Risk Simulator - Russian User Manual (2012)

RISK SIMULATOR
User Manual
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REAL OPTIONS VALUATION, INC.
This manual, and the software described in it, are furnished under license and may only be used or copied in accordance
with the terms of the end user license agreement. Information in this document is provided for informational purposes
only, is subject to change without notice, and does not represent a commitment as to merchantability or fitness for a
particular purpose by Real Options Valuation, Inc. No part of this manual may be reproduced or transmitted in any form
or by any means, electronic or mechanical, including photocopying and recording, for any purpose without the express
written permission of Real Options Valuation, Inc. Materials based on copyrighted publications by Dr. Johnathan Mun,
Ph.D., MBA, MS, BS, CRM, CFC, FRM, MIFC, Founder and CEO, Real Options Valuation, Inc., and creator of the
software. Written, designed, and published in the United States of America. Microsoft® is a registered trademark of
Microsoft Corporation in the U.S. and other countries. Other product names mentioned herein may be trademarks
and/or registered trademarks of the respective holders.
© Copyright 2005-2012 Dr. Johnathan Mun. All rights reserved.
Real Options Valuation, Inc.
4101F Dublin Blvd., Ste. 425
Dublin, California 94568 U.S.A.
Phone 925.271.4438 • Fax 925.369.0450
admin@realoptionsvaluation.com
www.risksimulator.com
www.realoptionsvaluation.com
Table of Contents
1. ВВЕДЕНИЕ .............................................................................................. 1 1.1 Добро пожаловать в программу Risk Simulator ................................................................................1 1.2 Требования к установке и процедурам ..................................................................................................2 1.3 Лицензирование ......................................................................................................................................2 1.4 ЧТО НОВОГО В ВЕРСИИ 2012 .................................................................................................5 1.4.1 General Capabilities ..................................................................................................................................... 5 1.4.2 Simulation Module....................................................................................................................................... 6 1.4.3 Forecasting Module ..................................................................................................................................... 7 1.4.4 Optimization Module.................................................................................................................................. 7 1.4.5 Analytical Tools Module ............................................................................................................................ 8 1.4.6 Statistics and BizStats Module................................................................................................................... 9 2. Моделирование по методу Монте-Карло ............................................ 11 2.1 Что такое Монте-Карло? ................................................................................................................ 11 2.2 Приступая к работе с Risk Simulator............................................................................................... 12 2.2.1 A High-Level Overview of the Software ............................................................................................. 12 2.2.2 Running a Monte Carlo Simulation ....................................................................................................... 13 Starting a New Simulation Profile ..........................................................................................................13 Defining Input Assumptions ...................................................................................................................15 Defining Output Forecasts .......................................................................................................................18 Running the Simulation ...........................................................................................................................19 Interpreting the Forecast Results...............................................................................................................19 Forecast Chart Tabs ................................................................................................................................20 Using Forecast Charts and Confidence Intervals .....................................................................................23 2.3 Корреляции иКонтроль точности ................................................................................................... 26 2.3.1 The Basics of Correlations ....................................................................................................................... 26 2.3.2 Applying Correlations in Risk Simulator .............................................................................................. 27 2.3.3 The Effects of Correlations in Monte Carlo Simulation .................................................................. 28 2.3.4 Precision and Error Control.................................................................................................................... 29 2.3.5 Понимание Статистического Прогнозирования ........................................................................ 31 Measuring the Center of the Distribution––the First Moment................................................................31 Measuring the Spread of the Distribution––the Second Moment ............................................................32 Measuring the Skew of the Distribution––the Third Moment................................................................33 Measuring the Catastrophic Tail Events in a Distribution––the Fourth Moment.................................34 The Functions of Moments ......................................................................................................................34 2.3.6 Понимание распределения вероятностей для моделирования Методом МонтеКарло......................................................................................................................................................... 36 2.4 Дискретные распределения ................................................................................................................. 39 Bernoulli or Yes/No Distribution...........................................................................................................39 Binomial Distribution..............................................................................................................................39 Discrete Uniform .....................................................................................................................................40 Geometric Distribution ............................................................................................................................41 Hypergeometric Distribution ....................................................................................................................41 Negative Binomial Distribution...............................................................................................................42 Pascal Distribution ..................................................................................................................................43 Poisson Distribution ................................................................................................................................44 2.5 Непрерывные распределения ............................................................................................................... 46 Arcsine Distribution ................................................................................................................................46 Beta Distribution .....................................................................................................................................46 Beta 3 and Beta 4 Distributions .............................................................................................................47 Cauchy Distribution, or Lorentzian or Breit-Wigner Distribution .........................................................48 Chi-Square Distribution..........................................................................................................................48 Cosine Distribution..................................................................................................................................49 Double Log Distribution .........................................................................................................................49 Erlang Distribution .................................................................................................................................50 Exponential Distribution ........................................................................................................................51 Exponential 2 Distribution.....................................................................................................................51 Extreme Value Distribution, or Gumbel Distribution ..........................................................................52 F Distribution, or Fisher-Snedecor Distribution......................................................................................52 Gamma Distribution (Erlang Distribution) ...........................................................................................53 Laplace Distribution................................................................................................................................54 Logistic Distribution ................................................................................................................................55 Lognormal Distribution...........................................................................................................................55 Lognormal 3 Distribution .......................................................................................................................56 Normal Distribution ...............................................................................................................................57 Parabolic Distribution .............................................................................................................................57 Pareto Distribution ..................................................................................................................................58 Pearson V Distribution...........................................................................................................................59 Pearson VI Distribution .........................................................................................................................59 PERT Distribution ................................................................................................................................60 Power Distribution...................................................................................................................................61 Power 3 Distribution ...............................................................................................................................61 Student’s t Distribution ...........................................................................................................................62 Triangular Distribution ...........................................................................................................................62 Uniform Distribution ..............................................................................................................................63 Weibull Distribution (Rayleigh Distribution) .........................................................................................64 Weibull 3 Distribution ............................................................................................................................65 3. ПРОГНОЗИРОВАНИЕ....................................................................... 66 3.1 Различные типы методов прогнозирования ..................................................................................... 67 3.2 Запуск инструмента прогнозирования рисков в Risk Simulator ...................................................... 70 3.3 Анализ временных рядов ................................................................................................................... 71 3.4 Многомерные регрессии ....................................................................................................................... 75 3.5 Стохастическое прогнозирование ...................................................................................................... 79 3.6 Нелинейная экстраполяция................................................................................................................ 81 3.7 ARIMA временные ряды Бокса-Дженкинса ................................................................................... 83 3.8 AUTO ARIMA (Усложнённые ARIMA временные ряды Бокса-Дженкинса) ......................... 88 3.9 Базовая эконометрика ........................................................................................................................ 89 3.10 Прогнозы J-S Кривых ...................................................................................................................... 90 3.11 Прогнозы волатильности GARCH ............................................................................................. 92 3.11.1 GARCH Equations..................................................................................................................93 3.12 Цепи Маркова .................................................................................................................................. 94 3.13 Ограниченные зависимые переменные: логит, пробит, тобит. Использование максимального
приближения к популяции ......................................................................................................................... 95 3.14 Сплайн (кубических сплайн-интерполяции и экстраполяции) ..................................................... 98 4. ОПТИМИЗАЦИЯ............................................................................... 100 4.1 Методологии оптимизации ............................................................................................................. 100 4.2 Оптимизация с непрерывными переменными решений ................................................................ 102 4.3 Оптимизация с дискретными целочисленными переменными.................................................... 106 4.4 Кривая Эффективности и дополнительные настройки оптимизации ...................................... 110 4.5 Стохастическая оптимизация........................................................................................................ 112 5. АНАЛИТИЧЕСКИЕ ИНСТРУМЕНТЫ RISK SIMULATOR .....117 5.1 Торнадо и Инструменты чувствительности в моделировании .................................................. 117 5.2 Анализ чувствительности .............................................................................................................. 124 5.3 Распределительная установка с одной или несколькими переменными ....................................... 127 5.4 Bootstrap Моделирование .................................................................................................................. 132 5.5 Проверка гипотезы ........................................................................................................................... 134 5.6 Извлечение данных и сохранение результатов моделирования .................................................... 135 5.7 Создать отчет.................................................................................................................................. 137 5.8 Диагностический инструменты Регрессии и Прогнозирования .................................................... 138 5.9 Инструмент статистического анализа......................................................................................... 145 5.10 Инструмент анализа распределений ............................................................................................ 149 5.11 Инструмент анализ сценариев ..................................................................................................... 152 5.12 Инструмент Сегментации и Кластеризации ............................................................................ 154 5.13 RISK SIMULATOR 2011/2012 Новые инструменты ........................................................ 156 5.14 Генератор случайных чисел. Метод Монте-Карло по сравнению с методом Латинского
гиперкуба и методом Корреляционной Связки ...................................................................................... 156 5.15 удаление сесонности и тренда данных........................................................................................... 157 5.16 Анализ основных компонентов..................................................................................................... 158 5.17 Анализ структурных разрывов.................................................................................................... 159 5.18 Прогнозы Трендов........................................................................................................................... 160 5.19 Инструмент проверки моделей..................................................................................................... 161 5.20 Инструмент установки процентных распределений.................................................................. 162 5.21 Распределительные диаграммы и таблиц: инструмент распределения вероятностей............ 164 5.22 ROV BizStats................................................................................................................................. 167 5.23 Нейронные сети и Комбинаторные методологии прогнозирования нечеткой логики .............. 171 5.24 Оптимизатор поиска цели ............................................................................................................ 173 5.25 Оптимизатор поиска цели ............................................................................................................ 174 5.26 оптимизация Генетического алгоритма ...................................................................................... 175 5.27 ROV Модуль Дерева Решений ..................................................................................................... 176 5.27.1 Дерево Решений .................................................................................................................................. 176 5.27.2 Симулятивное Моделирование...................................................................................................... 179 5.27.3 Байесовский Анализ .......................................................................................................................... 179 5.27.4 Ожидаемое значение идеальной информации, Minimax и Maximin Анализ,
Профилирование Риска и стоимость несовершенства информации ........................... 180 5.27.5 Чувствительность ................................................................................................................................ 180 5.27.6 Таблицы сценариев ............................................................................................................................ 180 5.27.7 Генерирование утилитарной функции....................................................................................... 181 6. Полезные советы и приемы................................................................ 189 СОВЕТЫ: Предположения (Установка входных данных и интерфейса пользователя) ..... 189 СОВЕТЫ: копирование и вставка ............................................................................................................ 189 СОВЕТЫ: Корреляции ................................................................................................................................ 190 СОВЕТЫ: Диагностика данных и статистический анализ ............................................................. 190 СОВЕТЫ: Дистрибутивный анализ, графики и таблицы вероятностей................................... 190 СОВЕТЫ: Кривая Эффективности......................................................................................................... 191 СОВЕТЫ: Клетки Прогнозов .................................................................................................................... 191 СОВЕТЫ: Чарты Прогнозов ..................................................................................................................... 191 СОВЕТЫ: Прогнозирование ..................................................................................................................... 191 СОВЕТЫ: прогнозирование: ARIMA .................................................................................................... 191 СОВЕТЫ: прогнозирование: Базовая эконометрика........................................................................ 192 СОВЕТЫ: прогнозирование: логит, пробит, и тобит ...................................................................... 192 СОВЕТЫ: прогнозирование: случайные процессы .......................................................................... 192 СОВЕТЫ: прогнозирование: тренд графика (кривой) .................................................................... 192 СОВЕТЫ: Вызов функций ......................................................................................................................... 192 СОВЕТЫ: Приступая к работе. Упражнения и начало работы (видеоматериалы) ............... 192 СОВЕТЫ: Hardware ID ................................................................................................................................ 193 СОВЕТЫ: Метод Латинский гиперкуба выборки (LHS) по сравнению с Монте-Карло
(MCS) ....................................................................................................................................................... 193 СОВЕТЫ: Интернет-ресурсы .................................................................................................................... 193 СОВЕТЫ: Оптимизация..............................................................................................................................193 СОВЕТЫ: Профили ..................................................................................................................................... 193 СОВЕТЫ: Сочетания клавиш и меню правой кнопкой мыши .................................................... 194 СОВЕТЫ: Сохранить.................................................................................................................................... 194 СОВЕТЫ: Отбор проб и методы моделирования ............................................................................. 194 СОВЕТЫ: Software Development Kit (SDK) и DLL-библиотеки ................................................. 194 СОВЕТЫ: Начиная работу с Risk Simulator в Excel ........................................................................... 195 СОВЕТЫ: Моделирование на сверхскоростях.................................................................................... 195 СОВЕТЫ: Анализ Торнадо ........................................................................................................................ 195 СОВЕТЫ: Устранение неполадок............................................................................................................ 196 INDEX ....................................................................................................... 197 R I S K
S I M U L A T O R
1
1. ВВЕДЕНИЕ
1.1 Добро пожаловать в программу
Risk Simulator
T
he Risk Simulator is a Monte Carlo simulation, Forecasting, and Optimization
software. The software is written in Microsoft .NET C# and functions together with
Excel as an add-in. This software is also compatible and often used with the Real
Options Super Lattice Solver (SLS) software and Employee Stock Options Valuation Toolkit
(ESOV) software, also developed by Real Options Valuation, Inc. Note that although we
attempt to be thorough in this user manual, the manual is absolutely not a substitute for the
Training DVD, live training courses, and books written by the software’s creator (e.g., Dr.
Johnathan Mun’s Real Options Analysis, 2nd Edition, Wiley Finance, 2005; Modeling Risk:
Applying Monte Carlo Simulation, Real Options Analysis, Forecasting, and Optimization, 2nd
Edition, Wiley Finance, 2010; and Valuing Employee Stock Options (2004 FAS 123R), Wiley
Finance, 2004). Please visit our website at www.realoptionsvaluation.com for more information
about these items.
The Risk Simulator software has the following modules:

Monte Carlo Simulation (runs parametric and nonparametric simulation of 42
probability distributions with different simulation profiles, truncated and correlated
simulations, customizable distributions, precision and error-controlled simulations, and
many other algorithms)

Forecasting (runs Box-Jenkins ARIMA, multiple regression, nonlinear extrapolation,
stochastic processes, and time-series analysis)

Optimization Under Uncertainty (runs optimizations using discrete integer and
continuous variables for portfolio and project optimization with and without
simulation)

Modeling and Analytical Tools (runs tornado, spider, and sensitivity analysis, as well as
bootstrap simulation, hypothesis testing, distributional fitting, etc.)

ROV BizStats (over 130 business statistics and analytical models)

ROV Decision Tree (decision tree models, Monte Carlo risk simulation on decision
trees, sensitivity analysis, scenario analysis, Bayesian joint and posterior probability
updating, expected value of information, MINIMAX, MAXIMIN, risk profiles)
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Real Options SLS software is used for computing simple and complex options and includes the
ability to create customizable option models. This software has the following modules:

Single Asset SLS (for solving abandonment, chooser, contraction, deferment, and
expansion options, as well as for solving customized options)

Multiple Asset and Multiple Phase SLS (for solving multiphase sequential options,
options with multiple underlying assets and phases, combination of multiphase
sequential with abandonment, chooser, contraction, deferment, expansion, and
switching options; it can also be used to solve customized options)

Multinomial SLS (for solving trinomial mean-reverting options, quadranomial jumpdiffusion options, and pentanomial rainbow options)

Excel Add-In Functions (for solving all the above options plus closed-form models
and customized options in an Excel-based environment)
1.2 Требования к установке и процедурам
To install the software, follow the on-screen instructions. The minimum requirements for this
software are:

Pentium IV processor or later (dual core recommended)

Windows XP, Vista, Windows 7, Windows 8, or later

Microsoft Excel XP, 2003, 2007, 2010, or later

Microsoft .NET Framework 2.0 or later (versions 3.0, 3.5, and so forth)

500 MB free space

2GB RAM minimum (2–4GB recommended)

Administrative rights to install software
Most new computers come with Microsoft .NET Framework 2.0/3.0 already installed.
However, if an error message pertaining to requiring .NET Framework occurs during the
installation of Risk Simulator, exit the installation. Then, install the relevant .NET Framework
software included in the CD (choose your own language). Complete the .NET installation,
restart the computer, and then reinstall the Risk Simulator software.
There is a default 10-day trial license file that comes with the software. To obtain a full
corporate
license,
please
contact
Real
Options
Valuation,
Inc.,
at
admin@realoptionsvaluation.com or call +1 (925) 271-4438, or visit our website at
www.realoptionsvaluation.com. Please visit this website and click on DOWNLOAD to obtain the
latest software release, or click on the FAQ link to obtain any updated information on licensing
or installation issues and fixes.
1.3 Лицензирование
If you have installed the software and have purchased a full license to use the software, you will
need to e-mail us your Hardware ID so that we can generate a license file for you. Follow the
instructions below:
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
Start Excel XP/2003/2007/2010, click on the License icon or Risk Simulator │ License
and copy down and e-mail your 11 to 20 digit and alphanumeric HARDWARE ID
that starts with the prefix “RS” (you can also select the Hardware ID and do a rightclick copy or click on the e-mail Hardware ID link) to admin@realoptionsvaluation.com.
Once we have obtained this ID, a newly generated permanent license will be e-mailed
to you. Once you obtain this license file, simply save it to your hard drive (if it is a
zipped file, first unzip its contents and save them to your hard drive). Start Excel, click
on Risk Simulator │ License or click on the License icon and click on Install License and
point to this new license file. Restart Excel and you are done. The entire process will
take less than a minute and you will be fully licensed.

Once installation is complete, start Microsoft Excel and if the installation was
successful, you should see an additional “Risk Simulator” item on the menu bar in
Excel XP/2003 or under the new icon group in Excel 2007/2010, and a new icon bar
on Excel as seen in Figure 1.1. In addition, a splash screen will appear as seen in Figure
1.2, indicating that the software is functioning and loaded into Excel. Figure 1.3 also
shows the Risk Simulator toolbar. If these items exist in Excel, you are now ready to
start using the software. The remainder of this user manual provides step-by-step
instructions for using the software.
Figure 1.1 – Risk Simulator Menu and Icon Bar in Excel 2007/2010
Website:
www.realoptionsvaluation.com
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Figure 1.2 – Risk Simulator Splash Screen
Figure 1.3 – Risk Simulator Icon Toolbars in Excel 2007/2010
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1.4 ЧТО НОВОГО В ВЕРСИИ 2012
The following lists the main capabilities of Risk Simulator, where the highlighted items indicate
the latest additions to version 2011/2012.
1.4.1 General Capabilities
1. Available in 11 languages––English, French, German, Italian, Japanese, Korean,
Portuguese, Russian, Spanish, Simplified Chinese, and Traditional Chinese.
2. ROV Decision Tree module is included in the latest version and is used to create and value
decision tree models. Additional advanced methodologies and analytics are also included:

Decision Tree Models

Monte Carlo risk simulation

Sensitivity Analysis

Scenario Analysis

Bayesian (Joint and Posterior Probability Updating)

Expected Value of Information

MINIMAX

MAXIMIN

Risk Profiles
3. Books––analytical theory, application, and case studies are supported by 10 books.
4. Commented Cells––turn cell comments on or off and decide if you wish to show cell
comments on all input assumptions, output forecasts, and decision variables.
5. Detailed Example Models––24 example models in Risk Simulator and over 300 models in
Modeling Toolkit.
6. Detailed Reports––all analyses come with detailed reports.
7. Detailed User Manual––step-by-step user manual.
8. Flexible Licensing––certain functionalities can be turned on or off to allow you to
customize your risk analysis experience. For instance, if you are only interested in the
forecasting tools in Risk Simulator, you may be able to obtain a special license that activates
only the forecasting tools and leaves the other modules deactivated, thereby saving some
costs on the software.
9. Flexible Requirements––works in Window 7, Vista, and XP; integrates with Excel 2010,
2007, 2003; and works in MAC operating systems running virtual machines.
10. Fully customizable colors and charts––tilt, 3D, color, chart type, and much more!
11. Hands-on Exercises––detailed step-by-step guide to running Risk Simulator, including
guides on interpreting the results.
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12. Multiple Cell Copy and Paste––allows assumptions, decision variables, and forecasts to be
copied and pasted.
13. Profiling––allows multiple profiles to be created in a single model (different scenarios of
simulation models can be created, duplicated, edited, and run in a single model).
14. Revised Icons in Excel 2007/2010––a completely reworked icon toolbar that is more
intuitive and user friendly. There are four sets of icons that fit most screen resolutions
(1280 x 760 and above).
15. Right-Click Shortcuts––access all of Risk Simulator's tools and menus using a mouse rightclick.
16. ROV Software Integration––works well with other ROV software including Real Options
SLS, Modeling Toolkit, Basel Toolkit, ROV Compiler, ROV Extractor and Evaluator,
ROV Modeler, ROV Valuator, ROV Optimizer, ROV Dashboard, ESO Valuation
Toolkit, and others!
17. RS Functions in Excel––insert RS functions for setting assumptions and forecasts, and
right-click support in Excel.
18. Troubleshooter—allows you to re-enable the software, check for your system
requirements, obtain the Hardware ID, and others.
19. Turbo Speed Analysis—runs forecasts and other analyses tools at blazingly fast speeds
(enhanced in version 5.2). The analyses and results remain the same but are now computed
very quickly; reports are generated very quickly as well.
20. Web Resources, Case Studies, and Videos––download free models, getting-started videos,
case studies, whitepapers, and other materials from our website.
1.4.2 Simulation Module
21. 6 random number generators––ROV Advanced Subtractive Generator, Subtractive
Random Shuffle Generator, Long Period Shuffle Generator, Portable Random Shuffle
Generator, Quick IEEE Hex Generator, and Basic Minimal Portable Generator.
22. 2 sampling methods––Monte Carlo and Latin Hypercube.
23. 3 Correlation Copulas––applying Normal Copula, T Copula, and Quasi-Normal Copula
for correlated simulations.
24. 42 probability distributions––arcsine, Bernoulli, beta, beta 3, beta 4, binomial, Cauchy, chisquare, cosine, custom, discrete uniform, double log, Erlang, exponential, exponential 2, F
distribution, gamma, geometric, Gumbel max, Gumbel min, hypergeometric, Laplace,
logistic, lognormal (arithmetic) and lognormal (log), lognormal 3 (arithmetic) and
lognormal 3 (log), negative binomial, normal, parabolic, Pareto, Pascal, Pearson V, Pearson
VI, PERT, Poisson, power, power 3, Rayleigh, t and t2, triangular, uniform, Weibull,
Weibull 3.
25. Alternate Parameters––using percentiles as an alternate way of inputting parameters.
26. Custom Nonparametric Distribution––make your own distributions for running historical
simulations, and applying the Delphi method.
27. Distribution Truncation––enabling data boundaries.
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28. Excel Functions––set assumptions and forecasts using functions inside Excel
29. Multidimensional Simulation––simulation of uncertain input parameters.
30. Precision Control––determines if the number of simulation trials run is sufficient.
31. Super Speed Simulation––runs 100,000 trials in a few seconds.
1.4.3 Forecasting Module
32. ARIMA––autoregressive integrated moving average models ARIMA (P,D,Q).
33. Auto ARIMA––runs the most common combinations of ARIMA to find the best-fitting
model.
34. Auto Econometrics––runs thousands of model combinations and permutations to obtain
the best-fitting model for existing data (linear, nonlinear, interacting, lag, leads, rate,
difference).
35. Basic Econometrics––econometric and linear/nonlinear and interacting regression models.
36. Combinatorial Fuzzy Logic Forecasts––time-series forecast methods
37. Cubic Spline––nonlinear interpolation and extrapolation.
38. GARCH––volatility projections using generalized autoregressive conditional
heteroskedasticity models: GARCH, GARCH-M, TGARCH, TGARCH-M, EGARCH,
EGARCH-T, GJR-GARCH, and GJR-TGARCH.
39. J-Curve––exponential J curves.
40. Limited Dependent Variables––Logit, Probit, and Tobit.
41. Markov Chains––two competing elements over time and market share predictions.
42. Multiple Regression––regular linear and nonlinear regression, with stepwise methodologies
(forward, backward, correlation, forward-backward).
43. Neural Network Forecasts––linear, nonlinear logistic, hyperbolic tangent, and cosine
44. Nonlinear Extrapolation––nonlinear time-series forecasting.
45. S Curve––logistic S curves.
46. Time-Series Analysis––8 time-series decomposition models for predicting levels, trends,
and seasonalities.
47. Trendlines––forecasting and fitting using linear, nonlinear polynomial, power, logarithmic,
exponential, and moving averages with goodness of fit.
1.4.4 Optimization Module
48. Linear Optimization––multiphasic optimization and general linear optimization.
49. Nonlinear Optimization––detailed results including Hessian matrices, LaGrange functions,
and more.
50. Static Optimization––quick runs for continuous, integers, and binary optimizations.
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51. Dynamic Optimization––simulation with optimization.
52. Stochastic Optimization––quadratic, tangential, central, forward, and convergence criteria.
53. Efficient Frontier––combinations of stochastic and dynamic optimizations on multivariate
efficient frontiers.
54. Genetic Algorithms––used for a variety of optimization problems.
55. Multiphasic Optimization––testing for local versus global optimum allowing better control
over how the optimization is run, and increases the accuracy and dependency of the results.
56. Percentiles and Conditional Means––additional statistics for stochastic optimization,
including percentiles as well as conditional means, which are critical in computing
conditional value at risk measures.
57. Search Algorithm––simple, fast, and efficient search algorithms for basic single decision
variable and goal seek applications.
58. Super Speed Simulation in Dynamic and Stochastic Optimization––runs simulation at
super speed while integrated with optimization.
1.4.5 Analytical Tools Module
59. Check Model––tests for the most common mistakes in your model.
60. Correlation Editor––allows large correlation matrices to be directly entered and edited.
61. Create Report––automates report generation of assumptions and forecasts in a model.
62. Create Statistics Report––generates comparative report of all forecast statistics.
63. Data Diagnostics––runs tests on heteroskedasticity, micronumerosity, outliers,
nonlinearity, autocorrelation, normality, sphericity, nonstationarity, multicollinearity, and
correlations.
64. Data Extraction and Export––extracts data to Excel or flat text files and Risk Sim files,
runs statistical reports and forecast result reports.
65. Data Open and Import––retrieves previous simulation run results.
66. Deseasonalization and Detrending––deseasonalizes and detrends your data.
67. Distributional Analysis––computes exact PDF, CDF, and ICDF of all 42 distributions and
generates probability tables.
68. Distributional Designer––allows you to create custom distributions.
69. Distributional Fitting (Multiple)–– runs multiple variables simultaneously, accounts for
correlations and correlation significance.
70. Distributional Fitting (Single)––Kolmogorov-Smirnov and chi-square tests on continuous
distributions, complete with reports and distributional assumptions.
71. Hypothesis Testing––tests if two forecasts are statistically similar or different.
72. Nonparametric Bootstrap––simulation of the statistics to obtain the precision and accuracy
of the results.
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73. Overlay Charts––fully customizable overlay charts of assumptions and forecasts together
(CDF, PDF, 2D/3D chart types).
74. Principal Component Analysis––tests the best predictor variables and ways to reduce the
data array.
75. Scenario Analysis––hundreds and thousands of static two-dimensional scenarios.
76. Seasonality Test––tests for various seasonality lags.
77. Segmentation Clustering––groups data into statistical clusters for segmenting your data.
78. Sensitivity Analysis––dynamic sensitivity (simultaneous analysis).
79. Structural Break Test––tests if your time-series data has statistical structural breaks.
80. Tornado Analysis––static perturbation of sensitivities, spider and tornado analysis, and
scenario tables.
1.4.6 Statistics and BizStats Module
81. Percentile Distributional Fitting––using percentiles and optimization to find the best-fitting
distribution.
82. Probability Distributions’ Charts and Tables––run 45 probability distributions, their four
moments, CDF, ICDF, PDF, charts, and overlay multiple distributional charts, and
generate probability distribution tables.
83. Statistical Analysis––descriptive statistics, distributional fitting, histograms, charts, nonlinear
extrapolation, normality test, stochastic parameters estimation, time-series forecasting,
trendline projections, etc.
84. ROV BIZSTATS––over 130 business statistics and analytical models:
Absolute Values, ANOVA: Randomized Blocks Multiple Treatments, ANOVA: Single Factor
Multiple Treatments, ANOVA: Two Way Analysis, ARIMA, Auto ARIMA, Autocorrelation
and Partial Autocorrelation, Autoeconometrics (Detailed), Autoeconometrics (Quick), Average,
Combinatorial Fuzzy Logic Forecasting, Control Chart: C, Control Chart: NP, Control Chart:
P, Control Chart: R, Control Chart: U, Control Chart: X, Control Chart: XMR, Correlation,
Correlation (Linear, Nonlinear), Count, Covariance, Cubic Spline, Custom Econometric
Model, Data Descriptive Statistics, Deseasonalize, Difference, Distributional Fitting,
Exponential J Curve, GARCH, Heteroskedasticity, Lag, Lead, Limited Dependent Variables
(Logit), Limited Dependent Variables (Probit), Limited Dependent Variables (Tobit), Linear
Interpolation, Linear Regression, LN, Log, Logistic S Curve, Markov Chain, Max, Median, Min,
Mode, Neural Network, Nonlinear Regression, Nonparametric: Chi-Square Goodness of Fit,
Nonparametric: Chi-Square Independence, Nonparametric: Chi-Square Population Variance,
Nonparametric: Friedman’s Test, Nonparametric: Kruskal-Wallis Test, Nonparametric:
Lilliefors Test, Nonparametric: Runs Test, Nonparametric: Wilcoxon Signed-Rank (One Var),
Nonparametric: Wilcoxon Signed-Rank (Two Var), Parametric: One Variable (T) Mean,
Parametric: One Variable (Z) Mean, Parametric: One Variable (Z) Proportion, Parametric: Two
Variable (F) Variances, Parametric: Two Variable (T) Dependent Means, Parametric: Two
Variable (T) Independent Equal Variance, Parametric: Two Variable (T) Independent Unequal
Variance, Parametric: Two Variable (Z) Independent Means, Parametric: Two Variable (Z)
Independent Proportions, Power, Principal Component Analysis, Rank Ascending, Rank
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Descending, Relative LN Returns, Relative Returns, Seasonality, Segmentation Clustering,
Semi-Standard Deviation (Lower), Semi-Standard Deviation (Upper), Standard 2D Area,
Standard 2D Bar, Standard 2D Line, Standard 2D Point, Standard 2D Scatter, Standard 3D
Area, Standard 3D Bar, Standard 3D Line, Standard 3D Point, Standard 3D Scatter, Standard
Deviation (Population), Standard Deviation (Sample), Stepwise Regression (Backward),
Stepwise Regression (Correlation), Stepwise Regression (Forward), Stepwise Regression
(Forward-Backward), Stochastic Processes (Exponential Brownian Motion), Stochastic
Processes (Geometric Brownian Motion), Stochastic Processes (Jump Diffusion), Stochastic
Processes (Mean Reversion with Jump Diffusion), Stochastic Processes (Mean Reversion),
Structural Break, Sum, Time-Series Analysis (Auto), Time-Series Analysis (Double Exponential
Smoothing), Time-Series Analysis (Double Moving Average), Time-Series Analysis (HoltWinter’s Additive), Time-Series Analysis (Holt-Winter’s Multiplicative), Time-Series Analysis
(Seasonal Additive), Time-Series Analysis (Seasonal Multiplicative), Time-Series Analysis (Single
Exponential Smoothing), Time-Series Analysis (Single Moving Average), Trend Line
(Difference Detrended), Trend Line (Exponential Detrended), Trend Line (Exponential),
Trend Line (Linear Detrended), Trend Line (Linear), Trend Line (Logarithmic Detrended),
Trend Line (Logarithmic), Trend Line (Moving Average Detrended), Trend Line (Moving
Average), Trend Line (Polynomial Detrended), Trend Line (Polynomial), Trend Line (Power
Detrended), Trend Line (Power), Trend Line (Rate Detrended), Trend Line (Static Mean
Detrended), Trend Line (Static Median Detrended), Variance (Population), Variance (Sample),
Volatility: EGARCH, Volatility: EGARCH-T, Volatility: GARCH, Volatility: GARCH-M,
Volatility: GJR GARCH, Volatility: GJR TGARCH, Volatility: Log Returns Approach,
Volatility: TGARCH, Volatility: TGARCH-M, Yield Curve (Bliss), and Yield Curve (NelsonSiegel).
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2
2. Моделирование по
методу Монте-Карло
M
onte Carlo risk simulation, named for the famous gambling capital of Monaco, is a
very potent methodology. For the practitioner, simulation opens the door for
solving difficult and complex but practical problems with great ease. Monte Carlo
creates artificial futures by generating thousands and even millions of sample paths of outcomes
and looks at their prevalent characteristics. For analysts in a company, taking graduate-level
advanced math courses is just not logical or practical. A brilliant analyst would use all available
tools at his or her disposal to obtain the same answer the easiest and most practical way
possible. And in all cases, when modeled correctly, Monte Carlo simulation provides similar
answers to the more mathematically elegant methods. So, what is Monte Carlo simulation and
how does it work?
2.1 Что такое Монте-Карло?
Monte Carlo simulation in its simplest form is a random number generator that is useful for
forecasting, estimation, and risk analysis. A simulation calculates numerous scenarios of a model
by repeatedly picking values from a user-predefined probability distribution for the uncertain
variables and using those values for the model. As all those scenarios produce associated results
in a model, each scenario can have a forecast. Forecasts are events (usually with formulas or
functions) that you define as important outputs of the model. These usually are events such as
totals, net profit, or gross expenses.
Simplistically, think of the Monte Carlo simulation approach as repeatedly picking golf balls out
of a large basket with replacement. The size and shape of the basket depend on the
distributional input assumption (e.g., a normal distribution with a mean of 100 and a standard
deviation of 10, versus a uniform distribution or a triangular distribution) where some baskets
are deeper or more symmetrical than others, allowing certain balls to be pulled out more
frequently than others. The number of balls pulled repeatedly depends on the number of trials
simulated. For a large model with multiple related assumptions, imagine a very large basket
wherein many smaller baskets reside. Each small basket has its own set of golf balls that are
bouncing around. Sometimes these small baskets are linked with each other (if there is a
correlation between the variables) and the golf balls are bouncing in tandem, while other times
the balls are bouncing independently of one another. The balls that are picked each time from
these interactions within the model (the large central basket) are tabulated and recorded,
providing a forecast output result of the simulation.
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2.2 Приступая к работе с Risk Simulator
2.2.1 A High-Level Overview of the Software
The Risk Simulator software has several different applications including Monte Carlo
simulation, forecasting, optimization, and risk analytics.

The Simulation Module allows you to run simulations in your existing Excel-based
models, generate and extract simulation forecasts (distributions of results), perform
distributional fitting (automatically finding the best-fitting statistical distribution),
compute correlations (maintain relationships among simulated random variables),
identify sensitivities (creating tornado and sensitivity charts), test statistical hypotheses
(finding statistical differences between pairs of forecasts), run bootstrap simulation
(testing the robustness of result statistics), and run custom and nonparametric
simulations (simulations using historical data without specifying any distributions or
their parameters for forecasting without data or applying expert opinion forecasts).

The Forecasting Module can be used to generate automatic time-series forecasts (with
and without seasonality and trend), multivariate regressions (modeling relationships
among variables), nonlinear extrapolations (curve fitting), stochastic processes
(random walks, mean-reversions, jump-diffusion, and mixed processes), Box-Jenkins
ARIMA (econometric forecasts), Auto ARIMA, basic econometrics and auto
econometrics (modeling relationships and generating forecasts), exponential J curves,
logistic S curves, GARCH models and their multiple variations (modeling and
forecasting volatility), maximum likelihood models for limited dependent variables
(logit, tobit, and probit models), Markov chains, trendlines, spline curves, and others.

The Optimization Module is used for optimizing multiple decision variables subject to
constraints to maximize or minimize an objective, and can be run either as a static
optimization, dynamic, and stochastic optimization under uncertainty together with
Monte Carlo simulation, or as a stochastic optimization with super speed simulations.
The software can handle linear and nonlinear optimizations with binary, integer, and
continuous variables, as well as generate Markowitz efficient frontiers.

The Analytical Tools Module allows you to run segmentation clustering, hypothesis
testing, statistical tests of raw data, data diagnostics of technical forecasting
assumptions (e.g., heteroskedasticity, multicollinearity, and the like), sensitivity and
scenario analyses, overlay chart analysis, spider charts, tornado charts, and many other
powerful tools.

ROV BizStats (over 130 business statistics and analytical models).

ROV Decision Tree (decision tree models, Monte Carlo risk simulation on decision
trees, sensitivity analysis, scenario analysis, Bayesian joint and posterior probability
updating, expected value of information, MINIMAX, MAXIMIN, risk profiles).

The Real Options Super Lattice Solver is a software that complements Risk Simulator,
used for solving simple to complex real options problems.
The following sections walk you through the basics of the Simulation Module in Risk
Simulator, while future chapters provide more details about the applications of other modules.
To follow along, make sure you have Risk Simulator installed on your computer to proceed.
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In fact, it is highly recommended that you first watch the getting started videos on the web
(www.realoptionsvaluation.com/risksimulator.html) or attempt the step-by-step exercises at the
end of this chapter before coming back and reviewing the text in this chapter. This approach is
recommended because the videos will get you started immediately, as will the exercises, whereas
the text in this chapter focuses more on the theory and detailed explanations of the properties
of simulation.
2.2.2 Running a Monte Carlo Simulation
Typically, to run a simulation in your existing Excel model, the following steps have to
be performed:
1. Start a new simulation profile or open an existing profile.
2. Define input assumptions in the relevant cells.
3. Define output forecasts in the relevant cells.
4. Run simulation.
5. Interpret the results.
If desired, and for practice, open the example file called Basic Simulation Model and follow
along with the examples below on creating a simulation. The example file can be found either
on the start menu at Start | Real Options Valuation | Risk Simulator | Examples or accessed
directly through Risk Simulator | Example Models.
Starting a New
Simulation Profile
To start a new simulation, you will first need to create a simulation profile. A simulation profile
contains a complete set of instructions on how you would like to run a simulation. That is, all
the assumptions, forecasts, run preferences, and so forth. Having profiles facilitates creating
multiple scenarios of simulations. That is, using the same exact model, several profiles can be
created, each with its own specific simulation properties and requirements. The same person
can create different test scenarios using different distributional assumptions and inputs or
multiple persons can test their own assumptions and inputs on the same model.

