRH
dR
ϕ1
dRH
dRH
{z
}
|
>0
because e
a∗ =
φ1
ϕ1
<1<
dφ1
dRH
dϕ1
1−RH H
dR
1−RH
and
dϕ1
dRH
<
dφ1
.
dRH
For a numerical example see Figure 4.
A.4 Proof of Proposition 3:
In the situation with high uncertainty about the fund’s prospects after low first-period
return associated with low θ, the both performance fee parameter a∗ and e
a∗ are below
the boundary value pRηH . With increase on θ both parameter increase also:
daI(θ) (1−p)ε(RLI+RH(1+RL)) L
H
L
L
H
L
(1+R
)(η+(p−ε)(R
−R
))ϕ
−(2p−ε)(R
I+R
(1+R
))φ
.
=
I
I
|
{z
}
dθ
ϕ2I
>0,
due
to
(RL+RH(1+RL))≥0
That leads to the simultaneously decrease in kI∗ = η − a∗I pRH . For θ equal to 1 the
maximal level on a∗ equals
86
lim a∗ =
θ→1
p2RH(1 + RH)2 η+(1−p)(1 +RL)2 RHpη
η
=
.
p3 (RH(1+RH))2 +(1−p)(RH(1+RL))2p2
pRH
Additionally we conclude that k ∗ = η − a∗ pRH = η −
way:
η
pRH
pRH
= 0. Rewrite e
a∗ in a new
L
ε(R
ε
H 2
L R
L
η p(1+R ) +(1−p)(1+R )(RH +(1+R ))(θ+(1− p )(1−θ)(1−
∗
e
a =
L
pRH
p(1+RH)2 +(1−p)(RRH +(1+RL))2 (θ+(1− pε )2 (1−θ))
|
{z
:=Υ(θ)
H−RL)
η
)) .
}
The value of e
a∗ is equal to pRηH for Υ(θ) ≡ 1. Thus, the critical value θ◦ ∈ (0, 1),
below which the management fee restriction, k ≥ 0, is binding, satisfies the condition:
Υ(θ◦ ) ≤ 1. For Υ > 1 and increasing θ > θ◦ the value of e
a∗ exceeds pRηH and for θ = 1
the performance fee rate with high water mark provision e
a∗ equals to
L
ξ + (1 − p) RRH (1 + RL )
η lim e
a =
.
θ→1
pRH ξ + (1 − p) RRHL ( RRHL + 2(1 + RL ))
|
{z
}
∗
>1,
due
to
(RL +RH(1+RL))≥0
The investor’s participation constraint with higher level of e
a∗ can only be satisfied with
negative value of e
k ∗ (compare Figure 16, first part). With the assumption about the
87
non-negativity of the management fee, e
k ∗ ≡ 0, for values of θ ∈ (θ◦ , 1) we calculate the
new level of performance fee rate e
a∗k=0 using modified45 investor’s participation constraint
(3). The solution can be described as:
p
Ψ2 +4(1−p)(RL+RH(1+RL))2 (p2 θ+(p−ε)2 (1−θ))(1+p(1+RH))ηωmax
=
,
2(1−p)(RL+RH(1+RL))2 (p2 θ+(p−ε)2 (1−θ))
with Ψ := (1−p)(1+RL)(RL+RH(1+RL)) pθη+(p−ε)(1−θ)(η−ε(RH−RL)) −pRH (1+p(1+RH)).
e
a∗k=0
Ψ+
The new value of period performance fee rate e
a∗k=0 does not achieve the maximum of
the surplus-function and is smaller than e
a∗ (compare Figure 16, second part). For θ ≥
θb ∈ (θ◦ , 1) the value of the expected fund’s surplus with respect to the new rate e
a∗k=0 is
smaller than the counterpart surplus in the optimal contract with period performance fee.
The difference increases for θ = 1:
S(e
a∗k=0 ) − S(a∗ ) =
2
L
ξ + (1 − p) RRH (1 + RL )
ηξ
−1
< 0.
2(1 + RH ) ξ + (1 − p) RRHL ( RRHL + 2(1 + RL ))
{z
}
|
<1
Thus, the expected surplus as a function of θ is continuous at θ ∈ (0, 1), and
b and S(e
S(e
a∗ ) − S(a∗ ) > 0 for θ ∈ (0, θ)
a∗ ) − S(a∗ ) < 0 for θ = 1, we have a suffucuent
condition for existence of θb ∈ (θ◦ , 1).
