Mathematics and statistics for financial risk management

Mathematics and Statistics for Financial Risk Management is a practical guide to modern financial risk management for both practitioners and academics.

Now in its second edition with more topics, more sample problems and more real world examples, this popular guide to financial risk management introduces readers to practical quantitative techniques for analyzing and managing financial risk.

In a concise and easy-to-read style, each chapter introduces a different topic in mathematics or statistics. As different techniques are introduced, sample problems and application sections demonstrate how these techniques can be applied to actual risk management problems. Exercises at the end of each chapter and the accompanying solutions at the end of the book allow readers to practice the techniques they are learning and monitor their progress. A companion Web site includes interactive Excel spreadsheet examples and templates.

Mathematics and Statistics for Financial Risk Management is an indispensable reference for today's financial risk professional.

Author Notes

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	p. ix
What's New in the Second Edition	p. xi
Acknowledgments	p. xiii
Chapter 1 Some Basic Math	p. 1
Logarithms	p. 1
Log Returns	p. 2
Compounding	p. 3
Limited Liability	p. 4
Graphing Log Returns	p. 5
Continuously Compounded Returns	p. 6
Combinatorics	p. 8
Discount Factors	p. 9
Geometric Series	p. 9
Problems	p. 14
Chapter 2 Probabilities	p. 15
Discrete Random Variables	p. 15
Continuous Random Variables	p. 15
Mutually Exclusive Events	p. 21
Independent Events	p. 22
Probability Matrices	p. 22
Conditional Probability	p. 24
Problems	p. 26
Chapter 3 Basic Statistics	p. 29
Averages	p. 29
Expectations	p. 34
Variance and Standard Deviation	p. 39
Standardized Variables	p. 41
Covariance	p. 42
Correlation	p. 43
Application: Portfolio Variance and Hedging	p. 44
Moments	p. 47
Skewness	p. 48
Kurtosis	p. 51
Coskewness and Cokurtosis	p. 53
Best Linear Unbiased Estimator (BLUE)	p. 57
Problems	p. 58
Chapter 4 Distributions	p. 61
Parametric Distributions	p. 61
Uniform Distribution	p. 61
Bernoulli Distribution	p. 63
Binomial Distribution	p. 65
Poisson Distribution	p. 68
Normal Distribution	p. 69
Lognormal Distribution	p. 72
Central Limit Theorem	p. 73
Application: Monte Carlo Simulations Part I: Creating Normal Random Variables	p. 76
Chi-Squared Distribution	p. 77
Student's t Distribution	p. 78
JF-Distribution	p. 79
Triangular Distribution	p. 81
Beta Distribution	p. 82
Mixture Distributions	p. 83
Problems	p. 86
Chapter 5 Multivariate Distributions and Copulas	p. 89
Multivariate Distributions	p. 89
Copulas	p. 97
Problems	p. 111
Chapter 6 Bayesian Analysis	p. 113
Overview	p. 113
Bayes' Theorem	p. 113
Bayes versus Frequentists	p. 119
Many-State Problems	p. 120
Continuous Distributions	p. 124
Bayesian Networks	p. 128
Bayesian Networks versus Correlation Matrices	p. 130
Problems	p. 132
Chapter 7 Hypothesis Testing and Confidence Intervals	p. 135
Sample Mean Revisited	p. 135
Sample Variance Revisited	p. 137
Confidence Intervals	p. 137
Hypothesis Testing	p. 139
Chebyshev's Inequality	p. 142
Application: VaR	p. 142
Problems	p. 152
Chapter 8 Matrix Algebra	p. 155
Matrix Notation	p. 155
Matrix Operations	p. 156
Application: Transition Matrices	p. 163
Application: Monte Carlo Simulations Part II: Cholesky Decomposition	p. 165
Problems	p. 168
Chapter 9 Vector Spaces	p. 169
Vectors Revisited	p. 169
Orthogonality	p. 172
Rotation	p. 177
Principal Component Analysis	p. 181
Application: The Dynamic Term Structure of Interest Rates	p. 185
Application: The Structure of Global Equity Markets	p. 191
Problems	p. 193
Chapter 10 Linear Regression Analysis	p. 195
Linear Regression (One Regressor)	p. 195
Linear Regression (Multivariate)	p. 203
Application: Factor Analysis	p. 208
Application: Stress Testing	p. 211
Problems	p. 212
Chapter 11 Time Series Models	p. 215
Random Walks	p. 215
Drift-Diffusion Model	p. 216
Autoregression	p. 217
Variance and Autocorrelation	p. 222
Stationarity	p. 223
Moving Average	p. 227
Continuous Models	p. 228
Application: GARCH	p. 230
Application: Jump-Diffusion Model	p. 232
Application: Interest Rate Models	p. 232
Problems	p. 234
Chapter 12 Decay Factors	p. 237
Mean	p. 237
Variance	p. 243
Weighted Least Squares	p. 244
Other Possibilities	p. 245
Application: Hybrid VaR	p. 245
Problems	p. 247
Appendix A Binary Numbers	p. 249
Appendix B Taylor Expansions	p. 251
Appendix C Vector Spaces	p. 253
Appendix D Greek Alphabet	p. 255
Appendix E Common Abbreviations	p. 257
Appendix F Copulas	p. 259
Answers	p. 283
References	p. 303
About the Author	p. 305
About the Companion Website	p. 307
Index	p. 309

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