Start Excel and create a new model or open an existing one (you can use the Basic
Simulation Model example to follow along).

Click on Risk Simulator | New Simulation Profile.

Specify a title for your simulation as well as all other pertinent information (Figure 2.1).
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Figure 2.1 – New Simulation Profile

Title: Specifying a simulation title allows you to create multiple simulation profiles in a
single Excel model. Thus you can now save different simulation scenario profiles
within the same model without having to delete existing assumptions and changing
them each time a new simulation scenario is required. You can always change the
profile’s name later (Risk Simulator | Edit Profile).

Number of trials: This is where the number of simulation trials required is entered.
That is, running 1,000 trials means that 1,000 different iterations of outcomes based on
the input assumptions will be generated. You can change this number as desired, but
the input has to be positive integers. The default number of runs is 1,000 trials. You
can use precision and error control later in this chapter to automatically help determine
how many simulation trials to run (see the section on precision and error control for
details).

Pause simulation on error: If checked, the simulation stops every time an error is
encountered in the Excel model. That is, if your model encounters a computation
error (e.g., some input values generated in a simulation trial may yield a divide by zero
error in one of your spreadsheet cells), the simulation stops. This function is important
to help audit your model to make sure there are no computational errors in your Excel
model. However, if you are sure the model works, then there is no need for this
preference to be checked.

Turn on correlations: If checked, correlations between paired input assumptions will
be computed. Otherwise, correlations will all be set to zero, and a simulation is run
assuming no cross-correlations between input assumptions. As an example, applying
correlations will yield more accurate results if, indeed, correlations exist, and will tend
to yield a lower forecast confidence if negative correlations exist. After turning on
correlations here, you can later set the relevant correlation coefficients on each
assumption generated (see the section on correlations for more details).

Specify random number sequence: Simulation by definition will yield slightly
different results every time a simulation is run. This characteristic is by virtue of the
random number generation routine in Monte Carlo simulation and is a theoretical fact
in all random number generators. However, when making presentations, sometimes
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you may require the same results (especially when the report being presented shows
one set of results and during a live presentation you would like to show the same
results being generated, or when you are sharing models with others and would like
the same results to be obtained every time), so you would then check this preference
and enter in an initial seed number. The seed number can be any positive integer.
Using the same initial seed value, the same number of trials, and the same input
assumptions, the simulation will always yield the same sequence of random numbers,
guaranteeing the same final set of results.
Note that once a new simulation profile has been created, you can come back later and modify
these selections. To do so, make sure that the current active profile is the profile you wish to
modify, otherwise, click on Risk Simulator | Change Simulation Profile, select the profile you wish
to change and click OK (Figure 2.2 shows an example where there are multiple profiles and
how to activate a selected profile). Then, click on Risk Simulator | Edit Simulation Profile and
make the required changes. You can also duplicate or rename an existing profile. When creating
multiple profiles in the same Excel model, make sure to provide each profile a unique name so
you can tell them apart later on. Also, these profiles are stored inside hidden sectors of the
Excel *.xls file and you do not have to save any additional files. The profiles and their contents
(assumptions, forecasts, etc.) are automatically saved when you save the Excel file. Finally, the
last profile that is active when you exit and save the Excel file will be the one that is opened the
next time the Excel file is accessed.
Figure 2.2 – Change Active Simulation
Defining Input
Assumptions
The next step is to set input assumptions in your model. Note that assumptions can only be
assigned to cells without any equations or functions—typed-in numerical values that are inputs
in a model—whereas output forecasts can only be assigned to cells with equations and
functions—outputs of a model. Recall that assumptions and forecasts cannot be set unless a
simulation profile already exists. Do the following to set new input assumptions in your model:

Make sure a Simulation Profile exists; open an existing profile or start a new profile
(Risk Simulator | New Simulation Profile).
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
Select the cell you wish to set an assumption on (e.g., cell G8 in the Basic Simulation
Model example).

Click on Risk Simulator | Set Input Assumption or click on the set input assumption icon
in the Risk Simulator icon toolbar.

Select the relevant distribution you want, enter the relevant distribution parameters (e.g., Triangular
distribution with 1, 2, 2.5 as the minimum, most likely, and maximum values), and hit
OK to insert the input assumption into your model (Figure 2.3).
Figure 2.3 – Setting an Input Assumption
Note that you can also set assumptions by selecting the cell you wish to set the assumption on
and using the mouse right-click, access the shortcut Risk Simulator menu to set an input
assumption. In addition, for expert users, you can set input assumptions using the Risk
Simulator RS Functions: select the cell of choice, click on Excel’s Insert, Function, select the All
Category, and scroll down to the RS functions list (we do not recommend using RS functions
unless you are an expert user). For the examples going forward, we suggest following the basic
instructions in accessing menus and icons.
As shown in Figure 2.4, there are several key areas in the Assumption Properties worthy of
mention.

Assumption Name: This is an optional area to allow you to enter in unique names
for the assumptions to help track what each of the assumptions represents. Good
modeling practice is to use short but precise assumption names.

Distribution Gallery: This area to the left shows all of the different distributions
available in the software. To change the views, right-click anywhere in the gallery and
select large icons, small icons, or list. There are over two dozen distributions available.
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
Input Parameters: Depending on the distribution selected, the required relevant
parameters are shown. You may either enter the parameters directly or link them to
specific cells in your worksheet. Hard coding or typing the parameters is useful when
the assumption parameters are assumed not to change. Linking to worksheet cells is
useful when the input parameters need to be visible or are allowed to be changed (click
on the link icon to link an input parameter to a worksheet cell).

Enable Data Boundary: These are typically not used by the average analyst but exist
for truncating the distributional assumptions. For instance, if a normal distribution is
selected, the theoretical boundaries are between negative infinity and positive infinity.
However, in practice, the simulated variable exists only within some smaller range, and
this range can then be entered to truncate the distribution appropriately.

Correlations: Pairwise correlations can be assigned to input assumptions here. If
correlations are required, remember to check the Turn on Correlations preference by
clicking on Risk Simulator │Edit Simulation Profile. See the discussion on correlations
later in this chapter for more details about assigning correlations and the effects
correlations will have on a model. Notice that you can either truncate a distribution or
correlate it to another assumption, but not both.

Short Descriptions: These exist for each of the distributions in the gallery. The short
descriptions explain when a certain distribution is used as well as the input parameter
requirements. See the section in Understanding Probability Distributions for Monte
Carlo Simulation for details on each distribution type available in the software.

Regular Input and Percentile Input: This option allows the user to perform a quick
due diligence test of the input assumption. For instance, if setting a normal distribution
with some mean and standard deviation inputs, you can click on the percentile input to
see what the corresponding 10th and 90th percentiles are.

Enable Dynamic Simulation: This option is unchecked by default, but if you wish
to run a multidimensional simulation (i.e., if you link the input parameters of the
assumption to another cell that is itself an assumption, you are simulating the inputs,
or simulating the simulation), then remember to check this option. Dynamic
simulation will not work unless the inputs are linked to other changing input
assumptions.
Note: If you are following along with the example, continue by setting another assumption on
cell G9. This time use the Uniform distribution with a minimum value of 0.9 and a maximum
value of 1.1. Then, proceed to defining the output forecasts in the next step.
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Figure 2.4 – Assumption Properties
Defining Output
Forecasts
The next step is to define output forecasts in the model. Forecasts can only be defined on
output cells with equations or functions. The following describes the set forecast process:

Select the cell you wish to set a forecast (e.g., cell G10 in the Basic Simulation Model
example).

Click on Risk Simulator │ Set Output Forecast or click on the set output forecast icon on
the Risk Simulator icon toolbar (Figure 1.3).

Enter the relevant information and click OK.
Note that you can also set output forecasts by selecting the cell you wish to set the forecast on
and using the mouse right-click, access the shortcut Risk Simulator menu to set an output
forecast. Figure 2.5 illustrates the set forecast properties.

Forecast Name: Specify the name of the forecast cell. This is important because
when you have a large model with multiple forecast cells, naming the forecast cells
individually allows you to access the right results quickly. Do not underestimate the
importance of this simple step. Good modeling practice is to use short but precise
forecast names.

Forecast Precision: Instead of relying on a guesstimate of how many trials to run in
your simulation, you can set up precision and error controls. When an error-precision
combination has been achieved in the simulation, the simulation will pause and inform
you of the precision achieved, making the required number of simulation trials an
automated process rather than a guessing game. Review the section on error and
precision control later in this chapter for more specific details.

Show Forecast Window: Allows the user to show or not show a particular forecast
window. The default is to always show a forecast chart.
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Figure 2.5 – Set Output Forecast
Running the
Simulation
If everything looks right, simply click on Risk Simulator | Run Simulation or click on the Run icon
on the Risk Simulator toolbar and the simulation will proceed. You may also reset a simulation
after it has run to rerun it (Risk Simulator | Reset Simulation or the reset simulation icon on the
toolbar) or to pause it during a run. Also, the step function (Risk Simulator | Step Simulation or
the step simulation icon on the toolbar) allows you to simulate a single trial, one at a time, useful
for educating others on simulation (i.e., you can show that at each trial, all the values in the
assumption cells are being replaced and the entire model is recalculated each time). You can
also access the run simulation menu by right-clicking anywhere in the model and selecting Run
Simulation.
Risk Simulator also allows you to run the simulation at extremely fast speed, called Super Speed.
To do this, click on Risk Simulator │ Run Super Speed Simulation or use the run super speed icon.
Notice how much faster the super speed simulation runs. In fact, for practice, Reset Simulation
and then Edit Simulation Profile and change the Number of Trials to 100,000, and Run Super
Speed. It should only take a few seconds to run. However, please be aware that super speed
simulation will not run if the model has errors, VBA (visual basic for applications), or links to
external data sources or applications. In such situations, you will be notified and the regular
speed simulation will be run instead. Regular speed simulations are always able to run even with
errors, VBA, or external links.
Interpreting the
Forecast Results
The final step in Monte Carlo simulation is to interpret the resulting forecast charts. Figures 2.6
through 2.13 show the forecast chart and the corresponding statistics generated after running
the simulation. Typically, the following elements are important in interpreting the results of a
simulation:

Forecast Chart: The forecast chart shown in Figure 2.6 is a probability histogram that
shows the frequency counts of values occurring in the total number of trials simulated.
The vertical bars show the frequency of a particular x value occurring out of the total
number of trials, while the cumulative frequency (smooth line) shows the total
probabilities of all values at and below x occurring in the forecast.
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
S I M U L A T O R
Forecast Statistics: The forecast statistics shown in Figure 2.7 summarize the
distribution of the forecast values in terms of the four moments of a distribution. See
the Understanding the Forecast Statistics section later in this chapter for more details
on what some of these statistics mean. You can rotate between the histogram and
statistics tabs by depressing the space bar.
Figure 2.6 – Forecast Chart
Figure 2.7 – Forecast Statistics
Forecast Chart
Tabs

Preferences: The preferences tab in the forecast chart (Figure 2.8A) allows you to
change the look and feel of the charts. For instance, if Always On Top is selected, the
forecast charts will always be visible regardless of what other software are running on
your computer. Histogram Resolution allows you to change the number of bins of the
histogram, anywhere from 5 bins to 100 bins. Also, the Data Update feature allows you
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to control how fast the simulation runs versus how often the forecast chart is updated.
For example, viewing the forecast chart updated at almost every trial will slow down
the simulation as more memory is being allocated to updating the chart versus running
the simulation. This is merely a user preference and in no way changes the results of
the simulation, just the speed of completing the simulation. To further increase the
speed of the simulation, you can minimize Excel while the simulation is running,
thereby reducing the memory required to visibly update the Excel spreadsheet and
freeing up the memory to run the simulation. The Clear All and Minimize All controls
all the open forecast charts.

Options: As shown in Figure 2.8B, this forecast chart feature allows you to show all
the forecast data or to filter in/out values that fall within either some specified interval
or some standard deviation you choose. Also, the precision level can be set here for
this specific forecast to show the error levels in the statistics view. See the section on
error and precision control later in this chapter for more details. Show the following
statistic on histogram is a user preference for whether the mean, median, first quartile,
and fourth quartile lines (25th and 75th percentiles) should be displayed on the
forecast chart.

Controls: As shown in Figure 2.8C, this tab has all the functionalities in allowing you
to change the type, color, size, zoom, tilt, 3D, and other things in the forecast chart, as
well as to generate overlay charts (PDF, CDF) and run distributional fitting on your
forecast data (see the Data Fitting sections for more details on this methodology).

Global View versus Normal View: Figures 2.8A to 2.8C show the forecast chart’s
Normal View where the forecast chart user interface is divided into tabs, making it
small and compact. In contrast, Figure 2.9 shows the Global View where all elements
are located in a single interface. The results are identical in both views and selecting
which view is a matter of personal preference. You can switch between these two
views by clicking on the link, located at the top right corner, called Global View and
Local View.
Figure 2.8A – Forecast Chart Preferences
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Figure 2.8B – Forecast Chart Options
Figure 2.8C – Forecast Chart Controls
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Figure 2.9 – Forecast Chart Global View
Using Forecast
Charts and
Confidence
Intervals
In forecast charts, you can determine the probability of occurrence called confidence intervals.
That is, given two values, what are the chances that the outcome will fall between these two
values? Figure 2.10 illustrates that there is a 90% probability that the final outcome (in this case,
the level of income) will be between $0.2653 and $1.3230. The two-tailed confidence interval
can be obtained by first selecting Two-Tail as the type, entering the desired certainty value (e.g.,
90) and hitting TAB on the keyboard. The two computed values corresponding to the certainty
value will then be displayed. In this example, there is a 5% probability that income will be below
$0.2653 and another 5% probability that income will be above $1.3230. That is, the two-tailed
confidence interval is a symmetrical interval centered on the median, or 50th percentile, value.
Thus, both tails will have the same probability.
Alternatively, a one-tail probability can be computed. Figure 2.11 shows a left-tail selection at
95% confidence (i.e., choose Left-Tail ≤ as the type, enter 95 as the certainty level, and hit TAB
on the keyboard). This means that there is a 95% probability that the income will be below
$1.3230 or a 5% probability that income will be above $1.3230, corresponding perfectly with
the results seen in Figure 2.10.
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Figure 2.10 – Forecast Chart Two-Tail Confidence Interval
Figure 2.11 – Forecast Chart One-Tail Confidence Interval
In addition to evaluating what the confidence interval is (i.e., given a probability level and
finding the relevant income values), you can determine the probability of a given income value.
For instance, what is the probability that income will be less than or equal to $1? To obtain the
answer, select the Left-Tail ≤ probability type, enter 1 into the value input box, and hit TAB.
The corresponding certainty will then be computed (in this case, as shown in Figure 2.12, there
is a 67.70% probability income will be at or below $1).
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For the sake of completeness, you can select the Right-Tail > probability type, and enter the
value 1 in the value input box, and hit TAB. The resulting probability indicates the right-tail
probability past the value 1, that is, the probability of income exceeding $1 (in this case, as
shown in Figure 2.13, we see that there is a 32.30% probability of income exceeding $1). The
sum of 67.70% and 32.30% is, of course, 100%, the total probability under the curve.
Figure 2.12 – Forecast Chart Probability Evaluation
Figure 2.13 – Forecast Chart Probability Evaluation
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TIPS
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
The forecast window is resizable by clicking on and dragging the bottom right corner
of the forecast window.

It is also advisable that the current simulation be reset (Risk Simulator | Reset Simulation)
before rerunning a simulation.

Remember that you will need to hit TAB on the keyboard to update the chart and
results when you type in the certainty values or right- and left-tail values.

You can also hit the spacebar on the keyboard repeatedly to cycle among the
histogram to statistics, preferences, options, and control tabs.

In addition, if you click on Risk Simulator | Options you can access several different
options for Risk Simulator, including allowing Risk Simulator to start each time Excel
starts or to only start when you want it to (by going to Start | Programs | Real Options
Valuation | Risk Simulator | Risk Simulator), changing the cell colors of assumptions and
forecasts, and turning cell comments on and off (cell comments will allow you to see
which cells are input assumptions and which are output forecasts as well as their
respective input parameters and names). Do spend some time playing around with the
forecast chart outputs and various bells and whistles, especially the Controls tab.
2.3 Корреляции иКонтроль точности
2.3.1 The Basics of Correlations
The correlation coefficient is a measure of the strength and direction of the relationship
between two variables, and it can take on any value between –1.0 and +1.0. That is, the
correlation coefficient can be decomposed into its sign (positive or negative relationship
between two variables) and the magnitude or strength of the relationship (the higher the
absolute value of the correlation coefficient, the stronger the relationship).
The correlation coefficient can be computed in several ways. The first approach is to manually
compute the correlation, r, of two variables, x and y, using:
rx , y 
n xi y i   xi  y i
n xi2   xi 
2
n y i2   y i 
2
The second approach is to use Excel’s CORREL function. For instance, if the 10 data points
for x and y are listed in cells A1:B10, then the Excel function to use is CORREL (A1:A10,
B1:B10).
The third approach is to run Risk Simulator’s Multi-Fit Tool, and the resulting correlation
matrix will be computed and displayed.
It is important to note that correlation does not imply causation. Two completely unrelated
random variables might display some correlation but this does not imply any causation between
the two (e.g., sunspot activity and events in the stock market are correlated but there is no
causation between the two).
There are two general types of correlations: parametric and nonparametric correlations.
Pearson’s correlation coefficient is the most common correlation measure and is usually
referred to simply as the correlation coefficient. However, Pearson’s correlation is a parametric
measure, which means that it requires both correlated variables to have an underlying normal
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distribution and that the relationship between the variables is linear. When these conditions are
violated, which is often the case in Monte Carlo simulation, the nonparametric counterparts
become more important. Spearman’s rank correlation and Kendall’s tau are the two
alternatives. The Spearman correlation is most commonly used and is most appropriate when
applied in the context of Monte Carlo simulation––there is no dependence on normal
distributions or linearity, meaning that correlations between different variables with different
distribution can be applied. To compute the Spearman correlation, first rank all the x and y
variable values and then apply the Pearson’s correlation computation.
In the case of Risk Simulator, the correlation used is the more robust nonparametric
Spearman’s rank correlation. However, to simplify the simulation process, and to be consistent
with Excel’s correlation function, the correlation inputs required are the Pearson’s correlation
coefficient. Risk Simulator will then apply its own algorithms to convert them into Spearman’s
rank correlation, thereby simplifying the process. However, to simplify the user interface, we
allow users to enter the more common Pearson’s product-moment correlation (e.g., computed
using Excel’s CORREL function), while in the mathematical codes, we convert these simple
correlations into Spearman’s rank-based correlations for distributional simulations.
2.3.2 Applying Correlations in Risk Simulator
Correlations can be applied in Risk Simulator in several ways:

When defining assumptions (Risk Simulator │Set Input Assumption), simply enter the
correlations into the correlation matrix grid in the Distribution Gallery.

With existing data, run the Multi-Fit tool (Risk Simulator │ Tools │ Distributional Fitting
│ Multiple Variables) to perform distributional fitting and to obtain the correlation
matrix between pairwise variables. If a simulation profile exists, the assumptions fitted
will automatically contain the relevant correlation values.

With existing assumptions, you can click on Risk Simulator │Tools │Edit Correlations to
enter the pairwise correlations of all the assumptions directly in one user interface.
Note that the correlation matrix must be positive definite. That is, the correlation must be
mathematically valid. For instance, suppose you are trying to correlate three variables: grades of
graduate students in a particular year, the number of beers they consume a week, and the
number of hours they study a week. One would assume that the following correlation
relationships exist:
Grades and Beer: –
The more they drink, the lower the grades (no-show on exams)
Grades and Study: +
The more they study, the higher the grades
Beer and Study:
The more they drink, the less they study (drunk and partying)
–
However, if you input a negative correlation between Grades and Study, and assuming that the
correlation coefficients have high magnitudes, the correlation matrix will be nonpositive
definite. It would defy logic, correlation requirements, and matrix mathematics. However,
smaller coefficients can sometimes still work even with the bad logic. When a nonpositive or
bad correlation matrix is entered, Risk Simulator will automatically inform you, and offers to
adjust these correlations to something that is semipositive definite while still maintaining the
overall structure of the correlation relationship (the same signs as well as the same relative
strengths).
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2.3.3 The Effects of Correlations in Monte Carlo Simulation
Although the computations required to correlate variables in a simulation are complex, the
resulting effects are fairly clear. Figure 2.14 shows a simple correlation model (Correlation
Effects Model in the example folder). The calculation for revenue is simply price multiplied by
quantity. The same model is replicated for no correlations, positive correlation (+0.8), and
negative correlation (–0.8) between price and quantity.
Figure 2.14 – Simple Correlation Model
The resulting statistics are shown in Figure 2.15. Notice that the standard deviation of the
model without correlations is 0.1450, compared to 0.1886 for the positive correlation and
0.0717 for the negative correlation. That is, for simple models, negative correlations tend to
reduce the average spread of the distribution and create a tight and more concentrated forecast
distribution as compared to positive correlations with larger average spreads. However, the
mean remains relatively stable. This implies that correlations do little to change the expected
value of projects but can reduce or increase a project’s risk.
Figure 2.15 – Correlation Results
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Figure 2.16 illustrates the results after running a simulation, extracting the raw data of the
assumptions and computing the correlations between the variables. The figure shows that the
input assumptions are recovered in the simulation. That is, you enter +0.8 and –0.8 correlations
and the resulting simulated values have the same correlations.
Figure 2.16 – Correlations Recovered
2.3.4 Precision and Error Control
One very powerful tool in Monte Carlo simulation is that of precision control. For instance,
how many trials are considered sufficient to run in a complex model? Precision control takes
the guesswork out of estimating the relevant number of trials by allowing the simulation to stop
if the level of prespecified precision is reached.
The precision control functionality lets you set how precise you want your forecast to be.
Generally speaking, as more trials are calculated, the confidence interval narrows and the
statistics become more accurate. The precision control feature in Risk Simulator uses the
characteristic of confidence intervals to determine when a specified accuracy of a statistic has
been reached. For each forecast, you can set the specific confidence interval for the precision
level.
Make sure that you do not confuse three very different terms: error, precision, and confidence.
Although they sound similar, the concepts are significantly different from one another. A
simple illustration is in order. Suppose you are a taco shell manufacturer and are interested in
finding out how many broken taco shells there are on average in a box of 100 shells. One way
to do this is to collect a sample of prepackaged boxes of 100 taco shells, open them, and count
how many of them are actually broken. You manufacture 1 million boxes a day (this is your
population) but you randomly open only 10 boxes (this is your sample size, also known as your
number of trials in a simulation). The number of broken shells in each box is as follows: 24, 22,
4, 15, 33, 32, 4, 1, 45, and 2. The calculated average number of broken shells is 18.2. Based on
these 10 samples or trials, the average is 18.2 units, while based on the sample, the 80%
confidence interval is between 2 and 33 units (that is, 80% of the time, the number of broken
shells is between 2 and 33 based on this sample size or number of trials run). However, how
sure are you that 18.2 is the correct average? Are 10 trials sufficient to establish this? The
confidence interval between 2 and 33 is too wide and too variable. Suppose you require a more
accurate average value where the error is ±2 taco shells 90% of the time––this means that if
you open all 1 million boxes manufactured in a day, 900,000 of these boxes will have broken
taco shells on average at some mean unit ±2 taco shells. How many more taco shell boxes
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would you then need to sample (or trials run) to obtain this level of precision? Here, the 2 taco
shells is the error level while the 90% is the level of precision. If sufficient numbers of trials are
run, then the 90% confidence interval will be identical to the 90% precision level, where a more
precise measure of the average is obtained such that 90% of the time, the error and, hence, the
confidence will be ±2 taco shells. As an example, say the average is 20 units, then the 90%
confidence interval will be between 18 and 22 units with this interval being precise 90% of the
time, where in opening all 1 million boxes, 900,000 of them will have between 18 and 22
broken taco shells. The number of trials required to hit this precision is based on the sampling
error equation of x  Z
s
n
, where Z
s
n
is the error of 2 taco shells, x is the sample
average, Z is the standard-normal Z-score obtained from the 90% precision level, s is the
sample standard deviation, and n is the number of trials required to hit this level of error with
the specified precision. Figures 2.17 and 2.18 illustrate how precision control can be performed
on multiple simulated forecasts in Risk Simulator. This feature prevents the user from having to
decide how many trials to run in a simulation and eliminates all possibilities of guesswork.
Figure 2.17 illustrates the forecast chart with a 95% precision level set. This value can be
changed and will be reflected in the Statistics tab as shown in Figure 2.18.
Figure 2.17 – Setting the Forecast’s Precision Level
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Figure 2.18 – Computing the Error
2.3.5 Понимание Статистического Прогнозирования
Most distributions can be defined up to four moments. The first moment describes a
distribution’s location or central tendency (expected returns); the second moment describes its
width or spread (risks); the third moment, its directional skew (most probable events); and the
fourth moment, its peakedness or thickness in the tails (catastrophic losses or gains). All four
moments should be calculated in practice and interpreted to provide a more comprehensive
view of the project under analysis. Risk Simulator provides the results of all four moments in its
Statistics view in the forecast charts.
Measuring the
Center of the
Distribution––the
First Moment
The first moment of a distribution measures the expected rate of return on a particular project.
It measures the location of the project’s scenarios and possible outcomes on average. The
common statistics for the first moment include the mean (average), median (center of a
distribution), and mode (most commonly occurring value). Figure 2.19 illustrates the first
moment––where, in this case, the first moment of this distribution is measured by the mean
( or average, value.
1
1=2
2
1
1 ≠ 2
2
Skew = 0
KurtosisXS =
Figure 2.19 – First Moment
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Measuring the
Spread of the
Distribution––the
Second Moment
S I M U L A T O R
The second moment measures the spread of a distribution, which is a measure of risk. The
spread, or width, of a distribution measures the variability of a variable, that is, the potential that
the variable can fall into different regions of the distribution––in other words, the potential
scenarios of outcomes. Figure 2.20 illustrates two distributions with identical first moments
(identical means) but very different second moments or risks. The visualization becomes clearer
in Figure 2.21. As an example, suppose there are two stocks and the first stock’s movements
(illustrated by the darker line) with the smaller fluctuation is compared against the second
stock’s movements (illustrated by the dotted line) with a much higher price fluctuation. Clearly
an investor would view the stock with the wilder fluctuation as riskier because the outcomes of
the more risky stock are relatively more unknown than the less risky stock. The vertical axis in
Figure 2.21 measures the stock prices, thus, the more risky stock has a wider range of potential
outcomes. This range is translated into a distribution’s width (the horizontal axis) in Figure 2.20,
where the wider distribution represents the riskier asset. Hence, width, or spread, of a
distribution measures a variable’s risks.
Notice that in Figure 2.20, both distributions have identical first moments, or central
tendencies, but the distributions are clearly very different. This difference in the distributional
width is measurable. Mathematically and statistically, the width, or risk, of a variable can be
measured through several different statistics, including the range, standard deviation (),
variance, coefficient of variation, and percentiles.
2
1
Skew = 0
KurtosisXS =
1 = 2
Figure 2.20 – Second Moment
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Stock prices
Time
Figure 2.21 – Stock Price Fluctuations
Measuring the
Skew of the
Distribution––the
Third Moment
The third moment measures a distribution’s skewness, that is, how the distribution is pulled to
one side or the other. Figure 2.22 illustrates a negative-skew, or left-skew, where the tail of the
distribution points to the left. Figure 2.23 illustrates a positive-skew or right-skew, where the tail
of the distribution points to the right. The mean is always skewed toward the tail of the
distribution, while the median remains constant. Another way of seeing this relationship is that
the mean moves but the standard deviation, variance, or width may still remain constant. If the
third moment is not considered, then looking only at the expected returns (e.g., median or
mean) and risk (standard deviation), a positively skewed project might be incorrectly chosen!
For example, if the horizontal axis represents the net revenues of a project, then clearly a left, or
negatively, skewed distribution might be preferred because there is a higher probability of
greater returns (Figure 2.22) as compared to a higher probability for lower level returns (Figure
2.23). Thus, in a skewed distribution, the median is a better measure of returns, as the medians
for both Figures 2.22 and 2.23 are identical, risks are identical, and, hence, a project with a
negatively skewed distribution of net profits is a better choice. Failure to account for a project’s
distributional skewness may mean that the incorrect project could be chosen (e.g., two projects
may have identical first and second moments, that is, they both have identical returns and risk
profiles, but their distributional skews may be very different).
1 = 2
Skew < 0
KurtosisXS =
1
2
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Figure 2.22 – Third Moment (Left Skew)
1 = 2
Skew > 0
KurtosisXS =
1 ≠ 2
1
2
Figure 2.23 – Third Moment (Right Skew)
Measuring the
Catastrophic Tail
Events in a
Distribution––the
Fourth Moment
The fourth moment, or kurtosis, measures the peakedness of a distribution. Figure 2.24
illustrates this effect. The background (denoted by the dotted line) is a normal distribution with
a kurtosis of 3.0, or an excess kurtosis (KurtosisXS) of 0.0. Risk Simulator’s results show the
KurtosisXS value, using 0 as the normal level of kurtosis, which means that a negative
KurtosisXS indicates flatter tails (platykurtic distributions like the uniform distribution), while
positive values indicate fatter tails (leptokurtic distributions like the student’s t or lognormal
distributions). The distribution depicted by the bold line has a higher excess kurtosis, thus the
area under the curve is thicker at the tails with less area in the central body. This condition has
major impacts on risk analysis. As shown for the two distributions in Figure 2.24, the first three
moments (mean, standard deviation, and skewness) can be identical, but the fourth moment
(kurtosis) is different. This condition means that, although the returns and risks are identical,
the probabilities of extreme and catastrophic events (potential large losses or large gains)
occurring are higher for a high kurtosis distribution (e.g., stock market returns are leptokurtic,
or have high kurtosis). Ignoring a project’s kurtosis may be detrimental. Typically, a higher
excess kurtosis value indicates that the downside risks are higher (e.g., the Value at Risk of a
project might be significant).
1 = 2
Skew = 0
Kurtosis > 0
1 = 2
Figure 2.24 – Fourth Moment
The Functions of
Moments
Ever wonder why these risk statistics are called “moments”? In mathematical vernacular,
moment means raised to the power of some value. In other words, the third moment implies that
in an equation, three is most probably the highest power. In fact, the equations below illustrate
the mathematical functions and applications of some moments for a sample statistic. For
example, notice that the highest power for the first moment average is one, the second moment
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standard deviation is two, the third moment skew is three, and the highest power for the fourth
moment is four.
First Moment: Arithmetic Average or Simple Mean (Sample)
n
x
x
i 1
i
The Excel equivalent function is AVERAGE.
n
Second Moment: Standard Deviation (Sample)
n
s
( x
i
i 1
 x )2
n 1
The Excel equivalent function is STDEV for a sample standard deviation.
The Excel equivalent function is STDEVP for a population standard deviation.
Third Moment: Skew (Sample)
skew 
n
( xi  x ) 3
n
 s
( n  1 )( n  2 ) i 1
The Excel equivalent function is SKEW.
Fourth Moment: Kurtosis (Sample)
kurtosis 
n
( xi  x )4
n( n  1 )
3( n  1 )2

 s
( n  1 )( n  2 )( n  3 ) i 1
( n  2 )( n  3 )
The Excel equivalent function is KURT.
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2.3.6 Понимание распределения вероятностей для
моделирования Методом Монте-Карло
This section demonstrates the power of Monte Carlo simulation, but to get started with
simulation, one first needs to understand the concept of probability distributions. To begin to
understand probability, consider this example: You want to look at the distribution of
nonexempt wages within one department of a large company. First, you gather raw data––in
this case, the wages of each nonexempt employee in the department. Second, you organize the
data into a meaningful format and plot the data as a frequency distribution on a chart. To create
a frequency distribution, you divide the wages into group intervals and list these intervals on the
chart’s horizontal axis. Then you list the number or frequency of employees in each interval on
the chart’s vertical axis. Now you can easily see the distribution of nonexempt wages within the
department.
A glance at the chart illustrated in Figure 2.25 reveals that most of the employees
(approximately 60 out of a total of 180) earn from $7.00 to $9.00 per hour.
60
50
Number of
Employees
40
30
20
10
7.00 7.50 8.00 8.50 9.00
Hourly Wage Ranges in Dollars
Figure 2.25 – Frequency Histogram I
You can chart this data as a probability distribution. A probability distribution shows the
number of employees in each interval as a fraction of the total number of employees. To create
a probability distribution, you divide the number of employees in each interval by the total
number of employees and list the results on the chart’s vertical axis.
The chart in Figure 2.26 shows you the number of employees in each wage group as a fraction
of all employees; you can estimate the likelihood or probability that an employee drawn at
random from the whole group earns a wage within a given interval. For example, assuming the
same conditions exist at the time the sample was taken, the probability is 0.33 (a one in three
chance) that an employee drawn at random from the whole group earns between $8.00 and
$8.50 an hour.
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0.33
Probability
7.00 7.50 8.00 8.50 9.00
Hourly Wage Ranges in Dollars
Figure 2.26 – Frequency Histogram II
Probability distributions are either discrete or continuous. Discrete probability distributions
describe distinct values, usually integers, with no intermediate values and are shown as a series
of vertical bars. A discrete distribution, for example, might describe the number of heads in
four flips of a coin as 0, 1, 2, 3, or 4. Continuous distributions are actually mathematical
abstractions because they assume the existence of every possible intermediate value between
two numbers. That is, a continuous distribution assumes there is an infinite number of values
between any two points in the distribution. However, in many situations, you can effectively use
a continuous distribution to approximate a discrete distribution even though the continuous
model does not necessarily describe the situation exactly.
Selecting the Right
Probability
Distribution
Monte Carlo
Simulation
Plotting data is one guide to selecting a probability distribution. The following steps provide
another process for selecting probability distributions that best describe the uncertain variables
in your spreadsheets:

Look at the variable in question. List everything you know about the conditions
surrounding this variable. You might be able to gather valuable information about the
uncertain variable from historical data. If historical data are not available, use your own
judgment, based on experience, listing everything you know about the uncertain
variable.

Review the descriptions of the probability distributions.

Select the distribution that characterizes this variable. A distribution characterizes a
variable when the conditions of the distribution match those of the variable.
Monte Carlo simulation in its simplest form is a random number generator that is useful for
forecasting, estimation, and risk analysis. A simulation calculates numerous scenarios of a model
by repeatedly picking values from a user-predefined probability distribution for the uncertain
variables and using those values for the model. As all those scenarios produce associated results
in a model, each scenario can have a forecast. Forecasts are events (usually with formulas or
functions) that you define as important outputs of the model. These usually are events such as
totals, net profit, or gross expenses.
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Simplistically, think of the Monte Carlo simulation approach as repeatedly picking golf balls out
of a large basket with replacement. The size and shape of the basket depend on the
distributional input assumption (e.g., a normal distribution with a mean of 100 and a standard
deviation of 10, versus a uniform distribution or a triangular distribution) where some baskets
are deeper or more symmetrical than others, allowing certain balls to be pulled out more
frequently than others. The number of balls pulled repeatedly depends on the number of trials
simulated. For a large model with multiple related assumptions, imagine a very large basket
wherein many smaller baskets reside. Each small basket has its own set of golf balls that are
bouncing around. Sometimes these small baskets are linked with each other (if there is a
correlation between the variables) and the golf balls are bouncing in tandem, while other times
the balls are bouncing independent of one another. The balls that are picked each time from
these interactions within the model (the large central basket) are tabulated and recorded,
providing a forecast output result of the simulation.
With Monte Carlo simulation, Risk Simulator generates random values for each assumption’s
probability distribution that are totally independent. In other words, the random value selected
for one trial has no effect on the next random value generated. Use Monte Carlo sampling
when you want to simulate real-world what-if scenarios for your spreadsheet model.
The two following sections provide a detailed listing of the different types of discrete and
continuous probability distributions that can be used in Monte Carlo simulation.
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2.4 Дискретные распределения
Bernoulli or Yes/No
Distribution
The Bernoulli distribution is a discrete distribution with two outcomes (e.g., head or tails,
success or failure, 0 or 1). It is the binomial distribution with one trial and can be used to
simulate Yes/No or Success/Failure conditions. This distribution is the fundamental building
block of other more complex distributions. For instance:

Binomial distribution: a Bernoulli distribution with higher number of n total trials that
computes the probability of x successes within this total number of trials.

Geometric distribution: a Bernoulli distribution with higher number of trials that
computes the number of failures required before the first success occurs.

Negative binomial distribution: a Bernoulli distribution with higher number of trials
that computes the number of failures before the Xth success occurs.
The mathematical constructs for the Bernoulli distribution are as follows:
1  p for x  0
P ( n)  
for x  1
p
or
P (n)  p x (1  p )1 x
Mean  p
Standard Deviation 
Skewness =
p (1  p )
1 2p
p (1  p )
2
Excess Kurtosis = 6 p  6 p  1
p(1  p )
Probability of success (p) is the only distributional parameter. Also, it is important to note that
there is only one trial in the Bernoulli distribution, and the resulting simulated value is either 0
or 1.
Input requirements:
Probability of success > 0 and < 1 (i.e., 0.0001 ≤ p ≤ 0.9999).
Binomial
Distribution
The binomial distribution describes the number of times a particular event occurs in a fixed
number of trials, such as the number of heads in 10 flips of a coin or the number of defective
items out of 50 items chosen.
Conditions
The three conditions underlying the binomial distribution are:

For each trial, only two outcomes are possible that are mutually exclusive.

