QUANTITATIVE FINANCIAL RISK MANAGEMENT Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States With offices in North America, Europe, Australia and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding The Wiley Finance series contains books written specifically for finance and investment professionals as well as sophisticated individual investors and their financial advisors Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation and financial instrument analysis, as well as much more For a list of available titles, visit our Web site at www.WileyFinance.com QUANTITATIVE FINANCIAL RISK MANAGEMENT Michael B Miller Copyright © 2019 by Michael B Miller All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No 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print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley com For more information about Wiley products, visit www.wiley.com Library of Congress Cataloging-in-Publication Data Names: Miller, Michael B (Michael Bernard), 1973- author Title: Quantitative financial risk management / Michael B Miller Description: Hoboken, New Jersey : Wiley, [2019] | Series: Wiley finance series | Includes bibliographical references and index | Identifiers: LCCN 2018033207 (print) | LCCN 2018044462 (ebook) | ISBN 9781119522232 (Adobe PDF) | ISBN 9781119522263 (ePub) | ISBN 9781119522201 | ISBN 9781119522201 (hardcover) | ISBN 9781119522232 (ePDF) | ISBN 9781119522263 (ePub) Subjects: LCSH: Financial risk management Classification: LCC HD61 (ebook) | LCC HD61 M5373 2019 (print) | DDC 332—dc23 LC record available at https://lccn.loc.gov/2018033207 Cover Design: Wiley Cover Images: © Sergey Nivens/Shutterstock; © whiteMocca/Shutterstock Printed in the United States of America 10 CONTENTS Preface vii About the Author ix Overview of Financial Risk Management Market Risk: Standard Deviation 15 Market Risk: Value at Risk 51 Market Risk: Expected Shortfall, and Extreme Value Theory 73 Market Risk: Portfolios and Correlation 91 Market Risk: Beyond Correlation 119 Market Risk: Risk Attribution 151 Credit Risk 167 Liquidity Risk 189 10 Bayesian Analysis 205 11 Behavioral Economics and Risk 231 Appendix A Maximum Likelihood Estimation 247 Appendix B Copulas 253 Answers to End-of-Chapter Questions 257 References 295 Index 297 v PREFACE My first book on financial risk management, Mathematics and Statistics for Financial Risk Management, grew out of my experience working in the hedge fund industry and my involvement with the Global Association of Risk Professionals It was written for practitioners who may not have had the opportunity to take the advanced courses in mathematics— especially those courses in statistics—that are necessary for a deeper understanding of modern financial risk management It was also for practitioners who had taken these courses but may have forgotten what they learned To be honest, I often use the first book as a reference myself Even authors forget As a result of that first book, I was asked to teach a graduate-level course in risk management I realized that my students had the opposite problem of my colleagues in the hedge fund industry My students came to the course with a very strong foundation in mathematics, but knew less about the workings of financial markets or the role of risk managers within a financial firm This book was written for them, and I have been teaching with the material that this book is based on for a number of years now There is considerable overlap between the two books Indeed, there are some sections that are almost identical While the first book was organized around topics in mathematics, however, this book is organized around topics in risk management In each chapter we explore a particular topic in risk management along with various mathematical tools that can be used to understand that topic As with the first book, I have tried to provide a large number of sample problems and practical end-of-chapter questions I firmly believe that the best way to understand financial models is to work through actual problems This book assumes that the reader is