A.5 Proof of Proposition 4.
At first we recalculate the level on performance fee parameter a∗ in the optimal contract
with intermittent redemption by the investor, if the manager’s outside income in period
2 is uniformly distributed on 0, 1 + RH η , and show that the maximal level on a∗ that
the investor is willing to pay to the manager is equal to pRηH . Investor’s participation
constraint is:
H
H
V0 ≤ −k + p − f +F(ω )(p(V
HH
−f
HH
)+(1−p)V
HL
H
)+(1−F (ω ))V
H
+(1−p)V L
a2 p3 (RH )2 (1 + RH )
⇐⇒
− apRH (1 − p(1 + RH )) − η + k ≤ 0
η
q
η 1 + p2 (1 + RH )2 + 2p
(1 + RH )(η − 2k) − (1 − p(1 + RH ))
η
a≤
2p2 RH (1 + RH )
45
For performance fee rate larger than
η
,
pRH
the probability of fund continuation in state H is equal to
one.
88
The value of the performance fee parameter a increases if k decreases, so it is optimal to
set k = 0. That leads to the result
η
a=
.
pRH
(The second analytical value a =
−η
p2 RH (1+RH )
< 0 is not feasible.)
We proof now that the optimal performance fee that maximizes the expected surplus with
respect to the manager’s optimal closing policy is also equal to a∗ = pRηH :
H
H
H
H
H
S (ω) = −1−E(ω)+p F(ω )(1+R )(η+1)+(1−F (ω ))(1+R +E(ω|ω ≥ ω )) +(1−p)V L
⇔ η − E(ω) + p F(ω H )(1+RH)η+(1−F (ω H ))E(ω|ω ≥ ω H )
rd
⇔ η−
p3(RH (1+RH))2
p2RH (1+RH)2 η
(1 − p)ωmax
+a
− a2
−→ M ax
2
ωmax
2ωmax
p2RH (1+RH)2 η
p3(RH (1+RH))2
η
dS(a)
=
−a
= 0 ⇒ a∗ =
da
ωmax
ωmax
pRH
The second order condition leads to
d2 S(a)
da
< 0.
With use of propositions 1 and 2 it is sufficient to show that the expected surplus without
redemptions S(a∗ ), with the choice of the performance fee a∗ = pRηH , is at least as high
the expected surplus with redemptions by the investor S rd (a∗ ). This is equal to the
consideration that the difference of both surpluses is positive:
S(a∗ ) − S rd (a∗ ) = −V L +
◦
◦
◦ +θ F (ω L )V L (η+ 1) +(1−F (ω L )) V L + E(ω|ω ≥ ω L ) +
L−
L
H
L
L−
L
L−
+(1−θ) F (ω )V (η+1−ε(R −R ))+(1−F (ω )) V +E(ω|ω ≥ ω ) > 0
(1+RL )2 2
η(1+RH )
H
L
+ 2
p
θη+(p−ε)
(1−θ)
(η
(p+ε)
−2pε(R
−R
))
>0 ⇔
|
{z
}
2
2p (1+RH)
>0,
with
(A1)
A.6 Proof of Proposition 5: A first we consider the change in the model parameter,
if A > 0.
Optimal Contract With a Period Performance Fee: Denote the performance fee by aA ≥ 0,
aA is charged on the fraction (1 − A). That leads to the following manager’s period
performance fees in each period, in the each of the states :
89
f H = aA (1 − A)RH , f HH = aA (1 − A)RH (1 + RH ) and f LH = aA (1 − A)RH (1 + RL )RH .
Manager needs to consider not only the possible profits after the positive return but
also the possible losses on her fraction A. This implies closing thresholds for ω equal
to (1 + RH )(aA (1 − A)pRH + Aη) = ω H (aA ) in state H, (1 + RL )(aA (1 − A)pRH + Aη) =
−
◦
ω L (aA ) in state L◦ and (1 + RL )(aA (1 − A)(p − ε)RH + A(η − ε(RH − RL )) = ω L (aA )
in state L− .
Optimal Contract With a High Water Mark Provision: Consider now the contract that
specifies a linear performance fee, e
aA ≥ 0, with a high water mark provision. At date 0 the
historic maximum of the fund value V0 = 1 describes the first high water mark, the fee level
is f H = e
aA (1−A)RH . The same applies to V H and therefore f HH = e
aA (1−A)(1+RH )RH .