The trials are independent––what happens in the first trial does not affect the next
trial.
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
S I M U L A T O R
The probability of an event occurring remains the same from trial to trial.
The mathematical constructs for the binomial distribution are as follows:
P ( x) 
n!
p x (1  p) ( n  x ) for n  0; x  0, 1, 2, ... n; and 0  p  1
x!(n  x)!
Mean  np
Standard Deviation  np (1  p )
Skewness =
1 2 p
np(1  p )
2
Excess Kurtosis = 6 p  6 p  1
np(1  p)
Probability of success (p) and the integer number of total trials (n) are the distributional
parameters. The number of successful trials is denoted x. It is important to note that probability
of success (p) of 0 or 1 are trivial conditions that do not require any simulations and, hence, are
not allowed in the software.
Input requirements:
Probability of success > 0 and < 1 (i.e., 0.0001 ≤ p ≤ 0.9999).
Number of trials ≥ 1 or positive integers and ≤ 1000 (for larger trials, use the normal
distribution with the relevant computed binomial mean and standard deviation as the normal
distribution’s parameters).
Discrete Uniform
The discrete uniform distribution is also known as the equally likely outcomes distribution, where
the distribution has a set of N elements and each element has the same probability. This
distribution is related to the uniform distribution but its elements are discrete and not
continuous.
The mathematical constructs for the discrete uniform distribution are as follows:
P( x) 
1
N
N 1
Mean = 2 ranked value
Standard Deviation =
( N  1)( N  1)
12
ranked value
Skewness = 0 (i.e., the distribution is perfectly symmetrical)
 6( N 2  1)
Excess Kurtosis = 5( N  1)( N  1) ranked value
Input requirements:
Minimum < maximum and both must be integers (negative integers and zero are allowed).
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Geometric
Distribution
S I M U L A T O R
The geometric distribution describes the number of trials until the first successful occurrence,
such as the number of times you need to spin a roulette wheel before you win.
Conditions
The three conditions underlying the geometric distribution are:

The number of trials is not fixed.

The trials continue until the first success.

The probability of success is the same from trial to trial.
The mathematical constructs for the geometric distribution are as follows:
P( x)  p(1  p) x 1 for 0  p  1 and x  1, 2, ..., n
Mean 
1
1
p
Standard Deviation 
1 p
p2
Skewness = 2  p
1 p
2
Excess Kurtosis = p  6 p  6
1 p
Probability of success (p) is the only distributional parameter. The number of successful trials
simulated is denoted x, which can only take on positive integers.
Input requirements:
Probability of success > 0 and < 1 (i.e., 0.0001 ≤ p ≤ 0.9999). It is important to note that
probability of success (p) of 0 or 1 are trivial conditions that do not require any simulations and,
hence, are not allowed in the software.
Hypergeometric
Distribution
The hypergeometric distribution is similar to the binomial distribution in that both describe the
number of times a particular event occurs in a fixed number of trials. The difference is that
binomial distribution trials are independent, whereas hypergeometric distribution trials change
the probability for each subsequent trial and are called “trials without replacement.” For
example, suppose a box of manufactured parts is known to contain some defective parts. You
choose a part from the box, find it is defective, and remove the part from the box. If you
choose another part from the box, the probability that it is defective is somewhat lower than
for the first part because you have already removed a defective part. If you had replaced the
defective part, the probabilities would have remained the same, and the process would have
satisfied the conditions for a binomial distribution.
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Conditions
The three conditions underlying the hypergeometric distribution are:

The total number of items or elements (the population size) is a fixed number, a finite
population. The population size must be less than or equal to 1,750.

The sample size (the number of trials) represents a portion of the population.

The known initial probability of success in the population changes after each trial.
The mathematical constructs for the hypergeometric distribution are as follows:
( N x )!
( N  N x )!
x!( N x  x)! (n  x)!( N  N x  n  x)!
for x  Max(n  ( N  N x ),0), ..., Min(n, N x )
P ( x) 
N!
n!( N  n)!
Mean = N x n
N
Standard Deviation =
Skewness =
( N  N x ) N x n( N  n)
N 2 ( N  1)
N 1
( N  N x ) N x n ( N  n)
Excess Kurtosis = complex function
The number of items in the population or Population Size (N), trials sampled or Sample Size
(n), and number of items in the population that have the successful trait or Population
Successes (Nx) are the distributional parameters. The number of successful trials is denoted x.
Input requirements:
Population Size ≥ 2 and integer.
Sample Size > 0 and integer.
Population Successes > 0 and integer.
Population Size > Population Successes.
Sample Size < Population Successes.
Population Size < 1750.
Negative Binomial
Distribution
The negative binomial distribution is useful for modeling the distribution of the number of
additional trials required in addition to the number of successful occurrences required (R). For
instance, in order to close a total of 10 sales opportunities, how many extra sales calls would
you need to make above 10 calls given some probability of success in each call? The x-axis
shows the number of additional calls required or the number of failed calls. The number of
trials is not fixed, the trials continue until the Rth success, and the probability of success is the
same from trial to trial. Probability of success (p) and number of successes required (R) are the
distributional parameters. It is essentially a superdistribution of the geometric and binomial
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distributions. This distribution shows the probabilities of each number of trials in excess of R to
produce the required success R.
Conditions
The three conditions underlying the negative binomial distribution are:

The number of trials is not fixed.

The trials continue until the rth success.

The probability of success is the same from trial to trial.
The mathematical constructs for the negative binomial distribution are as follows:
P( x) 
( x  r  1)! r
p (1  p) x for x  r , r  1, ...; and 0  p  1
(r  1)! x!
Mean 
r (1  p)
p
Standard Deviation 
Skewness =
r (1  p)
p2
2 p
r (1  p )
2
Excess Kurtosis = p  6 p  6
r (1  p)
Probability of success (p) and required successes (R) are the distributional parameters.
Input requirements:
Successes required must be positive integers > 0 and < 8000.
Probability of success > 0 and < 1 (that is, 0.0001 ≤ p ≤ 0.9999). It is important to note that
probability of success (p) of 0 or 1 are trivial conditions that do not require any simulations and,
hence, are not allowed in the software.
Pascal Distribution
The Pascal distribution is useful for modeling the distribution of the number of total trials
required to obtain the number of successful occurrences required. For instance, to close a total
of 10 sales opportunities, how many total sales calls would you need to make given some
probability of success in each call? The x-axis shows the total number of calls required, which
includes successful and failed calls. The number of trials is not fixed, the trials continue until the
Rth success, and the probability of success is the same from trial to trial. Pascal distribution is
related to the negative binomial distribution. Negative binomial distribution computes the
number of events required in addition to the number of successes required given some
probability (in other words, the total failures), whereas the Pascal distribution computes the
total number of events required (in other words, the sum of failures and successes) to achieve
the successes required given some probability. Successes required and probability, are the two
distributional parameters.
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Conditions
The three conditions underlying the negative binomial distribution are:

The number of trials is not fixed.

The trials continue until the rth success.

The probability of success is the same from trial to trial.
The mathematical constructs for the Pascal distribution are shown below:
 ( x  1)!
p S (1  p ) X  S for all x  s

f ( x)   ( x  s )!( s  1)!
0 otherwise

( x  1)!
 k
p S (1  p ) X  S for all x  s

F ( x)   x 1 ( x  s )!( s  1)!
0 otherwise

Mean 
s
p
Standard Deviation  s (1  p ) p 2
Skewness =
2 p
r (1  p )
Excess Kurtosis =
p2  6 p  6
r (1  p)
Successes Required and Probability are the distributional parameters.
Input requirements:
Successes required > 0 and is an integer.
0 ≤ Probability ≤ 1.
Poisson
Distribution
The Poisson distribution describes the number of times an event occurs in a given interval,
such as the number of telephone calls per minute or the number of errors per page in a
document.
Conditions
The three conditions underlying the Poisson distribution are:

The number of possible occurrences in any interval is unlimited.

The occurrences are independent. The number of occurrences in one interval does not
affect the number of occurrences in other intervals.

The average number of occurrences must remain the same from interval to interval.
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The mathematical constructs for the Poisson are as follows:
P( x) 
e  x
for x and   0
x!
Mean  
Standard Deviation =
Skewness =

1

Excess Kurtosis =
1

Rate, or Lambda (), is the only distributional parameter.
Input requirements:
Rate > 0 and ≤ 1000 (i.e., 0.0001 ≤ rate ≤ 1000).
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2.5 Непрерывные распределения
Arcsine
Distribution
The arcsine distribution is U-shaped and is a special case of the beta distribution when both
shape and scale are equal to 0.5. Values close to the minimum and maximum have high
probabilities of occurrence whereas values between these two extremes have very small
probabilities of occurrence. Minimum and maximum are the distributional parameters.
The mathematical constructs for the Arcsine distribution are shown below. The probability
density function (PDF) is denoted f(x) and the cumulative distribution function (CDF) is
denoted F(x).
1

for 0  x  1

f ( x )    x (1  x)
0
otherwise

x0
0
2

F ( x)   sin 1 ( x ) for 0  x  1

x 1
1
Mean 
Min  Max
2
Standard Deviation 
( Max  Min) 2
8
Skewness = 0 for all inputs
Excess Kurtosis = 1.5 for all inputs
Minimum and maximum are the distributional parameters.
Input requirements:
Maximum > minimum (either input parameter can be positive, negative, or zero).
Beta Distribution
The beta distribution is very flexible and is commonly used to represent variability over a fixed
range. One of the more important applications of the beta distribution is its use as a conjugate
distribution for the parameter of a Bernoulli distribution. In this application, the beta
distribution is used to represent the uncertainty in the probability of occurrence of an event. It
is also used to describe empirical data and predict the random behavior of percentages and
fractions, as the range of outcomes is typically between 0 and 1.
The value of the beta distribution lies in the wide variety of shapes it can assume when you vary
the two parameters, alpha and beta. If the parameters are equal, the distribution is symmetrical.
If either parameter is 1 and the other parameter is greater than 1, the distribution is J-shaped. If
alpha is less than beta, the distribution is said to be positively skewed (most of the values are
near the minimum value). If alpha is greater than beta, the distribution is negatively skewed
(most of the values are near the maximum value).
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The mathematical constructs for the beta distribution are as follows:
f ( x) 
Mean 
x ( 1) 1  x (  1)
 ( )(  ) 
 (   ) 


for   0;   0; x  0

 
Standard Deviation 

(   ) (1     )
2
Skewness = 2(    ) 1    
(2     ) 
2
Excess Kurtosis = 3(    1)[ (    6)  2(   ) ]  3
 (    2)(    3)
Alpha () and beta () are the two distributional shape parameters, and  is the Gamma
function.
Conditions
The two conditions underlying the beta distribution are:

The uncertain variable is a random value between 0 and a positive value.

The shape of the distribution can be specified using two positive values.
Input requirements:
Alpha and beta both > 0 and can be any positive value.
Beta 3 and Beta 4
Distributions
The original Beta distribution only takes two inputs, Alpha and Beta shape parameters.
However, the output of the simulated value is between 0 and 1. In the Beta 3 distribution, we
add an extra parameter called Location or Shift, where we are not free to move away from this
0 to 1 output limitation, therefore the Beta 3 distribution is also known as a Shifted Beta
distribution. Similarly, the Beta 4 distribution adds two input parameters, Location or Shift, and
Factor. The original beta distribution is multiplied by the factor and shifted by the location, and,
therefore the Beta 4 is also known as the Multiplicative Shifted Beta distribution.
The mathematical constructs for the Beta 3 and Beta 4 distributions are based on those in the
Beta distribution, with the relevant shifts and factorial multiplication (e.g., the PDF and CDF
will be adjusted by the shift and factor, and some of the moments, such as the mean, will
similarly be affected; the standard deviation, in contrast, is only affected by the factorial
multiplication, whereas the remaining moments are not affected at all).
Input requirements:
Location >=< 0 (location can take on any positive or negative value including zero).
Factor > 0.
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Cauchy
Distribution, or
Lorentzian or BreitWigner Distribution
S I M U L A T O R
The Cauchy distribution, also called the Lorentzian or Breit-Wigner distribution, is a continuous
distribution describing resonance behavior. It also describes the distribution of horizontal
distances at which a line segment tilted at a random angle cuts the x-axis.
The mathematical constructs for the cauchy or Lorentzian distribution are as follows:
f ( x) 
1
 /2
 ( x  m) 2   2 / 4
The Cauchy distribution is a special case because it does not have any theoretical moments
(mean, standard deviation, skewness, and kurtosis) as they are all undefined.
Mode location () and scale () are the only two parameters in this distribution. The location
parameter specifies the peak or mode of the distribution, while the scale parameter specifies the
half-width at half-maximum of the distribution. In addition, the mean and variance of a
Cauchy, or Lorentzian, distribution are undefined.
In addition, the Cauchy distribution is the Student’s T distribution with only 1 degree of
freedom. This distribution is also constructed by taking the ratio of two standard normal
distributions (normal distributions with a mean of zero and a variance of one) that are
independent of one another.
Input requirements:
Location (Alpha) can be any value.
Scale (Beta) > 0 and can be any positive value.
Chi-Square
Distribution
The chi-square distribution is a probability distribution used predominantly in hypothesis
testing, and is related to the gamma and standard normal distributions. For instance, the sum of
independent normal distributions is distributed as a chi-square () with k degrees of freedom:
d
Z 12  Z 22  ...  Z k2 ~  k2
The mathematical constructs for the chi-square distribution are as follows:
f ( x) 
0.5  k / 2 k / 21  x / 2 for all x > 0
x
e
(k / 2)
Mean = k
Standard Deviation = 2k
Skewness = 2 2
k
Excess Kurtosis = 12
k
 is the gamma function. Degrees of freedom, k, is the only distributional parameter.
The chi-square distribution can also be modeled using a gamma distribution by setting the
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Shape parameter equal to
k
and the scale equal to 2S 2 where S is the scale.
2
Input requirements:
Degrees of freedom > 1 and must be an integer < 300.
Cosine Distribution
The cosine distribution looks like a logistic distribution where the median value between the
minimum and maximum have the highest peak or mode, carrying the maximum probability of
occurrence, while the extreme tails close to the minimum and maximum values have lower
probabilities. Minimum and maximum are the distributional parameters.
The mathematical constructs for the Cosine distribution are shown below:
1
xa
for min  x  max
 cos 
f ( x)   2b
 b 
0
otherwise

min  max
max  min
where a 
and b 

2
1 
 x  a 
 1  sin 
  for min  x  max
F ( x)   2 
 b 
1
for x > max

Mean 
Min  Max
2
Standard Deviation =
( Max  Min)2 ( 2  8)
4 2
Skewness is always equal to 0
Excess Kurtosis =
6(90   4 )
5( 2  6) 2
Minimum and maximum are the distributional parameters.
Input requirements:
Maximum > minimum (either input parameter can be positive, negative, or zero).
Double Log
Distribution
The double log distribution looks like the Cauchy distribution where the central tendency is
peaked and carries the maximum value probability density but declines faster the further it gets
away from the center, creating a symmetrical distribution with an extreme peak in between the
minimum and maximum values. Minimum and maximum are the distributional parameters.
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The mathematical constructs for the Double Log distribution are shown below:
 1  x  a 
 ln 
 for min  x  max
f ( x)   2b  b 

otherwise
0
where a 
min  max
max  min
and b 
2
2
1 
 
2 
F ( x)  
1 
2  


Mean =
 x  a 
xa 
 1  ln 
  for min  x  a
2b  
 b  
xa 
 x  a 
 1  ln 
  for a  x  max
2b  
 b  
Min  Max
2
Standard Deviation =
( Max  Min) 2
36
Skewness is always equal to 0
Excess Kurtosis is a complex function and not easily represented
Minimum and maximum are the distributional parameters.
Input requirements:
Maximum > minimum (either input parameter can be positive, negative, or zero).
Erlang Distribution
The Erlang distribution is the same as the Gamma distribution with the requirement that the
Alpha or shape parameter must be a positive integer. An example application of the Erlang
distribution is the calibration of the rate of transition of elements through a system of
compartments. Such systems are widely used in biology and ecology (e.g., in epidemiology, an
individual may progress at an exponential rate from being healthy to becoming a disease carrier,
and continue exponentially from being a carrier to being infectious). Alpha (also known as
shape) and Beta (also known as scale) are the distributional parameters.
The mathematical constructs for the Erlang distribution are shown below:
  x  1  x / 
  e
 
f ( x)    
for x  0
  (  1)
0
otherwise

 1

( x /  )i
 x/ 
for x  0
1  e 
F ( x)  
i!
i 0
0
otherwise

Mean  
Standard Deviation   2
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2
Skew 

Excess Kurtosis 
6

3
Alpha and Beta are the distributional parameters.
Input requirements:
Alpha (Shape) > 0 and is an Integer
Beta (Scale) > 0
Exponential
Distribution
The exponential distribution is widely used to describe events recurring at random points in
time, such as the time between failures of electronic equipment or the time between arrivals at a
service booth. It is related to the Poisson distribution, which describes the number of
occurrences of an event in a given interval of time. An important characteristic of the
exponential distribution is the “memoryless” property, which means that the future lifetime of a
given object has the same distribution regardless of the time it existed. In other words, time has
no effect on future outcomes.
Conditions
The condition underlying the exponential distribution is:

The exponential distribution describes the amount of time between occurrences.
The mathematical constructs for the exponential distribution are as follows:
f ( x)  e  x for x  0;   0
Mean =
1

Standard Deviation = 1

Skewness = 2 (this value applies to all success rate inputs)
Excess Kurtosis = 6 (this value applies to all success rate inputs)
Success rate () is the only distributional parameter. The number of successful trials is denoted
x.
Input requirements:
Rate > 0.
Exponential 2
Distribution
The Exponential 2 distribution uses the same constructs as the original Exponential distribution
but adds a Location or Shift parameter. The Exponential distribution starts from a minimum
value of 0, whereas this Exponential 2 or Shifted Exponential, distribution shifts the starting
location to any other value.
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Rate, or Lambda, and Location, or Shift, are the distributional parameters.
Input requirements:
Rate (Lambda) > 0.
Location can be any positive or negative value including zero.
Extreme Value
Distribution, or
Gumbel
Distribution
The extreme value distribution (Type 1) is commonly used to describe the largest value of a
response over a period of time, for example, in flood flows, rainfall, and earthquakes. Other
applications include the breaking strengths of materials, construction design, and aircraft loads
and tolerances. The extreme value distribution is also known as the Gumbel distribution.
The mathematical constructs for the extreme value distribution are as follows:
f ( x) 
1

x 
ze
Z
where z  e

for   0; and any value of x and 
Mean =   0.577215
Standard Deviation =
1 2 2
 
6
Skewness = 12 6 (1.2020569)  1.13955 (this applies for all values of mode and scale)
3
Excess Kurtosis = 5.4 (this applies for all values of mode and scale)
Mode () and scale () are the distributional parameters.
Calculating Parameters
There are two standard parameters for the extreme value distribution: mode and scale. The
mode parameter is the most likely value for the variable (the highest point on the probability
distribution). After you select the mode parameter, you can estimate the scale parameter. The
scale parameter is a number greater than 0. The larger the scale parameter, the greater the
variance.
Input requirements:
Mode Alpha can be any value.
Scale Beta > 0.
F Distribution, or
Fisher-Snedecor
Distribution
The F distribution, also known as the Fisher-Snedecor distribution, is another continuous
distribution used most frequently for hypothesis testing. Specifically, it is used to test the
statistical difference between two variances in analysis of variance tests and likelihood ratio
tests. The F distribution with the numerator degree of freedom n and denominator degree of
freedom m is related to the chi-square distribution in that:
 n2 / n d
~ Fn ,m
 m2 / m
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Mean =
m
m2
2
Standard Deviation = 2m (m  n  2) for all m > 4
n ( m  2) 2 ( m  4)
Skewness = 2(m  2n  2)
m6
Excess Kurtosis =
2( m  4)
n ( m  n  2)
12(16  20m  8m 2  m 3  44n  32mn  5m 2 n  22n 2  5mn 2
n(m  6)(m  8)(n  m  2)
The numerator degree of freedom n and denominator degree of freedom m are the only
distributional parameters.
Input requirements:
Degrees of freedom numerator & degrees of freedom denominator must both be integers > 0
Gamma
Distribution
(Erlang
Distribution)
The gamma distribution applies to a wide range of physical quantities and is related to other
distributions: lognormal, exponential, Pascal, Erlang, Poisson, and chi-square. It is used in
meteorological processes to represent pollutant concentrations and precipitation quantities. The
gamma distribution is also used to measure the time between the occurrence of events when
the event process is not completely random. Other applications of the gamma distribution
include inventory control, economic theory, and insurance risk theory.
Conditions
The gamma distribution is most often used as the distribution of the amount of time until the
rth occurrence of an event in a Poisson process. When used in this fashion, the three
conditions underlying the gamma distribution are:

The number of possible occurrences in any unit of measurement is not limited to a
fixed number.

The occurrences are independent. The number of occurrences in one unit of
measurement does not affect the number of occurrences in other units.

The average number of occurrences must remain the same from unit to unit.
The mathematical constructs for the gamma distribution are as follows:
 1
x

x
  e 

f ( x)   
( ) 
with any value of   0 and   0
Mean = 
Standard Deviation =  2
Skewness = 2

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Excess Kurtosis = 6

Shape parameter alpha () and scale parameter beta () are the distributional parameters, and 
is the Gamma function. When the alpha parameter is a positive integer, the gamma distribution
is called the Erlang distribution, used to predict waiting times in queuing systems, where the
Erlang distribution is the sum of independent and identically distributed random variables each
having a memoryless exponential distribution. Setting n as the number of these random
variables, the mathematical construct of the Erlang distribution is:
f ( x) 
x n 1e  x for all x > 0 and all positive integers of n
(n  1)!
Input requirements:
Scale beta > 0 and can be any positive value.
Shape alpha ≥ 0.05 and any positive value.
Location can be any value.
Laplace
Distribution
The Laplace distribution is also sometimes called the double exponential distribution because it
can be constructed with two exponential distributions (with an additional location parameter)
spliced together back-to-back, creating an unusual peak in the middle. The probability density
function of the Laplace distribution is reminiscent of the normal distribution. However,
whereas the normal distribution is expressed in terms of the squared difference from the mean,
the Laplace density is expressed in terms of the absolute difference from the mean, making the
Laplace distribution’s tails fatter than those of the normal distribution. When the location
parameter is set to zero, the Laplace distribution’s random variable is exponentially distributed
with an inverse of the scale parameter. Alpha (also known as location) and Beta (also known as
scale) are the distributional parameters.
The mathematical constructs for the Laplace distribution are shown below:
 x  
1
exp  

2
 

1
 x  
 exp 
 when x  
2

  
F ( x)  
1  1 exp   x    when x  

 2
 


f ( x) 
Mean  
Standard Deviation  1.4142 
Skewness is always equal to 0 as it is a symmetrical distribution
Excess Kurtosis is always equal to 3
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Input requirements:
Alpha (Location) can take on any positive or negative value including zero.
Beta (Scale) > 0.
Logistic
Distribution
The logistic distribution is commonly used to describe growth, that is, the size of a population
expressed as a function of a time variable. It also can be used to describe chemical reactions and
the course of growth for a population or individual.
The mathematical constructs for the logistic distribution are as follows:
e
f ( x) 

 x

 1  e

 x




for any value of 
2
and 
Mean  
Standard Deviation 
1 2 2
 
3
Skewness = 0 (this applies to all mean and scale inputs)
Excess Kurtosis = 1.2 (this applies to all mean and scale inputs)
Mean () and scale () are the distributional parameters.
Calculating Parameters
There are two standard parameters for the logistic distribution: mean and scale. The mean
parameter is the average value, which for this distribution is the same as the mode because this
is a symmetrical distribution. After you select the mean parameter, you can estimate the scale
parameter. The scale parameter is a number greater than 0. The larger the scale parameter, the
greater the variance.
Input requirements:
Scale Beta > 0 and can be any positive value.
Mean Alpha can be any value.
Lognormal
Distribution
The lognormal distribution is widely used in situations where values are positively skewed, for
example, in financial analysis for security valuation or in real estate for property valuation, and
where values cannot fall below zero.
Stock prices are usually positively skewed rather than normally (symmetrically) distributed.
Stock prices exhibit this trend because they cannot fall below the lower limit of zero but might
increase to any price without limit. Similarly, real estate prices illustrate positive skewness as
property values cannot become negative.
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Conditions
The three conditions underlying the lognormal distribution are:

The uncertain variable can increase without limits but cannot fall below zero.

The uncertain variable is positively skewed, with most of the values near the lower
limit.

The natural logarithm of the uncertain variable yields a normal distribution.
Generally, if the coefficient of variability is greater than 30%, use a lognormal distribution.
Otherwise, use the normal distribution.
The mathematical constructs for the lognormal distribution are as follows:
f ( x) 

1
x 2 ln( )
[ln( x )  ln(  )]2
e
for x  0;   0 and   0
2[ln( )]2

2 
Mean  exp   

2 

2
2
Standard Deviation = exp  2  exp    1
Skewness =
 exp
2
  1 (2  exp(
2
))
2
2
2
Excess Kurtosis = exp4   2 exp3   3 exp2   6
Mean () and standard deviation () are the distributional parameters.
Input requirements:
Mean and standard deviation both > 0 and can be any positive value.
Lognormal Parameter Sets
By default, the lognormal distribution uses the arithmetic mean and standard deviation. For
applications for which historical data are available, it is more appropriate to use either the
logarithmic mean and standard deviation, or the geometric mean and standard deviation.
Lognormal 3
Distribution
The Lognormal 3 distribution uses the same constructs as the original Lognormal distribution
but adds a Location, or Shift, parameter. The Lognormal distribution starts from a minimum
value of 0, whereas this Lognormal 3, or Shifted Lognormal distribution shifts the starting
location to any other value.
Mean, Standard Deviation, and Location (Shift) are the distributional parameters.
Input requirements:
Mean > 0.
Standard Deviation > 0.
Location can be any positive or negative value including zero.
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Normal
Distribution
S I M U L A T O R
The normal distribution is the most important distribution in probability theory because it
describes many natural phenomena, such as people’s IQs or heights. Decision makers can use
the normal distribution to describe uncertain variables such as the inflation rate or the future
price of gasoline.
Conditions
The three conditions underlying the normal distribution are:

Some value of the uncertain variable is the most likely (the mean of the distribution).

The uncertain variable could as likely be above the mean as it could be below the mean
(symmetrical about the mean).

The uncertain variable is more likely to be in the vicinity of the mean than further
away.
The mathematical constructs for the normal distribution are as follows:
f ( x) 
1
2 

e
( x )2
2 2
for all values of x
and ; while  > 0
Mean  
Standard Deviation  
Skewness = 0 (this applies to all inputs of mean and standard deviation)
Excess Kurtosis = 0 (this applies to all inputs of mean and standard deviation)
Mean () and standard deviation ) are the distributional parameters.
Input requirements:
Standard deviation > 0 and can be any positive value.
Mean can take on any value.
Parabolic
Distribution
The parabolic distribution is a special case of the beta distribution when Shape = Scale = 2.
Values close to the minimum and maximum have low probabilities of occurrence, whereas
values between these two extremes have higher probabilities or occurrence. Minimum and
maximum are the distributional parameters.
The mathematical constructs for the Parabolic distribution are shown below:
f ( x) 
x ( 1) 1  x (  1)
 ( )(  ) 
 (   ) 


for   0;   0; x  0
Where the functional form above is for a Beta distribution, and for a Parabolic function, we set
Alpha = Beta = 2 and a shift of location in Minimum, with a multiplicative factor of (Maximum
– Minimum).
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Mean =
Min  Max
2
( Max  Min) 2
20
Standard Deviation =
Skewness = 0
Excess Kurtosis = –0.8571
Minimum and Maximum are the distributional parameters.
Input requirements:
Maximum > minimum (either input parameter can be positive, negative, or zero).
Pareto Distribution
The Pareto distribution is widely used for the investigation of distributions associated with such
empirical phenomena as city population sizes, the occurrence of natural resources, the size of
companies, personal incomes, stock price fluctuations, and error clustering in communication
circuits.
The mathematical constructs for the Pareto are as follows:
f ( x) 
mean 
 L
x (1  )
for x  L
L
 1
standard deviation 
skewness =
L2
(   1) 2 (   2)
  2  2(   1) 
    3 
6(  3   2  6   2)
excess kurtosis =  (   3)(  4)
Shape () and Location () are the distributional parameters.
Calculating Parameters
There are two standard parameters for the Pareto distribution: location and shape. The location
parameter is the lower bound for the variable. After you select the location parameter, you can
estimate the shape parameter. The shape parameter is a number greater than 0, usually greater
than 1. The larger the shape parameter, the smaller the variance and the thicker the right tail of
the distribution.
Input requirements:
Location > 0 and can be any positive value
Shape ≥ 0.05.
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Pearson V
Distribution
S I M U L A T O R
The Pearson V distribution is related to the Inverse Gamma distribution, where it is the
reciprocal of the variable distributed according to the Gamma distribution. Pearson V
distribution is also used to model time delays where there is almost certainty of some minimum
delay and the maximum delay is unbounded, for example, delay in arrival of emergency services
and time to repair a machine. Alpha (also known as shape) and Beta (also known as scale) are
the distributional parameters.
The mathematical constructs for the Pearson V distribution are shown below:
x  ( 1) e  / x
  ( )
( ,  / x)
F ( x) 
( )
f ( x) 
Mean 

 1
Standard Deviation 
Skew 
2
(  1)2 (  2)
4  2
 3
Excess Kurtosis 
30  66
3
(  3)(  4)
Input requirements:
Alpha (Shape) > 0.
Beta (Scale) > 0.
Pearson VI
Distribution
The Pearson VI distribution is related to the Gamma distribution, where it is the rational
function of two variables distributed according to two Gamma distributions. Alpha 1 (also
known as shape 1), Alpha 2 (also known as shape 2), and Beta (also known as scale) are the
distributional parameters.
The mathematical constructs for the Pearson VI distribution are shown below:
f ( x) 
( x /  )1 1
 (1 ,  2 )[1  ( x /  )]1  2
 x 
F ( x)  FB 

 x 
59 | P a g e
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Mean 
1
2 1
Standard Deviation =
Skew  2
 21 (1   2  1)
( 2  1)2 ( 2  2)
 21   2  1 
2  2
1 (1   2  1)   2  3 
Excess Kurtosis 

3( 2  2)  2( 2  1) 2
 ( 2  5)   3

( 2  3)( 2  4)  1 (1   2  1)

Input requirements:
Alpha 1 (Shape 1) > 0.
Alpha 2 (Shape 2) > 0.
Beta (Scale) > 0.
PERT Distribution
The PERT distribution is widely used in project and program management to define the worstcase, nominal-case, and best-case scenarios of project completion time. It is related to the Beta
and Triangular distributions. PERT distribution can be used to identify risks in project and cost
models based on the likelihood of meeting targets and goals across any number of project
components using minimum, most likely, and maximum values, but it is designed to generate a
distribution that more closely resembles realistic probability distributions. The PERT
distribution can provide a close fit to the normal or lognormal distributions. Like the triangular
distribution, the PERT distribution emphasizes the "most likely" value over the minimum and
maximum estimates. However, unlike the triangular distribution, the PERT distribution
constructs a smooth curve that places progressively more emphasis on values around (near) the
most likely value, in favor of values around the edges. In practice, this means that we "trust" the
estimate for the most likely value, and we believe that even if it is not exactly accurate (as
estimates seldom are), we have an expectation that the resulting value will be close to that
estimate. Assuming that many real-world phenomena are normally distributed, the appeal of the
PERT distribution is that it produces a curve similar to the normal curve in shape, without
knowing the precise parameters of the related normal curve. Minimum, Most Likely, and
Maximum are the distributional parameters.
The mathematical constructs for the PERT distribution are shown below:
f ( x) 
( x  min) A11 (max  x ) A 21
B ( A1, A2)(max  min) A1 A 21
min  4(likely)  max 
 min  4(likely)  max


 min 

 max 

6
6
where A1  6 
 and A2  6 

max  min
max  min








and B is the Beta function
60 | P a g e
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Mean 
Min  4Mode  Max
6
Standard Deviation 
Skew 
(   Min)( Max   )
7
7
 Min  Max  2 


(   Min)( Max   ) 
4

Input requirements:
Minimum ≤ Most Likely ≤ Maximum and can be positive, negative, or zero.
Power Distribution
The Power distribution is related to the exponential distribution in that the probability of small
outcomes is large but exponentially decreases as the outcome value increases. Alpha (also
known as shape) is the only distributional parameter.
The mathematical constructs for the Power distribution are shown below:
f ( x)   x 1
F ( x )  x
Mean 

1 
Standard Deviation 
Skew 

(1   ) 2 (2   )
  2  2(  1) 
    3 
Excess Kurtosis is a complex function and cannot be readily computed
Input requirements:
Alpha > 0.
Power 3
Distribution
The Power 3 distribution uses the same constructs as the original Power distribution but adds a
Location, or Shift, parameter, and a multiplicative Factor parameter. The Power distribution
starts from a minimum value of 0, whereas this Power 3, or Shifted Multiplicative Power,
distribution shifts the starting location to any other value.
Alpha, Location or Shift, and Factor are the distributional parameters.
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Input requirements:
Alpha > 0.05.
Location, or Shift, can be any positive or negative value including zero.
Factor > 0.
Student’s t
Distribution
The Student’s t distribution is the most widely used distribution in hypothesis test. This
distribution is used to estimate the mean of a normally distributed population when the sample
size is small to test the statistical significance of the difference between two sample means or
confidence intervals for small sample sizes.
The mathematical constructs for the t distribution are as follows:
f (t ) 
 [(r  1) / 2]
r  [r / 2]
(1  t 2 / r ) ( r 1) / 2
Mean = 0 (this applies to all degrees of freedom r except if the distribution is shifted to another
nonzero central location)
Standard Deviation =
r
r2
Skewness = 0 (this applies to all degrees of freedom r)
Excess Kurtosis =
where t 
6
for all r  4
r4
xx
and  is the gamma function.
s
Degrees of freedom r is the only distributional parameter.
The t distribution is related to the F distribution as follows: the square of a value of t with r
degrees of freedom is distributed as F with 1 and r degrees of freedom. The overall shape of the
probability density function of the t distribution also resembles the bell shape of a normally
distributed variable with mean 0 and variance 1, except that it is a bit lower and wider or is
leptokurtic (fat tails at the ends and peaked center). As the number of degrees of freedom
grows (say, above 30), the t distribution approaches the normal distribution with mean 0 and
variance 1.
Input requirements:
Degrees of freedom ≥ 1 and must be an integer.
Triangular
Distribution
The triangular distribution describes a situation where you know the minimum, maximum, and
most likely values to occur. For example, you could describe the number of cars sold per week
when past sales show the minimum, maximum, and usual number of cars sold.
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Conditions
The three conditions underlying the triangular distribution are:

The minimum number of items is fixed.

The maximum number of items is fixed.

The most likely number of items falls between the minimum and maximum values,
forming a triangular-shaped distribution, which shows that values near the minimum
and maximum are less likely to occur than those near the most-likely value.
The mathematical constructs for the triangular distribution are as follows:
2( x  Min)

 ( Max  Min)( Likely  min) for Min  x  Likely
f ( x)  
2( Max  x)

for Likely  x  Max
 ( Max  Min)( Max  Likely )
Mean = 1 ( Min  Likely  Max)
3
Standard Deviation =
Skewness =
1
( Min 2  Likely 2  Max 2  Min Max  Min Likely  Max Likely)
18
2 ( Min  Max  2 Likely )( 2 Min  Max  Likely )( Min  2 Max  Likely )
5( Min 2  Max 2  Likely 2  MinMax  MinLikely  MaxLikely ) 3 / 2
Excess Kurtosis = –0.6 (this applies to all inputs of Min, Max, and Likely)
Minimum value (Min), most-likely value (Likely), and maximum value (Max) are the
distributional parameters.
Input requirements:
Min ≤ Most Likely ≤ Max and can take any value.
However, Min < Max and can take any value.
Uniform
Distribution
With the uniform distribution, all values fall between the minimum and maximum and occur
with equal likelihood.
Conditions
The three conditions underlying the uniform distribution are:

The minimum value is fixed.

The maximum value is fixed.