familiar with basic calculus, linear algebra, and statistics When a particular topic in mathematics is central to a topic in risk management, I review the basics and introduce notation, but the pace can be quick For example, in the first chapter we review standard deviation, but we only spend one section on what would likely be an entire chapter in an introductory book on statistics vii viii Preface Risk management in practice often requires building models using spreadsheets or other financial software Many of the topics in this book are accompanied by an icon, shown here: These icons indicate that Excel examples can be found at John Wiley & Sons’ companion website for Quantitative Financial Risk Management, www.wiley.com/go/millerfinancialrisk REFERENCES Artzner, P., F Delbaen, J.-M Eber, and D Heath (1999) “Coherent Measures of Risk,” Mathematical Finance, 9(3): 203–228 Benartzi, S., and R H Thaler (1995) “Myopic Loss Aversion and the Equity Premium Puzzle,” Quarterly Journal of Economics, 110(1), 73–92 Henrich, J., S Heine, and A Norenzayan (2011) “The WEIRDest people in the world?” Behavioral and Brain Sciences, 33, 61–135 Kahneman, D and A Tversky (1974) “Judgment under Uncertainty: Heuristics and Biases,” Science, 185, 1124–1131 Kahneman, D and A Tversky (1984) “Choices, Values, and Frames,” American Psychologist, 34, 341–350 Koenker, R and G Bassett, Jr (1978) “Regression Quantiles,” Econometrica, 46, No 1, January 1978, 33–50 Levitt, S D and J A List (2007) “What Do Laboratory Experiments Measuring Social Preferences Reveal About the Real World?” Journal of Economic Perspectives, 21, 153–174 Mehra, R and E C Prescott (1985) “The Equity Premium: A Puzzle,” Journal of Monetary Economics, 15, 145–161 Meucci, A (2009) “Managing Diversification,” Risk, 22, May, 74–79 Miller, M B (2014) Mathematics and Statistics for Financial Risk Management 2nd ed Hoboken, NJ: John Wiley & Sons 295 INDEX A Absolute risk, 2–3 Actual default estimates, risk-neutral default estimates (contrast), 169–170 Alpha, 104 Anchoring, 243–244 Annualization, 46–47 Artzner, Philippe, 73 Autoregressive conditional heteroskedasticity (ARCH) model, 37 Availability bias, 242–243 Averages, 16–21 Bid-offer spread, 189 Binomial distribution, 65, 68, 180, 216, 221 Black Monday, 11–12, 18, 110 Bonds covenants, 184–185 default, 170, 180 default probability, sample problem, 181–182 expected value, determination (sample problem), 23–24 five-year corporate bond default rate, 173t Monte Carlo simulation, 182–183 plain-vanilla bonds, 168 portfolio, sample problem, 19 pricing, 167–168 ratings, 172t value, 168, 170–171 zero-coupon bonds, 169 Bootstrapping, 60 B Backtesting, 65–68, 79 Bayesian analysis, 205–229 Bayesian networks, 222–223, 224–227 Bayesians, frequentists (contrast), 213–214 Bayes, Thomas, 207 Bayes theorem, 208–213 Behavioral economics, risk (relationship), 231 Bernoulli, Daniel/Nicolas, 109 Beta distribution, 221f, 222 Beta, regression analysis 97 Biases and heuristics, 241–245 Bid-ask spread, 189, 197–198 C Cash-flow uses, 190 Cash flow value at risk (CFVaR), 191 Causal reasoning, 223 CDF See Cumulative distribution function Central moments, 62 297 298 Cholesky decomposition, 113–116 sample problem, 116 Clayton copulas, 253 Clearinghouse 186 Coefficient of determination, 103 Coherent risk measures, 73–78 Coin flip, expected value, 25 Cokurtosis, 119–123 Concordance, 142 Conditional probability, 78, 205–207 Conditional risk, 2–3 Conditional VaR (cVaR), 78 Confidence interval, construction, 81 Confidence level, 65 Conjugate distributions, 222 Continuous distributions, 125–126, 218–222 sample problem, 126, 219–220, 221–222 Continuous joint distribution, 124 Continuous random variables, 19–20 median, 20 mode, 20 Contour graph, usage, 127 Copulas, 132–146, 253–255 Clayton copulas, 133f, 253 data sets, comparison, 141t defining, 132–137 density function, 136 Farlie-Gumbel-Morgenstern (FGM) copula, 140, 253 Frank’s copula, 134, 138f, 254 graphing, 137–139 Gumbel copula, 133–134, 254 independent copula, 150, 255, 278 Index Joe copula, 255 parameterization, 140–146 sample