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