All values between the minimum and maximum occur with equal likelihood.
The mathematical constructs for the uniform distribution are as follows:
f ( x) 
1
for all values such that Min  Max
Max  Min
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Mean 
S I M U L A T O R
Min  Max
2
( Max  Min) 2
12
Standard Deviation 
Skewness = 0 (this applies to all inputs of Min and Max)
Excess Kurtosis = –1.2 (this applies to all inputs of Min and Max)
Maximum value (Max) and minimum value (Min) are the distributional parameters.
Input requirements:
Min < Max and can take any value.
Weibull
Distribution
(Rayleigh
Distribution)
The Weibull distribution describes data resulting from life and fatigue tests. It is commonly
used to describe failure time in reliability studies as well as the breaking strengths of materials in
reliability and quality control tests. Weibull distributions are also used to represent various
physical quantities, such as wind speed.
The Weibull distribution is a family of distributions that can assume the properties of several
other distributions. For example, depending on the shape parameter you define, the Weibull
distribution can be used to model the exponential and Rayleigh distributions, among others.
The Weibull distribution is very flexible. When the Weibull shape parameter is equal to 1.0, the
Weibull distribution is identical to the exponential distribution. The Weibull location parameter
lets you set up an exponential distribution to start at a location other than 0.0. When the shape
parameter is less than 1.0, the Weibull distribution becomes a steeply declining curve. A
manufacturer might find this effect useful in describing part failures during a burn-in period.
The mathematical constructs for the Weibull distribution are as follows:

f ( x) 

x
 
 
 1
e
x
 


Mean   (1   1 )
Standard Deviation   2  (1  2 1 )   2 (1   1 ) 
3
1
1
1
1
Skewness = 2 (1   )  3(1   )(1  2 )  (1  3 )
3
/
2
(1  2 1 )   2 (1   1 )
Excess Kurtosis =
 6 4 (1   1 )  12 2 (1   1 )(1  2 1 )  3 2 (1  2 1 )  4(1   1 )(1  3 1 )  (1  4 1 )
(1  2
1
)   2 (1   1 )

2
Shape () and central location scale () are the distributional parameters, and  is the Gamma
function.
Input requirements:
Shape Alpha ≥ 0.05.
Scale Beta > 0 and can be any positive value.
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Weibull 3
Distribution
S I M U L A T O R
The Weibull 3 distribution uses the same constructs as the original Weibull distribution but
adds a Location, or Shift, parameter. The Weibull distribution starts from a minimum value of
0, whereas this Weibull 3, or Shifted Weibull, distribution shifts the starting location to any
other value.
Alpha, Beta, and Location or Shift are the distributional parameters.
Input requirements:
Alpha (Shape) ≥ 0.05.
Beta (Central Location Scale) > 0 and can be any positive value.
Location can be any positive or negative value including zero.
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3
3. ПРОГНОЗИРОВАНИЕ
F
orecasting is the act of predicting the future. It can be based on historical data or
speculation about the future when no history exists. When historical data exist, a
quantitative or statistical approach is best, but if no historical data exist, then potentially
a qualitative or judgmental approach is usually the only recourse. Figure 3.1 lists the most
common methodologies for forecasting.
FORECASTING
QUANTITATIVE
CROSS-SECTIONAL
Econometric Models
Monte Carlo Simulation
Multiple Regression
Statistical Probabilities
QUALITATIVE
Use Risk Simulator’s
Forecast Tool for ARIMA,
Classical Decomposition,
Multivariate Regressions,
Nonlinear Regressions, Simulations
and Stochastic Processes
MIXED PANEL
ARIMA(X)
Multiple Regression
TIME-SERIES
ARIMA
Classical Decomposition
(8 Time-Series Models)
Multivariate Regression
Nonlinear Extrapolation
Stochastic Processes
Figure 3.1 – Forecasting Methods
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Delphi Method
Expert Opinions
Management Assumptions
Market Research
Polling Data
Surveys
Use Risk Simulator
to run Monte Carlo
Simulations (use
distributional fitting
or nonparametric
custom distributions)
R I S K
S I M U L A T O R
3.1 Различные типы методов прогнозирования
Generally, forecasting can be divided into quantitative and qualitative approaches. Qualitative
forecasting is used when little to no reliable historical, contemporaneous, or comparable data
are available. Several qualitative methods exist such as the Delphi, or expert opinion, approach
(a consensus-building forecast by field experts, marketing experts, or internal staff members),
management assumptions (target growth rates set by senior management), and market research
or external data or polling and surveys (data obtained from third-party sources, industry and
sector indexes, or active market research). These estimates can be either single-point estimates
(an average consensus) or a set of forecast values (a distribution of forecasts). The latter can be
entered into Risk Simulator as a custom distribution and the resulting forecasts can be
simulated, that is, a nonparametric simulation using the estimated data points themselves as the
distribution.
On the quantitative side of forecasting, the available data or data that need to be forecasted can
be divided into time-series (values that have a time element to them, such as revenues at
different years, inflation rates, interest rates, market share, failure rates), cross-sectional (values
that are time-independent, such as the grade point average of sophomore students across the
nation in a particular year, given each student’s levels of SAT scores, IQ, and number of
alcoholic beverages consumed per week), or mixed panel (mixture between time-series and
panel data, e.g., predicting sales over the next 10 years given budgeted marketing expenses and
market share projections, which means that the sales data is time series but exogenous variables,
such as marketing expenses and market share, exist to help to model the forecast predictions).
The Risk Simulator software provides the user several forecasting methodologies:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
ARIMA (Autoregressive Integrated Moving Average)
Auto ARIMA
Auto Econometrics
Basic Econometrics
Combinatorial Fuzzy Logic
Cubic Spline Curves
Custom Distributions
GARCH (Generalized Autoregressive Conditional Heteroskedasticity)
J Curve
Markov Chain
Maximum Likelihood (Logit, Probit, Tobit)
Multivariate Regression
Neural Network Forecasts
Nonlinear Extrapolation
S Curve
Stochastic Processes
Time-Series Analysis and Decomposition
Trendlines
The analytical details of each forecasting method fall outside the purview of this user manual.
For more details, please review Modeling Risk: Applying Monte Carlo Simulation, Real Options
Analysis, Stochastic Forecasting, and Portfolio Optimization, Second Edition, by Dr. Johnathan Mun
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(Wiley Finance, 2010), who is also the creator of the Risk Simulator software. Nonetheless, the
following illustrates some of the more common approaches and several quick getting started
examples in using the software. More detailed descriptions and example models of each of
these techniques are found throughout this chapter and the next. All other forecasting
approaches are fairly easy to apply within Risk Simulator.
ARIMA
Autoregressive integrated moving average (ARIMA, also known as Box-Jenkins ARIMA) is an
advanced econometric modeling technique. ARIMA looks at historical time-series data and
performs backfitting optimization routines to account for historical autocorrelation (the
relationship of one value versus another in time) and the stability of the data to correct for the
nonstationary characteristics of the data, and this predictive model learns over time by
correcting its forecasting errors. Advanced knowledge in econometrics is typically required to
build good predictive models using this approach.
Auto ARIMA
The Auto ARIMA module automates some of the traditional ARIMA modeling by
automatically testing multiple permutations of model specifications and returns the best-fitting
model. Running the Auto ARIMA is similar to regular ARIMA forecasts. The difference being
that the P, D, Q inputs are no longer required and different combinations of these inputs are
automatically run and compared.
Basic
Econometrics
Econometrics refers to a branch of business analytics, modeling, and forecasting techniques for
modeling the behavior of or forecasting certain business, economic, finance, physics,
manufacturing, operations, and any other variables. Running the Basic Econometrics models
are similar to regular regression analysis except that the dependent and independent variables
are allowed to be modified before a regression is run.
Auto Econometrics
Similar to basic econometrics, but Auto Econometrics allows thousands of linear, nonlinear,
interacting, lagged, and mixed variables to be automatically run on your data to determine the
best-fitting econometric model that describes the behavior of the dependent variable. It is
useful for modeling the effects of the variables and for forecasting future outcomes, while not
requiring the analyst to be an expert econometrician.
Combinatorial
Fuzzy Logic
In contrast, the term fuzzy logic is derived from fuzzy set theory to deal with reasoning that is
approximate rather than accurate––as opposed to crisp logic, where binary sets have binary
logic, fuzzy logic variables may have a truth value that ranges between 0 and 1 and is not
constrained to the two truth values of classic propositional logic. This fuzzy weighting schema
is used together with a combinatorial method to yield time-series forecast results.
Cubic Spline
Curves
Sometimes there are missing values in a time-series data set. For instance, interest rates for years
1 to 3 may exist, followed by years 5 to 8, and then year 10. Spline curves can be used to
interpolate the missing years’ interest rate values based on the data that exist. Spline curves can
also be used to forecast or extrapolate values of future time periods beyond the time period of
available data. The data can be linear or nonlinear.
Custom
Distributions
Using Risk Simulator, expert opinions can be collected and a customized distribution can be
generated. This forecasting technique comes in handy when the data set is small or the
goodness of fit is bad when applied to a distributional fitting routine.
GARCH
The generalized autoregressive conditional heteroskedasticity (GARCH) model is used to
model historical and forecast future volatility levels of a marketable security (e.g., stock prices,
commodity prices, and oil prices). The data set has to be a time series of raw price levels.
GARCH will first convert the prices into relative returns and then run an internal optimization
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to fit the historical data to a mean-reverting volatility term structure, while assuming that the
volatility is heteroskedastic in nature (changes over time according to some econometric
characteristics). Several variations of this methodology are available in Risk Simulator, including
EGARCH, EGARCH-T, GARCH-M, GJR-GARCH, GJR-GARCH-T, IGARCH, and TGARCH.
J Curve
The J curve, or exponential growth curve, is where the growth of the next period depends on
the current period’s level and the increase is exponential. This means that over time, the values
will increase significantly from one period to another. This model is typically used in forecasting
biological growth and chemical reactions over time.
Markov Chain
A Markov chain exists when the probability of a future state depends on a previous state and
when linked together form a chain that reverts to a long-run steady state level. This approach is
typically used to forecast the market share of two competitors. The required inputs are the
starting probability of a customer in the first store (the first state) will return to the same store in
the next period versus the probability of switching to a competitor’s store in the next state.
Maximum
Likelihood on
Logit, Probit, and
Tobit
Maximum likelihood estimation (MLE) is used to forecast the probability of something
occurring given some independent variables. For instance, MLE is used to predict if a credit
line or debt will default given the obligor’s characteristics (30 years old, single, salary of $100,000
per year, and having a total credit card debt of $10,000); or the probability a patient will have
lung cancer if the person is a male between the ages of 50 and 60, smokes 5 packs of cigarettes
per month, and so forth. In these circumstances, the dependent variable is limited (i.e., limited
to being binary 1 and 0 for default/die and no default/live, or limited to integer values like 1, 2,
3,etc.), and the desired outcome of the model is to predict the probability of an event occurring.
Traditional regression analysis will not work in these situations (the predicted probability is
usually less than zero or greater than one, and many of the required regression assumptions are
violated, such as independence and normality of the errors, and the errors will be fairly large).
Multivariate
Regression
Multivariate regression is used to model the relationship structure and characteristics of a
certain dependent variable as it depends on other independent exogenous variables. Using the
modeled relationship, we can forecast the future values of the dependent variable. The accuracy
and goodness of fit for this model can also be determined. Linear and nonlinear models can be
fitted in the multiple regression analysis.
Neural Network
Forecast
The term Neural Network is often used to refer to a network or circuit of biological neurons,
while modern usage of the term often refers to artificial neural networks comprising artificial
neurons, or nodes, recreated in a software environment. Such networks attempt to mimic the
neurons in the human brain in ways of thinking and identifying patterns and, in our situation,
identifying patterns for the purposes of forecasting time-series data.
Nonlinear
Extrapolation
The underlying structure of the data to be forecasted is assumed to be nonlinear over time. For
instance, a data set such as 1, 4, 9, 16, 25 is considered to be nonlinear (these data points are
from a squared function).
S Curve
The S curve or logistic growth curve starts off like a J curve, with exponential growth rates.
Over time, the environment becomes saturated (e.g., market saturation, competition,
overcrowding), the growth slows, and the forecast value eventually ends up at a saturation or
maximum level. This model is typically used in forecasting market share or sales growth of a
new product from market introduction until maturity and decline, population dynamics, and
other naturally occurring phenomenon.
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Stochastic
Processes
Sometimes variables cannot be readily predicted using traditional means, and these variables are
said to be stochastic. Nonetheless, most financial, economic, and naturally occurring
phenomena (e.g., motion of molecules through the air) follow a known mathematical law or
relationship. Although the resulting values are uncertain, the underlying mathematical structure
is known and can be simulated using Monte Carlo risk simulation. The processes supported in
Risk Simulator include Brownian motion random walk, mean-reversion, jump-diffusion, and
mixed processes, useful for forecasting nonstationary time-series variables.
Time-Series
Analysis and
Decomposition
In well-behaved time-series data (typical examples include sales revenues and cost structures of
large corporations), the values tend to have up to three elements: a base value, trend, and
seasonality. Time-series analysis uses these historical data and decomposes them into these
three elements, and recomposes them into future forecasts. In other words, this forecasting
method, like some of the others described, first performs a back-fitting (backcast) of historical
data before it provides estimates of future values (forecasts).
Trendlines
Trendlines can be used to determine if a set of time-series data follows any appreciable trend.
Trends can be linear or nonlinear (such as exponential, logarithmic, moving average, power,
polynomial, or power).
3.2 Запуск инструмента прогнозирования
рисков в Risk Simulator
In general, to create forecasts, several quick steps are required:

Start Excel and enter in or open your existing historical data.

Select the data, and click on Simulation and select Forecasting.

Select the relevant sections (ARIMA, Multivariate Regression, Nonlinear
Extrapolation, Stochastic Forecasting, Time-Series Analysis) and enter the relevant
inputs.
Figure 3.2 illustrates the Forecasting tool and the various methodologies and the following
provides a quick review of the selected methodology and several quick getting started examples
in using the software. The example file can be found either on the start menu at Start | Real
Options Valuation | Risk Simulator | Examples or accessed directly through Risk Simulator |
Example Models.
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Figure 3.2 – Risk Simulator’s Forecasting Methods
3.3 Анализ временных рядов
No Trend
Figure 3.3 lists the eight most common time-series models, segregated by seasonality and trend.
For instance, if the data variable has no trend or seasonality, then a single moving-average
model or a single exponential-smoothing model would suffice. However, if seasonality exists
but no discernible trend is present, either a seasonal additive or seasonal multiplicative model
would be better, and so forth.
With Trend
Theory
No Seasonality
With Seasonality
Single Moving Average
Seasonal
Additive
Single Exponential
Smoothing
Seasonal
Multiplicative
Double Moving
Average
Holt-Winter's
Additive
Double Exponential
Smoothing
Holt-Winter's
Multiplicative
Figure 3.3 – The Eight Most Common Time-Series Methods
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Procedure
Results
Interpretation
S I M U L A T O R

Start Excel and open your historical data if required (the example below uses the
Time-Series Forecasting file in the examples folder).

Select the historical data (data should be listed in a single column).

Select Risk Simulator | Forecasting | Time-Series Analysis.

Choose the model to apply, enter the relevant assumptions (Figure 3.4), and click OK
Figure 3.5 illustrates the sample results generated by using the Forecasting tool and a HoltWinter’s multiplicative model. The model-fitting and forecast chart indicates that the trend and
seasonality are picked up nicely by the Holt-Winter’s multiplicative model. The time-series
analysis report provides the relevant optimized alpha, beta, and gamma parameters; the error
measurements; fitted data; forecast values; and fitted-forecast graph. The parameters are simply
for reference. Alpha captures the memory effect of the base level changes over time, and beta is
the trend parameter that measures the strength of the trend, while gamma measures the
seasonality strength of the historical data. The analysis decomposes the historical data into these
three elements and then recomposes them to forecast the future. The fitted data illustrates the
historical data, and it uses the recomposed model and shows how close the forecasts are in the
past (a technique called backcasting). The forecast values are either single-point estimates or
assumptions (if the option to automatically generate assumptions is chosen and if a simulation
profile exists). The graph illustrates these historical, fitted, and forecast values. The chart is a
powerful communication and visual tool to see how good the forecast model is.
Figure 3.4 – Time-Series Analysis
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Notes
S I M U L A T O R
This time-series analysis module contains the eight time-series models seen in Figure 3.3. You
can choose the specific model to run based on the trend and seasonality criteria or choose the
Auto Model Selection, which will automatically iterate through all eight methods, optimize the
parameters, and find the best-fitting model for your data. Alternatively, if you choose one of the
eight models, you can also unselect the optimize checkboxes and enter your own alpha, beta, and
gamma parameters. Refer to Dr. Johnathan Mun’s Modeling Risk: Applying Monte Carlo Simulation,
Real Options Analysis, Forecasting, and Optimization, Second Edition (Wiley Finance, 2010) for more
details on the technical specifications of these parameters. In addition, you would need to enter
the relevant seasonality periods if you choose the automatic model selection or any of the
seasonal models. The seasonality input has to be a positive integer (e.g., if the data is quarterly,
enter 4 as the number of seasons or cycles a year, or enter 12 if monthly data). Next, enter the
number of periods to forecast. This value also has to be a positive integer. The maximum
runtime is set at 300 seconds. Typically, no changes are required. However, when forecasting
with a significant amount of historical data, the analysis might take slightly longer, and if the
processing time exceeds this runtime, the process will be terminated. You can also elect to have
the forecast automatically generate assumptions. That is, instead of single-point estimates, the
forecasts will be assumptions. Finally, the polar parameters option allows you to optimize the
alpha, beta, and gamma parameters to include zero and one. Certain forecasting software allows
these polar parameters while others do not. Risk Simulator allows you to choose which to use.
Typically, there is no need to use polar parameters.
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Figure 3.5 – Example Holt-Winter’s Forecast Report
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3.4 Многомерные регрессии
It is assumed that the user is sufficiently knowledgeable about the fundamentals of regression
analysis. The general bivariate linear regression equation takes the form of Y   0  1 X   ,
Theory
where 0 is the intercept, 1 is the slope, and  is the error term. It is bivariate as there are only
two variables: a Y, or dependent, variable and an X, or independent, variable, where X is also
known as the regressor (sometimes a bivariate regression is also known as a univariate
regression as there is only a single independent variable X). The dependent variable is so named
because it depends on the independent variable; for example, sales revenue depends on the
amount of marketing costs expended on a product’s advertising and promotion, making the
dependent variable sales and the independent variable marketing costs. An example of a
bivariate regression is seen as simply inserting the best-fitting line through a set of data points in
a two-dimensional plane as seen on the left panel in Figure 3.6. In other cases, a multivariate
regression can be performed, where there are multiple, or n number of, independent X
variables, where the general regression equation will now take the form of
Y   0  1 X 1   2 X 2   3 X 3 ...   n X n   . In this case, the best-fitting line will be within an
n + 1 dimensional plane.
Y
Y
Y1


Y2
X
Figure 3.6 – Bivariate Regression
However, fitting a line through a set of data points in a scatter plot as in Figure 3.6 may result in
numerous possible lines. The best-fitting line is defined as the single unique line that minimizes
the total vertical errors, that is, the sum of the absolute distances between the actual data points
(Yi) and the estimated line ( Yˆ ) as shown on the right panel of Figure 3.6. To find the bestfitting line that minimizes the errors, a more sophisticated approach is required, that is,
regression analysis. Regression analysis, therefore, finds the unique best-fitting line by requiring
that the total errors be minimized, or by calculating
n
Min  (Yi  Yˆi ) 2
i 1
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where only one unique line minimizes this sum of squared errors. The errors (vertical distance
between the actual data and the predicted line) are squared to avoid the negative errors
canceling out the positive errors. Solving this minimization problem with respect to the slope
and intercept requires calculating a first derivative and setting them equal to zero:
d
d 0
n
d n
 Yˆi ) 2  0 and
 (Yi  Yˆi ) 2  0
d1 i 1
 (Y
i
i 1
which yields the bivariate regression’s least squares equations:
n
n
n
(X  X)(Y Y ) X Y 
i
1  i1
i

n
(X  X)
2
i
i 1
n
X Y
i
i 1
i i
i 1
i
i 1
n
2


Xi 
n
Xi2   i1 

n
i 1
n
0  Y  1X
For multivariate regression, the analogy is expanded to account for multiple independent
variables, where Yi   1   2 X 2,i   3 X 3,i   i and the estimated slopes can be calculated
by:
Y X  X  Y X  X
 X  X   X X 
Y X  X  Y X  X

 X  X   X X 
ˆ 2 
ˆ3
i
2
3, i
2 ,i
2
2 ,i
i
i
3, i
2
2 ,i
2 ,i
i
2
3, i
2 ,i
X 3, i
2
2
3, i
2
2 ,i
3, i
3 ,i
2 ,i
2 ,i
X 3, i
2
2 ,i
3 ,i
In running multivariate regressions, great care has to be taken to set up and interpret the results.
For instance, a good understanding of econometric modeling is required (e.g., identifying
regression pitfalls such as structural breaks, multicollinearity, heteroskedasticity, autocorrelation,
specification tests, nonlinearities, etc.) before a proper model can be constructed. See Modeling
Risk: Applying Monte Carlo Simulation, Real Options Analysis, Forecasting, and Optimization, Second
Edition (Wiley Finance, 2010) by Dr. Johnathan Mun for more detailed analysis and discussion
of multivariate regression as well as how to identify these regression pitfalls.
Procedure
Results
Interpretation

Start Excel and open your historical data if required (the illustration below uses the file
Multiple Regression in the examples folder).

Check to make sure that the data is arranged in columns, select the entire data area
including the variable name, and select Risk Simulator | Forecasting | Multiple Regression.

Select the dependent variable and check the relevant options (lags, stepwise regression,
nonlinear regression, etc.), and click OK.
Figure 3.8 illustrates a sample multivariate regression result report. The report comes complete
with all the regression results, analysis of variance results, fitted chart, and hypothesis test
results. The technical details of interpreting these results are beyond the scope of this user
manual. See Modeling Risk: Applying Monte Carlo Simulation, Real Options Analysis, Forecasting, and
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Optimization, Second Edition (Wiley Finance, 2010) by Dr. Johnathan Mun for more detailed
analysis and discussion of multivariate regression as well as the interpretation of regression
reports.
Figure 3.7 – Running a Multivariate Regression
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Figure 3.8 – Multivariate Regression Results
3.5 Стохастическое прогнозирование
Theory
A stochastic process is nothing but a mathematically defined equation that can create a series of
outcomes over time, outcomes that are not deterministic in nature, that is, an equation or
process that does not follow any simple discernible rule such as price will increase X percent
every year or revenues will increase by this factor of X plus Y percent. A stochastic process is
by definition nondeterministic, and one can plug numbers into a stochastic process equation
and obtain different results every time. For instance, the path of a stock price is stochastic in
nature, and one cannot reliably predict the stock price path with any certainty. However, the
price evolution over time is enveloped in a process that generates these prices. The process is
fixed and predetermined, but the outcomes are not. Hence, by stochastic simulation, we create
multiple pathways of prices, obtain a statistical sampling of these simulations, and make
inferences on the potential pathways that the actual price may undertake given the nature and
parameters of the stochastic process used to generate the time series. Three basic stochastic
processes are included in Risk Simulator’s Forecasting tool, including geometric Brownian motion
or random walk, which is the most common and prevalently used process due to its simplicity
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and wide-ranging applications. The other two stochastic processes are the mean-reversion
process and the jump-diffusion process.
The interesting thing about stochastic process simulation is that historical data are not
necessarily required. That is, the model does not have to fit any sets of historical data. Simply
compute the expected returns and the volatility of the historical data or estimate them using
comparable external data or make assumptions about these values. See Modeling Risk: Applying
Monte Carlo Simulation, Real Options Analysis, Forecasting, and Optimization, Second Edition (Wiley
Finance, 2010) by Dr. Johnathan Mun for more details on how each of the inputs are
computed (e.g., mean-reversion rate, jump probabilities, volatility, etc.).
Procedure
Results
Interpretation

Start the module by selecting Risk Simulator | Forecasting | Stochastic Processes.

Select the desired process, enter the required inputs, click on Update Chart a few times
to make sure the process is behaving the way you expect it to, and click OK (Figure
3.9).
Figure 3.10 shows the results of a sample stochastic process. The chart shows a sample set of
the iterations while the report explains the basics of stochastic processes. In addition, the
forecast values (mean and standard deviation) for each time period are provided. Using these
values, you can decide which time period is relevant to your analysis and set assumptions based
on these mean and standard deviation values using the normal distribution. These assumptions
can then be simulated in your own custom model.
Figure 3.9 – Stochastic Process Forecasting
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Figure 3.10 – Stochastic Forecast Result
3.6 Нелинейная экстраполяция
Theory
Extrapolation involves making statistical projections by using historical trends that are projected
for a specified period of time into the future. It is only used for time-series forecasts. For crosssectional or mixed panel data (time-series with cross-sectional data), multivariate regression is
more appropriate. Extrapolation is useful when major changes are not expected, that is, causal
factors are expected to remain constant or when the causal factors of a situation are not clearly
understood. It also helps discourage introduction of personal biases into the process.
Extrapolation is fairly reliable, relatively simple, and inexpensive. However, extrapolation, which
assumes that recent and historical trends will continue, produces large forecast errors if
discontinuities occur within the projected time period. That is, pure extrapolation of time series
assumes that all we need to know is contained in the historical values of the series that is being
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forecasted. If we assume that past behavior is a good predictor of future behavior, extrapolation
is appealing. This makes it a useful approach when all that is needed are many short-term
forecasts.
This methodology estimates the f(x) function for any arbitrary x value by interpolating a smooth
nonlinear curve through all the x values and, using this smooth curve, extrapolates future x
values beyond the historical data set. The methodology employs either the polynomial
functional form or the rational functional form (a ratio of two polynomials). Typically, a
polynomial functional form is sufficient for well-behaved data, however, rational functional
forms are sometimes more accurate (especially with polar functions, i.e., functions with
denominators approaching zero).
Procedure

Start Excel and open your historical data if required (the illustration shown next uses
the file Nonlinear Extrapolation from the examples folder).

Select the time-series data and select Risk Simulator | Forecasting | Nonlinear Extrapolation.

Select the extrapolation type (automatic selection, polynomial function, or rational
function) and enter the number of forecast period desired (Figure 3.11), and click OK.
Results
Interpretation
The results report shown in Figure 3.12 shows the extrapolated forecast values, the error
measurements, and the graphical representation of the extrapolation results. The error
measurements should be used to check the validity of the forecast and are especially important
when used to compare the forecast quality and accuracy of extrapolation versus time-series
analysis.
Notes
When the historical data is smooth and follows some nonlinear patterns and curves,
extrapolation is better than time-series analysis. However, when the data patterns follow
seasonal cycles and a trend, time-series analysis will provide better results.
Figure 3.11 – Running a Nonlinear Extrapolation
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Figure 3.12 – Nonlinear Extrapolation Results
3.7 ARIMA временные ряды Бокса-Дженкинса
Theory
One very powerful advanced times-series forecasting tool is the ARIMA, or Auto Regressive
Integrated Moving Average, approach. ARIMA forecasting assembles three separate tools into a
comprehensive model. The first tool segment is the autoregressive (AR) term, which
corresponds to the number of lagged value of the residual in the unconditional forecast model.
In essence, the model captures the historical variation of actual data to a forecasting model and
uses this variation or residual to create a better predicting model. The second tool segment is
the integration order (I) term. This integration term corresponds to the number of differencing
the time series to be forecasted goes through. This element accounts for any nonlinear growth
rates existing in the data. The third tool segment is the moving average (MA) term, which is
essentially the moving average of lagged forecast errors. By incorporating this lagged forecast
errors term, the model in essence learns from its forecast errors or mistakes and corrects for
them through a moving-average calculation. The ARIMA model follows the Box-Jenkins
methodology with each term representing steps taken in the model construction until only
random noise remains. Also, ARIMA modeling uses correlation techniques in generating
forecasts. ARIMA can be used to model patterns that may not be visible in plotted data. In
addition, ARIMA models can be mixed with exogenous variables, but make sure that the
exogenous variables have enough data points to cover the additional number of periods to
forecast. Finally, be aware that due to the complexity of the models, this module may take
longer to run.
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There are many reasons why an ARIMA model is superior to common time-series analysis and
multivariate regressions. The common finding in time-series analysis and multivariate regression
is that the error residuals are correlated with their own lagged values. This serial correlation
violates the standard assumption of regression theory that disturbances are not correlated with
other disturbances. The primary problems associated with serial correlation are:

Regression analysis and basic time-series analysis are no longer efficient among the
different linear estimators. However, as the error residuals can help to predict current
error residuals, we can take advantage of this information to form a better prediction
of the dependent variable using ARIMA.

Standard errors computed using the regression and time-series formula are not correct,
and are generally understated, and if there are lagged-dependent variables set as the
regressors, regression estimates are biased and inconsistent but can be fixed using
ARIMA.
ARIMA(p,d,q) models are the extension of the AR model that uses three components for
modeling the serial correlation in the time series data. The first component is the autoregressive
(AR) term. The AR(p) model uses the p lags of the time series in the equation. An AR(p) model
has the form: yt = a1yt-1 + ... + apyt-p + et. The second component is the integration (d) order term.
Each integration order corresponds to differencing the time series. I(1) means differencing the
data once; I(d) means differencing the data d times. The third component is the moving
average (MA) term. The MA(q) model uses the q lags of the forecast errors to improve the
forecast. An MA(q) model has the form: yt = et + b1et-1 + ... + bqet-q. Finally, an ARIMA(p,q)
model has the combined form: yt = a1 yt-1 + ... + a p yt-p + et + b1 et-1 + ... + bq et-q.
Procedure

Start Excel and enter your data or open an existing worksheet with historical data to
forecast (the illustration shown next uses the file example file Time-Series ARIMA).

Select the time-series data and select Risk Simulator | Forecasting | ARIMA.

Enter the relevant P, D, Q parameters (positive integers only), enter the number of
forecast period desired, and click OK.
Notes
For ARIMA and Auto ARIMA, you can model and forecast future periods by either using only
the dependent variable (Y), that is, the Time Series Variable by itself, or you can add in exogenous
variables (X1, X2,…, Xn) just like in a regression analysis where you have multiple independent
variables. You can run as many forecast periods as you wish if you use only the time-series
variable (Y). However, if you add exogenous variables (X), note that your forecast period is
limited to the number of exogenous variables’ data periods minus the time-series variable’s data
periods. For example, you can only forecast up to 5 periods if you have time-series historical
data of 100 periods and only if you have exogenous variables of 105 periods (100 historical
periods to match the time-series variable and 5 additional future periods of independent
exogenous variables to forecast the time-series dependent variable).
Results
Interpretation
In interpreting the results of an ARIMA model, most of the specifications are identical to the
multivariate regression analysis (see Modeling Risk: Applying Monte Carlo Simulation, Real Options
Analysis, Stochastic Forecasting, and Portfolio Optimization, Second Edition, by Dr. Johnathan Mun for
more technical details about interpreting the multivariate regression analysis and ARIMA
models). There are however, several additional sets of results specific to the ARIMA analysis as
seen in Figure 3.14. The first is the addition of Akaike information criterion (AIC) and Schwarz
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criterion (SC), which are often used in ARIMA model selection and identification. That is, AIC
and SC are used to determine if a particular model with a specific set of p, d, and q parameters
is a good statistical fit. SC imposes a greater penalty for additional coefficients than the AIC but,
generally, the model with the lowest the AIC and SC values should be chosen. Finally, an
additional set of results called the autocorrelation (AC) and partial autocorrelation (PAC)
statistics are provided in the ARIMA report.
For instance, if autocorrelation AC(1) is nonzero, it means that the series is first-order serially
correlated. If AC dies off more or less geometrically with increasing lags, it implies that the
series follows a low-order autoregressive process. If AC drops to zero after a small number of
lags, it implies that the series follows a low-order moving-average process. In contrast, PAC
measures the correlation of values that are k periods apart after removing the correlation from
the intervening lags. If the pattern of autocorrelation can be captured by an autoregression of
order less than k, then the partial autocorrelation at lag k will be close to zero. The Ljung-Box
Q-statistics and their p-values at lag k are also provided, where the null hypothesis being tested
is such that there is no autocorrelation up to order k. The dotted lines in the plots of the
autocorrelations are the approximate two standard error bounds. If the autocorrelation is within
these bounds, it is not significantly different from zero at approximately the 5% significance
level. Finding the right ARIMA model takes practice and experience. These AC, PAC, SC, and
AIC diagnostic tools are highly useful in helping to identify the correct model specification.
Figure 3.13 – Box-Jenkins ARIMA Forecast Tool
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Figure 3.14 – Box-Jenkins ARIMA Forecast Report
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3.8 AUTO ARIMA (Усложнённые ARIMA
временные ряды Бокса-Дженкинса)
Theory
Procedure
Notes
While the analyses are identical, AUTO ARIMA differs from ARIMA in automating some of
the traditional ARIMA modeling. It automatically tests multiple permutations of model
specifications and returns the best-fitting model. Running the Auto ARIMA is similar to regular
ARIMA forecasting, with the difference being that the P, D, Q inputs are no longer required
and different combinations of these inputs are automatically run and compared.

Start Excel and enter your data or open an existing worksheet with historical data to
forecast (the illustration shown in Figure 3.15 uses the example file Advanced
Forecasting Models in the Examples menu of Risk Simulator).

In the Auto ARIMA worksheet, select the data and click on Risk Simulator | Forecasting
| AUTO-ARIMA. You can also access this method through the forecasting icons
ribbon, or right-clicking anywhere in the model and selecting the forecasting shortcut
menu.

Click on the link icon and link to the existing time-series data, enter the number of
forecast periods desired, and click OK.
For ARIMA and Auto ARIMA, you can model and forecast future periods by either using only
the dependent variable (Y), that is, the Time Series Variable by itself or you can add in exogenous
variables (X1, X2,…, Xn) just like in a regression analysis where you have multiple independent
variables. You can run as many forecast periods as you wish if you use only the time-series
variable (Y). However, if you add exogenous variables (X), note that your forecast period is
limited to the number of exogenous variables’ data periods minus the time-series variable’s data
periods. For example, you can only forecast up to 5 periods if you have time-series historical
data of 100 periods and only if you have exogenous variables of 105 periods (100 historical
periods to match the time-series variable and 5 additional future periods of independent
exogenous variables to forecast the time-series dependent variable).
Figure 3.15 – AUTO ARIMA Module
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3.9 Базовая эконометрика
Theory
Procedure
Econometrics refers to a branch of business analytics, modeling, and forecasting techniques for
modeling the behavior or forecasting certain business or economic variables. Running the Basic
Econometrics models is similar to regular regression analysis except that the dependent and
independent variables are allowed to be modified before a regression is run. The report
generated and its interpretation is the same as shown in the Multivariate Regression section
presented earlier.

Start Excel and enter your data or open an existing worksheet with historical data to
forecast (the illustration shown in Figure 3.16 uses the file example file Advanced
Forecasting Models in the Examples menu of Risk Simulator).

Select the data in the Basic Econometrics worksheet and select Risk Simulator | Forecasting
| Basic Econometrics.

Enter the desired dependent and independent variables (see Figure 3.16 for examples)
and click OK to run the model and report, or click on Show Results to view the results
before generating the report in case you need to make any changes to the model
Figure 3.16 – Basic Econometrics Module
Notes

To run an econometric model, simply select the data (B5:G55) including headers and
click on Risk Simulator | Forecasting | Basic Econometrics. You can then type in the
variables and their modifications for the dependent and independent variables (Figure
3.16). Note that only one variable is allowed as the Dependent Variable (Y), whereas
multiple variables are allowed in the Independent Variables (X) section, separated by a
semicolon (;), and that basic mathematical functions can be used (e.g., LN, LOG,
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LAG, +, -, /, *, TIME, RESIDUAL, DIFF). Click on Show Results to preview the
computed model and click OK to generate the econometric model report.

You can also automatically generate Multiple Models by entering a sample model and
using the predefined INTEGER(N) variable as well as Shifting Data up or down
specific rows repeatedly. For instance, if you use the variable LAG(VAR1,
INTEGER1) and you set INTEGER1 to be between MIN = 1 and MAX = 3, then
the following three models will be run: LAG(VAR1,1), then LAG(VAR1,2), and,
finally, LAG(VAR1,3). Also, sometimes you might want to test if the time-series data
has structural shifts or if the behavior of the model is consistent over time by shifting
the data and then running the same model. For example, if you have 100 months of
data listed chronologically, you can shift it down 3 months at a time for 10 times (i.e.,
the model will be run on months 1–100, 4–100, 7–100, etc.). Using this Multiple Models
section in Basic Econometrics, you can run hundreds of models by simply entering a
single model equation if you use these predefined integer variables and shifting
methods.
3.10 Прогнозы J-S Кривых
Theory
Procedure
The J curve, or exponential growth curve, is one where the growth of the next period depends
on the current period’s level and the increase is exponential. This means that over time, the
values will increase significantly, from one period to another. This model is typically used in
forecasting biological growth and chemical reactions over time.

Start Excel and select Risk Simulator | Forecasting | JS Curves.

Select the J or S curve type, enter the required input assumptions (see Figures 3.17 and 3.18
for examples), and click OK to run the model and report.
The S curve, or logistic growth curve, starts off like a J curve, with exponential growth rates.
Over time, the environment becomes saturated (e.g., market saturation, competition,
overcrowding), the growth slows, and the forecast value eventually ends up at a saturation or
maximum level. This model is typically used in forecasting market share or sales growth of a
new product from market introduction until maturity and decline, population dynamics, growth
of bacterial cultures, and other naturally occurring variables. Figure 3.18 illustrates a sample S
curve.
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Figure 3.17 – J-Curve Forecast
Figure 3.18 – S-Curve Forecast
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3.11 Прогнозы волатильности GARCH
Theory
Procedure
The generalized autoregressive conditional heteroskedasticity (GARCH) model is used to
model historical and forecast future volatility levels of a marketable security (e.g., stock prices,
commodity prices, oil prices, etc.). The data set has to be a time series of raw price levels.
GARCH will first convert the prices into relative returns and then run an internal optimization
to fit the historical data to a mean-reverting volatility term structure, while assuming that the
volatility is heteroskedastic in nature (changes over time according to some econometric
characteristics). The theoretical specifics of a GARCH model are outside the purview of this
user manual. For more details on GARCH models, please refer to Advanced Analytical Models, by
Dr. Johnathan Mun (Wiley Finance, 2008).

Start Excel and open the example file Advanced Forecasting Model, go to the GARCH
worksheet, select the data and click on Risk Simulator | Forecasting | GARCH.