problems, 134–136 Cornish-Fisher VaR, 61–64, 81 sample problem, 63–64 Corporate bond default rate, 173t 173f Correlation, 91, 92–92, 112, 119, 128–130 matrices, Bayesian networks (contrast), 224–227 sample problem, 93 Coskewness, 119–123 Counterparty credit risk, Counterparty risk, Country risk, 151 Covariance, 91–92 calculation, 92, 115 defining, 91, 112 matrices, 113, 226f Crash of 1929, 11 Credit risk, 7–8, 167–188 Cross-central moments, 119 Cumulative distribution function (CDF), 82, 86, 125, 131, 136–138 marginal CDF, calculation, 139 Cumulative distributions, 134 D Data-generation process, constancy, 87 Decay factor, 29–36 Default Merton model, 177f modeling, 183 n defaults, probability, 179–182 ratings approach, 171–174 recovery, relationship, 168–169 Index Default probability application, 174–175 calculation, sample problem, 178 determination, 171–179 quantitative approach, 175–179 sample problem, 181–182 Default risk, 7, 169–171 Delta, 63 Delta-gamma approximation, accuracy, 64 Delta-normal approach See Delta-normal VaR Delta-normal VaR, 55–58, 112–113 sample problem, 56 Density function, 129 calculation, 136–137 De Ratiociniis in Ludo Aleae (Huygens), 10, 22 Diagnostic reasoning, 223 Discordance, 141 Discount rate, 168 Discrete distributions, 123–125 Discrete random variables, 18–19 Distance-to-default model, 175–177 Distributions beta distribution, 221f continuous distributions, 125–126, 218–222 cumulative distributions, 134 discrete distributions, 123–125 extreme value theory distributions, 83t, 85f Fréchet distribution, 83 generalized Pareto distribution, 83t joint standard normal distribution, 129f, 130 marginal distributions, 129–132 multivariate distributions, 123–132 299 posterior distributions, 220f prior distributions, 220f univariate distributions, 133–134 Weibull distribution, 84 Disturbance terms, 37–38 Diversification, 96, 151, 158–161, 169–170 diworsification, contrast, 159f index, 161 score, 160–161 Diworsification, 159–160 Dollar standard deviation, 44–46 sample problem, 45–46 E Economic agents, utility maximizers, 231 Elasticity, equation, 198 Endogenous liquidity models, 198–200 Endowment effect, 244 Enterprise risk, 8–9 Enterprise value, 177–178 Equities index, expected return, 29 market risk, 151 Equity premium puzzle, 237–238 ESS See Explained sum of squares Evidence, term (usage), 213 EVT See Extreme value theory Exceedances, 66–67 Excess kurtosis, 43 Exogenous liquidity models, 197–198 Expectations, 21–26 operator, usage, 23–24 Expected shortfall, 73, 78–81 example, 80f sample problem, 79 Expected value, sample problem, 25–26 300 Explained sum of squares (ESS), 103 calculation, 108–109 Exponentially weighted moving average (EWMA), 31–32 characterization, half-life (usage), 33 rectangular window, contrast, 32f, 33f weights, example, 32t Exposure-adjusted Black-Scholes-Merton Greeks, 62 Extreme values, distributions, 82f Extreme value theory (EVT), 73, 81–88, 164 distributions, 82, 83t, 85f results, interpretation, 86 sample problem, 85–86 Extrinsic risk, 3–4 F Factor analysis, 151–154 Factor exposures, addition, 153t Farlie-Gumbel-Morgenstern (FGM) copula, 140, 144–146, 254 F-distribution, 109 F-distribution critical values, 110t FGM See Farlie-Gumbel-Morgenstern Financial risk management defining, 4–6 Financial Risk Manager (FRM) Exam, 13 enrollment, 14f Fitch (rating agency), 172 Frank’s copula, 134, 138f, 254 Frank’s joint standard uniform PDF, 135f, 136–137 Fréchet distribution, 83 Frequentists, Bayesians (contrast), 213–214 F-statistic, 109 Index G Generalized autoregressive conditional heteroskedasticity (GARCH), 36–38, 123 model, 43–44, 100 Generalized Pareto distribution, 83t Global Association of Risk Professionals (GARP), founding, 13 Goodwill, 175–176 Great Depression, 11 Great Recession, 11 Gumbel copula, 84f, 133–134, 254 H Half-life, usage, 33 Hedged portfolio, variance, 95 Hedge ratio, 95 Hedging, 93–96, 169–170 Heuristic biases, 241–245 Higher-order cross moments, 119, 123 Historical simulation, 81 Historical VaR, 56–58 Huygens, Christiaan, 10 Hybrid VaR, 58–59 I Idiosyncratic risk, 98, 105 