Mathematics and Statistics for Financial Risk Management

Preface ix Acknowledgments xi CHAPTER 1 Some Basic Math 1 Logarithms 1 Log Returns 3 Compounding 4 Limited Liability 5 Graphing Log Returns 5 Continuously Compounded Returns 7 Combinatorics 9 Discount Factors 10 Geometric Series 11 Problems 16 CHAPTER 2 Probabilities 19 Discrete Random Variables 19 Continuous Random Variables 20 Mutually Exclusive Events 26 Independent Events 27 Probability Matrices 28 Conditional Probability 30 Bayes' Theorem 31 Problems 36 CHAPTER 3 Basic Statistics 39 Averages 39 Expectations 46 Variance and Standard Deviation 51 Standardized Variables 54 Covariance 54 Correlation 56 Application: Portfolio Variance and Hedging 57 Moments 60 Skewness 60 Kurtosis 64 Coskewness and Cokurtosis 67 Best Linear Unbiased Estimator (BLUE) 71 Problems 72 CHAPTER 4 Distributions 75 Parametric Distributions 75 Uniform Distribution 75 Bernoulli Distribution 78 Binomial Distribution 79 Poisson Distribution 83 Normal Distribution 84 Lognormal Distribution 88 Central Limit Theorem 90 Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 92 Chi-Squared Distribution 94 Student's t Distribution 95 F-Distribution 97 Mixture Distributions 99 Problems 102 CHAPTER 5 Hypothesis Testing & Confidence Intervals 105 The Sample Mean Revisited 105 Sample Variance Revisited 107 Confidence Intervals 108 Hypothesis Testing 109 Chebyshev's Inequality 113 Application: VaR 114 Problems 124 CHAPTER 6 Matrix Algebra 127 Matrix Notation 127 Matrix Operations 129 Application: Transition Matrices 136 Application: Monte Carlo Simulations Part II: Cholesky Decomposition 138 Problems 141 CHAPTER 7 Vector Spaces 143 Vectors Revisited 143 Orthogonality 146 Rotation 152 Principal Component Analysis 157 Application: The Dynamic Term Structure of Interest Rates 162 Application: The Structure of Global Equity Markets 167 Problems 171 CHAPTER 8 Linear Regression Analysis 173 Linear Regression (One Regressor) 173 Linear Regression (Multivariate) 183 Application: Factor Analysis 188 Application: Stress Testing 192 Problems 194 CHAPTER 9 Time Series Models 197 Random Walks 197 Drift-Diffusion 199 Autoregression 200 Variance and Autocorrelation 205 Stationarity 206 Moving Average 212 Continuous Models 212 Application: GARCH 215 Application: Jump-Diffusion 217 Application: Interest Rate Models 218 Problems 220 CHAPTER 10 Decay Factors 223 Mean 223 Variance 230 Weighted Least Squares 231 Other Possibilities 232 Application: Hybrid VaR 233 Problems 234 APPENDIX A Binary Numbers 237 APPENDIX B Taylor Expansions 239 APPENDIX C Vector Spaces 241 APPENDIX D Greek Alphabet 242 APPENDIX E Common Abbreviations 243 Answers 245 References 283 About the Author 285 Index 287