Click on the link icon, select the Data Location, enter the required input assumptions
(see Figure 3.19), and click OK to run the model and report.
Note: The typical volatility forecast situation requires P = 1, Q = 1, Periodicity = number of
periods per year (12 for monthly data, 52 for weekly data, 252 or 365 for daily data), Base =
minimum of 1 and up to the periodicity value, and Forecast Periods = number of annualized
volatility forecasts you wish to obtain. There are several GARCH models available in Risk
Simulator, including EGARCH, EGARCH-T, GARCH-M, GJR-GARCH, GJR-GARCH-T,
IGARCH, and T-GARCH. See the chapter in Modeling Risk, Second Edition, by Dr. Johnathan
Mun (Wiley Finance, 2010), on GARCH modeling for more details on what each specification
is for.
Figure 3.19 – GARCH Volatility Forecast
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3.11.1 GARCH
Equations
S I M U L A T O R
The accompanying table lists some of the GARCH specifications used in Risk Simulator with
two underlying distributional assumptions: one for normal distribution and the other for the t
distribution.
GARCH-M
Variance in
zt ~ Normal Distribution
zt ~ T-Distribution
yt  c   t2   t
yt  c   t2   t
Mean
Equation
 t   t zt
 t   t zt
 t2     t21   t21
 t2     t21   t21
GARCH-M
yt  c   t   t
yt  c   t   t
Standard
Deviation
 t   t zt
in Mean
Equation
GARCH-M
Log
Variance
in Mean
Equation
GARCH
 t   t zt
    
2
t
2
t 1
 
2
t 1
 t2     t21   t21
yt  c   ln( t2 )   t
yt  c   ln( t2 )   t
 t   t zt
 t   t zt
    
2
t
2
t 1
 
2
t 1
 t2     t21   t21
yt  xt    t
yt   t
 t2     t21   t21
 t   t zt
 t2     t21   t21
EGARCH
yt   t
yt   t
 t   t zt
 t   t zt
ln  t2       ln  t21  
  t 1

  t 1
E( t ) 


 E (  t )   r t 1
 t 1

2

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ln  t2       ln  t21  

  t 1


 E (  t )   r t 1
 t 1

E( t ) 
2   2 ((  1) / 2)
(  1)( / 2) 
  t 1
R I S K
GJRGARCH
S I M U L A T O R
yt   t
yt   t
 t   t zt
    
 t   t zt

 t2     t21 
r t21dt 1   t21
r t21dt 1   t21
 1if  t 1  
dt 1  
0otherwise
 1if  t 1  
dt 1  
0otherwise
2
t
2
t 1
For the GARCH-M models, the conditional variance equations are the same in the six
variations but the mean questions are different and assumption on zt can be either normal
distribution or t distribution. The estimated parameters for GARCH-M with normal
distribution are those five parameters in the mean and conditional variance equations. The
estimated parameters for GARCH-M with the t distribution are those five parameters in the
mean and conditional variance equations plus another parameter, the degrees of freedom for
the t distribution. In contrast, for the GJR models, the mean equations are the same in the six
variations and the differences are that the conditional variance equations and the assumption on
zt can be either a normal distribution or t distribution. The estimated parameters for
EGARCH and GJR-GARCH with normal distribution are those four parameters in the
conditional variance equation. The estimated parameters for GARCH, EARCH, and GJRGARCH with t distribution are those parameters in the conditional variance equation plus the
degrees of freedom for the t distribution. More technical details of GARCH methodologies fall
outside of the scope of this book.
3.12 Цепи Маркова
Theory
Procedure
Notes
A Markov chain exists when the probability of a future state depends on a previous state and
when linked together form a chain that reverts to a long-run steady state level. This approach is
typically used to forecast the market share of two competitors. The required inputs are the
starting probability of a customer in the first store (the first state) will return to the same store in
the next period versus the probability of switching to a competitor’s store in the next state.

Start Excel and select Risk Simulator | Forecasting | Markov Chain.

Enter in the required input assumptions (see Figure 3.20 for an example) and click OK to run
the model and report.
Set both probabilities to 10% and rerun the Markov chain and you will see the effects of
switching behaviors very clearly in the resulting chart.
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Figure 3.20 – Markov Chains (Switching Regimes)
3.13 Ограниченные зависимые переменные:
логит, пробит, тобит. Использование
максимального приближения к популяции
Theory
The term Limited Dependent Variables describes the situation where the dependent variable
contains data that are limited in scope and range, such as binary responses (0 or 1) or truncated,
ordered, or censored data. For instance, given a set of independent variables (e.g., age, income,
education level of credit card or mortgage loan holders), we can model the probability of
default using maximum likelihood estimation (MLE). The response, or dependent variable Y, is
binary. That is, it can have only two possible outcomes that we denote as 1 and 0 (e.g., Y may
represent presence/absence of a certain condition, defaulted/not defaulted on previous loans,
success/failure of some device, answer yes/no on a survey, etc.). We also have a vector of
independent variable regressors X, which are assumed to influence the outcome Y. A typical
ordinary least squares regression approach is invalid because the regression errors are
heteroskedastic and non-normal, and the resulting estimated probability estimates will return
nonsensical values of above 1 or below 0. MLE analysis handles these problems using an
iterative optimization routine to maximize a log likelihood function when the dependent
variables are limited.
A Logit or Logistic regression, is used for predicting the probability of occurrence of an event
by fitting data to a logistic curve. It is a generalized linear model used for binomial regression,
and, like many forms of regression analysis, it makes use of several predictor variables that may
be either numerical or categorical. MLE applied in a binary multivariate logistic analysis is used
to model dependent variables to determine the expected probability of success of belonging to
a certain group. The estimated coefficients for the Logit model are the logarithmic odds ratios
and cannot be interpreted directly as probabilities. A quick computation is first required and the
approach is simple.
Specifically, the Logit model is specified as Estimated Y = LN[Pi/(1–Pi)] or, conversely, Pi =
EXP(Estimated Y)/(1+EXP(Estimated Y)), and the coefficients βi are the log odds ratios. So,
taking the antilog, or EXP(βi), we obtain the odds ratio of Pi/(1–Pi). This means that with an
increase in a unit of βi, the log odds ratio increases by this amount. Finally, the rate of change is
the probability dP/dX = βiPi(1–Pi). The standard error measures how accurate the predicted
coefficients are, and the t-statistics are the ratios of each predicted coefficient to its standard
error and are used in the typical regression hypothesis test of the significance of each estimated
parameter. To estimate the probability of success of belonging to a certain group (e.g.,
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predicting if a smoker will develop chest complications given the amount smoked per year),
simply compute the Estimated Y value using the MLE coefficients. For example, if the model is
Y = 1.1 + 0.005 (Cigarettes), then someone smoking 100 packs per year has an Estimated Y of 1.1
+ 0.005(100) = 1.6. Next, compute the inverse antilog of the odds ratio by EXP(Estimated
Y)/[1 + EXP(Estimated Y)] = EXP(1.6)/(1+ EXP(1.6)) = 0.8320. So, such a person has an
83.20% chance of developing some chest complications in his or her lifetime.
A Probit model (sometimes also known as a Normit model) is a popular alternative
specification for a binary response model, which employs a probit function estimated using
maximum likelihood estimation and the approach is called probit regression. The Probit and
Logistic regression models tend to produce very similar predictions where the parameter
estimates in a logistic regression tend to be 1.6 to 1.8 times higher than they are in a
corresponding Probit model. The choice of using a Probit or Logit is entirely up to
convenience, and the main distinction is that the logistic distribution has a higher kurtosis (fatter
tails) to account for extreme values. For example, suppose that house ownership is the decision
to be modeled, and this response variable is binary (home purchase or no home purchase) and
depends on a series of independent variables Xi such as income, age, and so forth, such that Ii
= β0 + β1X1 +...+ βnXn, where the larger the value of Ii, the higher the probability of home
ownership. For each family, a critical I* threshold exists where, if exceeded, the house is
purchased, otherwise, no home is purchased, and the outcome probability (P) is assumed to be
normally distributed such that Pi = CDF(I) using a standard normal cumulative distribution
function (CDF). Therefore, using the estimated coefficients exactly like those of a regression
model and using the Estimated Y value, apply a standard normal distribution (you can use
Excel’s NORMSDIST function or Risk Simulator's Distributional Analysis tool by selecting
Normal distribution and setting the mean to be 0 and standard deviation to be 1). Finally, to
obtain a Probit or probability unit measure, set Ii + 5 (because whenever the probability Pi <
0.5, the estimated Ii is negative, due to the fact that the normal distribution is symmetrical
around a mean of zero).
The Tobit Model (Censored Tobit) is an econometric and biometric modeling method used to
describe the relationship between a non-negative dependent variable Yi and one or more
independent variables Xi. The dependent variable in a Tobit econometric model is censored; it
is censored because values below zero are not observed. The Tobit model assumes that there is
a latent unobservable variable Y*. This variable is linearly dependent on the Xi variables via a
vector of βi coefficients that determine their interrelationships. In addition, there is a normally
distributed error term Ui to capture random influences on this relationship. The observable
variable Yi is defined to be equal to the latent variables whenever the latent variables are above
zero and is assumed to be zero otherwise. That is, Yi = Y* if Y* > 0 and Yi = 0 if Y* = 0. If the
relationship parameter βi is estimated by using ordinary least squares regression of the observed
Yi on Xi, the resulting regression estimators are inconsistent and yield downward-biased slope
coefficients and an upward-biased intercept. Only MLE would be consistent for a Tobit model.
In the Tobit model, there is an ancillary statistic called sigma, which is equivalent to the standard
error of estimate in a standard ordinary least squares regression, and the estimated coefficients
are used the same way as a regression analysis.
Procedure

Start Excel and open the example file Advanced Forecasting Model, go to the MLE
worksheet, select the data set including the headers, and click on Risk Simulator |
Forecasting | Maximum Likelihood.

Select the dependent variable from the drop-down list (see Figure 3.21) and click OK to run
the model and report.
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Figure 3.21 – Maximum Likelihood Module
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3.14 Сплайн (кубических сплайн-интерполяции
и экстраполяции)
Theory
Sometimes there are missing values in a time-series data set. For instance, interest rates for years
1 to 3 may exist, followed by years 5 to 8, and then year 10. Spline curves can be used to
interpolate the missing years’ interest rate values based on the data that exist. Spline curves can
also be used to forecast or extrapolate values of future time periods beyond the time period of
available data. The data can be linear or nonlinear. Figure 3.22 illustrates how a cubic spline is
run and Figure 3.23 shows the resulting forecast report from this module. The Known X
values represent the values on the x-axis of a chart (in our example, this is Years of the known
interest rates, and, usually, the x-axis values are those that are known in advance such as time or
years) and the Known Y values represent the values on the y-axis (in our case, the known
Interest Rates). The y-axis variable is typically the variable you wish to interpolate missing values
from or extrapolate the values into the future.
Figure 3.22 – Cubic Spline Module
Procedure

Start Excel and open the example file Advanced Forecasting Model, go to the Cubic Spline
worksheet, select the data set excluding the headers, and click on Risk Simulator |
Forecasting | Cubic Spline.

The data location is automatically inserted into the user interface if you first select the
data, or you can also manually click on the link icon and link the Known X values and
Known Y values (see Figure 3.22 for an example), then enter in the required Starting and
Ending values to extrapolate and interpolate, as well as the required Step Size between
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these starting and ending values. Click OK to run the model and report (see Figure
3.23).
Figure 3.23 – Spline Forecast Results
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4
4. ОПТИМИЗАЦИЯ
T
his chapter looks at the optimization process and methodologies in more detail in
connection with using Risk Simulator. These methodologies include the use of
continuous versus discrete integer optimization, as well as static versus dynamic and
stochastic optimizations.
4.1 Методологии оптимизации
Many algorithms exist to run optimization, and many different procedures exist when
optimization is coupled with Monte Carlo simulation. In Risk Simulator, there are three distinct
optimization procedures and optimization types as well as different decision variable types. For
instance, Risk Simulator can handle Continuous Decision Variables (1.2535, 0.2215, etc.) as well
as Integers Decision Variables (1, 2, 3, 4, etc.), Binary Decision Variables (1 and 0 for go and
no-go decisions), and Mixed Decision Variables (both integers and continuous variables). On
top of that, Risk Simulator can handle Linear Optimization (i.e., when both the objective and
constraints are all linear equations and functions) as well as Nonlinear Optimizations (i.e., when
the objective and constraints are a mixture of linear and nonlinear functions and equations).
As far as the optimization process is concerned, Risk Simulator can be used to run a Discrete
Optimization, that is, an optimization that is run on a discrete or static model, where no
simulations are run. In other words, all the inputs in the model are static and unchanging. This
optimization type is applicable when the model is assumed to be known and no uncertainties
exist. Also, a discrete optimization can be first run to determine the optimal portfolio and its
corresponding optimal allocation of decision variables before more advanced optimization
procedures are applied. For instance, before running a stochastic optimization problem, a
discrete optimization is first run to determine if there exist solutions to the optimization
problem before a more protracted analysis is performed.
Next, Dynamic Optimization is applied when Monte Carlo simulation is used together with
optimization. Another name for such a procedure is Simulation-Optimization. That is, a
simulation is first run, then the results of the simulation are then applied in the Excel model,
and then an optimization is applied to the simulated values. In other words, a simulation is run
for N trials, and then an optimization process is run for M iterations until the optimal results are
obtained or an infeasible set is found. That is, using Risk Simulator’s optimization module, you
can choose which forecast and assumption statistics to use and replace in the model after the
simulation is run. Then, these forecast statistics can be applied in the optimization process. This
approach is useful when you have a large model with many interacting assumptions and
forecasts, and when some of the forecast statistics are required in the optimization. For
example, if the standard deviation of an assumption or forecast is required in the optimization
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model (e.g., computing the Sharpe ratio in asset allocation and optimization problems where we
have mean divided by standard deviation of the portfolio), then this approach should be used.
The Stochastic Optimization process, in contrast, is similar to the dynamic optimization
procedure with the exception that the entire dynamic optimization process is repeated T times.
That is, a simulation with N trials is run, and then an optimization is run with M iterations to
obtain the optimal results. Then the process is replicated T times. The results will be a forecast
chart of each decision variable with T values. In other words, a simulation is run and the
forecast or assumption statistics are used in the optimization model to find the optimal
allocation of decision variables. Then, another simulation is run, generating different forecast
statistics, and these new updated values are then optimized, and so forth. Hence, the final
decision variables will each have their own forecast chart, indicating the range of the optimal
decision variables. For instance, instead of obtaining single-point estimates in the dynamic
optimization procedure, you can now obtain a distribution of the decision variables and, hence,
a range of optimal values for each decision variable, also known as a stochastic optimization.
Finally, an Efficient Frontier optimization procedure applies the concepts of marginal
increments and shadow pricing in optimization. That is, what would happen to the results of
the optimization if one of the constraints were relaxed slightly? Say, for instance, the budget
constraint is set at $1 million. What would happen to the portfolio’s outcome and optimal
decisions if the constraint were now $1.5 million, or $2 million, and so forth? This is the
concept of the Markowitz efficient frontiers in investment finance, whereby one can determine
what additional returns the portfolio will generate if the portfolio standard deviation is allowed
to increase slightly. This process is similar to the dynamic optimization process with the
exception that one of the constraints is allowed to change, and with each change, the simulation
and optimization process is run. This process is best applied manually using Risk Simulator.
That is, run a dynamic or stochastic optimization, then rerun another optimization with a
constraint, and repeat that procedure several times. This manual process is important because
by changing the constraint, the analyst can determine if the results are similar or different, and,
hence, whether it is worthy of any additional analysis, or the analyst can determine how far a
marginal increase in the constraint should be to obtain a significant change in the objective and
decision variables.
One item is worthy of consideration. There exist other software products that supposedly
perform stochastic optimization but, in fact, they do not. For instance, after a simulation is run,
then one iteration of the optimization process is generated, and then another simulation is run,
then the second optimization iteration is generated and so forth. This approach is simply a
waste of time and resources. That is, in optimization, the model is put through a rigorous set of
algorithms, where multiple iterations (ranging from several to thousands of iterations) are
required to obtain the optimal results. Hence, generating one iteration at a time is a waste of
time and resources. The same portfolio can be solved using Risk Simulator in under a minute as
compared to multiple hours using such a backward approach. Also, such a simulationoptimization approach will typically yield bad results, and it is not a stochastic optimization
approach. Be extremely careful of such methodologies when applying optimization to your
models.
The next two sections provide examples of optimization problems. One uses continuous
decision variables while the other uses discrete integer decision variables. In either model, you
can apply discrete optimization, dynamic optimization, stochastic optimization, or even the
efficient frontiers with shadow pricing. Any of these approaches can be used for these two
examples. Therefore, for simplicity, only the model setup is illustrated and it is up to the user to
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decide which optimization process to run. Also, the continuous model uses the nonlinear
optimization approach (because the portfolio risk computed is a nonlinear function, and the
objective is a nonlinear function of portfolio returns divided by portfolio risks) and integer
optimization is an example of a linear optimization model (its objective and all of its constraints
are linear). Therefore, these two examples encapsulate all of the procedures aforementioned.
4.2 Оптимизация с непрерывными
переменными решений
Figure 4.1 illustrates the sample continuous optimization model. The example here uses the
Continuous Optimization file found either on the start menu at Start | Real Options Valuation
| Risk Simulator | Examples or accessed directly through Risk Simulator | Example Models.
In this example, there are 10 distinct asset classes (e.g., different types of mutual funds, stocks,
or assets) where the idea is to most efficiently and effectively allocate the portfolio holdings
such that the best bang for the buck is obtained; that is, to generate the best portfolio returns
possible given the risks inherent in each asset class. To truly understand the concept of
optimization, we will have to delve deeply into this sample model to see how the optimization
process can best be applied.
As mentioned, the model shows the 10 asset classes each with its own set of annualized returns
and annualized volatilities. These return and risk measures are annualized values such that they
can be consistently compared across different asset classes. Returns are computed using the
geometric average of the relative returns, while the risks are computed using the logarithmic
relative stock returns approach.
Figure 4.1 – Continuous Optimization Model
Referring to Figure 4.1, column E (Allocation Weights) holds the decision variables, which are
the variables that need to be tweaked and tested such that the total weight is constrained at
100% (cell E17). Typically, to start the optimization, we set these cells to a uniform value, where
in this case, cells E6 to E15 are set at 10% each. In addition, each decision variable may have
specific restrictions in its allowed range. In this example, the lower and upper allocations
allowed are 5% and 35%, as seen in columns F and G. This means that each asset class may
have its own allocation boundaries. Next, column H shows the return to risk ratio, which is
simply the return percentage divided by the risk percentage, where the higher this value, the
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higher the bang for the buck. Columns I through L show the individual asset class rankings by
returns, risk, return to risk ratio, and allocation. In other words, these rankings show at a glance
which asset class has the lowest risk, or the highest return, and so forth.
The portfolio’s total returns in cell C17 is SUMPRODUCT(C6:C15, E6:E15), that is, the sum
of the allocation weights multiplied by the annualized returns for each asset class. In other
words, we have RP   A R A   B RB  C RC   D RD , where RP is the return on the
portfolio, RA,B,C,D are the individual returns on the projects, and A,B,C,D are the respective
weights, or capital allocation, across each project.
In addition, the portfolio’s diversified risk in cell D17 is computed by taking:
P 
i
n
i 1
i 1
 i2 i2  
m

j 1
2 i  j  i , j  i j .
Here, i,j are the respective cross-correlations between the asset classes––hence, if the crosscorrelations are negative, there are risk diversification effects, and the portfolio risk decreases.
However, to simplify the computations here, we assume zero correlations among the asset
classes through this portfolio risk computation, but assume the correlations when applying
simulation on the returns as will be seen later. Therefore, instead of applying static correlations
among these different asset returns, we apply the correlations in the simulation assumptions
themselves, creating a more dynamic relationship among the simulated return values.
Finally, the return to risk ratio, or Sharpe ratio, is computed for the portfolio. This value is seen
in cell C18, and represents the objective to be maximized in this optimization exercise. To
summarize, we have the following specifications in this example model:
Procedure
Objective:
Maximize Return to Risk Ratio (C18)
Decision Variables:
Allocation Weights (E6:E15)
Restrictions on Decision Variables:
Minimum and Maximum Required (F6:G15)
Constraints:
Total Allocation Weights Sum to 100% (E17)
Open the example file and start a new profile by clicking on Risk Simulator | New Profile and
provide it a name.

The first step in optimization is to set the decision variables. Select cell E6, set the first
decision variable (Risk Simulator | Optimization | Set Decision), and click on the link icon
to select the name cell (B6), as well as the lower bound and upper bound values at cells
F6 and G6. Then, using Risk Simulator’s copy, copy this cell E6 decision variable and
paste it to the remaining cells in E7 to E15.

The second step in optimization is to set the constraint. There is only one constraint
here, that is, the total allocation in the portfolio must sum to 100%. So, click on Risk
Simulator | Optimization | Constraints… and select ADD to add a new constraint. Then,
select the cell E17 and make it equal (=) to 100%. Click OK when done.

The final step in optimization is to set the objective function and start the optimization
by selecting the objective cell C18 and Risk Simulator | Optimization | Run Optimization
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and then selecting the optimization of choice (Static Optimization, Dynamic Optimization,
or Stochastic Optimization). To get started, select Static Optimization. Check to make sure
the objective cell is set for C18 and select Maximize. You can now review the decision
variables and constraints if required, or click OK to run the static optimization.
Once the optimization is complete, you may select Revert to revert back to the original values
of the decision variables as well as the objective, or select Replace to apply the optimized
decision variables. Typically, Replace is chosen after the optimization is done.
Figure 4.2 shows the screen shots of these procedural steps. You can add simulation
assumptions on the model’s returns and risk (columns C and D) and apply the dynamic
optimization and stochastic optimization for additional practice.
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Figure 4.2 – Running Continuous Optimization in Risk Simulator
Results
Interpretation
The optimization’s final results are shown in Figure 4.3, where the optimal allocation of assets
for the portfolio is seen in cells E6:E15. That is, given the restrictions of each asset fluctuating
between 5% and 35%, and where the sum of the allocation must equal 100%, the allocation
that maximizes the return to risk ratio can be identified from the data provided in Figure 4.3.
A few important things have to be noted when reviewing the results and optimization
procedures performed thus far:

The correct way to run the optimization is to maximize the bang for the buck, or
returns to risk Sharpe ratio, as we have done.

If instead we maximized the total portfolio returns, the optimal allocation result is
trivial and does not require optimization to obtain. That is, simply allocate 5% (the
minimum allowed) to the lowest eight assets, 35% (the maximum allowed) to the
highest returning asset, and the remaining (25%) to the second-best returns asset.
Optimization is not required. However, when allocating the portfolio this way, the risk
is a lot higher as compared to when maximizing the returns to risk ratio, although the
portfolio returns by themselves are higher.

In contrast, one can minimize the total portfolio risk, but the returns will now be less.
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Table 4.1 illustrates the results from the three different objectives being optimized and shows
that the best approach is to maximize the returns to risk ratio, that is, for the same amount of
risk, this allocation provides the highest amount of return. Conversely, for the same amount of
return, this allocation provides the lowest amount of risk possible. This approach of bang for
the buck, or returns to risk ratio, is the cornerstone of the Markowitz efficient frontier in
modern portfolio theory. That is, if we constrained the total portfolio risk level and successively
increased it over time, we will obtain several efficient portfolio allocations for different risk
characteristics. Thus, different efficient portfolio allocations can be obtained for different
individuals with different risk preferences.
Portfolio
Returns
Portfolio
Risk
Portfolio
Returns to
Risk Ratio
Maximize Returns to Risk Ratio
12.69%
4.52%
2.8091
Maximize Returns
13.97%
6.77%
2.0636
Minimize Risk
12.38%
4.46%
2.7754
Objective
Table 4.1 – Optimization Results
Figure 4.3 – Continuous Optimization Results
4.3 Оптимизация с дискретными
целочисленными переменными
Sometimes, the decision variables are not continuous but are discrete integers (e.g., 0 and 1). We
can use optimization with discrete integer variables as on-off switches or go/no-go decisions.
Figure 4.4 illustrates a project selection model with 12 projects listed. The example here uses the
Discrete Optimization file found either on the start menu at Start | Real Options Valuation | Risk
Simulator | Examples or accessed directly through Risk Simulator | Example Models. Each
project has its own returns (ENPV and NPV, for expanded net present value and net present
value––the ENPV is simply the NPV plus any strategic real options values), costs of
implementation, risks, and so forth. If required, this model can be modified to include required
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full-time equivalences (FTE) and other resources of various functions, and additional
constraints can be set on these additional resources. The inputs into this model are typically
linked from other spreadsheet models. For instance, each project will have its own discounted
cash flow or returns on investment model. The application here is to maximize the portfolio’s
Sharpe ratio subject to some budget allocation. Many other versions of this model can be
created, for instance, maximizing the portfolio returns or minimizing the risks, or adding
constraints where the total number of projects chosen cannot exceed 6, and so forth and so on.
All of these items can be run using this existing model.
Procedure
Open the example file and start a new profile by clicking on Risk Simulator | New Profile and
provide it a name.

The first step in optimization is to set up the decision variables. Set the first decision
variable by selecting cell J4, select Risk Simulator | Optimization | Set Decision, click on
the link icon to select the name cell (B4), and select the Binary variable. Then, using
Risk Simulator’s copy, copy this cell J4 decision variable and paste the decision variable to
the remaining cells in J5 to J15. This is the best method if you have only several
decision variables and you can name each decision variable with a unique name for
identification later.

The second step in optimization is to set the constraint. There are two constraints
here: the total budget allocation in the portfolio must be less than $5,000 and the total
number of projects must not exceed 6. So, click on Risk Simulator | Optimization |
Constraints… and select ADD to add a new constraint. Then, select the cell D17 and
make it less than or equal to (<=) 5000. Repeat by setting cell J17 <= 6.

The final step in optimization is to set the objective function and start the optimization
by selecting cell C19 and Risk Simulator | Optimization | Set Objective. Then run the
optimization using Risk Simulator | Optimization | Run Optimization and selecting the
optimization of choice (Static Optimization, Dynamic Optimization, or Stochastic
Optimization). To get started, select Static Optimization. Check to make sure that the
objective cell is either the Sharpe ratio or portfolio returns to risk ratio and select Maximize.
You can now review the decision variables and constraints if required, or click OK to
run the static optimization.
Figure 4.5 shows the screen shots of these procedural steps. You can add simulation
assumptions on the model’s ENPV and risk (columns C and E), and apply the dynamic
optimization and stochastic optimization for additional practice.
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Figure 4.4 – Discrete Integer Optimization Model
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Figure 4.5 – Running Discrete Integer Optimization in Risk Simulator
Results
Interpretation
Figure 4.6 shows a sample optimal selection of projects that maximizes the Sharpe ratio. In
contrast, one can always maximize total revenues, but, as before, this is a trivial process and
simply involves choosing the highest returning project and going down the list until you run out
of money or exceed the budget constraint. Doing so will yield theoretically undesirable projects
as the highest yielding projects typically hold higher risks. Now, if desired, you can replicate the
optimization using a stochastic or dynamic optimization by adding assumptions in the ENPV
and/or cost, and/or risk values.
For additional hands-on examples of optimization in action, see the case study in Chapter 11
on Integrated Risk Management in the book, Real Options Analysis: Tools and Techniques,
Second Edition (Wiley Finance, 2010), by Dr. Johnathan Mun. That case study illustrates how an
efficient frontier can be generated and how forecasting, simulation, optimization, and real
options can be combined into a seamless analytical process.
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Figure 4.6 – Optimal Selection of Projects That Maximizes the Sharpe Ratio
4.4 Кривая Эффективности и дополнительные
настройки оптимизации
The middle graphic in Figure 4.5 shows the constraints set for the example optimization.
Within this function, if you click on the Efficient Frontier button after you have set some
constraints, you can make the constraints changing. That is, each of the constraints can be
created to step through between some maximum and minimum value. As an example, the
constraint in cell J17 <= 6 can be set to run between 4 and 8 (Figure 4.7). Thus, five optimizations
will be run, each with the following constraints: J17 <= 4, J17 <= 5, J17 <= 6, J17 <= 7, and
J17 <= 8. The optimal results will then be plotted as an efficient frontier and the report will be
generated (Figure 4.8). Specifically, here are the steps required to create a changing constraint:

In an optimization model (i.e., a model with Objective, Decision Variables, and
Constraints already set up), click on Risk Simulator | Optimization | Constraints and click
on Efficient Frontier.

Select the constraint you want to change or step (e.g., J17), enter in the parameters for
Min, Max, and Step Size (Figure 4.7), click ADD, and then click OK and OK again. You
should deselect D17 <= 5000 constraint before running.

Run Optimization as usual (Risk Simulator | Optimization | Run Optimization). You can
choose static, dynamic, or stochastic.

The results will be shown as a user interface (Figure 4.8). Click on Create Report to
generate a report worksheet with all the details of the optimization runs.
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Figure 4.7 – Generating Changing Constraints in an Efficient Frontier
Figure 4.8 – Efficient Frontier Results
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4.5 Стохастическая оптимизация
This example illustrates the application of stochastic optimization using a sample model with
four asset classes each with different risk and return characteristics. The idea here is to find the
best portfolio allocation such that the portfolio’s bang for the buck, or returns to risk ratio, is
maximized. That is, the goal is to allocate 100% of an individual’s investment among several
different asset classes (e.g., different types of mutual funds or investment styles: growth, value,
aggressive growth, income, global, index, contrarian, momentum, etc.). This model is different
from others in that there exists several simulation assumptions (risk and return values for each
asset in columns C and D), as seen in Figure 4.9.
A simulation is run, then optimization is executed, and the entire process is repeated multiple
times to obtain distributions of each decision variable. The entire analysis can be automated
using Stochastic Optimization. To run an optimization, several key specifications on the model
have to be identified first:
Objective:
Maximize Return to Risk Ratio (C12)
Decision Variables:
Allocation Weights (E6:E9)
Restrictions on Decision Variables:
Minimum and Maximum Required (F6:G9)
Constraints:
Portfolio Total Allocation Weights 100%
(E11 is set to 100%)
Simulation Assumptions:
Return and Risk Values (C6:D9)
The model shows the various asset classes. Each asset class has its own set of annualized
returns and annualized volatilities. These return and risk measures are annualized values such
that they can be consistently compared across different asset classes. Returns are computed
using the geometric average of the relative returns, while the risks are computed using the
logarithmic relative stock returns approach.
In Figure 4.9, column E (Allocation Weights) holds the decision variables, which are the
variables that need to be tweaked and tested such that the total weight is constrained at 100%
(cell E11). Typically, to start the optimization, we set these cells to a uniform value. In this case,
cells E6 to E9 are set at 25% each. In addition, each decision variable may have specific
restrictions in its allowed range. In this example, the lower and upper allocations allowed are
10% and 40%, as seen in columns F and G. This setting means that each asset class may have
its own allocation boundaries.
Next, column H shows the return to risk ratio, which is simply the return percentage divided by
the risk percentage for each asset, where the higher this value, the higher the bang for the buck.
The remaining parts of the model show the individual asset class rankings by returns, risk,
return to risk ratio, and allocation. In other words, these rankings show at a glance which asset
class has the lowest risk, or the highest return, and so forth.
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Figure 4.9 – Asset Allocation Model Ready for Stochastic Optimization
Procedure
To run this model, simply click on Risk Simulator | Optimization | Run Optimization. Alternatively,
and for practice, you can set up the model using the following steps illustrated in Figure 4.10:
1. Start a new profile (Risk Simulator | New Profile). For stochastic optimization, set
distributional assumptions on the risk and returns for each asset class. That is, select cell
C6, set an assumption (Risk Simulator | Set Input Assumption), and designate your own
assumption as required. Repeat for cells C7 to D9.
2. Select cell E6, and define the decision variable (Risk Simulator | Optimization | Set Decision or
click on the Set Decision D icon) and make it a Continuous Variable. Then link the decision
variable’s name and minimum/maximum required to the relevant cells (B6, F6, G6).
3. Then use Risk Simulator’s copy on cell E6, select cells E7 to E9, and use Risk Simulator’s
paste (Risk Simulator | Copy Parameter and Risk Simulator | Paste Parameter or use the copy
and paste icons). Remember not to use Excel’s regular copy and paste functions.
4. Next, set up the optimization’s constraints by selecting Risk Simulator | Optimization |
Constraints, selecting ADD, and selecting the cell E11 and making it equal 100% (total
allocation, and do not forget the % sign).
5. Select cell C12, the objective to be maximized, and make it the objective: Risk Simulator |
Optimization | Set Objective or click on the O icon. Run the optimization by going to Risk
Simulator | Optimization | Run Optimization. Review the different tabs to make sure that all
the required inputs in steps 2 and 3 are correct. Select Stochastic Optimization and let it run
for 500 trials repeated 20 times. Click OK when the simulation completes and a detailed
stochastic optimization report will be generated along with forecast charts of the decision
variables.
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Figure 4.10 – Setting Up the Stochastic Optimization Problem
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Results
Interpretation
Stochastic optimization is performed when a simulation is run first and then the optimization is
run. Then the whole analysis is repeated multiple times. As shown in Figure 4.11 for the
example optimization, the result is a distribution of each decision variable rather than a singlepoint estimate. This means that instead of saying you should invest 30.69% in Asset 1, the
results show that the optimal decision is to invest between 30.35% and 31.04% as long as the
total portfolio sums to 100%. This way, the results provide management or decision makers a
range of flexibility in the optimal decisions while accounting for the risks and uncertainties in
the inputs.
Notes
Super Speed Simulation with Optimization. You can also run stochastic optimization with
super speed simulation. To do this, first reset the optimization by resetting all four decision
variables back to 25%. Next, Run Optimization, click on the Advanced button (Figure 4.10),
and select the checkbox for Run Super Speed Simulation. Then, in the run optimization user
interface, select Stochastic Optimization on the Method tab and set it to run 500 trials and 20
optimization runs, and click OK. This approach will integrate the super speed simulation with
optimization. Notice how much faster the stochastic optimization runs. You can now quickly
rerun the optimization with a higher number of simulation trials.
Simulation Statistics for Stochastic and Dynamic Optimization. Notice that if there are
input simulation assumptions in the optimization model (i.e., these input assumptions are
required in order to run the dynamic or stochastic optimization routines), the Statistics tab is
now populated in the Run Optimization user interface. You can select from the drop-down list
the statistics you want, such as average, standard deviation, coefficient of variation, conditional
mean, conditional variance, a specific percentile, and so forth. This means that if you run a
stochastic optimization, a simulation of thousands of trials will first run, then the selected
statistic will be computed and this value will be temporarily placed in the simulation assumption
cell, then an optimization will be run based on this statistic, and then the entire process is
repeated multiple times. This method is important and useful for banking applications in
computing conditional Value at Risk, or conditional VaR.
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Figure 4.11 – Simulated Results from the Stochastic Optimization Approach
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5
5. АНАЛИТИЧЕСКИЕ
ИНСТРУМЕНТЫ RISK SIMULATOR
T
his chapter covers Risk Simulator’s analytical tools, providing detailed discussions of
the applicability of each tool and through example applications, complete with stepby-step illustrations. These tools are very valuable to analysts working in the realm of
risk analysis.
5.1 Торнадо и Инструменты чувствительности в
моделировании
Theory
Tornado analysis is a powerful simulation tool that captures the static impacts of each variable
on the outcome of the model. That is, the tool automatically perturbs each variable in the
model a preset amount, captures the fluctuation on the model’s forecast or final result, and lists
the resulting perturbations ranked from the most significant to the least. Figures 5.1 through 5.6
illustrate the application of a tornado analysis. For instance, Figure 5.1 is a sample discounted
cash flow model where the input assumptions in the model are shown. The question is what
are the critical success drivers that affect the model’s output the most? That is, what really
drives the net present value of $96.63 or which input variable impacts this value the most?
The tornado chart tool can be accessed through Risk Simulator | Tools | Tornado Analysis. To
follow along the first example, open the Tornado and Sensitivity Charts (Linear) file in the
examples folder. Figure 5.2 shows this sample model where cell G6 containing the net present
value is chosen as the target result to be analyzed. The target cell’s precedents in the model are
used in creating the tornado chart. Precedents are all the input and intermediate variables that
affect the outcome of the model. For instance, if the model consists of A = B + C, and where
C = D + E, then B, D, and E are the precedents for A (C is not a precedent as it is only an
intermediate calculated value). Figure 5.2 also shows the testing range of each precedent
variable used to estimate the target result. If the precedent variables are simple inputs, then the
testing range will be a simple perturbation based on the range chosen (e.g., the default is
±10%). Each precedent variable can be perturbed at different percentages if required. A wider
range is important as it is better able to test extreme values rather than smaller perturbations
around the expected values. In certain circumstances, extreme values may have a larger, smaller,
or unbalanced impact (e.g., nonlinearities may occur where increasing or decreasing economies
of scale and scope creep in for larger or smaller values of a variable) and only a wider range will
capture this nonlinear impact.
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Procedure
S I M U L A T O R

Select the single output cell (i.e., a cell with a function or equation) in an Excel model
(e.g., cell G6 is selected in our example).

Select Risk Simulator | Tools | Tornado Analysis.

