COVER
PREFACE
ABOUT THE AUTHOR
1 OVERVIEW OF FINANCIAL RISK MANAGEMENT
1. WHAT IS RISK?
2. WHAT IS FINANCIAL RISK MANAGEMENT?
3. TYPES OF FINANCIAL RISK
4. WHAT DOES A RISK MANAGER DO?
5. A VERY BRIEF HISTORY OF RISK MANAGEMENT
6. THE FUTURE OF RISK MANAGEMENT
7. NOTE
2 MARKET RISK: STANDARD DEVIATION
1. RISK AND STANDARD DEVIATION
2. AVERAGES
3. EXPECTATIONS
4. VARIANCE AND STANDARD DEVIATION
5. STANDARD DEVIATION WITH DECAY
6. GARCH
7. MOMENTS
8. SKEWNESS
9. KURTOSIS
10. JUMP-DIFFUSION MODEL
11. DOLLAR STANDARD DEVIATION
12. ANNUALIZATION
13. END-OF-CHAPTER QUESTIONS
14. NOTES
3 MARKET RISK: VALUE AT RISK
1. WHAT IS VALUE AT RISK?
2. DELTA-NORMAL VAR
3. HISTORICAL VAR
4. HYBRID VAR
5. MONTE CARLO SIMULATION
6. CORNISH-FISHER VAR
7. BACKTESTING
8. END-OF-CHAPTER QUESTIONS
4 MARKET RISK: EXPECTED SHORTFALL, AND EXTREME VALUE THEORY
1. COHERENT RISK MEASURES
2. EXPECTED SHORTFALL
3. EXTREME VALUE THEORY
4. END-OF-CHAPTER QUESTIONS
5 MARKET RISK: PORTFOLIOS AND CORRELATION
1. COVARIANCE
2. CORRELATION
3. PORTFOLIO VARIANCE AND HEDGING
4. LINEAR REGRESSION (UNIVARIATE)
5. LINEAR REGRESSION (MULTIVARIATE)
6. STRESS TESTING
7. DELTA-NORMAL MODEL
8. CHOLESKY DECOMPOSITION AND MONTE CARLO SIMULATIONS
9. END-OF-CHAPTER QUESTIONS
6 MARKET RISK: BEYOND CORRELATION
1. COSKEWNESS AND COKURTOSIS
2. MULTIVARIATE DISTRIBUTIONS
3. COPULAS
4. INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES
5. END-OF-CHAPTER QUESTIONS
7 MARKET RISK: RISK ATTRIBUTION
1. FACTOR ANALYSIS
2. INCREMENTAL VR
3. DIVERSIFICATION
4. RISK-ADJUSTED PERFORMANCE
5. CHOOSING STATISTICS
6. END-OF-CHAPTER QUESTIONS
7. NOTE
8 CREDIT RISK
1. DEFAULT RISK AND PRICING
2. DETERMINING THE PROBABILITY OF DEFAULT
3. PORTFOLIO CREDIT RISK
4. REDUCING CREDIT RISK
5. END-OF-CHAPTER QUESTIONS
9 LIQUIDITY RISK
1. WHAT IS LIQUIDITY RISK?
2. SIMPLE LIQUIDITY MEASURES
3. LIQUIDITY COST MODELS
4. OPTIMAL LIQUIDATION
5. END-OF-CHAPTER QUESTIONS
6. NOTES
10 BAYESIAN ANALYSIS
1. CONDITIONAL PROBABILITY
2. OVERVIEW OF BAYESIAN ANALYSIS
3. BAYES' THEOREM
4. BAYESIANS VERSUS FREQUENTISTS
5. MANY-STATE PROBLEMS
6. CONTINUOUS DISTRIBUTIONS
7. BAYESIAN NETWORKS
8. BAYESIAN NETWORKS VERSUS CORRELATION MATRICES
9. END-OF-CHAPTER QUESTIONS
10. NOTE
11 BEHAVIORAL ECONOMICS AND RISK
1. UTILITY FUNCTIONS
2. UTILITY UNDER UNCERTAINTY
3. HEURISTICS AND BIASES
4. A WORD OF CAUTION
5. END-OF-CHAPTER QUESTIONS
APPENDIX A MAXIMUM LIKELIHOOD ESTIMATION
APPENDIX B COPULAS
ANSWERS TO END-OF-CHAPTER QUESTIONS
1. CHAPTER 2
2. CHAPTER 3
3. CHAPTER 4
4. CHAPTER 5
5. CHAPTER 6
6. CHAPTER 7
7. CHAPTER 8
8. CHAPTER 9
9. CHAPTER 10
10. CHAPTER 11
REFERENCES
INDEX
END USER LICENSE AGREEMENT

