Mathematics and Statistics for Financial Risk Management

Michael B. Miller studied economics at the American University of Paris and the University of Oxford before starting a career in finance. He is currently the CEO of Northstar Risk Corp. Before that, he was the Chief Risk Officer of Tremblant Capital Group, and prior to that, Head of Quantitative Risk Management at Fortress Investment Group. Mr. Miller is also a certified FRM and an adjunct professor at Rutgers Business School.

• Preface ixWhat’s New in the Second Edition xiAcknowledgments xiiiChapter 1 Some Basic Math 1Logarithms 1Log Returns 2Compounding 3Limited Liability 4Graphing Log Returns 5Continuously Compounded Returns 6Combinatorics 8Discount Factors 9Geometric Series 9Problems 14Chapter 2 Probabilities 15Discrete Random Variables 15Continuous Random Variables 15Mutually Exclusive Events 21Independent Events 22Probability Matrices 22Conditional Probability 24Problems 26Chapter 3 Basic Statistics 29Averages 29Expectations 34Variance and Standard Deviation 39Standardized Variables 41Covariance 42Correlation 43Application: Portfolio Variance and Hedging 44Moments 47Skewness 48Kurtosis 51Coskewness and Cokurtosis 53Best Linear Unbiased Estimator (BLUE) 57Problems 58Chapter 4 Distributions 61Parametric Distributions 61Uniform Distribution 61Bernoulli Distribution 63Binomial Distribution 65Poisson Distribution 68Normal Distribution 69Lognormal Distribution 72Central Limit Theorem 73Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 76Chi-Squared Distribution 77Student’s t Distribution 78F-Distribution 79Triangular Distribution 81Beta Distribution 82Mixture Distributions 83Problems 86Chapter 5 Multivariate Distributions and Copulas 89Multivariate Distributions 89Copulas 97Problems 111Chapter 6 Bayesian Analysis 113Overview 113Bayes’ Theorem 113Bayes versus Frequentists 119Many-State Problems 120Continuous Distributions 124Bayesian Networks 128Bayesian Networks versus Correlation Matrices 130Problems 132Chapter 7 Hypothesis Testing and Confidence Intervals 135Sample Mean Revisited 135Sample Variance Revisited 137Confidence Intervals 137Hypothesis Testing 139Chebyshev’s Inequality 142Application: VaR 142Problems 152Chapter 8 Matrix Algebra 155Matrix Notation 155Matrix Operations 156Application: Transition Matrices 163Application: Monte Carlo Simulations Part II: Cholesky Decomposition 165Problems 168Chapter 9 Vector Spaces 169Vectors Revisited 169Orthogonality 172Rotation 177Principal Component Analysis 181Application: The Dynamic Term Structure of Interest Rates 185Application: The Structure of Global Equity Markets 191Problems 193Chapter 10 Linear Regression Analysis 195Linear Regression (One Regressor) 195Linear Regression (Multivariate) 203Application: Factor Analysis 208Application: Stress Testing 211Problems 212Chapter 11 Time Series Models 215Random Walks 215Drift-Diffusion Model 216Autoregression 217Variance and Autocorrelation 222Stationarity 223Moving Average 227Continuous Models 228Application: GARCH 230Application: Jump-Diffusion Model 232Application: Interest Rate Models 232Problems 234Chapter 12 Decay Factors 237Mean 237Variance 243Weighted Least Squares 244Other Possibilities 245Application: Hybrid VaR 245Problems 247Appendix A Binary Numbers 249Appendix B Taylor Expansions 251Appendix C Vector Spaces 253Appendix D Greek Alphabet 255Appendix E Common Abbreviations 257Appendix F Copulas 259Answers 263References 303About the Author 305About the Companion Website 307Index 309

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