PEARSON ALWAYS LEARNING Financial Risk Manager (FRM®) Exam Part I Quantitative Analysis Fifth Custom Edition for Global Association of Risk Professionals 2015 @GARP Global Association of Risk Professionals Excerpts taken from: Introduction to Econometrics, Brief Edition, by James H Stock and Mark W Watson Options, Futures, and Other Derivatives, Ninth Edition, by John C Hull Excerpts taken from: Introduction to Econometrics, Brief Edition by James H Stock and Mark W Watson Copyright © 2008 by Pearson Education, Inc Published by Addison Wesley Boston, Massachusetts 02116 Options, Futures, and Other Derivatives, Ninth Edition by John C Hull Copyright © 2015, 2012, 2009, 2006, 2003, 2000 by Pearson Education, Inc Upper Saddle River, New Jersey 07458 Copyright © 2015, 2014, 2013, 2012, 2011 by Pearson Learning Solutions All rights reserved This copyright covers material written expressly for this volume by the editor/s as well as the compilation itself It does not cover the individual selections herein that first appeared elsewhere Permission to reprint these has been obtained by Pearson Learning Solutions for this edition only Further reproduction by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, must be arranged with the individual copyright holders noted Grateful acknowledgment is made to the following sources for permission to reprint material copyrighted or controlled by them: Chapters 2, 3,4, 6, and from Mathematics and Statistics for Financial Risk Management, Second Edition (2013), by Michael Miller, by permission of John Wiley & Sons, Inc "Correlations and Copulas," by John Hull, reprinted from Risk Management and Financial Institutions, Third Edition (2012), by permission of John Wiley & Sons, Inc Chapters 5, 7, and from Elements of Forecasting, Fourth Edition (2006), by Francis X Diebold, Cengage Learning "Simulation Modeling," by Dessislava A Pachamanova and Frank Fabozzi, reprinted from Simulation and Optimization in Finance + Web Site (2010), by permission of John Wiley & Sons, Inc Learning Objectives provided by the Global Association of Risk Professionals All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116 A Pearson Education Company www.pearsoned.com Printed in the United States of America 23456789 10 VOll 19 18 17 16 15 000200010271930510 JH/KE PEARSON ISBN 10: 1-323-01120-X ISBN 13: 978-1-323-01120-1 CHAPTER PROBABILITIES Standardized Variables 18 Covariance 19 Discrete Random Variables Correlation 19 Continuous Random Variables Probability Density Functions Cumulative Distribution Functions Inverse Cumulative Distribution Functions Application: Portfolio Variance and Hedging 20 Moments 21 Skewness 21 Mutually Exclusive Events Kurtosis 23 Independent Events Coskewness and Cokurtosis 24 Probability Matrices Conditional Probability Best Linear Unbiased Estimator (BLUE) 26 CHAPTER BASIC STATISTICS Averages Population and Sample Data Discrete Random Variables Continuous Random Variables 11 12 12 13 13 Expectations 14 Variance and Standard Deviation 17 CHAPTER DISTRIBUTIONS 29 Parametric Distributions 30 Uniform Distribution 30 Bernoulli Distribution 31 Binomial Distribution 31 Poisson Distribution 33 iii Normal Distribution 34 Lognormal Distribution 36 Central Limit Theorem 36 Application: Monte Carlo Simulations Part 1: Creating Normal Random Variables 38 Chi-Squared Distribution 39 Student's t Distribution 39 F-Distribution 40 Triangular Distribution 41 Beta Distribution 42 Mixture Distributions 42 CHAPTER4 BAVESIAN ANALVSIS 47 Overview 48 Bayes' Theorem 48 Bayes versus Frequentists 51 Many-State Problems 52 CHAPTER HVPOTHESIS TESTING AND CONFIDENCE INTERVALS 57 Sample Mean Revisited 58 Sample Variance Revisited 59 Confidence Intervals 59 Hypothesis Testing 60 iv • Contents Which Way to Test? One Tailor Two? The Confidence Level Returns 60 61 61 Chebyshev's Inequality 62 Application: VaR 62 Backtesting Subadditivity Expected Shortfall CHAPTER 64 65 66 CORRELATIONS AND COPULAS 69 Definition of Correlation 70 Correlation vs Dependence 70 Monitoring Correlation 71 EWMA GARCH Consistency Condition for Covariances 71 72 72 Multivariate Normal Distributions 73 Generating Random Samples from Normal Distributions Factor Models