Incremental VaR (iVaR), 155–158 Independence, testing, 131 Independent and identically distributed (i.i.d.) random variables, 47, 95, 147–148 Independent copula, 150, 255, 278 Infinite series, weight, 34 Interest-rate risk, 151 Intrinsic risk, 3–4 Isolines/isoquants, 128 Index J Joe copula, 255 Joint cumulative distribution, 125 function, 132 Joint distribution, shape, 164 Joint probabilities, 124t matrix, 125t Joint probability density function (joint PDF), 275f defining, 126 sample problem, 131–132 Joint standard normal distribution negative correlation, combination, 130f positive correlation, combination, 129f Joint uniform probability density function (joint uniform PDF), 127f JP Morgan, RiskMetrics spinoff, 12–13 Jump-diffusion model, 43–44 Junk bonds, 172 K Kendall’s tau, 141 calculation, 142–143 sample problem, 145–146 KMV, history, 179 Kurtosis, 41–43 excess kurtosis, 43 L Likelihood See Maximum likelihood estimation Likelihood, Bayesian, 213, 215 Linear regression analysis See Regression analysis Liquidity cost models, 196–200 demand, 190–191 301 endogenous liquidity models, 198–200 exogenous liquidity models, 197–198 internal demand, 190 measures, 192–196 schedule, 196 sources, modeling difficulty, 191–192 supply, 191–192 Liquidity-adjusted value at risk (LVaR), 196, 197 Liquidity at risk (LaR), 196 Liquidity risk See liquidity London Interbank Offered Rate (LIBOR), 110, 168 Long-Term Capital Management (LTCM), 12 Loss aversion, 234–235, 235f Loss given default (LGD), 169, 171 Low-probability events, perception, 238–239 LVaR See Liquidity-adjusted value at risk M Marginal CDF, calculation, 139 Marginal distributions, 129–132 Marginal PDF, calculation, 131 Marginal utility, 233 Marked to model assets, Market portfolio theory (MPT), 15 Market risk, 6–7, 15–150, 196 correlation, 91, 119 expected shortfall, 73 extreme value theory, 73 portfolios, 91 risk attribution, 151 standard deviation, 15 stress testing, 73 value at risk, 51 302 Markowitz, Harry, 11, 15 Maxima, and extreme value theory, 87 Maximum likelihood estimation (MLE), 83, 140, 247–251 Mean calculation, 17 estimation, refinement, 34 estimator, 35 Median, calculation, 17 Meriwether, John, 12 Merton distance to default model See Distance to default model Merton, Robert, 12, 175, 177, 184 Mode, 17, 20 Modern Portfolio Theory (MPT), 11 Moments, 38 Momentum, 151 Monotonicity, 73–75 sample problem, 74–75 Monte Carlo simulation, 59–61, 78, 113–116, 182–183 creation, 113–114 power, 609 reduction, 61 sample problem, 116 Moody’s (rating agency), 172 Motorola, Six Sigma usage, Multicollinearity, 106–108 Multi-period returns, generation, 60 Multivariate distributions, 123–132 Multivariate linear regression, 106–110 evaluation, 108–110 parameters, estimation, 108 Multivariate regression, 106 analysis, usage, 154 Multivariate regressors, 97 Municipal bonds, 167 Index N Negative correlation, joint standard normal distribution (combination), 130f Negative skewness, 39f Networks See Bayesian networks Non-parametric distribution, 197 Non-stationary variables, 46 Nontrivial covariances, calculation, 113 Nontrivial cross moments, 123t Normal variables, uniform variables transformation (correlation), 184f NORM.S.DIST function, usage, 138 O OLS See Ordinary least squares One-year ratings transition matrix, 174t Operational risk, Optimal hedging, example, 101 Optimal liquidation, 200–202, 201f Options Black-Scholes-Merton option pricing formula, usage, 177 exposure-adjusted Black-Scholes-Merton Greeks, 62 option-implied standard deviation, quoting, 47 Ordinary least squares (OLS), 98–102 assumptions, 106–108, 112 Osband, Kent, 159 Outlier, inclusion, 30f, 31f P Parametric distribution, 87, 197 Parsimony, 109 Partial supply and demand curves, 199f Peaks-over-threshold (POT) approach, 81, 85 Index Pearson’s correlation, 142 Plateauing, EWMA (usage), 33–34 Population data, 16–18 standard deviation, 43 Portfolios, 91 credit risk, 179–184 diversification, 78, 160 factor exposures, addition, 153t hedged portfolio, variance, 95 managers, performance, 242 risk, coskewness/cokurtosis (impact), 133 total VaR, 