[eBook Code] Mathematics and Statistics for Financial Risk Management (eBook Code, 2nd)

Preface ix

What’s New in the Second Edition xi

Acknowledgments xiii

Chapter 1 Some Basic Math 1

Logarithms 1

Log Returns 2

Compounding 3

Limited Liability 4

Graphing Log Returns 5

Continuously Compounded Returns 6

Combinatorics 8

Discount Factors 9

Geometric Series 9

Problems 14

Chapter 2 Probabilities 15

Discrete Random Variables 15

Continuous Random Variables 15

Mutually Exclusive Events 21

Independent Events 22

Probability Matrices 22

Conditional Probability 24

Problems 26

Chapter 3 Basic Statistics 29

Averages 29

Expectations 34

Variance and Standard Deviation 39

Standardized Variables 41

Covariance 42

Correlation 43

Application: Portfolio Variance and Hedging 44

Moments 47

Skewness 48

Kurtosis 51

Coskewness and Cokurtosis 53

Best Linear Unbiased Estimator (BLUE) 57

Problems 58

Chapter 4 Distributions 61

Parametric Distributions 61

Uniform Distribution 61

Bernoulli Distribution 63

Binomial Distribution 65

Poisson Distribution 68

Normal Distribution 69

Lognormal Distribution 72

Central Limit Theorem 73

Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 76

Chi-Squared Distribution 77

Student’s t Distribution 78

F-Distribution 79

Triangular Distribution 81

Beta Distribution 82

Mixture Distributions 83

Problems 86

Chapter 5 Multivariate Distributions and Copulas 89

Multivariate Distributions 89

Copulas 97

Problems 111

Chapter 6 Bayesian Analysis 113

Overview 113

Bayes’ Theorem 113

Bayes versus Frequentists 119

Many-State Problems 120

Continuous Distributions 124

Bayesian Networks 128

Bayesian Networks versus Correlation Matrices 130

Problems 132

Chapter 7 Hypothesis Testing and Confidence Intervals 135

Sample Mean Revisited 135

Sample Variance Revisited 137

Confidence Intervals 137

Hypothesis Testing 139

Chebyshev’s Inequality 142

Application: VaR 142

Problems 152

Chapter 8 Matrix Algebra 155

Matrix Notation 155

Matrix Operations 156

Application: Transition Matrices 163

Application: Monte Carlo Simulations Part II: Cholesky Decomposition 165

Problems 168

Chapter 9 Vector Spaces 169

Vectors Revisited 169

Orthogonality 172

Rotation 177

Principal Component Analysis 181

Application: The Dynamic Term Structure of Interest Rates 185

Application: The Structure of Global Equity Markets 191

Problems 193

Chapter 10 Linear Regression Analysis 195

Linear Regression (One Regressor) 195

Linear Regression (Multivariate) 203

Application: Factor Analysis 208

Application: Stress Testing 211

Problems 212

Chapter 11 Time Series Models 215

Random Walks 215

Drift-Diffusion Model 216

Autoregression 217

Variance and Autocorrelation 222

Stationarity 223

Moving Average 227

Continuous Models 228

Application: GARCH 230

Application: Jump-Diffusion Model 232

Application: Interest Rate Models 232

Problems 234

Chapter 12 Decay Factors 237

Mean 237

Variance 243

Weighted Least Squares 244

Other Possibilities 245

Application: Hybrid VaR 245

Problems 247

Appendix A Binary Numbers 249

Appendix B Taylor Expansions 251

Appendix C Vector Spaces 253

Appendix D Greek Alphabet 255

Appendix E Common Abbreviations 257

Appendix F Copulas 259

Answers 263

References 303

About the Author 305

About the Companion Website 307

Index 309