Review the precedents and rename them as needed (renaming the precedents to
shorter names allows a more visually pleasing tornado and spider chart), and click OK.
Figure 5.1 – Sample Model
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Figure 5.2 – Running Tornado Analysis
Results
Interpretation
Figure 5.3 shows the resulting tornado analysis report, which indicates that capital investment
has the largest impact on net present value, followed by tax rate, average sale price, quantity
demanded of the product lines, and so forth. The report contains four distinct elements:
A statistical summary listing the procedure performed.
A sensitivity table (Figure 5.4) showing the starting NPV base value of 96.63 and how each
input is changed (e.g., Investment is changed from $1,800 to $1,980 on the upside with a +10%
swing, and from $1,800 to $1,620 on the downside with a –10% swing. The resulting upside
and downside values on NPV is –$83.37 and $276.63, with a total change of $360, making
investment the variable with the highest impact on NPV.) The precedent variables are ranked
from the highest impact to the lowest impact.
A spider chart (Figure 5.5) illustrating the effects graphically. The y-axis is the NPV target value
while the x-axis depicts the percentage change on each of the precedent values (the central
point is the base case value at 96.63 at 0% change from the base value of each precedent). A
positively sloped line indicates a positive relationship or effect, while negatively sloped lines
indicate a negative relationship (e.g., Investment is negatively sloped, which means that the
higher the investment level, the lower the NPV). The absolute value of the slope indicates the
magnitude of the effect (a steep line indicates a higher impact on the NPV y-axis given a change
in the precedent x-axis).
A tornado chart illustrating the effects in another graphical manner, where the highest
impacting precedent is listed first. The x-axis is the NPV value, with the center of the chart
being the base case condition. Green bars in the chart indicate a positive effect, while red bars
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indicate a negative effect. Therefore, for investments, the red bar on the right side indicates a
negative effect of investment on higher NPV––in other words, capital investment and NPV are
negatively correlated. The opposite is true for price and quantity of products A to C (their green
bars are on the right side of the chart).
Figure 5.3 – Tornado Analysis Report
Notes
Remember that tornado analysis is a static sensitivity analysis applied on each input variable in
the model––that is, each variable is perturbed individually and the resulting effects are tabulated.
This approach makes tornado analysis a key component to execute before running a
simulation. One of the very first steps in risk analysis is capturing and identifying the most
important impact drivers in the model. The next step is to identify which of these important
impact drivers are uncertain. These uncertain impact drivers are the critical success drivers of a
project, where the results of the model depend on these critical success drivers. These variables
are the ones that should be simulated. Do not waste time simulating variables that are neither
uncertain nor have little impact on the results. Tornado charts assist in identifying these critical
success drivers quickly and easily. Following this example, it might be that price and quantity
should be simulated, assuming that the required investment and effective tax rate are both
known in advance and unchanging.
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Figure 5.4 – Sensitivity Table
Figure 5.5 – Spider Chart
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Figure 5.6 – Tornado Chart
Although the tornado chart is easier to read, the spider chart is important for determining if
there are any nonlinearities in the model. For instance, Figure 5.7 shows another spider chart
where nonlinearities are fairly evident (the lines on the graph are not straight but curved). The
model used is Tornado and Sensitivity Charts (Nonlinear), which uses the Black-Scholes option
pricing model as an example. Such nonlinearities cannot be ascertained from a tornado chart
and may be important information in the model or provide decision makers with important
insight into the model’s dynamics.
Notes
Figure 5.2 shows the Tornado analysis tool’s user interface. Notice that there are a few new
enhancements starting in Risk Simulator version 4 and beyond. Here are some tips on running
Tornado analysis and details on the new enhancements:

Tornado analysis should never be run just once. It is meant as a model diagnostic tool,
which means that it should ideally be run several times on the same model. For
instance, in a large model, Tornado can be run the first time using all of the default
settings and all precedents should be shown (select Show All Variables). The result
may be a large report and long (and potentially unsightly) Tornado charts.
Nonetheless, this analysis provides a great starting point to determine how many of
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the precedents are considered critical success factors. For example, the Tornado chart
may show that the first 5 variables have high impact on the output, while the
remaining 200 variables have little to no impact, in which case, a second Tornado
analysis is run showing fewer variables. For example, select the Show Top 10 Variables
if the first 5 are critical, thereby creating a nice report and Tornado chart that shows a
contrast between the key factors and less critical factors. (You should never show a
Tornado chart with only the key variables. You need to show some less critical
variables as a contrast to their effects on the output). Finally, the default testing points
can be increased from the ±10% of the parameter to some larger value to test for
nonlinearities (the Spider chart will show nonlinear lines and Tornado charts will be
skewed to one side if the precedent effects are nonlinear).

Selecting Use Cell Address is always a good idea if your model is large, as it allows you to
identify the location (worksheet name and cell address) of a precedent cell. If this
option is not selected, the software will apply its own fuzzy logic in an attempt to
determine the name of each precedent variable (in a large model, the names might
sometimes end up being confusing, with repeated variables or the names that are too
long, possibly making the Tornado chart unsightly).

The Analyze This Worksheet and Analyze All Worksheets options allow you to control
whether the precedents should only be part of the current worksheet or include all
worksheets in the same workbook. This option comes in handy when you are only
attempting to analyze an output based on values in the current sheet versus
performing a global search of all linked precedents across multiple worksheets in the
same workbook.

Selecting Use Global Setting is useful when you have a large model and wish to test all
the precedents at, say, ±50% instead of the default 10%. Instead of having to change
each precedent’s test values one at a time, you can select this option, change one
setting and click somewhere else in the user interface to change the entire list of the
precedents. Deselecting this option will allow you the control to change test points one
precedent at a time.

Ignore Zero or Empty Values is an option turned on by default where precedent cells with
zero or empty values will not be run in the Tornado analysis. This is the typical setting.

Highlight Possible Integer Values is an option that quickly identifies all possible precedent
cells that currently have integer inputs. This function is sometimes important if your
model uses switches (e.g., functions such as IF a cell is 1. then something happens, and
IF a cell has a 0 value, something else happens, or integers such as 1, 2, 3, etc., which
you do not wish to test). For instance, ±10% of a flag switch value of 1 will return a
test value of 0.9 and 1.1, both of which are irrelevant and incorrect input values in the
model, and Excel may interpret the function as an error. This option, when selected,
will quickly highlight potential problem areas for Tornado analysis, and then you can
determine which precedents to turn on or off manually, or you can use the Ignore
Possible Integer Values function to turn all of them off simultaneously.
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Figure 5.7 – Nonlinear Spider Chart
5.2 Анализ чувствительности
Theory
While tornado analysis (tornado charts and spider charts) applies static perturbations before a
simulation run, sensitivity analysis applies dynamic perturbations created after the simulation
run. Tornado and spider charts are the results of static perturbations, meaning that each
precedent or assumption variable is perturbed a preset amount one at a time, and the
fluctuations in the results are tabulated. In contrast, sensitivity charts are the results of dynamic
perturbations in the sense that multiple assumptions are perturbed simultaneously and their
interactions in the model and correlations among variables are captured in the fluctuations of
the results. Tornado charts, therefore, identify which variables drive the results the most and,
hence, are suitable for simulation, whereas sensitivity charts identify the impact to the results
when multiple interacting variables are simulated together in the model. This effect is clearly
illustrated in Figure 5.8. Notice that the ranking of critical success drivers similar to the tornado
chart in the previous examples. However, if correlations are added between the assumptions, a
very different picture results, as shown in Figure 5.9. Notice, for instance, that price erosion had
little impact on NPV, but when some of the input assumptions are correlated, the interaction
that exists between these correlated variables makes price erosion have more impact.
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Figure 5.8 – Sensitivity Chart Without Correlations
Figure 5.9 – Sensitivity Chart With Correlations
Procedure

Open or create a model, define assumptions and forecasts, and run the simulation (the
example here uses the Tornado and Sensitivity Charts (Linear) file).

Select Risk Simulator | Tools | Sensitivity Analysis.

Select the forecast of choice to analyze and click OK (Figure 5.10)
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Figure 5.10 – Running Sensitivity Analysis
Results
Interpretation
The results of the sensitivity analysis comprise a report and two key charts. The first is a
nonlinear rank correlation chart (Figure 5.11) that ranks from highest to lowest the assumptionforecast correlation pairs. These correlations are nonlinear and nonparametric, making them
free of any distributional requirements (i.e., an assumption with a Weibull distribution can be
compared to another with a beta distribution). The results from this chart are fairly similar to
that of the tornado analysis seen previously (of course, without the capital investment value,
which we decided was a known value and, hence, was not simulated), with one special
exception: Tax rate was relegated to a much lower position in the sensitivity analysis chart
(Figure 5.11) as compared to the tornado chart (Figure 5.6). This is because by itself, tax rate
will have a significant impact, but once the other variables are interacting in the model, it
appears that tax rate has less of a dominant effect (because tax rate has a smaller distribution as
historical tax rates tend not to fluctuate too much, and also because tax rate is a straight
percentage value of the income before taxes, where other precedent variables have a larger
effect on). This example proves that performing sensitivity analysis after a simulation run is
important to ascertain if there are any interactions in the model and if the effects of certain
variables still hold. The second chart (Figure 5.12) illustrates the percent variation explained.
That is, of the fluctuations in the forecast, how much of the variation can be explained by each
of the assumptions after accounting for all the interactions among variables? Notice that the
sum of all variations explained is usually close to 100% (there are sometimes other elements
that impact the model but that cannot be captured here directly), and if correlations exist, the
sum may sometimes exceed 100% (due to the interaction effects that are cumulative).
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Figure 5.11 – Rank Correlation Chart
Figure 5.12 – Contribution to Variance Chart
Notes
Tornado analysis is performed before a simulation run, while sensitivity analysis is performed
after a simulation run. Spider charts in tornado analysis can consider nonlinearities, while rank
correlation charts in sensitivity analysis can account for nonlinear and distributional-free
conditions.
5.3 Распределительная установка с одной или
несколькими переменными
Theory
Another powerful simulation tool is distributional fitting. That is, determining which
distribution to use for a particular input variable in a model and what the relevant distributional
parameters are. If no historical data exist, then the analyst must make assumptions about the
variables in question. One approach is to use the Delphi method where a group of experts are
tasked with estimating the behavior of each variable. For instance, a group of mechanical
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engineers can be tasked with evaluating the extreme possibilities of a spring coil’s diameter
through rigorous experimentation or guesstimates. These values can be used as the variable’s
input parameters (e.g., uniform distribution with extreme values between 0.5 and 1.2). When
testing is not possible (e.g., market share and revenue growth rate), management can still make
estimates of potential outcomes and provide the best-case, most-likely case, and worst-case
scenarios.
However, if reliable historical data are available, distributional fitting can be accomplished.
Assuming that historical patterns hold and that history tends to repeat itself, then historical data
can be used to find the best-fitting distribution with their relevant parameters to better define
the variables to be simulated. Figures 5.13, 5.14, and 5.15 illustrate a distributional-fitting
example. This illustration uses the Data Fitting file in the examples folder.
Procedure
Results
Interpretation

Open a spreadsheet with existing data for fitting.

Select the data you wish to fit (data should be in a single column with multiple rows).

Select Risk Simulator | Tools | Distributional Fitting (Single-Variable).

Select the specific distributions you wish to fit to or keep the default where all
distributions are selected and click OK (Figure 5.13).

Review the results of the fit, choose the relevant distribution you want, and click OK
(Figure 5.14).
The null hypothesis being tested is such that the fitted distribution is the same distribution
as the population from which the sample data to be fitted comes. Thus, if the computed pvalue is lower than a critical alpha level (typically 0.10 or 0.05), then the distribution is the
wrong distribution. Conversely, the higher the p-value, the better the distribution fits the
data. Roughly, you can think of p-value as a percentage explained; that is, if the p-value is
0.9727 (Figure 5.14), then setting a normal distribution with a mean of 99.28 and a
standard deviation of 10.17 explains about 97.27% of the variation in the data, indicating
an especially good fit. Both the results (Figure 5.14) and the report (Figure 5.15) show the
test statistic, p-value, theoretical statistics (based on the selected distribution), empirical
statistics (based on the raw data), the original data (to maintain a record of the data used),
and the assumption complete with the relevant distributional parameters (i.e., if you
selected the option to automatically generate assumption and if a simulation profile already
exists). The results also rank all the selected distributions and how well they fit the data.
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Figure 5.13 – Single Variable Distributional Fitting
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Figure 5.14 – Distributional Fitting Result
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Figure 5.15 – Distributional Fitting Report
For fitting multiple variables, the process is fairly similar to fitting individual variables. However,
the data should be arranged in columns (i.e., each variable is arranged as a column) and all the
variables are fitted one at a time.
Procedure
Notes

Open a spreadsheet with existing data for fitting.

Select the data you wish to fit (data should be in a multiple columns with multiple rows).

Select Risk Simulator | Tools | Distributional Fitting (Multi-Variable).

Review the data, choose the relevant types of distribution you want and click OK.
Notice that the statistical ranking methods used in the distributional fitting routines are the chisquare test and Kolmogorov-Smirnov test. The former is used to test discrete distributions and
the latter, continuous distributions. Briefly, a hypothesis test coupled with an internal
optimization routine is used to find the best-fitting parameters on each distribution tested, and
the results are ranked from the best fit to the worst fit.
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5.4 Bootstrap Моделирование
Theory
Procedure
Bootstrap simulation is a simple technique that estimates the reliability or accuracy of forecast
statistics or other sample raw data. Essentially, bootstrap simulation is used in hypothesis
testing. Classical methods used in the past relied on mathematical formulas to describe the
accuracy of sample statistics. These methods assume that the distribution of a sample statistic
approaches a normal distribution, making the calculation of the statistic’s standard error or
confidence interval relatively easy. However, when a statistic’s sampling distribution is not
normally distributed or easily found, these classical methods are difficult to use or are invalid. In
contrast, bootstrapping analyzes sample statistics empirically by repeatedly sampling the data
and creating distributions of the different statistics from each sampling.

Run a simulation.

Select Risk Simulator | Tools | Nonparametric Bootstrap.

Select only one forecast to bootstrap, select the statistic(s) to bootstrap, enter the number
of bootstrap trials, and click OK (Figure 5.16).
Figure 5.16 – Nonparametric Bootstrap Simulation
Results
Interpretation
In essence, nonparametric bootstrap simulation can be thought of as simulation based on a
simulation. Thus, after running a simulation, the resulting statistics are displayed, but the
accuracy of such statistics and their statistical significance are sometimes in question. For
instance, if a simulation run’s skewness statistic is –0.10, is this distribution truly negatively
skewed or is the slight negative value attributable to random chance? What about –0.15, –0.20,
and so forth? That is, how far is far enough such that this distribution is considered to be
negatively skewed? The same question can be applied to all the other statistics. Is one
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distribution statistically identical to another distribution with regard to some computed statistics
or are they significantly different? Suppose for instance, the 90% confidence for the skewness
statistic is between –0.0189 and 0.0952, such that the value 0 falls within this confidence,
indicating that on a 90% confidence, the skewness of this forecast is not statistically significantly
different from 0, or that this distribution can be considered as symmetrical and not skewed.
Conversely, if the value 0 falls outside of this confidence, then the opposite is true, and the
distribution is skewed (positively skewed if the forecast statistic is positive, and negatively
skewed if the forecast statistic is negative). Figure 5.17 illustrates some sample bootstrap results.
Figure 5.17 – Bootstrap Simulation Results
Notes
The term bootstrap comes from the saying, “to pull oneself up by one’s own bootstraps,” and
is applicable because this method uses the distribution of statistics themselves to analyze the
statistics’ accuracy. Nonparametric simulation is simply randomly picking golf balls from a large
basket with replacement where each golf ball is based on a historical data point. Suppose there
are 365 golf balls in the basket (representing 365 historical data points). Imagine that the value
of each golf ball picked at random is written on a large whiteboard. The results of the 365 balls
picked with replacement are written in the first column of the board with 365 rows of numbers.
Relevant statistics (e.g., mean, median, standard deviation, etc.) are calculated on these 365
rows. The process is then repeated, say, five thousand times. The whiteboard will now be filled
with 365 rows and 5,000 columns. Hence, 5,000 sets of statistics (i.e., there will be 5,000 means,
5,000 medians, 5,000 standard deviations, etc.) are tabulated and their distributions shown. The
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relevant statistics of the statistics are then tabulated, where from these results one can ascertain
how confident the simulated statistics are. In other words, in a simple 10,000-trial simulation,
say the resulting forecast average is found to be $5.00. How certain is the analyst of the results?
Bootstrapping allows the user to ascertain the confidence interval of the calculated mean
statistic, indicating the distribution of the statistics. Finally, bootstrap results are important
because according to the Law of Large Numbers and the Central Limit Theorem in statistics,
the mean of the sample means is an unbiased estimator and approaches the true population
mean when the sample size increases.
5.5 Проверка гипотезы
Theory
Procedure
A hypothesis test is performed when testing the means and variances of two distributions to
determine if they are statistically identical or statistically different from one another; that is,
whether the differences are based on random chance or if they are, in fact, statistically
significant.

Run a simulation.

Select Risk Simulator | Tools | Hypothesis Testing.

Select only two forecasts to test at a time, select the type of hypothesis test you wish to
run, and click OK (Figure 5.18).
Figure 5.18 – Hypothesis Testing
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Results
Interpretation
S I M U L A T O R
A two-tailed hypothesis test is performed on the null hypothesis (H0) such that the two
variables' population means are statistically identical to one another. The alternative hypothesis
(Ha) is such that the population means are statistically different from one another. If the
calculated p-values are less than or equal to 0.01, 0.05, or 0.10, this means that the null
hypothesis is rejected, which implies that the forecast means are statistically significantly
different at the 1%, 5%, and 10% significance levels. If the null hypothesis is not rejected when
the p-values are high, the means of the two forecast distributions are statistically similar to one
another. The same analysis is performed on variances of two forecasts at a time using the
pairwise F-test. If the p-values are small, then the variances (and standard deviations) are
statistically different from one another; otherwise, for large p-values, the variances are
statistically identical to one another.
Figure 5.19 – Hypothesis Testing Results
Notes
The two-variable t-test with unequal variances (the population variance of forecast 1 is expected
to be different from the population variance of forecast 2) is appropriate when the forecast
distributions are from different populations (e.g., data collected from two different geographical
locations or two different operating business units). The two-variable t-test with equal variances
(the population variance of forecast 1 is expected to be equal to the population variance of
forecast 2) is appropriate when the forecast distributions are from similar populations (e.g., data
collected from two different engine designs with similar specifications). The paired dependent
two-variable t-test is appropriate when the forecast distributions are from the exact same
population (e.g., data collected from the same group of customers but on different occasions).
5.6 Извлечение данных и сохранение
результатов моделирования
A simulation’s raw data can be very easily extracted using Risk Simulator’s Data Extraction
routine. Both assumptions and forecasts can be extracted, but a simulation must first be run.
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The extracted data can then be used for a variety of other analysis.
Procedure

Open or create a model, define assumptions and forecasts, and run the simulation.

Select Risk Simulator | Tools | Data Extraction.

Select the assumptions and/or forecasts you wish to extract the data from and click OK.
The data can be extracted to various formats:

Raw data in a new worksheet where the simulated values (both assumptions and
forecasts) can then be saved or further analyzed as required

Flat text file where the data can be exported into other data analysis software

Risk Simulator file where the results (both assumptions and forecasts) can be retrieved
at a later time by selecting Risk Simulator | Tools | Data Open/Import
The third option is the most popular selection, that is, to save the simulated results as a
*.risksim file where the results can be retrieved later and a simulation does not have to be rerun
each time. Figure 5.20 shows the dialog box for extracting or exporting and saving the
simulation results.
Figure 5.20 – Sample Simulation Report
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5.7 Создать отчет
After a simulation is run, you can generate a report of the assumptions and forecasts used in the
simulation run, as well as the results obtained during the simulation run.
Procedure

Open or create a model, define assumptions and forecasts, and run the simulation.

Select Risk Simulator | Create Report (Figure 5.21).
Симуляция - Example Simulation
Общий
Число попыток
Остановка симуляции при
Случайный источник
Включить корреляции
1000
Нет
999
Да
Допущения
Имя
G8: Доход
Включено
Да
Ячейка
$G$8
Нет
Динамическая симуляция
Имя
G9: Затраты
Включено
Да
Ячейка
$G$9
Динамическая симуляция
Нет
Диапазон
Минимум
Максимум
Диапазон
Минимум
Максимум
-Infinity
Infinity
Треугольное
Распределение
Минимальное
1.5
Наиболее вероятное
2
Максимальное
2.25
Распределение
Минимальное
Максимальное
-Infinity
Infinity
Равномерное
0.85
1.25
Прогнозы
Имя
Включено
Ячейка
Выручка
Да
$G$10
Точность прогноза
Уровень точности
Уровень ошибок
-----
Число точек данных
Среднее
Медиана
Стандартное отклонение
Дисперсия
Коэффициент вариативно
Максимум
Минимум
Диапазон
Асимметрия
Куртозис
25% процентиль
75% процентиль
Случайная ошибка на 95%
1000
0.8626
0.8674
0.1933
0.0374
0.2241
1.3570
0.3019
1.0551
-0.1157
-0.4480
0.7269
1.0068
0.0139
Матрица корреляции
G8: Доход
G9: Затраты
G8: Доход 9: Затраты
1.00
0.00
1.00
Figure 5.21 – Sample Simulation Report
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5.8 Диагностический инструменты Регрессии и
Прогнозирования
The regression and forecasting Diagnostic tool in Risk Simulator is an advanced analytical tool
used to determine the econometric properties of your data. The diagnostics include checking
the data for heteroskedasticity, nonlinearity, outliers, specification errors, micronumerosity,
stationarity and stochastic properties, normality and sphericity of the errors, and
multicollinearity. Each test is described in more detail in its respective report in the model.
Procedure

Open the example model (Risk Simulator | Examples | Regression Diagnostics), go to the
Time-Series Data worksheet, and select the data, including the variable names (cells
C5:H55).

Click on Risk Simulator | Tools | Diagnostic Tool.

Check the data and select from the Dependent Variable Y drop-down menu. Click OK
when finished (Figure 5.22).
Figure 5.22 – Running the Data Diagnostic Tool
A common violation in forecasting and regression analysis is heteroskedasticity, that is, the
variance of the errors increases over time (see Figure 5.23 for test results using the Diagnostic
tool). Visually, the width of the vertical data fluctuations increases, or fans out, over time, and,
typically, the coefficient of determination (R-squared coefficient) drops significantly when
heteroskedasticity exists. If the variance of the dependent variable is not constant, then the
error’s variance will not be constant. Unless the heteroskedasticity of the dependent variable is
pronounced, its effect will not be severe: The least-squares estimates will still be unbiased, and
the estimates of the slope and intercept will either be normally distributed if the errors are
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normally distributed, or at least normally distributed asymptotically (as the number of data
points becomes large) if the errors are not normally distributed. The estimate for the variance of
the slope and overall variance will be inaccurate, but the inaccuracy is not likely to be substantial
if the independent-variable values are symmetric about their mean.
If the number of data points is small (micronumerosity), it may be difficult to detect
assumption violations. With small sample sizes, assumption violations such as non-normality or
heteroskedasticity of variances are difficult to detect even when they are present. With a small
number of data points, linear regression offers less protection against violation of assumptions.
With few data points, it may be hard to determine how well the fitted line matches the data, or
whether a nonlinear function would be more appropriate. Even if none of the test assumptions
are violated, a linear regression on a small number of data points may not have sufficient power
to detect a significant difference between the slope and zero, even if the slope is nonzero. The
power depends on the residual error, the observed variation in the independent variable, the
selected significance alpha level of the test, and the number of data points. Power decreases as
the residual variance increases, decreases as the significance level is decreased (i.e., as the test is
made more stringent), increases as the variation in observed independent variable increases, and
increases as the number of data points increases.
Values may not be identically distributed because of the presence of outliers which are
anomalous values in the data. Outliers may have a strong influence over the fitted slope and
intercept, giving a poor fit to the bulk of the data points. Outliers tend to increase the estimate
of residual variance, lowering the chance of rejecting the null hypothesis (that is, creating higher
prediction errors). They may be due to recording errors, which may be correctable, or they may
be due to the dependent-variable values not all being sampled from the same population.
Apparent outliers may also be due to the dependent-variable values being from the same, but
non-normal, population. However, a point may be an unusual value in either an independent or
dependent variable without necessarily being an outlier in the scatter plot. In regression analysis,
the fitted line can be highly sensitive to outliers. In other words, least squares regression is not
resistant to outliers, thus, neither is the fitted-slope estimate. A point vertically removed from
the other points can cause the fitted line to pass close to it, instead of following the general
linear trend of the rest of the data, especially if the point is relatively far horizontally from the
center of the data.
However, great care should be taken when deciding if the outliers should be removed.
Although in most cases when outliers are removed, the regression results look better, a priori
justification must first exist. For instance, if one is regressing the performance of a particular
firm’s stock returns, outliers caused by downturns in the stock market should be included; these
are not truly outliers as they are inevitabilities in the business cycle. Forgoing these outliers and
using the regression equation to forecast one’s retirement fund based on the firm’s stocks will
yield incorrect results at best. In contrast, suppose the outliers are caused by a single
nonrecurring business condition (e.g., merger and acquisition) and such business structural
changes are not forecast to recur. These outliers, then, should be removed and the data
cleansed prior to running a regression analysis. The analysis here only identifies outliers and it is
up to the user to determine if they should remain or be excluded.
Sometimes, a nonlinear relationship between the dependent and independent variables is more
appropriate than a linear relationship. In such cases, running a linear regression will not be
optimal. If the linear model is not the correct form, then the slope and intercept estimates and
the fitted values from the linear regression will be biased, and the fitted slope and intercept
estimates will not be meaningful. Over a restricted range of independent or dependent
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variables, nonlinear models may be well approximated by linear models (this is, in fact, the basis
of linear interpolation), but for accurate prediction, a model appropriate to the data should be
selected. A nonlinear transformation should first be applied to the data before running a
regression. One simple approach is to take the natural logarithm of the independent variable
(other approaches include taking the square root or raising the independent variable to the
second or third power) and run a regression or forecast using the nonlinearly transformed data.
Figure 5.23 – Results from Tests of Outliers, Heteroskedasticity, Micronumerosity, and Nonlinearity
Another typical issue when forecasting time-series data is whether the independent-variable
values are truly independent of each other or are actually dependent. Dependent variable values
collected over a time series may be autocorrelated. For serially correlated dependent-variable
values, the estimates of the slope and intercept will be unbiased, but the estimates of their
forecast and variances will not be reliable and, hence, the validity of certain statistical goodnessof-fit tests will be flawed. For instance, interest rates, inflation rates, sales, revenues, and many
other time-series data are typically autocorrelated, where the value in the current period is
related to the value in a previous period, and so forth (clearly, the inflation rate in March is
related to February’s level, which, in turn, is related to January’s level, etc.). Ignoring such
blatant relationships will yield biased and less accurate forecasts. In such events, an
autocorrelated regression model, or an ARIMA model, may be better suited (Risk Simulator |
Forecasting | ARIMA). Finally, the autocorrelation functions of a series that is nonstationary
tend to decay slowly (see the nonstationary report in the model).
If autocorrelation AC(1) is nonzero, it means that the series is first-order serially correlated. If
AC(k) dies off more or less geometrically with increasing lag, it implies that the series follows a
low-order autoregressive process. If AC(k) drops to zero after a small number of lags, it implies
that the series follows a low-order moving-average process. Partial correlation PAC(k) measures
the correlation of values that are k periods apart after removing the correlation from the
intervening lags. If the pattern of autocorrelation can be captured by an autoregression of order
less than k, then the partial autocorrelation at lag k will be close to zero. Ljung-Box Q-statistics
and their p-values at lag k have the null hypothesis that there is no autocorrelation up to order
k. The dotted lines in the plots of the autocorrelations are the approximate two standard error
bounds. If the autocorrelation is within these bounds, it is not significantly different from zero
at the 5% significance level.
Autocorrelation measures the relationship to the past of the dependent Y variable to itself.
Distributive lags, in contrast, are time-lag relationships between the dependent Y variable and
different independent X variables. For instance, the movement and direction of mortgage rates
tend to follow the federal funds rate but at a time lag (typically 1 to 3 months). Sometimes, time
lags follow cycles and seasonality (e.g., ice cream sales tend to peak during the summer months
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and are, hence, related to last summer’s sales, 12 months in the past). The distributive lag
analysis (Figure 5.24) shows how the dependent variable is related to each of the independent
variables at various time lags, when all lags are considered simultaneously, to determine which
time lags are statistically significant and should be considered.
Figure 5.24 – Autocorrelation and Distributive Lag Results
Another requirement in running a regression model is the assumption of normality and
sphericity of the error term. If the assumption of normality is violated or outliers are present,
then the linear regression goodness-of-fit test may not be the most powerful or informative test
available, and this could mean the difference between detecting a linear fit or not. If the errors
are not independent and not normally distributed, it may indicate that the data might be
autocorrelated or suffer from nonlinearities or other more destructive errors. Independence of
the errors can also be detected in the heteroskedasticity tests (Figure 5.25).
The Normality test on the errors performed is a nonparametric test, which makes no
assumptions about the specific shape of the population, from which the samples are drawn,
allowing for smaller sample data sets to be analyzed. This test evaluates the null hypothesis of
whether the sample errors were drawn from a normally distributed population, versus an
alternate hypothesis that the data sample is not normally distributed. If the calculated D-statistic
is greater than or equal to the D-critical values at various significance values, then reject the null
hypothesis and accept the alternate hypothesis (the errors are not normally distributed).
Otherwise, if the D-statistic is less than the D-critical value, do not reject the null hypothesis
(the errors are normally distributed). The Normality test relies on two cumulative frequencies:
one derived from the sample data set and the second from a theoretical distribution based on
the mean and standard deviation of the sample data.
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Figure 5.25 – Test for Normality of Errors
Sometimes, certain types of time-series data cannot be modeled using any other methods
except for a stochastic process, because the underlying events are stochastic in nature. For
instance, you cannot adequately model and forecast stock prices, interest rates, price of oil, and
other commodity prices using a simple regression model because these variables are highly
uncertain and volatile, and they do not follow a predefined static rule of behavior; in other
words, the process is not stationary. Stationarity is checked using the Runs Test function, while
another visual clue is found in the autocorrelation report (the ACF tends to decay slowly). A
stochastic process is a sequence of events or paths generated by probabilistic laws. That is,
random events can occur over time but are governed by specific statistical and probabilistic
rules. The main stochastic processes include random walk or Brownian motion, meanreversion, and jump-diffusion. These processes can be used to forecast a multitude of variables
that seemingly follow random trends but restricted by probabilistic laws. The processgenerating equation is known in advance but the actual results generated are unknown (Figure
5.26).
The Random Walk Brownian Motion process can be used to forecast stock prices, prices of
commodities, and other stochastic time-series data given a drift or growth rate and volatility
around the drift path. The Mean-Reversion process can be used to reduce the fluctuations of
the Random Walk process by allowing the path to target a long-term value, making it useful for
forecasting time-series variables that have a long-term rate such as interest rates and inflation
rates (these are long-term target rates by regulatory authorities or the market). The JumpDiffusion process is useful for forecasting time-series data when the variable can occasionally
exhibit random jumps, such as oil prices or price of electricity (discrete exogenous event shocks
can make prices jump up or down). These processes can also be mixed and matched as
required.
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A note of caution is required here. The stochastic parameters calibration shows all the
parameters for all processes and does not distinguish which process is better and which is
worse or which process is more appropriate to use. It is up to the user to make this
determination. For instance, if we see a 283% reversion rate, chances are, a mean-reversion
process is inappropriate; or a very high jump rate of, say, 100% most probably means that a
jump-diffusion process
is probably not appropriate; and so forth. Further, the analysis
cannot determine what the variable is and what the data source is. For instance, is the raw data
from historical stock prices or is it the historical prices of electricity or inflation rates or the
molecular motion of subatomic particles, and so forth. Only the user would know about the
raw data, and, hence, using a priori knowledge and theory, be able to pick the correct process to
use (e.g., stock prices tend to follow a Brownian motion random walk, whereas inflation rates
follow a mean-reversion process; or a jump-diffusion process is more appropriate should you
be forecasting the price of electricity).
Figure 5.26 – Stochastic Process Parameter Estimation
Multicollinearity exists when there is a linear relationship between the independent variables.
When this occurs, the regression equation cannot be estimated at all. In near collinearity
situations, the estimated regression equation will be biased and provide inaccurate results. This
situation is especially true when a stepwise regression approach is used, where the statistically
significant independent variables will be thrown out of the regression mix earlier than expected,
resulting in a regression equation that is neither efficient nor accurate. One quick test of the
presence of multicollinearity in a multiple regression equation is that the R-squared value is
relatively high, while the t-statistics are relatively low.
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Another quick test is to create a correlation matrix between the independent. A high crosscorrelation indicates a potential for autocorrelation. The rule of thumb is that a correlation with
an absolute value greater than 0.75 is indicative of severe multicollinearity.
Figure 5.27 – Multicollinearity Errors
The Correlation Matrix lists the Pearson’s Product Moment Correlations (commonly referred
to as the Pearson’s R) between variable pairs. The correlation coefficient ranges between –1.0
and + 1.0 inclusive. The sign indicates the direction of association between the variables, while
the coefficient indicates the magnitude or strength of association. The Pearson’s R only
measures a linear relationship and is less effective in measuring nonlinear relationships.
To test whether the correlations are significant, a two-tailed hypothesis test is performed and
the resulting p-value(s) is listed. In Figure 5.27 (top), P-values less than 0.10, 0.05, and 0.01 are
highlighted in blue to indicate statistical significance. In other words, a p-value for a correlation
pair that is less than a given significance value is statistically significantly different from zero,
indicating that there is significant a linear relationship between the two variables.
The Pearson’s R between two variables (x and y) is related to the covariance (cov) measure,
where: R x, y 
COV x, y
sxsy
. The benefit of dividing the covariance by the product of the two
variables’ standard deviation (s) is that the resulting correlation coefficient is bounded between
–1.0 and +1.0 inclusive. This makes the correlation a good relative measure to compare among
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different variables (particularly with different units and magnitude). The Spearman rank-based
nonparametric correlation is also included in the report. The Spearman’s R is related to the
Pearson’s R in that the data is first ranked and then correlated. The rank correlation provides a
better estimate of the relationship between two variables when one or both of them is
nonlinear.
It must be stressed that a significant correlation does not imply causation. Associations between
variables in no way imply that the change of one variable causes another variable to change.
When two variables that are moving independently of each other but in a related path, they may
be correlated but their relationship might be spurious (e.g., a correlation between sunspots and
the stock market might be strong, but one can surmise that there is no causality and that this
relationship is purely spurious).
Another test for multicollinearity is the use of the variance
inflation factor (VIF), obtained by regressing each independent variable to all the other
independent variables, obtaining the R-squared value, and calculating the VIF. A VIF exceeding
2.0 can be considered as severe multicollinearity. A VIF exceeding 10.0 indicates destructive
multicollinearity (Figure 5.27, bottom).
5.9 Инструмент статистического анализа
Another very powerful tool in Risk Simulator is the Statistical Analysis tool, which determines
the statistical properties of the data. The diagnostics run include checking the data for various
statistical properties, from basic descriptive statistics to testing for and calibrating the stochastic
properties of the data.
Procedure

Open the example model (Risk Simulator | Examples | Statistical Analysis), go to the
Data worksheet, and select the data including the variable names (cells C5:E55).

Click on Risk Simulator | Tools | Statistical Analysis (Figure 5.28).

Check the data type, whether the data selected are from a single variable or multiple
variables arranged in rows. In our example, we assume that the data areas selected are
from multiple variables. Click OK when finished.