If the first-period return is negative, the historic maximum of the fund value remains its
initial value V0 = 1. This implies that a high water mark provision is given by f LH
=e
aA (1 − A)(RH (1 + RL ) + RL ) > 0.
The corresponding closing thresholds for ω are defined as follows:
ω H (e
aA ) = e
aA (1 − A)pRH (1 + RH ) + Aη(1 + RH ),
◦
ω L (e
aA ) = e
aA (1 − A)p(RH (1 + RL ) + RL ) + A(1 + RL )η,
−
ω L (e
aA ) = e
aA (1 − A)(p − ε)(RH (1 + RL ) + RL ) + A(1 + RL )(η − ε(RH − RL )).
Recall that the the optimal period performance fee, as calculated in Proposition 2, is
given by aI = ϕφII for A = 0. Using new closing thresholds for ω fund’s expected surplus
(2) can be rewritten as
◦
−
S(ω H , ω L , ω L ) = η +
a2 (1 − A)2
A(2 − A)
aI(1 − A)2
φI − I
ϕI +
Const,
ωmax
2ωmax
2ωmax
where Const = p2(1 + RH)2 η 2 +(1−p)(1 +RL)2(pη2 +(1−θ)(η−ε(RH −RL))2 ). The first and
the second order conditions lead to the optimal performance fee parameter choice given
by:
φI
a∗I = .
ϕI
With the proposition 2 it holds that e
a∗ > a∗ , and also the fund’s expected surplus with
the choice of e
a∗ is larger than the corresponding surplus with the choice of a∗ .
Let us calculate the management fee e
kA∗ for A > 0. Using e
a∗ given by e
a∗ =
investor’s participation constraint (1) yields
90
φ1
ϕ1
and the
η−e
a∗ pRH
Ae
a∗ φ1
A
∗
e
kA>0
=
−
+
Const.
(1 − A)
(1 − A)ωmax ωmax
∗
Note that for A = 0 the above equation becomes e
kA=0
= η−e
a∗ pRH and presents the
management fee that the manager chooses in the optimal contract without wealth contribution, that was calculated in proposition 2.
◦
−
We now show that increasing in A > 0 results in increasing S(ω H , ω L , ω L ) and e
kA∗ .
Examining the term
◦
−
dS(ω H , ω L , ω L )(e
a∗ )
(1 − A) φ21 (1 − A)
(1 − A) φ2 =−
+
Const =
Const − 1
dA
ωmax ϕ1
ωmax
ωmax
ϕ1
{z
}
|
>0
◦
we observe that
−
dS(ω H ,ω L ,ω L )
dA
> 0. Identical steps lead to
∗
de
kA
dA
> 0 since
φ21
η−e
a∗ pRH Const
+
−
=
ωmax
ϕ1 ωmax
(1 − A)2
1 η−e
a∗ pRH
φ21 +
> 0
Const
−
ωmax
ϕ1
(1 − A)2
{z
}
|
| {z }
>0
∗
=e
kA=0
>0
Note that the term Const −
Const −
φ21
ϕ1
is positive since
φ21
=
ϕ1
(1 − θ)(1 − p)ε2 (RH (1 + RL )RL )2 θ(1 − p)(1 + RL )2 + p(1 + RH )2
=
> 0. ϕ1
Proof of Proposition 5: Recall that the optimal contract with the high water mark
provision in case A = 0 is given by (e
a∗ , e
k ∗ ) with e
a∗ =: e
a∗A=0 = ϕφ11 and
∗
e
k ∗ =: e
kA=0
= η − ϕφ11 pRH and is positive. We also use the previous denotation for the
fund’s expected return η = pRH + (1 − p)RL > 0, where RL < 0. The continuation
H
H )+Aη(1+RH )
.
probability in state H is given by F (ω H (e
aA )) = ea(1−A)pR (1+R
ωmax
φ
The continuation probability in state ω
Rearranging gives:
H
H
is given by F (ω (e
aA )) =
91
(1+RH)( ϕ1 (1−A)pRH+Aη)
1
ωmax
.
