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.

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.

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

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