Choose the statistical tests you wish to perform. The suggestion (and by default) is to
choose all the tests. Click OK when finished (Figure 5.29).
Spend some time going through the reports generated to get a better understanding of the
statistical tests performed (sample reports are shown in Figures 5.30 through 5.33).
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Figure 5.28 – Running the Statistical Analysis Tool
Figure 5.29 – Statistical Tests
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Figure 5.30 – Sample Statistical Analysis Tool Report
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Figure 5.31 – Sample Statistical Analysis Tool Report (Hypothesis Testing of One Variable)
Figure 5.32 – Sample Statistical Analysis Tool Report (Normality Test)
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Figure 5.33 – Sample Statistical Analysis Tool Report (Stochastic Parameter Estimation)
5.10 Инструмент анализа распределений
The Distributional Analysis tool is a statistical probability tool in Risk Simulator that is useful in
a variety of settings. It can be used to compute the probability density function (PDF), which is
also called the probability mass function (PMF) for discrete distributions (these terms are used
interchangeably), where given some distribution and its parameters, we can determine the
probability of occurrence given some outcome x. In addition, the cumulative distribution
function (CDF) can be computed, which is the sum of the PDF values up to this x value.
Finally, the inverse cumulative distribution function (ICDF) is used to compute the value x
given the cumulative probability of occurrence.
This tool is accessible via Risk Simulator | Tools | Distributional Analysis. As an example of its use,
Figure 5.34 shows the computation of a binomial distribution (i.e., a distribution with two
outcomes, such as the tossing of a coin, where the outcome is either Head or Tail, with some
prescribed probability of heads and tails). Suppose we toss a coin two times. Setting the
outcome Head as a success, we use the binomial distribution with Trials = 2 (tossing the coin
twice) and Probability = 0.50 (the probability of success, of getting Heads). Selecting the PDF
and setting the range of values x as from 0 to 2 with a step size of 1 (this means we are
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requesting the values 0, 1, 2 for x), the resulting probabilities, as well as the theoretical four
moments of the distribution, are provided in tabular and in graphical formats. As the outcomes
of the coin toss are Heads-Heads, Tails-Tails, Heads-Tails, and Tails-Heads, the probability of
getting exactly no Heads is 25%, one Head is 50%, and two Heads is 25%. Similarly, we can
obtain the exact probabilities of tossing the coin, say, 20 times, as seen in Figure 5.35.
Figure 5.34 – Distributional Analysis Tool (Binomial Distribution with 2 Trials)
Figure 5.36 shows the same binomial distribution for 20 trials, but now the CDF is computed.
The CDF is simply the sum of the PDF values up to the point x. For instance, in Figure 5.35,
we see that the probabilities of 0, 1, and 2 are 0.000001, 0.000019, and 0.000181, whose sum is
0.000201, which is the value of the CDF at x = 2 in Figure 5.36. Whereas the PDF computes
the probabilities of getting exactly 2 heads, the CDF computes the probability of getting no
more than 2 heads or up to 2 heads (or probabilities of 0, 1, and 2 heads). Taking the
complement (i.e., 1 – 0.00021) obtains 0.999799, or 99.9799%, which is the probability of
getting at least 3 heads or more.
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Figure 5.35 – Distributional Analysis Tool (Binomial Distribution with 20 Trials)
Using this Distributional Analysis tool in Risk Simulator, even more advanced distributions can
be analyzed, such as the gamma, beta, negative binomial, and many others. As further example
of the tool’s use in a continuous distribution and the ICDF functionality, Figure 5.37 shows the
standard normal distribution (normal distribution with a mean of zero and standard deviation
of one), where we apply the ICDF to find the value of x that corresponds to the cumulative
probability of 97.50% (CDF). That is, a one-tail CDF of 97.50% is equivalent to a two-tail 95%
confidence interval (there is a 2.50% probability in the right tail and 2.50% in the left tail,
leaving 95% in the center or confidence interval area, which is equivalent to a 97.50% area for
one tail). The result is the familiar Z-score of 1.96. Therefore, using this Distributional Analysis
tool, the standardized scores for other distributions and the exact and cumulative probabilities
of other distributions can all be obtained quickly and easily.
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Figure 5.36 – Distributional Analysis Tool (Binomial Distribution’s CDF with 20 Trials)
5.11 Инструмент анализ сценариев
The Scenario Analysis tool in Risk Simulator allows you to run multiple scenarios quickly and
effortlessly by changing one or two input parameters to determine the output of a variable.
Figure 5.38 illustrates how this tool works on the discounted cash flow sample model (Model 7
in Risk Simulator’s Example Models folder). In this example, cell G6 (net present value) is
selected as the output of interest, whereas cells C9 (effective tax rate) and C12 (product price)
are selected as inputs to perturb. You can set the starting and ending values to test, as well as
the step size, or the number of steps, to run between these starting and ending values. The
result is a scenario analysis table (Figure 5.39), where the row and column headers are the two
input variables and the body of the table shows the net present values.
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Figure 5.37 – Distributional Analysis Tool (Normal Distribution’s ICDF and Z-Score)
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Figure 5.38 – Scenario Analysis Tool
Figure 5.39 – Scenario Analysis Table
5.12 Инструмент Сегментации и Кластеризации
A final analytical technique of interest is that of segmentation clustering. Figure 5.40 illustrates a
sample dataset. You can select the data and run the tool through Risk Simulator | Tools |
Segmentation Clustering. Figure 5.40 shows a sample segmentation of two groups. That is, taking
the original data set, we run some internal algorithms (a combination or k-means hierarchical
clustering and other method of moments in order to find the best-fitting groups or natural
statistical clusters) to statistically divide, or segment, the original data set into two groups. You
can see the two-group memberships in Figure 5.40. Clearly you can segment this data set into as
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many groups as you wish. This technique is valuable in a variety of settings including marketing
(market segmentation of customers into various customer relationship management groups
etc.), physical sciences, engineering, and others.
Figure 5.40 – Segmentation Clustering Tool and Results
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5.13 RISK SIMULATOR 2011/2012 Новые инструменты
5.14 Генератор случайных чисел. Метод МонтеКарло по сравнению с методом Латинского
гиперкуба и методом Корреляционной Связки
Starting with version 2011/2012, there are 6 Random Number Generators, 3 Correlation
Copulas, and 2 Simulation Sampling Methods to choose from (Figure 5.41). These preferences
are set through the Risk Simulator | Options location.
The Random Number Generator (RNG) is at the heart of any simulation software. Based on
the random number generated, different mathematical distributions can be constructed. The
default method is the ROV Risk Simulator proprietary methodology, which provides the best
and most robust random numbers. As noted, there are 6 supported random number generators
and, in general, the ROV Risk Simulator default method and the Advanced Subtractive
Random Shuffle method are the two approaches recommended for use. Do not apply the
other methods unless your model or analytics specifically calls for their use, and even then, we
recommended testing the results against these two recommended approaches. The further
down the list of RNGs, the simpler the algorithm and the faster it runs, in comparison with the
more robust results from RNGs further up the list.
In the Correlations section, three methods are supported: the Normal Copula, T-Copula, and
Quasi-Normal Copula. These methods rely on mathematical integration techniques, and when
in doubt, the normal copula provides the safest and most conservative results. The t-copula
provides for extreme values in the tails of the simulated distributions, whereas the quasi-normal
copula returns results that are between the values derived by the other two methods.
In the Simulation methods section, Monte Carlo Simulation (MCS) and Latin Hypercube
Sampling (LHS) methods are supported. Note that Copulas and other multivariate functions
are not compatible with LHS because LHS can be applied to a single random variable but not
over a joint distribution. In reality, LHS has very limited impact on the model output's accuracy
the more distributions there are in a model since LHS only applies to distributions individually.
The benefit of LHS is also eroded if one does not complete the number of samples nominated
at the beginning, that is, if one halts the simulation run in mid-simulation. LHS also applies a
heavy burden on a simulation model with a large number of inputs because it needs to generate
and organize samples from each distribution prior to running the first sample from a
distribution. This can cause a long delay in running a large model without providing much more
additional accuracy. Finally, LHS is best applied when the distributions are well behaved and
symmetrical and without any correlations. Nonetheless, LHS is a powerful approach that yields
a uniformly sampled distribution, where MCS can sometimes generate lumpy distributions
(sampled data can sometimes be more heavily concentrated in one area of the distribution) as
compared to a more uniformly sampled distribution (every part of the distribution will be
sampled) when LHS is applied.
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Figure 5.41 – Risk Simulator Options
5.15 удаление сесонности и тренда данных
The data deseasonalization and detrending tool removes any seasonal and trending
components in your original data (Figure 5.42). In forecasting models, the process usually
includes removing the effects of accumulating data sets from seasonality and trend to show
only the absolute changes in values and to allow potential cyclical patterns to be identified after
removing the general drift, tendency, twists, bends, and effects of seasonal cycles of a set of
time-series data. For example, a detrended data set may be necessary to see a more accurate
account of a company's sales in a given year more clearly by shifting the entire data set from a
slope to a flat surface to better expose the underlying cycles and fluctuations.
Many time-series data exhibit seasonality where certain events repeat themselves after some
time period or seasonality period (e.g., ski resorts’ revenues are higher in winter than in summer,
and this predictable cycle will repeat itself every winter). Seasonality periods represent how
many periods would have to pass before the cycle repeats itself (e.g., 24 hours in a day, 12
months in a year, 4 quarters in a year, 60 minutes in an hour, etc.). For deseasonalized and
detrended data, a seasonal index greater than 1 indicates a high period or peak within the
seasonal cycle, and a value below 1 indicates a dip in the cycle.
Procedure
(Deseasonalization
and Detrending)

Select the data you wish to analyze (e.g., B9:B28) and click on Risk Simulator | Tools |
Data Deseasonalization and Detrending.

Select Deseasonalize Data and/or Detrend Data, select any detrending models you wish to
run, enter in the relevant orders (e.g., polynomial order, moving average order,
difference order, and rate order), and click OK.
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Procedure
(Seasonality Test)
S I M U L A T O R

Review the two reports generated for more details on the methodology, application,
and resulting charts and deseasonalized/detrended data.

Select the data you wish to analyze (e.g., B9:B28) and click on Risk Simulator | Tools |
Data Seasonality Test.

Enter in the maximum seasonality period to test. That is, if you enter 6, the tool will test the
following seasonality periods: 1, 2, 3, 4, 5, and 6. Period 1, of course, imply no
seasonality in the data.

Review the report generated for more details on the methodology, application, and
resulting charts and seasonality test results. The best seasonality periodicity is listed first
(ranked by the lowest RMSE error measure), and all the relevant error measurements
are included for comparison: root mean squared error (RMSE), mean squared error
(MSE), mean absolute deviation (MAD), and mean absolute percentage error
(MAPE).
Figure 5.42 – Deseasonalization and Detrending Data
5.16 Анализ основных компонентов
Principal Component Analysis is a way of identifying patterns in data and recasting the data in
such a way as to highlight their similarities and differences (Figure 5.43). Patterns of data are
very difficult to find in high dimensions when multiple variables exist, and higher dimensional
graphs are very difficult to represent and interpret. Once the patterns in the data are found, they
can be compressed, and the number of dimensions is now reduced. This reduction of data
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dimensions does not mean much reduction in loss of information. Instead, similar levels of
information can now be obtained with a smaller number of variables.
Procedure

Select the data to analyze (e.g., B11:K30), click on Risk Simulator | Tools | Principal
Component Analysis, and click OK.

Review the generated report for the computed results.
Figure 5.43 – Principal Component Analysis
5.17 Анализ структурных разрывов
A structural break tests whether the coefficients in different data sets are equal, and this test is
most commonly used in time-series analysis to test for the presence of a structural break (Figure
5.44). A time-series data set can be divided into two subsets. Structural break analysis is used to
test each subset individually and on one another and on the entire data set to statistically
determine if, indeed, there is a break starting at a particular time period. The structural break
test is often used to determine whether the independent variables have different impacts on
different subgroups of the population, such as to test if a new marketing campaign, activity,
major event, acquisition, divestiture, and so forth have an impact on the time-series data.
Suppose, for example, a data set has 100 time-series data points. You can set various
breakpoints to test, for instance, data points 10, 30, and 51. (This means that three structural
break tests will be performed: data points 1–9 compared with 10–100; data points 1–29
compared with 30–100; and 1–50 compared with 51–100 to see if there is a break in the
underlying structure at the start of data points 10, 30, and 51.). A one-tailed hypothesis test is
performed on the null hypothesis (H0) such that the two data subsets are statistically similar to
one another, that is, there is no statistically significant structural break. The alternative
hypothesis (Ha) is that the two data subsets are statistically different from one another,
indicating a possible structural break. If the calculated p-values are less than or equal to 0.01,
0.05, or 0.10, then the hypothesis is rejected, which implies that the two data subsets are
statistically significantly different at the 1%, 5%, and 10% significance levels. High p-values
indicate that there is no statistically significant structural break.
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Procedure
S I M U L A T O R

Select the data you wish to analyze (e.g., B15:D34), click on Risk Simulator | Tools |
Structural Break Test, enter in the relevant test points you wish to apply on the data (e.g.,
6, 10, 12), and click OK.

Review the report to determine which of these test points indicate a statistically
significant break point in your data and which points do not.
Figure 5.44 – Structural Break Analysis
5.18 Прогнозы Трендов
Trendlines can be used to determine if a set of time-series data follows any appreciable trend
(Figure 5.45). Trends can be linear or nonlinear (such as exponential, logarithmic, moving
average, power, polynomial, or power).
Procedure

Select the data you wish to analyze, click on Risk Simulator | Forecasting | Trendline, select
the relevant trendlines you wish to apply on the data (e.g., select all methods by
default), enter in the number of periods to forecast (e.g., 6 periods), and click OK.

Review the report to determine which of these test trendlines provide the best fit and
best forecast for your data.
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Figure 5.45 – Trendline Forecasts
5.19 Инструмент проверки моделей
After a model is created and after assumptions and forecasts have been set, you can run the
simulation as usual or run the Check Model tool (Figure 5.46) to test if the model has been set
up correctly. Alternatively, if the model does not run and you suspect that some settings may be
incorrect, run this tool from Risk Simulator | Tools | Check Model to identify where there might
be problems with your model. Note that while this tool checks for the most common model
problems as well as for problems in Risk Simulator assumptions and forecasts, it is in no way
comprehensive enough to test for all types of problems. It is still up to the model developer to
make sure the model works properly.
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Figure 5.46 – Model Checking Tool
5.20 Инструмент установки процентных
распределений
The Percentile Distributional Fitting tool (Figure 5.47) is another alternate way of fitting
probability distributions. There are several related tools and each has its own uses and
advantages:

Distributional Fitting (Percentiles)––using an alternate method of entry
(percentiles and first/second moment combinations) to find the best-fitting
parameters of a specified distribution without the need for having raw data. This
method is suitable for use when there are insufficient data, only when percentiles and
moments are available, or as a means to recover the entire distribution with only two
or three data points but the distribution type needs to be assumed or known.

Distributional Fitting (Single Variable)––using statistical methods to fit your raw
data to all 42 distributions to find the best fitting distribution and its input parameters.
Multiple data points are required for a good fit, and the distribution type may or may
not be known ahead of time.
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
Distributional Fitting (Multiple Variables)––using statistical methods to fit your
raw data on multiple variables at the same time. This method uses the same algorithms
as the single variable fitting, but incorporates a pairwise correlation matrix between the
variables. Multiple data points are required for a good fit, and the distribution type may
or may not be known ahead of time.

Custom Distribution (Set Assumption)––using nonparametric resampling
techniques to generate a custom distribution with the existing raw data and to simulate
the distribution based on this empirical distribution. Fewer data points are required,
and the distribution type is not known ahead of time.
Click on Risk Simulator | Tools | Distributional Fitting (Percentiles), choose the probability
distribution and types of inputs you wish to use, enter the parameters, and click Run to obtain
the results. Review the fitted R-square results and compare the empirical versus theoretical
fitting results to determine if your distribution is a good fit.
Figure 5.47 – Percentile Distributional Fitting Tool
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5.21 Распределительные диаграммы и таблиц:
инструмент распределения вероятностей
Distributional Charts and Tables is a new Probability Distribution tool that is a very powerful
and fast module used for generating distribution charts and tables (Figures 5.48 through 5.51).
Note that there are three similar tools in Risk Simulator but each does very different things:
Distributional Analysis––used to quickly compute the PDF, CDF, and ICDF of the 42
probability distributions available in Risk Simulator, and to return a probability table of these
values.
Distributional Charts and Tables––the Probability Distribution tool described here used to
compare different parameters of the same distribution (e.g., the shapes and PDF, CDF, ICDF
values of a Weibull distribution with Alpha and Beta of [2, 2], [3, 5], and [3.5, 8], and overlays
them on top of one another).
Overlay Charts––used to compare different distributions (theoretical input assumptions and
empirically simulated output forecasts) and to overlay them on top of one another for a visual
comparison.
Procedure

Run ROV BizStats at Risk Simulator | Distributional Charts and Tables, click on the Apply
Global Inputs button to load a sample set of input parameters or enter your own
inputs, and click Run to compute the results. The resulting four moments and CDF,
ICDF, PDF are computed for each of the 45 probability distributions (Figure 5.48).

Click on the Charts and Tables tab (Figure 5.49), select a distribution [A] (e.g.,
Arcsine), choose if you wish to run the CDF, ICDF, or PDF [B], enter the relevant
inputs, and click Run Chart or Run Table [C]. You can switch between the Charts and
Table tab to view the results as well as try out some of the chart icons [E] to see the
effects on the chart.

You can also change two parameters [H] to generate multiple charts and distribution
tables by entering the From/To/Step input or using the Custom inputs and then
hitting Run. For example, as illustrated in Figure 5.50, run the Beta distribution and
select PDF [G], select Alpha and Beta to change [H] using custom [I] inputs and enter
the relevant input parameters: 2;5;5 for Alpha and 5;3;5 for Beta [J], and click Run
Chart. This will generate three Beta distributions [K]: Beta (2,5), Beta (5,3), and Beta
(5,5) [L]. Explore various chart types, gridlines, language, and decimal settings [M], and
try rerunning the distribution using theoretical versus empirically simulated values [N].

Figure 5.51 illustrates the probability tables generated for a binomial distribution where
the probability of success and number of successful trials (random variable X) are
selected to vary [O] using the From/To/Step option. Try to replicate the calculation as
shown and click on the Table tab [P] to view the created probability density function
results. This example uses a binomial distribution with a starting input set of Trials =
20, Probability (of success) = 0.5, and Random X, or Number of Successful Trials, =
10, where the Probability of Success is allowed to change from 0., 0.25, …, 0.50 and is
shown as the row variable, and the Number of Successful Trials is also allowed to
change from 0, 1, 2, …, 8, and is shown as the column variable. PDF is chosen and,
hence, the results in the table show the probability that the given events occur. For
instance, the probability of getting exactly 2 successes when 20 trials are run where
each trial has a 25% chance of success is 0.0669, or 6.69%.
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Figure 5.48 – Probability Distribution Tool (45 Probability Distributions)
Figure 5.49 – ROV Probability Distribution (PDF and CDF Charts)
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Figure 5.50 – ROV Probability Distribution (Multiple Overlay Charts)
Figure 5.51 – ROV Probability Distribution (Distribution Tables)
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5.22 ROV BizStats
This new ROV BizStats tool is a very powerful and fast module in Risk Simulator that is used
for running business statistics and analytical models on your data. It covers more than 130
business statistics and analytical models (Figures 5.52 through 5.55). The following provides a
few quick getting started steps on running the module and details on each of the elements in
the software.
Procedure
Notes

Run ROV BizStats at Risk Simulator | ROV BizStats and click on Example to load a
sample data and model profile [A] or type in your data or copy/paste into the data grid
[D] (Figure 5.52). You can add your own notes or variable names in the first Notes
row [C].

Select the relevant model [F] to run in Step 2 and using the example data input settings
[G], enter in the relevant variables [H]. Separate variables for the same parameter using
semicolons and use a new line (hit Enter to create a new line) for different parameters.

Click Run [I] to compute the results [J]. You can view any relevant analytical results,
charts, or statistics from the various tabs in Step 3.

If required, you can provide a model name to save into the profile in Step 4 [L].
Multiple models can be saved in the same profile. Existing models can be edited or
deleted [M] and rearranged in order of appearance [N], and all the changes can be
saved [O] into a single profile with the file name extension *.bizstats.

The data grid size can be set in the menu, where the grid can accommodate up to
1,000 variable columns with 1 million rows of data per variable. The menu also allows
you to change the language settings and decimal settings for your data.

To get started, it is always a good idea to load the example file [A] that comes
complete with some data and precreated models [S]. You can double-click on any of
these models to run them and the results are shown in the report area [J], which
sometimes can be a chart or model statistics [T/U]. Using this example file, you can
now see how the input parameters [H] are entered based on the model description
[G], and you can proceed to create your own custom models.

Click on the variable headers [D] to select one or multiple variables at once, and then
right-click to add, delete, copy, paste, or visualize [P] the variables selected.

Models can also be entered using a Command console [V/W/X]. To see how this
works, double-click to run a model [S] and go to the Command console [V]. You can
replicate the model or create your own and click Run Command [X] when ready. Each
line in the console represents a model and its relevant parameters.

The entire *.bizstats profile (where data and multiple models are created and saved)
can be edited directly in XML [Z] by opening the XML Editor from the File menu.
Changes to the profile can be programmatically made here and takes effect once the
file is saved.

Click on the data grid’s column header(s) to select the entire column(s) or variable(s),
and once selected, you can right-click on the header to Auto Fit the column, Cut,
Copy, Delete, or Paste data. You can also click on and select multiple column headers
to select multiple variables and right-click and select Visualize to chart the data.
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
If a cell has a large value that is not completely displayed, click on and hover your
mouse over that cell and you will see a popup comment showing the entire value, or
simply resize the variable column (drag the column to make it wider, double click on
the column’s edge to auto fit the column, or right click on the column header and
select auto fit).

Use the up, down, left, right keys to move around the grid, or use the Home and End
keys on the keyboard to move to the far left and far right of a row. You can also use
combination keys such as: Ctrl+Home to jump to the top left cell, Ctrl+End to the
bottom right cell, Shift+Up/Down to select a specific area, and so forth.

You can enter short notes for each variable on the Notes row. Remember to make
your notes short and simple.

Try out the various chart icons on the Visualize tab to change the look and feel of the
charts (e.g., rotate, shift, zoom, change colors, add legend, and so forth).

The Copy button is used to copy the Results, Charts, and Statistics tabs in Step 3 after
a model is run. If no models are run, then the copy function will only copy a blank
page.

The Report button will only run if there are saved models in Step 4 or if there is data
in the grid, else the report generated will be empty. You will also need Microsoft Excel
to be installed to run the data extraction and results reports, and Microsoft
PowerPoint available to run the chart reports.

When in doubt about how to run a specific model or statistical method, start the
Example profile and review how the data is setup in Step 1 or how the input
parameters are entered in Step 2. You can use these as getting started guides and
templates for your own data and models.

The language can be changed in the Language menu. Note that currently there are 10
languages available in the software with more to be added later. However, sometimes
certain limited results will still be shown in English.

You can change how the list of models in Step 2 is shown by changing the View drop
list. You can list the models alphabetically, categorically, and by data input
requirements––note that in certain Unicode languages (e.g., Chinese, Japanese, and
Korean), there is no alphabetical arrangement and therefore the first option will be
unavailable.

The software can handle different regional decimal and numerical settings (e.g., one
thousand dollars and fifty cents can be written as 1,000.50 or 1.000,50 or 1’000,50 and
so forth). The decimal settings can be set in ROV BizStats’ menu Data | Decimal
Settings. However, when in doubt, please change the computer’s regional settings to
English USA and keep the default North America 1,000.50 in ROV BizStats (this
setting is guaranteed to work with ROV BizStats and the default examples).
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Figure 5.52 – ROV BizStats (Statistical Analysis)
Figure 5.53 – ROV BizStats (Data Visualization and Results Charts)
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Figure 5.54 – ROV BizStats (Command Console)
Figure 5.55 – ROV BizStats (XML Editor)
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5.23 Нейронные сети и Комбинаторные
методологии прогнозирования нечеткой логики
The term Neural Network is often used to refer to a network or circuit of biological neurons,
while modern usage of the term often refers to artificial neural networks comprising artificial
neurons, or nodes, recreated in a software environment. Such networks attempt to mimic the
neurons in the human brain in ways of thinking and identifying patterns and, in our situation,
identifying patterns for the purposes of forecasting time-series data. In Risk Simulator, the
methodology is found inside the ROV BizStats module located at Risk Simulator | ROV
BizStats | Neural Network as well as in Risk Simulator | Forecasting | Neural Network. Figure 5.56
shows the Neural Network forecast methodology.
Procedure

Click on Risk Simulator | Forecasting | Neural Network.

Start by either manually entering data or pasting some data from the clipboard (e.g.,
select and copy some data from Excel, start this tool, and paste the data by clicking on
the Paste button).

Select if you wish to run a Linear or Nonlinear Neural Network model, enter in the
desired number of Forecast Periods (e.g., 5), the number of hidden Layers in the
Neural Network (e.g., 3), and number of Testing Periods (e.g., 5).

Click Run to execute the analysis and review the computed results and charts. You can
also Copy the results and chart to the clipboard and paste it in another software
application.
Note that the number of hidden layers in the network is an input parameter and will need to be
calibrated with your data. Typically, the more complicated the data pattern, the higher the
number of hidden layers you would need and the longer it would take to compute. It is
recommended that you start at 3 layers. The testing period is simply the number of data points
used in the final calibration of the Neural Network model, and we recommend using at least
the same number of periods you wish to forecast as the testing period.
In contrast, the term fuzzy logic is derived from fuzzy set theory to deal with reasoning that is
approximate rather than accurate––as opposed to crisp logic, where binary sets have binary
logic, fuzzy logic variables may have a truth value that ranges between 0 and 1 and is not
constrained to the two truth values of classic propositional logic. This fuzzy weighting schema
is used together with a combinatorial method to yield time-series forecast results in Risk
Simulator as illustrated in Figure 5.57, and is most applicable when applied to time-series data
that has seasonality and trend. This methodology is found inside the ROV BizStats module in
Risk Simulator, at Risk Simulator | ROV BizStats | Combinatorial Fuzzy Logic as well as in Risk
Simulator | Forecasting | Combinatorial Fuzzy Logic.
Procedure

Click on Risk Simulator | Forecasting | Combinatorial Fuzzy Logic.

Start by either manually entering data or pasting some data from the clipboard (e.g.,
select and copy some data from Excel, start this tool, and paste the data by clicking on
the Paste button)
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
Select the variable you wish to run the analysis on from the drop-down list, and enter
in the seasonality period (e.g., 4 for quarterly data, 12 for monthly data, etc.) and the
desired number of Forecast Periods (e.g., 5).

Click Run to execute the analysis and review the computed results and charts. You can
also Copy the results and chart to the clipboard and paste it in another software
application.
Note that neither neural networks nor fuzzy logic techniques have yet been established as valid
and reliable methods in the business forecasting domain, on either a strategic, tactical, or
operational level. Much research is still required in these advanced forecasting fields.
Nonetheless, Risk Simulator provides the fundamentals of these two techniques for the
purposes of running time-series forecasts. We recommend that you do not use any of these
techniques in isolation, but, rather, in combination with the other Risk Simulator forecasting
methodologies to build more robust models.
Figure 5.56 – Neural Network Forecast
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Figure 5.57 – Fuzzy Logic Time-Series Forecast
5.24 Оптимизатор поиска цели
The Goal Seek tool is a search algorithm applied to find the solution of a single variable within
a model. If you know the result that you want from a formula or a model, but are not sure what
input value the formula needs to get that result, use the Risk Simulator | Tools | Goal Seek
feature. Note that Goal Seek works only with one variable input value. If you want to accept
more than one input value, use Risk Simulator’s advanced Optimization routines. Figure 5.58
shows how Goal Seek is applied to a simple model.
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Figure 5.58 – Goal Seek
5.25 Оптимизатор поиска цели
The Single Variable Optimizer tool is a search algorithm used to find the solution of a single
variable within a model, just like the goal seek routine discussed previously. If you want the
maximum or minimum possible result from a model but are not sure what input value the
formula needs to get that result, use the Risk Simulator | Tools | Single Variable Optimizer feature
(Figure 5.59). Note that this tool runs very quickly but is only applicable to finding one variable
input. If you want to accept more than one input value, use Risk Simulator’s advanced
Optimization routines. Note that this tool is included in Risk Simulator because if you require a
quick optimization computation for a single decision variable, this tool provides that capability
without having to set up an optimization model with profiles, simulation assumptions, decision
variables, objectives, and constraints.
Figure 5.59 – Single Variable Optimizer
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5.26 оптимизация Генетического алгоритма
Genetic Algorithms belong to the larger class of evolutionary algorithms that generate solutions
to optimization problems using techniques inspired by natural evolution, such as inheritance,
mutation, selection, and crossover. Genetic Algorithm is a search heuristic that mimics the
process of natural evolution and is routinely used to generate useful solutions to optimization
and search problems.
The genetic algorithm is available in Risk Simulator | Tools | Genetic Algorithm (Figure 5.60). Care
should be taken in calibrating the model’s inputs as the results will be fairly sensitive to the
inputs (the default inputs are provided as a general guide to the most common input levels), and
it is recommended that the Gradient Search Test option be chosen for a more robust set of
results (you can deselect this option to get started and then select this choice, rerun the analysis,
and compare the results).
Notes
In many problems, genetic algorithms may have a tendency to converge towards local optima
or even arbitrary points rather than the global optimum of the problem. This means that it does
not know how to sacrifice short-term fitness to gain longer-term fitness. For specific
optimization problems and problem instances, other optimization algorithms may find better
solutions than genetic algorithms (given the same amount of computation time). Therefore, it is
recommended that you first run the Genetic Algorithm and then rerun it by selecting the Apply
Gradient Search Test option (Figure 5.60) to check the robustness of the model. This gradient
search test will attempt to run combinations of traditional optimization techniques with Genetic
Algorithm methods and return the best possible solution. Finally, unless there is a specific
theoretical need to use Genetic Algorithm, we recommend using Risk Simulator’s Optimization
module, which allows you to run more advanced risk-based dynamic and stochastic
optimization routines, for more robust results.
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Figure 5.60 – Genetic Algorithm
5.27 ROV Модуль Дерева Решений
5.27.1 Дерево Решений
Risk Simulator | ROV Decision Tree runs the Decision Tree module (Figure 5.61). ROV Decision
Tree is used to create and value decision tree models. Additional advanced methodologies and
analytics are also included:

Decision Tree Models

Monte Carlo risk simulation

Sensitivity Analysis

Scenario Analysis

Bayesian (Joint and Posterior Probability Updating)

Expected Value of Information

MINIMAX

MAXIMIN

Risk Profiles
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The following are some main quick getting started tips and procedures for using this intuitive
tool:

There are 11 localized languages available in this module and the current language can
be changed through the Language menu.

Insert Option nodes or Insert Terminal nodes by first selecting any existing node and
then clicking on the option node icon (square) or terminal node icon (triangle), or use
the functions in the Insert menu.

Modify individual Option Node or Terminal Node properties by double-clicking on a
node. Sometimes when you click on a node, all subsequent child nodes are also
selected (this allows you to move the entire tree starting from that selected node). If
you wish to select only that node, you may have to click on the empty background and
click back on that node to select it individually. Also, you can move individual nodes
or the entire tree started from the selected node depending on the current setting
(right-click, or in the Edit menu, and select Move Nodes Individually or Move Nodes
Together).

The following are some quick descriptions of the things that can be customized and
configured in the node properties user interface. It is simplest to try different settings
for each of the following to see its effects in the Strategy Tree:

o
Name. Name shown above the node.
o
Value. Value shown below the node.
o
Excel Link. Links the value from an Excel spreadsheet’s cell.
o
Notes. Notes can be inserted above or below a node.
o
Show in Model. Show any combinations of Name, Value, and Notes.
o
Local Color versus Global Color. Node colors can be changed locally to a
node or globally.
o
Label Inside Shape. Text can be placed inside the node (you may need to
make the node wider to accommodate longer text).
o
Branch Event Name. Text can be placed on the branch leading to the node
to indicate the event leading to this node.
o
Select Real Options. A specific real option type can be assigned to the current
node. Assigning real options to nodes allows the tool to generate a list of
required input variables.
Global Elements are all customizable, including elements of the Strategy Tree’s
Background, Connection Lines, Option Nodes, Terminal Nodes, and Text Boxes. For
instance, the following settings can be changed for each of the elements:
o
Font settings on Name, Value, Notes, Label, Event names.
o
Node Size (minimum and maximum height and width).
o
Borders (line styles, width, and color).
o
Shadow (colors and whether to apply a shadow or not).
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o
Global Color.
o
Global Shape.

The Edit menu’s View Data Requirements Window command opens a docked
window on the right of the Strategy Tree such that when an option node or terminal
node is selected, the properties of that node will be displayed and can be updated
directly. This feature provides an alternative to double-clicking on a node each time.

Example Files are available in the File menu to help you get started on building
Strategy Trees.

Protect File from the File menu allows the Strategy Tree to be encrypted with up to a
256-bit password encryption. Be careful when a file is being encrypted because if the
password is lost, the file can no longer be opened.

Capturing the Screen or printing the existing model can be done through the File
menu. The captured screen can then be pasted into other software applications.

Add, Duplicate, Rename, and Delete a Strategy Tree can be performed through rightclicking the Strategy Tree tab or the Edit menu.

You can also Insert File Link and Insert Comment on any option or terminal node, or
Insert Text or Insert Picture anywhere in the background or canvas area.

You can Change Existing Styles, or Manage and Create Custom Styles of your Strategy
Tree (this includes size, shape, color schemes, and font size/color specifications of the
entire Strategy Tree).

Insert Decision, Insert Uncertainty, or Insert Terminal nodes by selecting any existing
node and then clicking on the decision node icon (square), uncertainty node icon
(circle), or terminal node icon (triangle), or use the functionalities in the Insert menu

Modify individual Decision, Uncertainty, or Terminal nodes’ properties by doubleclicking on a node. The following are some additional unique items in the Decision
Tree module that can be customized and configured in the node properties user
interface:

o
Decision Nodes: Custom Override or Auto Compute the value on a node.
The automatically compute option is set as default and when you click RUN
on a completed Decision Tree model, the decision nodes will be updated
with the results.
o
Uncertainty Nodes: Event Names, Probabilities, and Set Simulation
Assumptions. You can add probability event names, probabilities, and
simulation assumptions only after the uncertainty branches are created.
o
Terminal Nodes: Manual Input, Excel Link, and Set Simulation Assumptions.
The terminal event payoffs can be entered manually or linked to an Excel cell
(e.g., if you have a large Excel model that computes the payoff, you can link
the model to this Excel model’s output cell) or set probability distributional
assumptions for running simulations.
View Node Properties Window is available from the Edit menu and the selected
node’s properties will update when a node is selected.
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
S I M U L A T O R
The Decision Tree module also comes with the following advanced analytics:
o
Monte Carlo Simulation Modeling on Decision Trees
o
Bayes Analysis for obtaining posterior probabilities
o
Expected Value of Perfect Information, MINIMAX and MAXIMIN
Analysis, Risk Profiles, and Value of Imperfect Information
o
Sensitivity Analysis
o
Scenario Analysis
o
Utility Function Analysis
5.27.2 Симулятивное Моделирование
This tool runs Monte Carlo risk simulation on the decision tree (Figure 5.62). It allows you to
set probability distributions as input assumptions for running simulations. You can either set an
assumption for the selected node or set a new assumption and use this new assumption (or use
previously created assumptions) in a numerical equation or formula. For example, you can set a
new assumption called Normal (e.g., normal distribution with a mean of 100 and standard
deviation of 10) and run a simulation in the decision tree, or use this assumption in an equation
such as (100*Normal+15.25).
Create your own model in the numerical expression box. You can use basic computations or
add existing variables into your equation by double-clicking on the list of existing variables.
New variables can be added to the list as required either as numerical expressions or
assumptions.
5.27.3 Байесовский Анализ
This Bayesian analysis tool (Figure 5.63) can be used on any two uncertainty events that are
linked along a path. For instance, in the example on the right (Figure 5.63), uncertainties A and
B are linked, where event A occurs first in the timeline and event B occurs second. First Event
A is Market Research with 2 outcomes (Favorable or Unfavorable). Second Event B is Market
Conditions also with 2 outcomes (Strong and Weak). This tool is used to compute joint,
marginal, and Bayesian posterior updated probabilities by entering the prior probabilities and
reliability conditional probabilities; or reliability probabilities can be computed when you have
posterior updated conditional probabilities. Select the relevant analysis desired below and click
on Load Example to see the sample inputs corresponding to the selected analysis and the
results shown in the grid on the right, as well as which results are used as inputs in the decision
tree in the figure.
Procedure

STEP 1: Enter the names for the first and second uncertainty events and choose how
many probability events (states of nature or outcomes) each event has.

STEP 2: Enter the names of each probability event or outcome.

STEP 3: Enter the second event's prior probabilities and the conditional probabilities
for each event or outcome. The probabilities must sum to 100%.
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5.27.4 Ожидаемое значение идеальной информации, Minimax и
Maximin Анализ, Профилирование Риска и стоимость
несовершенства информации
This tool (Figure 5.64) computes the Expected Value of Perfect Information (EVPI),
MINIMAX and MAXIMIN Analysis, as well as the Risk Profile and the Value of Imperfect
Information. To get started, enter the number of decision branches or strategies under
consideration (e.g., build a large, medium, or small facility), the number of uncertain events or
states of nature outcomes (e.g., good market, bad market), and the expected payoffs under each
scenario.
The Expected Value of Perfect Information (EVPI), that is, assuming you had perfect foresight
and knew exactly what to do (through market research or other means to better discern the
probabilistic outcomes), computes if there is added value in such information (i.e., if market
research will add value) as compared to more naive estimates of the probabilistic states of
nature. To get started, enter the number of decision branches or strategies under consideration
(e.g., build a large, medium, or small facility) and the number of uncertain events or states of
nature outcomes (e.g., good market, bad market), and enter the expected payoffs under each
scenario.
MINIMAX (minimizing the maximum regret) and MAXIMIN (maximizing the minimum
payoff) are two alternate approaches to finding the optimal decision path. These two
approaches are not used often but still provide added insight into the decision-making process.
Enter the number of decision branches or paths that exist (e.g., building a large, medium, or
small facility), as well as the uncertainty events or states of nature under each path (e.g., good
economy vs. bad economy). Then, complete the payoff table for the various scenarios and
Compute the MINIMAX and MAXIMIN results. You can also click on Load Example to see
a sample calculation.
5.27.5 Чувствительность
Sensitivity analysis (Figure 5.65) on the input probabilities is performed to determine the impact
of inputs on the values of decision paths. First, select one Decision Node to analyze below, and
then select one probability event to test from the list. If there are multiple uncertainty events
with identical probabilities, they can be analyzed either independently or concurrently.
The sensitivity charts show the values of the decision paths under varying probability levels.
The numerical values are shown in the results table. The location of crossover lines, if any,
represents at what probabilistic events a certain decision path becomes dominant over another.
5.27.6 Таблицы сценариев
Scenario tables (Figure 5.66) can be generated to determine the output values given some
changes to the input. You can choose one or more Decision paths to analyze (the results of
each path chosen will be represented as a separate table and chart) and one or two Uncertainty
or Terminal nodes as input variables to the scenario table.
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Procedure
S I M U L A T O R

Select one or more Decision paths to analyze from the list below.

Select one or two Uncertainty Events or Terminal Payoffs to model.

Decide if you wish to change the event's probability on its own or all identical
probability events at once.