H
a∗A )) =
F (ω L (e
φ1
φ1
A
pRH (1 + RH ) + ωmax
(1 + RH ) (η − pRH )
ϕω
ϕ1
| 1 max {z
}
|
{z
}
=F (e
a∗A=0 )>0
=e
k∗ >0
Identical consideration we have in states L◦ and L−
φ1
φ1
φ1
◦
A
F (ω L (e
a∗A )) =
p(RL + RH (1+RH))+ωmax
(1+RL) (η− pRH) − pRL
ϕω
ϕ1
ϕ1
| 1 max
|
{z
}
{z
} | {z
}
=F (e
a∗A=0 )>0
−
F (ω L (e
a∗A )) =
=e
k∗ >0
>0
φ1
(p−ε)(RL+RH(1+RH)) −
ϕω
| 1 max
{z
}
=F (e
a∗A=0 )>0
A
− ωmax
φ
1
(p−ε)(RL+RH (1+RH))+(1+RL)(η−ε(RH−RL)) .
ϕ
| 1
{z
}
>0
92
Appendix B
Recall for all following proofs, that the critical belief below which withdrawals can occur
in t = 1 is µ1 ∈ (0, µ). The critical belief above which withdrawals of funds can occur in
t = 2 is µ2 ∈ (µ, 1).
B.1 Proof of Poposition 7
At first we prove that in state separating(u,d) there exist only one state in which the
u-type manager has an incentive to report v1 = 1.
If u-type manager deviates from the strategy profile separating(u,d) by reporting v1 = 1
in t = 1, we have to consider all possible investors’ beliefs α(u|1) in t = 1 and α(u|1, 1) in
t = 2 (if the second-period reported value of assets under managemenst will be v2 = 1).
The u-type manager’s expected fee payments, contingent on all possible investors’ beliefs
in t = 1, 2 are described as follows:
E u f |1, α(u|1), α(u|1, 1) =















E u f |u, 1 −f (1−p)(u−1)
u
E f |1, µ, α(u|1, 1) < µ1
(1−λ1 )E u f |1, α(u|1) ≥ µ1
(1−λ1 )E u f |1, µ, α(u|1, 1) < µ2
f or
α(u|1) ≥ µ1 and
α(u|1, 1) ≥ µ2
f or
α(u|1) ≥ µ1 and
α(u|1, 1) < µ2
f or
α(u|1) < µ1 and
α(u|1, 1) ≥ µ2
f or
α(u|1) < µ1 and
α(u|1, 1) < µ2 .
The value E u (f |u, 1) is described as (4), the value E u f |1, α(u|1) ≥ µ1 as (6) and the
u
value E f |1, µ, α(u|1, 1) < µ2 as (10). Thus, it appears that the u-type manager can
only have an incentive to deviate towards underreporting of her first-period return, if the
investors beliefs are α(u|1) ≥ µ1 in t = 1 and α(u|1, 1) < µ2 in t = 2. Exactly this beliefs
investors have in the strategy profile pooling(1,1). Based on the previous consideration
we can conclude, that the u-type manager has an incentive to underreport her first-period
return for η ≥ η u or, equaivalently, for λ2 ≥ λd2 , if investors beliefs are as described in the
strategy profile pooling(1,1).
Analogously, given the strategy profile separating(u,d), we consider the d-type manager’s
expected fee payments by deviating towards misreporting and by announcing v1 = 1,
contingent on all possible investors’ leliefs in t = 1, 2:
E f |1, α(u|1), α(u|1, 1) =
d
93















E d f |1, α(u|1) ≥ µ1
d
E f |1, µ, α(u|1, 1) < µ2
(1−λ1 )E d f |1, α(u|1) ≥ µ1
(1−λ1 )E d f |1, α(u|1, 1) < µ2
f or
α(u|1) ≥ µ1 and
α(u|1, 1) ≥ µ2
f or
α(u|1) ≥ µ1 and
α(u|1, 1) < µ2
f or
α(u|1) < µ1 and
α(u|1, 1) ≥ µ2
f or
α(u|1) < µ1 and
α(u|1, 1) < µ2 .
The value E d f |1, α(u|1) ≥ µ1 is described as (7), the value E d f |1, µ, α(u|1, 1) < µ2
d
as (11) and the value E f |1, α(u|1, 1) < µ2 as (9). Comparing the d-type manager’s
expected fee payments shows that the d-type manager can only have an incentive to
deviate towards overreporting of her first-period return if investors’ beliefs are α(u|1) ≥ µ1
in t = 1 and α(u|1, 1) < µ2 in t = 2. This is exactly what the strategy pooling(1,1)
describes. The d-type manager has an incentive to overreport her first-period return for
η ≥ η d or, equaivalently, for λ2 < λd2 , if investors’ beliefs are as in the strategy profile
pooling(1,1).