156 variance/hedging, 93–96 VaR measurement, sample problem, 155 volatility, reduction, 96 Positive homogeneity, 73, 75–76 Posterior distributions, 218, 219, 220f Posterior probabilities, 216 POT See Peaks-over-threshold Prior distributions, 220f Probability density function (PDF), 19–20, 21f, 181, 248 binomial probability density function, 182f bivariate standard normal PDF, 127f, 128f, 133f, 138f Frank’s joint standard uniform PDF, 135f Fréchet probability density functions (Fréchet PDFs), 83f Gumbel probability density functions, 84f joint uniform probability density function, 127f 303 marginal PDF, calculation, 131 sample problem, 20–21, 53–54 triangular PDF, 53f, 80f Weibull probability density functions (Weibull PDFs), 84f R RAND function, 183 Rating agencies, 171–172 Ratings approach, 171–174 Recovery, default, 168–169 Rectangular weights, EWMA (contrast), 32f Rectangular window, EWMA (contrast), 33f Regression analysis, 96–110 Relative risk, 2–3 Relative utility, 239–241 Representativeness, 241–242 Residual sum of squares (RSS), 102, 108–109 RiskMetrics, 12–13 R-squared, 103, 105–106, 109 RSS See Residual sum of squares S Sample data, 16–18 Sampling, with replacement, 60 Scholes, Myron, 12 Securities and Exchange Commission (SEC), establishment, 11 Settlement risk, Sharpe ratio, 162–163 Sharpe, William, 162 Skewness, 38–41 Sovereign bonds, default risk, 167 Spearman’s rho, 141, 146 304 Square-root rule, 94–95 Standard deviation, 4, 15–16, 26–28, 58 decay, impact, 29–36 dollar standard deviation, 44–46 Standard & Poor’s (rating agency), 172 Stationary variables, usage, 46 Stock-specific risk, 154 Stress testing, 73, 110–112 Student’s t-distribution, 83–84 Style risk, 151 Subadditive risk measures, 77 Subadditivity, 73, 77–78 sample problem, 77–78 Sum of squared residuals, 102 T Theta, 55, 63 Total sum of squares (TSS), 103 calculation, 108–109 Transition matrices, 174–175 five-year ratings transition matrix, 175t one-year ratings transition matrix, 174t Translation invariance, 73, 76–77 Triangular PDF, 53f, 80f TSS See Total sum of squares t-statistics, usage, 104–105 Twain, Mark, 111 U Uniform variables, correlation, 184f Univariate distributions, 133–134 Univariate linear regression, 96–106 parameters, estimation, 102–103 regression, evaluation, 103–106 Index Utility functions, 231–235, 232f loss aversion, 234–235 sample problem, 233–234 Utility under uncertainty, 236–241 sample problem, 236–237 V Value at risk (VaR), 51, 80f 95% historical VaR, example, 57t 95% hybrid VaR, example, 59t 95% VaR, example, 52f backtesting, 87 calculation, 74 conditional VaR (cVaR), 78 Cornish-Fisher VaR, 61–64 defining, 51–54 delta-normal VaR, 55–56 estimation, 113 exceedance, probability, 68 historical VaR, 56–58 hybrid VaR, 58–59 incremental VaR, 155–158 measurement, sample problem, 155 models, 67, 68 Variables continuous random variables, 19–20 discrete random variables, 18–19 distributions, 34–35 joint distribution, 225 linear combination, 108 product, expected value, 25 Variance calculation/equation, 28 long-run variance, 37 standard deviation, relationship, 26–28 standard estimator, 35 Vitruvius, Index Volatility (vol), 15–16, 44 low level, 61 reduction, 96 Volume-weighted average price (VWAP), 200 W Weibull distribution, 84 Weibull probability density functions (Weibull PDFs), 84f 305 Weighted less squares (WLS) regression, 112 Window length, 31 Y Yield, 170–171 Z Zero-coupon bonds, 169, 171 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... Author ix Overview of Financial Risk Management Market Risk: Standard Deviation 15 Market Risk: Value at Risk 51 Market Risk: Expected Shortfall, and Extreme Value Theory 73 Market Risk: Portfolios... step back and look at risk management more broadly Before delving into the models, we explore the following Quantitative Financial Risk Management questions: What is risk management? What is the... fundamental types of risk is important in financial risk management In practice, financial risk management is as much about reducing extrinsic risk as it is about managing intrinsic risk Risk and Standard