Enter the input scenario range.
5.27.7 Генерирование утилитарной функции
Utility functions (Figure 5.67), or U(x), are sometimes used in place of expected values of
terminal payoffs in a decision tree. U(x) can be developed two ways: using tedious and detailed
experimentation of every possible outcome or an exponential extrapolation method (used
here). They can be modeled for a decision maker who is risk-averse (downsides are more
disastrous or painful than an equal upside potential), risk-neutral (upsides and downsides have
equal attractiveness), or risk-loving (upside potential is more attractive). Enter the minimum
and maximum expected value of your terminal payoffs and the number of data points in
between to compute the utility curve and table.
If you had a 50:50 gamble where you either earn $X or lose -$X/2 versus not playing and
getting a $0 payoff, what would this $X be? For example, if you are indifferent between a bet
where you can win $100 or lose -$50 with equal probability compared to not playing at all, then
your X is $100. Enter the X in the Positive Earnings box below. Note that the larger X is, the
less risk-averse you are, whereas a smaller X indicates that you are more risk-averse.
Enter the required inputs, select the U(x) type, and click Compute Utility to obtain the results.
You can also apply the computed U(x) values to the decision tree to re-run it, or revert the tree
back to using expected values of the payoffs.
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Figure 5.61 – ROV Decision Tree (Decision Tree)
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Figure 5.62 – ROV Decision Tree (Simulation Results)
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Figure 5.63 – ROV Decision Tree (Bayes Analysis)
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Figure 5.64 – ROV Decision Tree (EVPI, MINIMAX, Risk Profile)
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Figure 5.65 – ROV Decision Tree (Sensitivity Analysis)
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Figure 5.66 – ROV Decision Tree (Scenario Tables)
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Figure 5.67 – ROV Decision Tree (Utility Functions)
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6
6. Полезные советы и
приемы
The following are some quick helpful tips and shortcut techniques for advanced users of Risk
Simulator. For details on using specific tools, refer to the relevant sections in this user manual.
СОВЕТЫ: Предположения
интерфейса пользователя)
(Установка
входных
данных
и

Quick Jump––select any distribution and type in any letter and it will jump to the first
distribution starting with that letter (e.g., click on Normal and type in W and it will take
you to the Weibull distribution).

Right-Click Views––select any distribution, right-click, and select the different views of
the distributions (large icons, small icons, list).

Tab to Update Charts––after entering some new input parameters (e.g., you type in a
new mean or standard deviation value), hit TAB on the keyboard or click anywhere on
the user interface away from the input box to see the distributional chart automatically
update.

Enter Correlations––enter pairwise correlations directly here (the columns are
resizable as needed), use the multiple distributional fitting tool to automatically
compute and enter all pairwise correlations, or, after setting some assumptions, use the
edit correlation tool to enter your correlation matrix.

Equations in an Assumption Cell––only empty cells or cells with static values can be
set as assumptions; however, there might be times when a function or equation is
required in an assumption cell, and this can be done by first entering the input
assumption in the cell and then typing in the equation or function (when the
simulation is being run, the simulated values will replace the function, and after the
simulation completes, the function or equation is again shown).
СОВЕТЫ: копирование и вставка

Copy and Paste using Escape––when you select a cell and use the Risk Simulator
Copy function, it copies everything into Windows clipboard, including the cell’s value,
equation, function, color, font, and size, as well as Risk Simulator assumptions,
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forecasts, or decision variables. Then, as you apply the Risk Simulator Paste function,
you have two options. The first option is to apply the Risk Simulator Paste directly,
and all cell values, color, font, equation, functions and parameters will be pasted into
the new cell. The second option is to first click Escape on the keyboard, and then
apply the Risk Simulator Paste. Escape tells Risk Simulator that you wish to paste only
the Risk Simulator assumption, forecast, or decision variable, and not the cell’s values,
color, equation, function, font, and so forth. Hitting Escape before pasting allows you
to maintain the target cell’s values and computations, and pastes only the Risk
Simulator parameters.

Copy and Paste on Multiple Cells––select multiple cells for copy and paste (with
contiguous and noncontiguous assumptions).
СОВЕТЫ: Корреляции

Set Assumption––set pairwise correlations using the set input assumption dialog (ideal
for entering only several correlations).

Edit Correlations––set up a correlation matrix by manually entering or pasting from
Windows clipboard (ideal for large correlation matrices and multiple correlations).

Multiple Distributional Fitting––automatically computes and enters pairwise
correlations (ideal when performing multiple variable fitting to automatically compute
the correlations for deciding what constitutes a statistically significant correlation).
СОВЕТЫ: Диагностика данных и статистический анализ

Stochastic Parameter Estimation––in the Statistical Analysis and Data Diagnostic
reports, there is a tab on stochastic parameter estimations that estimates the volatility,
drift, mean-reversion rate, and jump-diffusion rates based on historical data. Be aware
that these parameter results are based solely on historical data used, and the parameters
may change over time and depending on the amount of fitted historical data. Further,
the analysis results show all parameters and do not imply which stochastic process
model (e.g., Brownian Motion, Mean-Reversion, Jump-Diffusion, or mixed process) is
the best fit. It is up to the user to make this determination depending on the timeseries variable to be forecasted. The analysis cannot determine which process if best;
only the user can do this (e.g., Brownian Motion process is best for modeling stock
prices, but the analysis cannot determine that the historical data analyzed is from a
stock or some other variable, and only the user will know this). Finally, a good hint is
that if a certain parameter is out of the normal range, the process requiring this input
parameter is most probably not the correct process (e.g., if the mean-reversion rate is
110%, chances are, mean-reversion is not the correct process).
СОВЕТЫ: Дистрибутивный
вероятностей
анализ,
графики
и
таблицы

Distributional Analysis––used to quickly compute the PDF, CDF, and ICDF of the
42 probability distributions available in Risk Simulator, and to return a table of these
values.

Distributional Charts and Tables––used to compare different parameters of the same
distribution (e.g., takes the shapes and PDF, CDF, ICDF values of a Weibull
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distribution with Alpha and Beta of [2, 2], [3, 5], and [3.5, 8] and overlays them on top
of one another).

Overlay Charts––used to compare different distributions (theoretical input
assumptions and empirically simulated output forecasts) and overlay them on top of
one another for a visual comparison.
СОВЕТЫ: Кривая Эффективности

Efficient Frontier Variables––to access the frontier variables, first set the model’s
Constraints before setting efficient frontier variables.
СОВЕТЫ: Клетки Прогнозов

Forecast Cells with No Equations––you can set output forecasts on cells without any
equations or values (simply ignore the warning message) but be aware that the
resulting forecast chart will be empty. Output forecasts are typically set on empty cells
when there are macros that are being computed and the cell will be continually
updated.
СОВЕТЫ: Чарты Прогнозов

TAB versus Spacebar––hit TAB on the keyboard to update the forecast chart and to
obtain the percentile and confidence values after you enter some inputs, and hit the
Spacebar to rotate among the various tabs in the forecast chart.

Normal versus Global View––click on these views to rotate between a tabbed
interface and a global interface where all elements of the forecast charts are visible at
once.

Copy––copies the forecast chart or the entire global view depending on whether you
are in the normal or global view.
СОВЕТЫ: Прогнозирование

Cell Link Address––if you first select the data in the spreadsheet and then run a
forecasting tool, the cell address of the selected data will be automatically entered into
the user interface Otherwise, you will have to manually enter in the cell address or use
the link icon to link to the relevant data location.

Forecast RMSE––use as the universal error measure on multiple forecast models for
direct comparisons of the accuracy of each model.
СОВЕТЫ: прогнозирование: ARIMA

Forecast Periods––the number of exogenous data rows has to exceed the time-series
data rows by at least the desired forecast periods (e.g., if you wish to forecast 5 periods
into the future and have 100 time-series data points, you will need to have at least 105
or more data points on the exogenous variable). Otherwise, just run ARIMA without
the exogenous variable to forecast as many periods as you wish without any
limitations.
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СОВЕТЫ: прогнозирование: Базовая эконометрика

Variable Separation with Semicolons––separate independent variables using a
semicolon.
СОВЕТЫ: прогнозирование: логит, пробит, и тобит

Data Requirements––the dependent variables for running logit and probit models
must be binary only (0 and 1), whereas the Tobit model can take binary and other
numerical decimal values. The independent variables for all three models can take any
numerical value.
СОВЕТЫ: прогнозирование: случайные процессы

Default Sample Inputs––when in doubt, use the default inputs as a starting point to
develop your own model.

Statistical Analysis Tool for Parameter Estimation––use this tool to calibrate the input
parameters into the stochastic process models by estimating them from your raw data.

Stochastic Process Model––sometimes if the stochastic process user interface hangs
for a long time, chances are your inputs are incorrect and the model is not correctly
specified (e.g., if the mean-reversion rate is 110%, mean-reversion is probably not the
correct process). Try with different inputs or use a different model.
СОВЕТЫ: прогнозирование: тренд графика (кривой)

Forecast Results––scroll to the bottom of the report to see the forecasted values.
СОВЕТЫ: Вызов функций

RS Functions––there are functions that you can use inside your Excel spreadsheet to
set input assumption and get forecast statistics. To use these functions, you need to
first install RS Functions (which include Start, Programs, Real Options Valuation, Risk
Simulator, Tools, and Install Functions) and then run a simulation before setting the
RS functions inside Excel. Refer to the example model 24 for examples on how to use
these functions.
СОВЕТЫ: Приступая к работе. Упражнения и начало работы
(видеоматериалы)

Getting Started Exercises––there are multiple step-by-step hands-on examples and
results interpretation exercises available in the Start, Programs, Real Options
Valuation, Risk Simulator shortcut location. These exercises are meant to quickly get
you up to speed with the use of the software.

Getting Started Videos––these are all available for free on our website:
www.realoptionsvaluation.com/download.html
or
www.rovdownloads.com/download.html.
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СОВЕТЫ: Hardware ID

Right-Click HWID Copy––in the Install License user interface, select or double-click
on the HWID to select its value, right-click to copy or click on the E-mail HWID link
to generate an e-mail with the HWID.

Troubleshooter––run the Troubleshooter from the Start, Programs, Real Options
Valuation, Risk Simulator folder, and run the Get HWID tool to obtain your
computer’s HWID.
СОВЕТЫ: Метод Латинский гиперкуба выборки (LHS) по
сравнению с Монте-Карло (MCS)

Correlations––when setting pairwise correlations among input assumptions, we
recommend using the Monte Carlo setting in the Risk Simulator Options menu. Latin
Hypercube Sampling is not compatible with the correlated copula method for
simulation.

LHS Bins––a larger number of bins will slow down the simulation while providing a
more uniform set of simulation results.

Randomness––all of the random simulation techniques in the Options menu have
been tested and are all good simulators and approach the same levels of randomness
when larger number of trials are run.
СОВЕТЫ: Интернет-ресурсы

Books, Getting Started Videos, Models, White Papers––resources available on our
website:
www.realoptionsvaluation.com/download.html
or
www.rovdownloads.com/download.html.
СОВЕТЫ: Оптимизация

Infeasible Results––if the optimization run returns infeasible results, you can change
the constraints from an Equal (=) to an Inequality (>= or <=) and try again. This also
applies when you are running an efficient frontier analysis.
СОВЕТЫ: Профили

Multiple Profiles––create and switch among multiple profiles in a single model. This
allows you to run scenarios on simulation by being able to change input parameters or
distribution types in your model to see the effects on the results.

Profile Required––Assumptions, Forecasts, or Decision Variables cannot be created if
there is no active profile. However, once you have a profile, you no longer have to
keep creating new profiles each time. In fact, if you wish to run a simulation model by
adding additional assumptions or forecasts, you should keep the same profile.

Active Profile––the last profile used when you save Excel will be automatically opened
the next time the Excel file is opened.

Multiple Excel Files––when switching between several opened Excel models, the
active profile will be from the current and active Excel model.
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
Cross Workbook Profiles––be careful when you have multiple Excel files open
because if only one of the Excel files has an active profile and you accidentally switch
to another Excel file and set assumptions and forecasts on this file, the assumptions
and forecast will not run and will be invalid.

Deleting Profiles––you can clone existing profiles and delete existing profiles, but note
that at least one profile must exist in the Excel file if you delete profiles.

Profile Location––the profiles you create (containing the assumptions, forecasts,
decision variables, objectives, constraints, etc.) are saved as an encrypted hidden
worksheet. This is why the profile is automatically saved when you save the Excel
workbook file.
СОВЕТЫ: Сочетания клавиш и меню правой кнопкой мыши

Right-Click––you can open the Risk Simulator shortcut menu by right-clicking on a
cell anywhere in Excel.
СОВЕТЫ: Сохранить

Saving the Excel File––saves the profile settings, assumptions, forecasts, decision
variables, and your Excel model (including any Risk Simulator reports, charts, and data
extracted).

Saving the Chart Settings––saves the forecast chart settings such that the same settings
can be recovered and applied to future forecast charts (use the save and open icons in
the forecast charts).

Saving and Extracting Simulated Data in Excel––extracts a simulated run’s
assumptions and forecasts; the Excel file itself will still have to be saved in order to
save the data for retrieval later.

Saving Simulated Data and Charts in Risk Simulator––using the Risk Simulator Data
Extract and saving to a *.RiskSim file will allow you to reopen the dynamic and live
forecast chart with the same data in the future without having to rerun the simulation.

Saving and Generating Reports––simulation reports and other analytical reports are
extracted as separate worksheets in your workbook, and the entire Excel file will have
to be saved in order to save the data for future retrieval later.
СОВЕТЫ: Отбор проб и методы моделирования

Random Number Generator––there are six supported random number generators
(see the user manual for details) and, in general, the ROV Risk Simulator default
method and the Advanced Subtractive Random Shuffle method are the two
recommended approaches to use. Do not apply the other methods unless your model
or analytics specifically calls for their uses, and, even then, we recommended testing
the results against these two recommended approaches.
СОВЕТЫ: Software Development Kit (SDK) и DLL-библиотеки

SDK, DLL, and OEM––all of the analytics in Risk Simulator can be called outside of
this software and integrated in any user proprietary software. Contact
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admin@realoptionsvaluation.com for details on using our Software Development Kit
to access the Dynamic Link Library (DLL) analytics files.
СОВЕТЫ: Начиная работу с Risk Simulator в Excel

ROV Troubleshooter––run this troubleshooter to obtain your computer’s HWID for
licensing purposes, to view your computer settings and prerequisites, and to re-enable
Risk Simulator if it has been accidentally disabled.

Start Risk Simulator when Excel Starts––you can let Risk Simulator start automatically
when Excel starts each time or start it manually from the Start, Programs, Real
Options Valuation, Risk Simulator shortcut location. This preference can be set in the
Risk Simulator, Options menu.
СОВЕТЫ: Моделирование на сверхскоростях

Model Development––if you wish to run super speed in your model, test run a few
super speed simulations while the model is being constructed to make sure that the
final product will run the super speed simulation. Do not wait until the final model is
complete before testing super speed to avoid having to backtrack to identify where any
broken links or incompatible functions exist.

Regular Speed––when in doubt, regular speed simulation always works.
СОВЕТЫ: Анализ Торнадо

Tornado Analysis––the tornado analysis should never be run just once. It is meant as a
model diagnostic tool, which means that it should ideally be run several times on the
same model. For instance, in a large model, Tornado can be run the first time using all
of the default settings and all precedents should be shown (select Show All Variables).
This single analysis may result in a large report and long (and potentially unsightly)
Tornado charts. Nonetheless, it provides a great starting point to determine how many
of the precedents are considered critical success factors. For example, the Tornado
chart may show that the first 5 variables have high impact on the output, while the
remaining 200 variables have little to no impact, in which case, a second tornado
analysis is run showing fewer variables. For the second run, select Show Top 10
Variables if the first 5 are critical, thereby creating a nice report and a Tornado chart
that shows a contrast between the key factors and less critical factors. (You should
never show a Tornado chart with only the key variables without showing some less
critical variables as a contrast to their effects on the output.)

Default Values––the default testing points can be increased from the ±10% value to
some larger value to test for nonlinearities (the Spider chart will show nonlinear lines
and Tornado charts will be skewed to one side if the precedent effects are nonlinear).

Zero Values and Integers––inputs with zero or integer values only should be
deselected in the Tornado analysis before it is run. Otherwise, the percentage
perturbation may invalidate your model (e.g., if your model uses a lookup table where
Jan = 1, Feb = 2, Mar = 3, etc., perturbing the value 1 at a ±10% value yields 0.9 and
1.1, which makes no sense to the model).

Chart Options––try various chart options to find the best options to turn on or off for
your model.
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СОВЕТЫ: Устранение неполадок

ROV Troubleshooter––run this troubleshooter to obtain your computer’s HWID for
licensing purposes, to view your computer settings and prerequisites, and to re-enable
Risk Simulator if it has been accidentally disabled.
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INDEX
acquisition, 139, 158
allocation, 100, 101, 103, 104, 105, 106, 107, 112, 113
alpha, 45, 53, 71, 72, 128, 139
analysis, 1, 5, 9, 11, 12, 31, 33, 36, 51, 54, 68, 69, 71, 72, 74, 75, 78, 80, 82, 83, 95, 100, 101, 112,
115, 117, 119, 121, 122, 123, 124, 126, 127, 135, 136, 139, 141, 142, 145, 151, 158, 170,
171, 174, 178, 179, 185, 188, 191
annualized, 91, 102, 103, 112
approach, 11, 13, 26, 37, 65, 66, 67, 68, 74, 80, 82, 94, 95, 96, 100, 101, 102, 106, 112, 115, 121,
128, 140, 143, 155, 188
ARIMA, 1, 7, 9, 12, 66, 67, 69, 82, 83, 84, 86, 87, 88, 90, 91, 94, 97, 99, 140, 187
asset, 31, 101, 102, 103, 105, 106, 112, 113
asset classes, 102, 103, 112
assumption, 11, 14, 15, 16, 17, 19, 37, 82, 94, 100, 101, 113, 115, 124, 126, 128, 139, 141, 177,
184, 185, 187
assumptions, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 26, 27, 28, 37, 66, 68, 71, 72, 78, 90, 91, 92,
94, 100, 103, 104, 107, 109, 112, 113, 115, 117, 124, 125, 127, 128, 135, 136, 137, 139,
141, 160, 163, 173, 177, 178, 184, 185, 186, 188, 189
autocorrelation, 8, 67, 75, 83, 140, 142, 143
behavior, 45, 47, 67, 80, 88, 89, 128, 142
Beta, 45, 46, 47, 49, 50, 51, 53, 54, 56, 58, 59, 63, 64, 163, 186
binomial, 6, 38, 39, 40, 41, 42, 43, 95, 148, 149, 163
Binomial, 38, 41, 149, 150, 151
bootstrap, 1, 12, 132, 133
Bootstrap, 8, 132, 133
Box-Jenkins, 1, 12, 67, 82, 84, 86, 87
Brownian Motion, 10, 142, 185
causality, 144
center of, 31, 120, 139
coefficient of determination, 138
confidence interval, 23, 24, 29, 61, 132, 134, 150
constraints, 12, 100, 101, 102, 104, 107, 110, 113, 173, 188, 189
Continuous, 36, 45, 100, 102, 105, 106, 113
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correlation, 7, 8, 11, 14, 26, 27, 28, 37, 82, 83, 126, 140, 143, 144, 162, 184, 185
correlation coefficient, 14, 26, 27, 144
correlations, 8, 12, 14, 17, 26, 27, 28, 103, 124, 126, 144, 155, 184, 185, 188
cross-sectional, 66, 80
data, 6, 7, 8, 9, 12, 19, 21, 26, 27, 28, 35, 36, 45, 55, 63, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 78,
80, 82, 83, 87, 88, 89, 91, 95, 97, 98, 99, 105, 128, 131, 132, 133, 135, 136, 138, 139,
140, 141, 142, 144, 145, 153, 155, 156, 157, 158, 159, 161, 162, 166, 167, 170, 171, 179,
185, 186, 187, 189, 190
decision variable, 5, 6, 8, 12, 100, 101, 103, 104, 107, 112, 113, 115, 173, 184, 189
decision variables, 5, 6, 12, 100, 101, 103, 104, 107, 112, 113, 115, 173, 184, 189
decisions, 100, 107, 115
Delphi, 6, 66, 128
Delphi method, 6, 128
dependent variable, 12, 67, 68, 74, 75, 82, 83, 87, 95, 96, 97, 138, 139, 140, 141, 187
discrete, 1, 6, 36, 37, 38, 39, 100, 101, 107, 131, 142, 148
Discrete, 36, 38, 39, 100, 107, 108, 109
distribution, 6, 9, 11, 12, 15, 16, 17, 20, 26, 28, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 78, 92, 94,
96, 101, 115, 126, 128, 131, 132, 133, 141, 148, 149, 150, 155, 161,162, 163, 177, 184,
186, 189
Distribution, 6, 16, 27, 31, 32, 33, 36, 38, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63, 64, 93, 130, 131, 149, 150, 151, 152, 162, 163, 164, 165
distributional, 1, 8, 9, 11, 12, 13, 17, 21, 27, 31, 33, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 92, 113, 126, 127, 128,
131, 177, 184
distributions, 1, 6, 8, 9, 12, 16, 17, 26, 31, 33, 35, 36, 37, 38, 42, 46, 47, 52, 53, 57, 58, 59, 63,
112, 128, 131, 132, 133, 134, 135, 148, 150, 155, 161, 162, 163, 177, 184, 185, 186
e-mail, 2, 3, 188
equation, 30, 34, 74, 78, 82, 89, 94, 118, 139, 142, 143, 177, 184
Erlang, 6, 49, 52, 53
error, 1, 2, 14, 18, 21, 29, 57, 71, 74, 80, 82, 83, 96, 123, 132, 139, 140, 141, 157, 186
errors, 14, 19, 43, 67, 68, 74, 75, 80, 82, 95, 138, 139, 141
estimates, 59, 66, 69, 80, 82, 95, 96, 128, 132, 139, 140, 178, 185
Excel, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 16, 21, 26, 27, 34, 69, 71, 75, 80, 83, 87, 88, 90, 91, 94, 96,
97, 99, 100, 113, 118, 123, 167, 170, 175, 177, 187, 189, 190
excess kurtosis, 33, 38, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 61, 62, 63
extrapolation, 1, 7, 9, 80, 179
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first moment, 31, 34
Fisher-Snedecor, 51
flexibility, 115
fluctuations, 57, 124, 127, 138, 142, 156
forecast, 7, 8, 11, 14, 18, 19, 20, 21, 23, 25, 26, 28, 29, 30, 31, 36, 37, 66, 67, 68, 71, 72, 78, 80,
82, 83, 87, 88, 90, 91, 94, 98, 100, 101, 113, 117, 125, 126, 132, 133, 134, 135, 139, 140,
142, 159, 170, 185, 186, 187, 189, 190
forecast statistics, 8, 20, 100, 101, 132, 187
forecasting, 5, 7, 9, 11, 12, 36, 65, 66, 67, 68, 69, 72, 82, 87, 88, 90, 109, 138, 140, 142, 143, 156,
170, 171, 186
Forecasting, 1, 7, 9, 12, 65, 66, 69, 70, 71, 72, 75, 78, 79, 80, 83, 87, 88, 89, 90, 91, 94, 97, 99,
138, 140, 159, 170, 186, 187
forecasts, 5, 6, 7, 8, 9, 12, 13, 15, 17, 18, 26, 30, 66, 67, 69, 71, 72, 80, 82, 91, 100, 125, 134, 135,
136, 137, 140, 160, 163, 171, 184, 186, 189
fourth moment, 31, 33, 34
Frequency, 35, 36
functions, 1, 6, 7, 11, 15, 16, 18, 34, 36, 80, 89, 100, 107, 113, 123, 140, 155, 175, 179, 185, 187,
190
functions of, 140
gallery, 16, 17
gamma, 6, 47, 52, 53, 61, 71, 72, 150
Gamma, 46, 49, 52, 53, 58, 63
geometric, 6, 40, 41, 55, 78, 102, 112
Geometric, 10, 38, 40
geometric average, 102, 112
goodness-of-fit, 140, 141
goodness-of-fit tests, 140
growth, 54, 66, 68, 82, 90, 112, 128, 142
growth rate, 66, 68, 82, 90, 128, 142
heteroskedasticity, 7, 8, 12, 67, 75, 91, 138, 139, 140, 141
Histogram, 21, 35, 36
Holt-Winter, 10, 71, 73
hypergeometric, 6, 40, 41
Hypergeometric, 40
hypothesis, 1, 12, 47, 51, 61, 75, 83, 96, 131, 132, 134, 135, 141, 144, 158
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icon, 3, 6, 15, 16, 18, 19, 87, 91, 99, 103, 107, 113, 175, 177, 186
icons, 6, 16, 87, 113, 163, 167, 184, 189
independent variable, 67, 68, 74, 75, 83, 87, 88, 89, 95, 96, 139, 141, 143, 144, 158, 187
inflation, 56, 66, 140, 142, 144
inputs, 13, 15, 17, 27, 45, 46, 50, 54, 56, 62, 63, 67, 68, 69, 78, 87, 94, 100, 107, 113, 115, 117,
123, 151, 155, 162, 163, 174, 178, 179, 180, 186, 187, 191
installation, 2, 3
integer, 1, 12, 14, 39, 41, 43, 48, 49, 53, 61, 68, 72, 89, 100, 101, 107, 123, 191
interest, 66, 67, 98, 140, 142, 151, 153
interest rate, 66, 67, 98, 140, 142
investment, 101, 107, 112, 119, 120, 121, 126
jump-diffusion, 2, 12, 69, 78, 142, 185
Kolmogorov-Smirnov test, 131
kurtosis, 33, 47, 57, 96
lags, 9, 75, 82, 83, 140
least squares, 75, 95, 96, 139
least squares regression, 95, 96, 139
linear, 7, 12, 26, 67, 69, 74, 82, 95, 98, 100, 102, 139, 141, 143, 144, 159
Ljung-Box Q-statistics, 83, 140
logistic, 6, 7, 12, 48, 54, 68, 90, 95, 96
Lognormal, 54, 55
lower, 14, 27, 33, 40, 48, 54, 55, 57, 61, 103, 112, 119, 126, 128
management, 59, 66, 115, 128, 154
market, 7, 26, 33, 66, 68, 90, 94, 128, 139, 142, 144, 154, 178
matrix, 26, 27, 143, 162, 184, 185
mean, 2, 11, 12, 17, 20, 21, 28, 29, 31, 32, 33, 37, 39, 46, 47, 53, 54, 55, 56, 61, 68, 69, 78, 91, 94,
96, 101, 115, 128, 133, 139, 141, 142, 150, 157, 177, 184, 185, 187
Mean, 9, 34, 39, 41, 47, 50, 51, 52, 54, 55, 56, 61, 62, 93, 142, 185
mean-reversion, 12, 69, 78, 142, 185, 187
mix, 143
model, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 27, 28, 29, 36, 37, 58, 63, 66, 67, 68, 70, 71, 72, 75, 78,
82, 83, 87, 88, 89, 90, 91, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 107, 110, 112, 113,
115, 117, 118, 121, 122, 123, 124, 125, 126, 128, 136, 137, 138,140, 141, 142, 145, 151,
155, 160, 166, 167, 170, 172, 173, 174, 176, 177, 179, 185, 186, 187, 189, 190, 191
Model, 8, 9, 13, 15, 18, 27, 28, 72, 91, 96, 97, 99, 102, 108, 113, 118, 151, 160, 161, 176, 187, 190
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models, 2, 5, 6, 7, 9, 12, 14, 28, 59, 67, 68, 70, 72, 82, 83, 88, 89, 91, 94, 96, 101, 107, 140, 156,
166, 167, 171, 175, 186, 187, 189
Monte Carlo, 1, 5, 6, 11, 12, 13, 14, 17, 19, 26, 27, 29, 35, 36, 37, 66, 69, 72, 75, 78, 83, 100, 155,
175, 177, 188
multicollinearity, 8, 12, 75, 138, 143, 144
Multinomial SLS, 2
multiple, 1, 2, 6, 8, 9, 11, 12, 13, 14, 18, 30, 37, 67, 68, 74, 75, 78, 83, 87, 89, 101, 112, 115, 123,
124, 128, 131, 143, 145, 151, 157, 162, 163, 166, 179, 184, 185, 186, 188, 189
multiple regression, 1, 68, 143
multiple variables, 8, 89, 131, 145, 157, 162, 166
multivariate, 8, 12, 74, 75, 80, 82, 83, 95, 155
M
Muunn, 0, i, 1, 66, 72, 75, 76, 78, 83, 91, 109
negative binomial, 6, 41, 42, 43, 150
nonlinear, 1, 7, 9, 12, 67, 68, 69, 75, 80, 82, 98, 100, 102, 117, 123, 126, 127, 139, 144, 159, 191
normal, 6, 11, 17, 26, 30, 33, 37, 39, 47, 53, 55, 56, 59, 61, 78, 92, 94, 95, 96, 128, 132, 139, 150,
155, 177, 185, 186
Normal, 6, 21, 56, 93, 96, 152, 155, 177, 184, 186
null hypothesis, 83, 128, 135, 139, 140, 141, 158
objective, 12, 100, 101, 102, 103, 104, 107, 113
optimal, 100, 101, 105, 106, 109, 110, 115, 140, 178
optimal decision, 101, 115, 178
optimization, 1, 7, 8, 9, 12, 67, 91, 95, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 112, 113,
115, 131, 173, 174, 188
option, 2, 17, 71, 72, 122, 123, 128, 136, 163, 167, 174, 175, 176, 177, 184
outliers, 8, 138, 139, 140, 141
parameter, 16, 17, 38, 40, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 57, 60, 61, 63, 64, 71, 94, 96,
123, 143, 166, 170, 185
Parameter, 55, 113, 143, 148, 185, 187
pareto, 57
Pareto, 6, 57
pause, 18, 19
Pearson, 6, 26, 27, 58, 144
point estimate, 66, 71, 72, 101, 115
Poisson, 6, 43, 44, 50, 52
population, 29, 34, 41, 54, 57, 61, 68, 90, 128, 134, 135, 139, 141, 158
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portfolio, 1, 100, 101, 102, 103, 104, 105, 106, 107, 112, 115
precision, 1, 8, 14, 18, 21, 29
prediction, 82, 139, 140
price, 27, 31, 54, 56, 67, 78, 91, 119, 120, 121, 124, 142, 143, 151
probability, 1, 6, 8, 9, 11, 19, 23, 24, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 51, 53, 56,
59, 60, 61, 68, 94, 95, 96, 148, 149, 150, 161, 162, 163, 177, 178, 179, 185
Probability, 5, 9, 17, 25, 35, 36, 38, 39, 40, 41, 42, 43, 148, 163, 164, 165, 175, 185
profile, 13, 14, 15, 27, 71, 103, 107, 113, 128, 166, 167, 189
p-value, 83, 128, 135, 140, 144, 158
random, 6, 11, 12, 14, 26, 35, 36, 37, 45, 46, 47, 50, 52, 53, 69, 78, 82, 96, 132, 133, 134, 142,
143, 155, 163, 188, 190
random number, 6, 11, 14, 36, 155, 190
range, 17, 31, 45, 52, 95, 101, 103, 112, 115, 117, 140, 148, 179, 185
rank correlation, 26, 27, 126, 127, 144
rate, 7, 31, 44, 49, 50, 56, 67, 78, 96, 98, 119, 121, 126, 140, 141, 142, 151, 156, 185, 187
ratio, 47, 51, 80, 96, 101, 103, 105, 106, 107, 109, 112
regression, 7, 67, 68, 74, 75, 80, 82, 83, 87, 88, 95, 96, 138, 139, 140, 141, 142, 143
Regression, 7, 9, 66, 68, 69, 74, 75, 76, 77, 82, 88, 138
regression analysis, 67, 68, 74, 83, 87, 88, 95, 97, 138, 139
relative returns, 67, 91, 102, 112
report, 8, 14, 71, 75, 78, 80, 83, 88, 89, 90, 91, 94, 97, 98, 99, 110, 113, 119, 123, 126, 128, 137,
138, 140, 142, 144, 157, 158, 159, 166, 167, 187, 191
return, 31, 68, 94, 95, 102, 103, 105, 106, 112, 123, 163, 174, 185
returns, 31, 32, 33, 67, 78, 87, 101, 102, 103, 104, 106, 107, 112, 113, 139, 155, 188
risk, 5, 8, 11, 12, 28, 31, 32, 33, 34, 36, 52, 69, 102, 103, 104, 105, 106, 107, 109, 112, 113, 117,
121, 174, 175, 177, 179
Risk Simulator, 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16, 17, 18, 19, 25, 26, 27, 29, 30, 31, 33, 37, 66, 67,
68, 69, 70, 71, 72, 75, 78, 80, 83, 87, 88, 89, 90, 91, 92, 94, 96, 97, 99, 100, 101, 102,
103, 104, 105, 107, 109, 110, 113, 117, 118, 122, 125, 128, 131, 132, 134, 135, 136, 137,
138, 140, 145, 148, 150, 151, 153, 155, 156, 157, 158, 159, 160, 162, 163, 166, 170, 171,
172, 173, 174, 175, 184, 185, 187, 188, 189, 190, 191
running, 5, 6, 14, 19, 21, 28, 75, 89, 100, 110, 121, 122, 132, 139, 141, 155, 166, 171, 177, 187,
188
sales, 41, 42, 61, 66, 68, 69, 74, 90, 140, 141, 156
sample, 11, 29, 34, 35, 41, 61, 71, 75, 78, 89, 90, 102, 109, 112, 117, 128, 132, 133, 134, 139, 141,
145, 151, 153, 155, 163, 166, 178, 179
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save, 3, 14, 15, 136, 166, 189, 190
saving, 5, 136, 190
seasonality, 9, 12, 69, 70, 71, 72, 141, 156, 157, 170, 171
second moment, 31, 33, 34, 161
sensitivity, 1, 9, 12, 119, 121, 124, 126, 127, 179
Sensitivity, 5, 9, 117, 121, 122, 124, 125, 126, 175, 177, 179, 182
significance, 8, 61, 83, 96, 132, 135, 139, 140, 141, 144, 158
simulation, 1, 5, 6, 7, 8, 11, 12, 13, 14, 15, 17, 18, 19, 21, 25, 26, 27, 28, 29, 35, 36, 37, 66, 69, 71,
78, 100, 101, 103, 104, 107, 109, 112, 113, 115, 117, 121, 124, 125, 126, 127, 128, 132,
133, 134, 135, 136, 137, 155, 160, 173, 175, 177, 184, 187, 188, 189, 190
Simulation, 1, 6, 7, 8, 11, 12, 13, 14, 15, 17, 18, 19, 25, 27, 35, 36, 66, 69, 72, 75, 78, 83, 100, 112,
115, 117, 132, 133, 135, 136, 137, 155, 177, 180, 188, 190
single, 6, 8, 14, 19, 21, 66, 68, 70, 71, 72, 74, 89, 101, 115, 118, 128, 139, 145, 155, 162, 166, 172,
173, 189, 191
Single Asset SLS, 2
skew, 31, 32, 34
Skew, 32, 33, 34
skewness, 32, 33, 38, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 55, 56, 57, 61, 62, 63, 132
SLS, 1, 2, 6
Spearman, 26, 27, 144
specification errors, 138
spider, 1, 9, 12, 118, 119, 122, 124
spread, 28, 31
standard deviation, 11, 17, 21, 28, 30, 31, 32, 33, 34, 37, 39, 41, 44, 46, 47, 50, 51, 52, 55, 56, 61,
62, 78, 96, 100, 101, 115, 128, 133, 135, 141, 144, 150, 177, 184
static, 9, 12, 100, 103, 104, 107, 110, 117, 121, 124, 142, 184
statistics, 1, 8, 9, 12, 19, 20, 21, 26, 28, 29, 31, 34, 83, 100, 101, 115, 128, 132, 133, 145, 166
stochastic, 1, 8, 9, 12, 69, 78, 100, 101, 104, 107, 109, 110, 112, 113, 114, 115, 116, 138, 142,
145, 174, 185, 187
stochastic optimization, 8, 12, 100, 101, 104, 107, 112, 113, 114, 115, 116, 174
stock price, 31, 57, 67, 78, 91, 142, 185
symmetric, 139
t distribution, 61, 92, 94
third moment, 31, 32, 34
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S I M U L A T O R
time-series, 1, 7, 9, 12, 66, 67, 68, 69, 70, 71, 72, 78, 80, 82, 83, 87, 89, 98, 140, 142, 156, 158,
159, 170, 171, 185, 187
time-series data, 9, 67, 68, 69, 80, 83, 87, 89, 98, 140, 142, 156, 158, 159, 170, 187
title, 13, 14
toolbar, 3, 6, 15, 18, 19
tornado, 1, 9, 12, 117, 118, 119, 120, 121, 122, 124, 126, 127, 191
Tornado, 9, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 191
trends, 7, 80, 142
trials, 7, 11, 14, 18, 19, 29, 37, 38, 39, 40, 41, 42, 43, 50, 100, 101, 113, 115, 132, 149, 163, 188
triangular, 6, 11, 37, 59, 61, 62
Triangular, 15, 59, 61
t-statistic, 96, 143
types of, 26, 37, 102, 112, 131, 142, 160, 162
uniform, 6, 11, 33, 37, 39, 62, 103, 112, 128, 188
Uniform, 17, 39, 62
upper, 103, 112
validity of, 80, 140
value, 5, 8, 14, 17, 19, 23, 24, 26, 28, 29, 31, 33, 34, 36, 37, 38, 39, 45, 46, 47, 48, 50, 51, 53, 54,
55, 56, 57, 59, 60, 61, 62, 63, 64, 67, 68, 69, 72, 80, 82, 90, 91, 96, 103, 107, 110, 112,
115, 117, 119, 120, 123, 126, 128, 132, 133, 139, 140, 141, 142,143, 144, 148, 149, 150,
151, 156, 167, 170, 172, 173, 175, 177, 178, 179, 184, 187, 188, 191
values, 11, 14, 15, 19, 20, 21, 23, 24, 25, 27, 28, 33, 36, 37, 45, 46, 48, 51, 54, 55, 56, 59, 61, 62,
66, 67, 68, 69, 71, 78, 80, 82, 83, 90, 95, 96, 98, 99, 100, 101, 102, 103, 104, 107, 109,
112, 117, 119, 123, 128, 135, 136, 139, 140, 141, 144, 148, 149, 151, 155, 156, 158, 163,
170, 179, 180, 184, 185, 186, 187, 191
variance, 31, 32, 47, 51, 54, 57, 61, 75, 94, 115, 135, 138, 139, 144
volatility, 7, 12, 67, 78, 91, 142, 185
Weibull, 6, 63, 64, 126, 163, 184, 186
Yes/No, 38
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