In summary, the strategy profile separating(u,d) is an PBE for
(
η u f or λ1 > λ∗1
0≤η≤
η d f or λ1 ≤ λ∗1 .
B.2 Proof for the Nonexistence of separating(u,1) and separating(1,d) Equilibria in the Contract With the Period Performance Fee.
Consider at first the strategy profile separating(u,1). Given the contract specifies the
periodperformance fee, the d-type manager’s expected fee payment in this state is given
by E d f |1, α(u|1) < µ1 (9). Deviating towards thruthful first-period reporting of v1 = d
yields for the d-type manager an expected fee payment of E d (f |d, 0) given by (5). In both
cases, independent on whether the d-type manager reports her true first-period return or
overreports it, the investors have always a belief of 1 on the d-type in the predescribed
strategy profile.
Comparing the both expected fee payments shows, that the d-type manager has never an
incentive for underreporting, given investors’ beliefs in the strategy profile separating(u,1):
E d f |1, α(u|1) < µ1 > E d (f |d, 0) ⇔ 0 > pε(1−d) −→ @,
thus, the separating strategy profile separating(u,1) can not be supported as a PBE,
independent on whetrer the u-type manager has an incentive to deviate from the strategy
profile or not.
94
Analogously, consider the separating strategy profile separating(1,d). According to the
definition of this strategy profile, by observing the reported return of v1 = 1 the investors
believe with probability 1 to face an u-type manager. By observing v1 = d they believe
with probability 1 to face a d-type manager. Theu-type manager’s
expected fee payment
u
in strategy profile separating(1,d) is given by E f |1, α(u|1) ≥ µ1 (6). Deviating towards
thruthful fisrt-period reporting of v1 = u generates an expected fee payment of E u (f |u, 1)
(4) for the u-type manager. Comparing both terms shows, that the u-type managerhas
never an incentive for underreporting, given the investors’ beliefs in the strategy profile
separating(1,d):
E f |1, α(u|1) ≥ µ1 > E u (f |u, 1) ⇔
u
p(u2 − 1) > (u − 1) + p(u2 − u) ⇔ p > 1 −→ @,
thus, the strategy profile separating(1,d) can not be supported as a PBE, independent on
whetrer the d-type manager has an incentive to deviate from the strategy profile or not.
B.3 Search for the Remaining PBE Equilibria if the Contract Specifies the
Performance Fee With the High Water Mark Provision.
At first we proof, that the strategy profile separating(u,d) is a PBE.
If the u-type manager reports v1 = 1 in the strategy profile separating(u,d), investors’
beliefs α(u|1) in t = 1 and α(u|1, 1) in t = 2 (if the second- period reported value of assets
under managemenst is V2 = 1) have to be specified. The u-type manager’s expected fee
payments, contingent on investors’ beliefs are described as follows:
1) E u h|1, α(u|1) ≥ µ1 , α(u|1, 1) ≥ µ2 = hp u2−1+p(u−1)(u2+η) +h (1−p)(1−λ2 )pε(u−1)
< E u (h|u, 1), since 1 ≥ (1−λ2 )pε.
For the given investors’ beliefs, α(u|1) ≥ µ1 in t = 1 and α(u|1, 1) ≥ µ2 in t = 2, the u-type
manager reports v1 = u, since the condition:
E u (h|u, 1) ≥ E u h|1, α(u|1) ≥ µ1 , α(u|1, 1) ≥ µ2 ⇔ 1 ≥ (1−λ2 )pε
is always satisfied for each set of parameters.
2) E h|1, α(u|1) ≥ µ1 , α(u|1, 1) < µ2
u
95
described as in (15). For η ≥ ηeu the u-type manager has an incentive for underreporting.
3) E u h|1, α(u|1) < µ1 , α(u|1, 1) ≥ µ2 =
2
2
= h(1−λ1 ) p(u −1+p(u−1)(u +η))+(1−p)(1−λ2 )pε(u−1) .
The u-type manager’s expected fee in this case is even smaller than in 1). Thus, for given
investors’ beliefs α(u|1) < µ1 in t = 1 and α(u|1, 1) ≥ µ1 in t = 2, the manager has never
an incentive to underreport her first-period return.
4) E h|1, α(u|1) < µ1 , α(u|1, 1) < µ2 =
u
= h(1−λ1 ) p(u2 −1+p(u−1)(u2 +η))+(1−p)pε(1 + η)(u−1) .