Copulas Expressing the Approach Algebraically Other Copulas Tail Dependence Multivariate Copulas A Factor Copula Model Application to Loan Portfolios: Vasicek's Model Estimating PO and p Alternatives to the Gaussian Copula Summary 73 73 74 76 76 76 77 77 78 79 80 80 CHAPTER LINEAR REGRESSION WITH ONE REGRESSOR CHAPTER 83 The Linear Regression Model 84 Estimating the Coefficients of the Linear Regression Model 86 The Ordinary Least Squares Estimator OLS Estimates of the Relationship Between Test Scores and the Student-Teacher Ratio Why Use the OLS Estimator? Measures of Fit 87 88 89 90 The R2 The Standard Error of the Regression Application to the Test Score Data The Least Squares Assumptions Assumption #1: The Conditional Distribution of ui Given Xi Has a Mean of Zero Assumption #2: (Xi' ~), i 1, , n Are Independently and Identically Distributed Assumption #3: Large Outliers Are Unlikely Use of the Least Squares Assumptions 90 91 91 92 92 = Sampling Distribution of the OLS Estimators The Sampling Distribution of the OLS Estimators 93 94 95 95 97 Summary 97 Appendix A 98 Appendix B 98 98 Derivation of the OLS Estimators REGRESSION WITH A SINGLE REGRESSOR Testing Hypotheses about One of the Regression Coefficients 101 102 Two-Sided Hypotheses Concerning 131 102 One-Sided Hypotheses Concerning 131 104 Testing Hypotheses about the Intercept 13 105 Confidence Intervals for a Regression Coefficient 105 Regression When X Is a Binary Variable 107 Interpretation of the Regression Coefficients Heteroskedasticity and Homoskedasticity 107 108 What Are Heteroskedasticity and Homoskedasticity? Mathematical Implications of Homoskedasticity What Does This Mean in Practice? The Theoretical Foundations of Ordinary Least Squares Linear Conditionally Unbiased Estimators and the Gauss-Markov Theorem Regression Estimators Other than OLS 108 109 110 111 112 112 95 Conclusion The California Test Score Data Set 98 Using the t-Statistic in Regression When the Sample Size Is Small 113 The t-Statistic and the Student t Distribution Use of the Student t Distribution in Practice Conclusion - - - - - - - - - - _ _ - - - - - - - - - - - - - - - 113 114 114 Contents II v Summary 115 Appendix 115 The Gauss-Markov Conditions and a Proof of the Gauss-Markov Theorem The Gauss-Markov Conditions The Sample Average Is the Efficient Linear Estimator of E(y) CHAPTER 115 115 116 REGRESSORS Omitted Variable Bias Definition of Omitted Variable Bias A Formula for Omitted Variable Bias Addressing Omitted Variable Bias by Dividing the Data into Groups 119 120 120 121 122 The Multiple Regression Model 124 The Population Regression Line The Population Multiple Regression Model 124 The OLS Estimator in Multiple Regression The OLS Estimator Application to Test Scores and the Student-Teacher Ratio Measures of Fit in Multiple Regression The Standard Error of the Regression (SER) TheR2 The "Adjusted R2" Application to Test Scores Contents Assumption #1: The Conditional Distribution of u/ Given Xli' X 2i , , X kl Has a Mean of Zero Assumption #2: (Xli' X 2/ , ,Xkl , Y,), i = 1, , n Are i.i.d Assumption #3: Large Outliers Are Unlikely Assumption #4: No Perfect Multicollinearity 129 129 129 129 LINEAR REGRESSION WITH MULTIPLE vi • The Least Squares Assumptions 129 in Multiple Regression 124 The Distribution of the OLS Estimators in Multiple Regression Multicollinearity Conclusion 133 Summary 133 CHAPTER 10 HYPOTHESIS TESTS AND CONFIDENCE INTERVALS IN MULTIPLE REGRESSION 126 127 127 128 128 128 131 Examples of Perfect Multicollinearity 131 Imperfect Multicollinearity 132 126 126 130 137 Hypothesis Tests and Confidence Intervals for a Single Coefficient 138 Standard Errors for the OLS Estimators Hypothesis Tests for a Single Coefficient Confidence Intervals for a Single Coefficient Application to Test Scores and the Student-Teacher Ratio 138 138 139 139 Tests of Joint Hypotheses Testing Hypotheses on Two or More Coefficients The F-Statistic Application to Test Scores and the Student-Teacher Ratio The Homoskedasticity-Only F-Statistic 140 Covariance Stationary Time Series 162 143 White Noise 165 The Lag Operator 168 Wold's Theorem, the General Linear Process, and Rational Distributed Lags 168 145 146 146 147 