List of Tables

Chapter 2
1. TABLE 2.1 Example of EWMA Weights
Chapter 3
1. TABLE 3.1 95% Historical VaR Example
2. TABLE 3.2 95% Hybrid VaR Example
Chapter 4
1. TABLE 4.1 Functions and Restrictions of EVT Distributions and the Generalized Pareto Distribution
Chapter 5
1. TABLE 5.1 F-Distribution Critical Values
Chapter 6
1. TABLE 6.1 Fund Returns
2. TABLE 6.2 Combined Fund Returns
3. TABLE 6.3 Sample Coskewness
4. TABLE 6.4 Number of Nontrivial Cross Moments
5. TABLE 6.5 Probability Matrix
6. TABLE 6.6 Joint Probabilities Table
7. TABLE 6.7 Joint Distribution Matrix
8. TABLE 6.8 Probability Matrix with Subtotals
9. TABLE 6.9 Comparing Two Data Sets
10. TABLE 6.10 Scaling Factors for i.i.d Random Variables
Chapter 7
1. TABLE 7.1 Geographic Exposures
2. TABLE 7.2 Adding Factor Exposures across Portfolios
Chapter 8
1. TABLE 8.1 Bond Ratings
2. TABLE 8.2 Five-Year Corporate Bond Default Rate, 1920–2011.
3. TABLE 8.3 One-Year Ratings Transition Matrix
4. TABLE 8.4 Five-Year Ratings Transition Matrix
Chapter 9
1. TABLE 9.1 Liquidity Schedule
Chapter 10
1. TABLE 10.1 Probability Matrix
2. TABLE 10.2 Probabilities of Networks
Chapter 11
1. TABLE 11.1 Probability of Observing a Loss over Various Time Horizons

List of Illustrations

Chapter 1
1. FIGURE 1.1 RiskMetrics Annual Revenue
2. FIGURE 1.2 FRM Exam Enrollment
Chapter 2
1. FIGURE 2.1 Probability Density Function
2. FIGURE 2.2 Time Series with Outlier
3. FIGURE 2.3 Rolling Mean of Time Series with Outlier
4. FIGURE 2.4 EWMA versus Rectangular Weights
5. FIGURE 2.5 Rolling Mean, EWMA versus Rectangular Window
6. FIGURE 2.6 S&P 500, Annualized 60-Day Return Standard Deviation, 1928–2008
7. FIGURE 2.7 Positive Skewness
8. FIGURE 2.8 Negative Skewness
9. FIGURE 2.9 High Kurtosis
10. FIGURE 2.10 Low Kurtosis
Chapter 3
1. FIGURE 3.1 Example of 95% Value at Risk
2. FIGURE 3.2 Triangular Probability Density Function
3. FIGURE 3.3 Probability Density Function for Box Asset Management
4. FIGURE 3.4 Probability Density Function for Pyramid Asset Management
Chapter 4
1. FIGURE 4.1 Triangular PDF, VaR, and Expected Shortfall
2. FIGURE 4.2 Distributions of Extreme Values
3. FIGURE 4.3 Fréchet Probability Density Functions
4. FIGURE 4.4 Gumbel Probability Density Functions
5. FIGURE 4.5 Weibull Probability Density Functions
6. FIGURE 4.6 Determining the EVT Distribution
Chapter 5
1. FIGURE 5.1 Linear Regression Example
Chapter 6
1. FIGURE 6.1 Funds A and B
2. FIGURE 6.2 Funds C and D
3. FIGURE 6.3 Joint Uniform Probability Density Function
4. FIGURE 6.4 Bivariate Standard Normal PDF
5. FIGURE 6.5 Bivariate Standard Normal PDF Contour Graph
6. FIGURE 6.6 Joint Standard Normal Distribution with Positive Correlation
7. FIGURE 6.7 Joint Standard Normal Distribution with Negative Correlation
8. FIGURE 6.8 Bivariate Standard Normal PDF with Clayton Copula
9. FIGURE 6.9 Frank's Joint Standard Uniform PDF, α = 1
10. FIGURE 6.10 Bivariate Standard Normal PDF with Frank's Copula, α = 2
11. FIGURE 6.11 Example of Transformed Data
12. FIGURE 6.12 Contour Plots for Question #4
Chapter 7
1. FIGURE 7.1 Diversification or Diworsification
Chapter 8
1. FIGURE 8.1 Corporate Bond Default Rate, All Issuers, 1920–2011.
2. FIGURE 8.2 The Merton Model of Default
3. FIGURE 8.3 Binomial Probability Density Functions
4. FIGURE 8.4 Correlation when Transforming Normal Variables to Uniform Variables
Chapter 9
1. FIGURE 9.1 Partial Supply and Demand Curves
2. FIGURE 9.2 Optimal Liquidation
Chapter 10
1. FIGURE 10.1 Prior and Posterior Distributions
2. FIGURE 10.2 Beta Distribution
3. FIGURE 10.3 Bayesian Network with Two Nodes
4. FIGURE 10.4 Bayesian Network with Seven Nodes
5. FIGURE 10.5 Two Bayesian Networks with Three Nodes Each
6. FIGURE 10.6 Covariance Matrices
7. FIGURE 10.7 Three-State Network
Chapter 11
1. FIGURE 11.1 Utility Function
2. FIGURE 11.2 Loss Aversion
3. FIGURE 11.3 The Perception of Low-Probability Events
APPENDIX A
1. FIGURE A.1 Probability of Heads then Tails
ANSWERS TO END-OF-CHAPTER QUESTIONS
1. FIGURE EOCQ.1 Euler Fund PDF and VaR
2. FIGURE 6.A1 Joint Probability Density Function

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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. |

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

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.

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:

geni001 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.