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.
Table of Contents:
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
About the Author :
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.