147 Analysis of the Test Score Data Set 148 Conclusion 151 Summary 151 Appendix 152 Wold's Theorem Theorem The General Linear Process Rational Distributed Lags Estimation and Inference for the Mean, Autocorrelation, and Partial Autocorrelation Functions Sample Mean Sample Autocorrelations Sample Partial Autocorrelations Application: Characterizing Canadian Employment Dynamics 11 168 169 169 170 170 170 170 172 172 152 CHAPTER CHAPTER 161 CYCLES 143 Confidence Sets for Multiple Coefficients The Bonferroni Test of a Joint Hypothesis CHARACTERIZING 142 144 Omitted Variable Bias in Multiple Regression Model Specification in Theory and in Practice Interpreting the R2 and the Adjusted R2 in Practice 12 140 Testing Single Restrictions Involving Multiple Coefficients Model Specification for Multiple Regression CHAPTER 13 MODELING CYCLES: MA, AR, AND MODELING AND ARMAMoDELS 177 FORECASTING TREND Selecting Forecasting Models Using the Akaike and Schwarz Criteria 155 Moving Average (MA) Models The MA(l) Process The MA(q) Process 156 178 178 181 Autoregressive (AR) Models The AR(l) Process The AR(p) Process 182 182 184 Contents • vii Autoregressive Moving Average (ARMA) Models 186 Application: Specifying and Estimating Models for Employment Forecasting CHAPTER 14 15 SIMULATION MODELING 187 ESTIMATING VOLATILITIES AND CORRELATIONS CHAPTER 197 Monte Carlo Simulation: A Simple Example Selecting Probability Distributions for the Inputs Interpreting Monte Carlo Simulation Output Why Use Simulation? Estimating Volatility 198 Weighting Schemes 198 The Exponentially Weighted Moving Average Model 199 The GARCH(l, 1) Model 200 The Weights Mean Reversion 201 201 Choosing Between the Models 201 Maximum Likelihood Methods 201 Estimating a Constant Variance Estimating EWMA or GARCH(l 1) Parameters How Good Is the Model? 201 Using GARCH(l, 1) to Forecast Future Volatility Volatility Term Structures Impact of Volatility Changes Correlations 202 204 205 205 206 206 Consistency Condition for Covariances 207 Application of EWMA to Four-Index Example 208 Summary 209 viii III Contents Multiple Input Variables and Compounding Distributions Incorporating Correlations Evaluating Decisions Important Questions in Simulation Modeling How Many Scenarios? Estimator Bias Estimator Efficiency Random Number Generation Inverse Transform Method What Defines a "Good" Random Number Generator? Pseudorandom Number Generators Quasirandom (Low-Discrepancy) Sequences Stratified Sampling Summary 213 214 215 215 217 218 218 219 221 221 221 222 222 223 224 225 226 226 228 Sample Exam QuestionsQuantitative Analysis 231 Sample Exam Answers and ExplanationsQuantitative Analysis 235 Appendix Table 239 Index 241 2015 FRM COMMITTEE MEMBERS Dr Rene Stulz (Chairman) Ohio State University Dr Victor Ng Goldman Sachs & Co Richard Apostolik Global Association of Risk Professionals Dr Elliot Noma Garrett Asset Management Richard Brandt Citibank Dr Matthew Pritsker Federal Reserve Bank of Boston Dr Christopher Donohue Global Association of Risk Professionals Liu Ruixia Industrial and Commercial Bank of China Herve Geny London Stock Exchange Dr Til Schuermann Oliver Wyman Keith Isaac, FRM® TD Bank Nick Strange Bank of England, Prudential Regulation Authority Steve Lerit, CFA UBS Wealth Management Serge Sverdlov Redmond Analytics William May Global Association of Risk Professionals Alan Weindorf Visa Michelle McCarthy Nuveen Investments ix An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year, and 60% of policyholders who have only a homeowner policy will renew next year The company estimates that 80% of policyholders who have both an auto and a homeowner policy will renew at least one of those policies next year Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowner policy, and 15% of policyholders have both an auto and a homeowner policy Using the company's estimates, what is the percentage of policyholders that will renew at least one policy next year? A 20% B 29% C.41% D.53% The following graphs show the cumulative distribution function (CDF) of four different random variables The dotted vertical line indicates the mean of the distribution Assuming each random variable can only be values between -10 and 10, which distribution has the highest variance? A B ~ ::l '" , '" ~ ~ ,'" " 111 ~ "'" g "'" 10 !:! 