Mathematics and Statistics for Financial Risk Management

"The first edition of this work was clear, comprehensive, and up-to-date. The Second Edition is all that and includes important new material on Bayesian and classical methods. Extensive examples and problems make clear how these concepts are used in the world's top financial institutions. The book is perfect for self-study or classroom use." —Aaron Brown, author of Red-Blooded Risk, A World of Chance , and The Poker Face of Wall Street "Michael B. Miller provides a very accessible ride across the daunting waters of mathematics for quantitative risk management." —Attilio Meucci, founder, SYMMYS "At every turn, this book shows the relevance of mathematical and statistical concepts to risk management. They are no longer the desiccated notions found in most textbooks but assume a sense of vibrancy. So, if you're trying to hone your skills, this book is a great place to start." — Seeking Alpha 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. This incisive resource: Covers basic statistical concepts—from standard deviation and correlation to regression analysis and hypothesis testing Explores widely used risk models, including value at risk, factor analysis, Monte Carlo simulation, and stress testing Contains numerous sample problems and end-of-chapter exercises (with answers) Includes a companion website with Excel examples and templates Features two new chapters, which cover multivariate distributions, copulas, and Bayesian analysis Mathematics and Statistics for Financial Risk Management is an indispensable reference for today's financial risk professional.

Acknowledgments Chapter 1 Some Basic Math Logarithms Log Returns Compounding Limited Liability Graphing Log Returns Continuously Compounded Returns Combinatorics Discount Factors Geometric Series Problems Chapter 2 Probabilities Discrete Random Variables Continuous Random Variables Mutually Exclusive Events Independent Events Probability Matrices Conditional Probability Problems Chapter 3 Basic Statistics Averages Expectations Variance and Standard Deviation Standardized Variables Covariance Correlation Application: Portfolio Variance and Hedging Moments Skewness Kurtosis Coskewness and Cokurtosis Best Linear Unbiased Estimator (BLUE) Problems Chapter 4 Distributions Parametric Distributions Uniform Distribution Bernoulli Distribution Binomial Distribution Poisson Distribution Normal Distribution Lognormal Distribution Central Limit Theorem Application: Monte Carlo Simulations Part I: Creating Normal Random Variables Chi–Squared Distribution Student′s t Distribution F–Distribution Triangular Distribution Beta Distribution Mixture Distributions Problems Chapter 5 Multivariate Distributions and Copulas Multivariate Distributions Copulas Problems Chapter 6 Bayesian Analysis Overview Bayes’ Theorem Bayes vs. Frequentists Many State Problems Continuous Distributions Bayesian Networks Bayesian Networks versus Correlation Matrices Problems Chapter 7 Hypothesis Testing & Confidence Intervals The Sample Mean Revisited Sample Variance Revisited Confidence Intervals Hypothesis Testing Chebyshev′s Inequality Application: VaR Problems Chapter 8 Matrix Algebra Matrix Notation Matrix Operations Application: Transition Matrices Application: Monte Carlo Simulations Part II: Cholesky Decomposition Problems Chapter 9 Vector Spaces Vectors Revisited Orthogonality Rotation Principal Component Analysis Application: The Dynamic Term Structure of Interest Rates Application: The Structure of Global Equity Markets Problems Chapter 10 Linear Regression Analysis Linear Regression (One Regressor) Linear Regression (Multivariate) Application: Factor Analysis Application: Stress Testing Problems Chapter 11 Time Series Models Random Walks Drift–Diffusion Autoregression Variance and Autocorrelation Stationarity Moving Average Continuous Models Application: GARCH Application: Jump–Diffusion Application: Interest Rate Models Problems Chapter 12 Decay Factors Mean Variance Weighted Least Squares Other Possibilities Application: Hybrid VaR Problems Appendix A Binary Numbers Appendix B Taylor Expansions Appendix C Vector Spaces Appendix D Greek Alphabet Appendix E Common Abbreviations Appendix F Copulas References About the Author About the Companion Website Index

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.