Consider the strategy profile separating(u,d). The d-type manager’s expected fee payments by deviating towards announcing v1 = 1, contingent on all possible investors’
beliefs in this state are given by:
1)
2)
3)
4)
E h|1, α(u|1) ≥ µ1 , α(u|1, 1) ≥ µ2 = hp2 ε(1−λ2 )(u−1) < E d (h|d, 0).
d
E h|1, α(u|1) ≥ µ1 , α(u|1, 1) < µ2 = hp2 ε(1+η)(u−1) ≥ E d (h|d, 0).
E d h|1, α(u|1) < µ1 , α(u|1, 1) ≥ µ2 = h(1−λ1 )p2 ε(1−λ2 )(u−1) < E d (h|d, 0).
E d h|1, α(u|1) < µ1 , α(u|1, 1) < µ2 = h(1−λ1 )p2 ε(1+η)(u−1) ≡ E d (h|d, 0).
d
In summary, for pε ≤ 0.5 and η ∈ [0, η u ] the state separating(u,d) is an PBE.
Proof of Proposition 8.
Consider now the separating strategy profile separating(u,1). In this state in spite of
observed misreporting in the first period the investors are completely informed about the
menager’s types. Thus, by observing the reported value of V1 = 1 the investors’ have a
believ of 0 to face the u-type manager. The d-type manager’s expected pees are given by:
E d h|1, α(u|1) < µ1 , α(u|1, 1) < µ2 = h(1−λ1 )p2 ε(1+η)(u−1) ≡ E d (h|d, 0).
Obviously, the d-type manager receives the same expected fees by thrutful reporting of
v1 = d and overreporting of her first-period return in the given strategy profile. Using
96
the previous consideration, we know that the u-type manager never deviates towards
underreporting in the given strategy profile:
u
2
2
E h|1, α(u|1) < µ1 , α(u|1, 1) < µ2 = h(1−λ1 ) p(u −1+p(u−1)(u +η))+(1−p)pε(1+η)(u−1) .
Consider as last the separating strategy profile separating(1,d). In this state the u-type
manager underreports and the d-type manager thruthfully reports her first-period return.
By observing the reported value of v1 = 1 in t = 1 investors’ belief onu-type manager is 1. Therefore, the u-type manager’s expected fees in this state, E u h|1, α(u|1) ≥
µ1 , α(u|1, 1) ≥ µ2 is always smaller than her expected fees by thruthful reporting, E u (h|u, 1).
Hence, the strategy profile separating(1,d) can not be supported as a PBE. B.4 Proof of Corrolaries 4, 5 and Proposition 9
B.4.1 Proof of Corollary 4: If the contract specifies the period performance fee, we
have the following consideration:
dη s
d(1 − λ2 pε)/pε
1
=
= − 2 < 0.
dpε
dpε
pε
With increasing values of pε the threshold η u desreases. Consequently, the manager’s
incentive condition for misreporting is satisfied for larger numerical interval η u , 1 and
u
smaler values of η u . Analogously we have: dη
= −1 < 0.
dλ2
If the contract specifies performance fee with the high water mark provision, the following
condition are satisfied:
d(1 − pε)/pε
1
de
ηu
=
= − 2 < 0.
dpε
dpε
pε
B.4.2 Proof of Proposition 9: At first we use the Proposition 6 to define the restricted
domain D1 in which the strategy profile pooling(1,1) is a PBE, if the management contract
specifies the period performance fee:
n
D1 := pε, η, λ1 , λ2 | pε ∈ (
o
1
, 1], λ1 > λ∗1 , λ2 ∈ [0, 1], η ∈ (η u , 1] .
1 + λ2
As second we use the Proposition 8 to define the restricted domain D2 in which the
strategy profile pooling(1,1) is a PBE, if the management contract specifies performance
fee with the high water mark provision:
97
n
o
1
D2 := pε, η, λ1 , λ2 | pε ∈ ( , 1], λ1 ∈ (0, 1), λ2 ∈ [0, 1], η ∈ (e
η u , 1] .