4i D ~ '" '" w 111 ~ '"'" :;! C ~ :li , b ~ '" " ~ 11) 232 II Financial Risk Manager Exam Part I: Quantitative Analysis 4! ~ You are running a Monte Carlo simulation to price a portfolio of options When generating random numbers for use in the simulation: A The stratified sampling method eliminates extreme observations B A truly random number generator would avoid clustered observations C A congruential pseudorandom number generator creates sequences converging to a constant value D The Latin hypercube sampling method ensures that all strata are sufficiently well-represented A risk manager is calculating the VaR of a fund with a data set of 25 weekly returns The mean and standard deviation of weekly returns are 7% and 10%, respectively Assuming that weekly returns are independent and identically distributed, what is the standard deviation of the mean of the weekly returns? A 0.4% B.0.7% C.2.0% D 10.0% The recent performance of Prudent Fund, with USD 50 million in assets, has been weak and the institutional sales group is recommending that it be merged with Aggressive Fund, a USD 200 million fund The returns on Prudent Fund are normally distributed with a mean of 3% and a standard deviation of 7% and the returns on Aggressive Fund are normally distributed with a mean of 7% and a standard deviation of 15% Senior management has asked you to estimate the likelihood that returns on the combined portfolio will exceed 26% Assuming the returns on the two funds are independent, your estimate for the probability that the returns on the combined fund will exceed 26% is closest to: A 1.0% B 2.5% C 5.0% D 10.0% Which of the following four statements on models for estimating volatility is INCORRECT? A In the exponentially weighted moving average (EWMA) model, some positive weight is assigned to the long-run average variance B In the EWMA model, the weights assigned to observations decrease exponentially as the observations become older C In the GARCH (1, 1) model, a positive weight is estimated for the long-run average variance D In the GARCH (1, 1) model, the weights estimated for observations decrease exponentially as the observations become older Sample Exam Questions-Quantitative Analysis III 233 10 Based on 21 daily returns of an asset, a risk manager estimates the standard deviation of the asset's daily returns to be 2% Assuming that returns are normally distributed and that there are 260 trading days in a year, what is the appropriate Chi-square test statistic if the risk manager wants to test the null hypothesis that the true annual volatility is 25% at a 5% significance level? A 25.80 B 33.28 C 34.94 D 54.74 234 • Financial Risk Manager Exam Part I: Quantitative Analysis SAMPLE EXAM ANSWERS AND EXPLANATIONSQUANTITATIVE ANALYSIS Answer: Explanation: Here the t-reliability factor is used since the population variance is unknown Since there are 30 observations, the degrees of freedom are 30 - = 29 The t-test is a twotailed test So the correct critical t-value is t29 ,25 = 2.045, thus the 95% confidence interval for the mean return is: [4% - 2.045 ( ~ J 4% + 2,045 ( ~) = [-3.464%, 11.464%] Answer: Explanation: A binomial distribution is a probability distribution, and it refers to the various probabilities associated with the number of correct answers out of the total sample The correct approach is to find the cumulative probability for in a binomial distribution with N = 20 and p = 0.5 The cumulative probability is to be calculated on the basis of a binomial distribution with number of questions (n) equaling 20 and probability of a single event occurring being 50% (p = 0.5) Answer: B Explanation: Prob(mean - 2*0' < X < mean + 1.5*0') = (0.5 - 0.0228) + (0.5 - 0.0668) = 0.9104 Prob(mean - 2*0' > X or X > mean + 1.5*0') = - Prob(mean - 2*0' < X < mean + 1.5*0') = 0.0896 Answer: Explanation: Let: A = event that a policyholder has an auto policy H = event that a policyholder has a homeowners policy Then, based on the information given: p(AnH) = 0.15 p(AnHC) = peA) - p(AnH) = 0.65 - 0.15 = 0.5 p(NnH) = P(H) - p(AnH) = 0.5 - 0.15 = 0.35 Therefore, the proportion of policyholders that will renew at least one policy is shown below: 0.4 p(AnHC) + 0.6 p(NnH) + 0.8 p(AnH) = 0.4 x 0.5 + 0.6 x 0.35 + 0.8 x 0.15 = 0.53 Sample Exam Answers and Explanations-Quantitative Analysis II 235 Answer: D Explanation: Variance is a measure of the mean deviation In the above four graphs, it can be seen that (D) has the highest proportion of the distribution that deviates from the mean, and it also has a relatively higher density in both tails Hence, "D" has the highest variance Answer: D Explanation: The stratified sampling method in random number generation divides the population into groups, called strata, with similar characteristics and then attempts to distribute the group of random numbers across all the strata By doing so, the method tries to ensure that extreme observations will be incorporated into the sample The Latin hypercube sampling method is an enhanced version of the basic stratified sampling method which reduces the number of times each particular random number is generated and therefore, ensures that all strata included in the population are sufficiently well-represented Answer: C Explanation: In order to calculate the standard deviation of the mean of weekly returns, we must divide the standard deviation of the weekly returns by the square root of the sample size Therefore the correct answer is 10%/sqrt (25) = 2% Answer: C Explanation: Since these are independent normally distributed random variables, the combined expected mean return is: JL = 0.2 * 3% + 0.8 * 7% = 6.2% Combined volatility is: u= Z -statistic IS Z Th e appropnate = 26% - 6.2% = 164 12.1% and therefore P(Z>1.64) = - 095 = 05 = 5% Answer: A Explanation: The EWMA model does not involve the long-run average variance in updating volatility, in other words, the weight assigned to the long-run average variance is zero Only the current estimate of the variance is used The other statements are all correct.· 10 Answer: B Explanation: The formula for the Chi-squared test statistic is: (n - 1)* (sample variance/hypothesis variance) annualized Since we are given a daily standard deviation, we must first annualize it by multiplying it by the square root of the number of trading days Therefore: Sample volatility = sqrt (260) 2% = 32.25%, and the Chi-squared test statistic = (21 - 1)* 0.3225 2/0.25 = 33.28 236 III Financial Risk Manager Exam Part I: Quantitative Analysis APPENDIX TABLE Reference Table: let Z be a standard normal random variable z -3.00 -2.99 -2.98 -2.97 -2.96 -2.95 -2.94 -2.93 -2.92 -2.91 -2.90 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 -2.83 -2.82 -2.81 -2.80 -2.79 -2.78 -2.77 -2.76 -2.75 -2.74 -2.73 -2.72 -2.71 -2.70 -2.69 -2.68 -2.67 -2.66 -2.65 -2.64 -2.63 -2.62 -2.61 -2.60 -2.59 -2.58 -2.57 -2.56 -2.55 -2.54 -2.53 -2.52 -2.51 P(Z < z) 0.0013 0.0014 0.0014 0.0015 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019 0.0019 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0026 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037 0.0038 0.0039 0.0040 0.0041 0.0043 0.0044 0.0045 0.0047 0.0048 0.0049 0.0051 0.0052 0.0054 0.0055 0.0057 0.0059 0.0060 z -2.50 -2.49 -2.48 -2.47 -2.46 -2.45 -2.44 -2.43 -2.42 -2.41 -2.40 -2.39 -2.38 -2.37 -2.36 -2.35 -2.34 -2.33 -2.32 -2.31 -2.30 -2.29 -2.28 -2.27 -2.26 -2.25 -2.24 -2.23 -2.22 -2.21 -2.20 -2.19 -2.18 -2.17 -2.16 -2.15 -2.14 -2.13 -2.12 -2.11 -2.10 -2.09 -2.08 -2.07 -2.06 -2.05 -2.04 -2.03 -2.02 -2.01 P(Z < z) 0.0062 0.0064 0.0066 0.0068 0.0069 0.0071 0.0073 0.0075 0.0078 0.0080 0.0082 0.0084 0.0087 0.0089 0.0091 0.0094 0.0096 0.0099 0.0102 0.0104 0.0107 0.0110 0.0113 0.0116 0.0119 0.0122 0.0125 0.0129 0.0132 0.0136 0.0139 0.0143 0.0146 0.0150 0.0154 0.0158 0.0162 0.0166 0.0170 0.0174 0.0179 0.0183 0.0188 0.0192 0.0197 0.0202 0.0207 0.0212 0.0217 0.0222 z ! P(Z < z) -2.00 -1.99 -1.98 -1.97 -1.96 -1.95 -1.94 -1.93 -1.92 -1.91 -1.90 -1.89 -1.88 -1.87 -1.86 -1.85 -1.84 -1.83 -1.82 -1.81 -1.80 -1.79 -1.78 -1.77 -1.76 -1.75 -1.74 -1.73 -1.72 -1.71 -1.70 -1.69 -1.68 -1.67 -1.66 -1.65 -1.64 -1.63 -1.62 -1.61 -1.60 -1.59 -1.58 -1.57 -1.56 -1.55 -1.54 -1.53 -1.52 -1.51 0.0228 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359 0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 z P(Z