2
We consider additionally that ηeu ≤ η u ⇔ (η u , 1] ⊂ (e
η u , 1] to see that D1 ⊂ D2 . In
summary, there exist a non-empty (for numerical examples compare Table 4) subset of
parameters
o
n
1
1
u u
), λ1 ∈ (0, 1), λ2 ∈ [0, 1], η ∈ (e
η ,η )
D2 \ D1 = pε, η, λ1 , λ2 | pε ∈ ( ,
2 1 + λ2
for which pooling(1,1) equilibrium occurs only when the hedge fund management contract specifies performance fee with the high water mark provision. (To illustrate the
consideration in a simplified way compare Figure 17)
B.4.3 Proof of Corollary 5: If the contract specifies the period performance fee, the
1
pooling equilibrium occurs for pε ≥ 1+λ
(shown in Proposition 6). If the contract specifies
2
performance fee with the high water mark provision, the pooling equilibrium occurs for
1
> 12 . pε ≥ 0.5 (as shown in Proposition 8). For λ2 < 1 it is always satisfied 1+λ
2
B.5 New Thresholds in the Extended Model Version
We begin with the contract that specifies the period perfoemance fee. In the strategy
profile pooling(1,1) the u-type manager’s expected fee payment is described as follows:
E f |1, µ, α(u|1, 1) < µ2 =
u
98
= f p u2 −1+p(u−1)(u2 +η)+(1−p)ε(u−1)(ud+η) .
By reporting v1 = u the u-type manager credibly signals her type to the investors. In this
state the u-type manager’s expected fees are given by:
E u (f |u, 1) = f (u−1)+f p u2 −u+p(u−1)(u2 +η)+(1−p)(1−λ2 )εud(u−1) .
The u-type manager has an insentive to deviate towards thruthful first-period reporting
in t = 1 for the given investors’ beliefs of 1 on the strong-type manager for:
E u (f |u, 1) ≥ E u f |1, µ, α(u|1, 1) < µ2 ⇔
η≤
1 − pελ2 ud
u
:= ηnew
pε
⇔
λ2 ≤
1 − pεη
:= λu2 new .
pεud
In order to receive values η ∈ [0, 1], it should be satisfied:
pε ≥
1
.
1+λ2 ud
d-type manager’s expected fee payment in the strategy profile pooling(1,1) is given by:
E d f |1, µ, α(u|1, 1) < µ2 = f pε p(u−1)(ud+η)+(1−pε)(1−λ2 )d2 (u−1) .
Given strategy profile pooling(1,1). Thruthful reporting of v1 = d by the d-type manager
leads to a credible signal about her type. The expected fees of d-type manager in this
state is given by
d
2
E (f |d, 0) = f (1−λ1 )pε d(u−1)+p(ud+η∆)(u−1)+(1−pε)(1−λ2 )d (u−1) .
Comparing the d-type manager’s expected fee payments by thruthful reporting of v1 = d
and overreporting of her first-period return by announcing v1 = 1 leads to the following
thresholds:
E d (f |d, 0) ≥ E d f |1, µ, α(u|1, 1) < µ2 ⇔
1 − λ1
p(ud+η) + (1−pε)(1−λ2 )d2
≥
λ1
d
2
(1−λ1 )d−λ1 pud+(1−pε)(1−λ2 )d
⇔ η≤
:= η∆dnew
λ1 p
99
(1−λ1 )d−λ1 (p(ud+η)+(1−pε)d2 )
.
⇔ λd2 new =
λ1 (1−pε)d2
Analogously, if the contract specifies performance fee with the high water mark provision,
the u-type manager’s expected fee payment in the strategy profile pooling(1,1) is given
by:
u
E h|1, µ, α(u|1, 1) < µ2 =
= h p(u2 −1)+p2 (u2 +η)(u−1)+(1−p)(ud−1)+(1−p)pε(ud+η)(u−1) .
By deviating towards thruthful reporting of v1 = u the u-type manager’s expected fee
payment is given by:
E u (h|u, 1) = h(u−1)+hp u2 −u+p(u−1)(u2 +η)+(1−p)(1−λ2 )εu(ud−1) .
Comparing the u-type manager’s expected fees in both regimes leads to the following
thresholds:
E u (h|u, 1) ≥ E u h|1, µ, α(u|1, 1) < µ2 ⇔
η≤
u(1−d)(1−pε)−λ2 pεu(ud−1)
u
:= ηenew
pε(u−1)
⇔ λ2 ≤
u(1−d)(1−pε)−pεη(u−1)
eu new .
:= λ
2
pεu(ud−1)
In order to receive values η ∈ [0, 1], it should be satisfied:
ε≥
u(1−d)
.
u(1−d)+λ2 u(ud−1)+u−1
100
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