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Mathematics and Statistics for Financial Risk Management

Mathematics and Statistics for Financial Risk Management

Michael B. Miller

ISBN: 978-1-118-17062-5

Mar 2012

304 pages


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

The recent financial crisis and its impact on the broader economy underscore the importance of financial risk management in today's world. At the same time, financial products and investment strategies are becoming increasingly complex. Today, it is more important than ever that risk managers possess a sound understanding of mathematics and statistics.

In a concise and easy-to-read style, each chapter of this book 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 website includes interactive Excel spreadsheet examples and templates.

This comprehensive resource covers basic statistical concepts from volatility and Bayes' Law to regression analysis and hypothesis testing. Widely used risk models, including Value-at-Risk, factor analysis, Monte Carlo simulations, and stress testing are also explored. A chapter on time series analysis introduces interest rate modeling, GARCH, and jump-diffusion models. Bond pricing, portfolio credit risk, optimal hedging, and many other financial risk topics are covered as well.

If you're looking for a book that will help you understand the mathematics and statistics of financial risk management, look no further.

Preface ix

Acknowledgments xi

CHAPTER 1 Some Basic Math 1

Logarithms 1

Log Returns 3

Compounding 4

Limited Liability 5

Graphing Log Returns 5

Continuously Compounded Returns 7

Combinatorics 9

Discount Factors 10

Geometric Series 11

Problems 16

CHAPTER 2 Probabilities 19

Discrete Random Variables 19

Continuous Random Variables 20

Mutually Exclusive Events 26

Independent Events 27

Probability Matrices 28

Conditional Probability 30

Bayes’ Theorem 31

Problems 36

CHAPTER 3 Basic Statistics 39

Averages 39

Expectations 46

Variance and Standard Deviation 51

Standardized Variables 54

Covariance 54

Correlation 56

Application: Portfolio Variance and Hedging 57

Moments 60

Skewness 60

Kurtosis 64

Coskewness and Cokurtosis 67

Best Linear Unbiased Estimator (BLUE) 71

Problems 72

CHAPTER 4 Distributions 75

Parametric Distributions 75

Uniform Distribution 75

Bernoulli Distribution 78

Binomial Distribution 79

Poisson Distribution 83

Normal Distribution 84

Lognormal Distribution 88

Central Limit Theorem 90

Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 92

Chi-Squared Distribution 94

Student’s t Distribution 95

F-Distribution 97

Mixture Distributions 99

Problems 102

CHAPTER 5 Hypothesis Testing & Confidence Intervals 105

The Sample Mean Revisited 105

Sample Variance Revisited 107

Confidence Intervals 108

Hypothesis Testing 109

Chebyshev’s Inequality 113

Application: VaR 114

Problems 124

CHAPTER 6 Matrix Algebra 127

Matrix Notation 127

Matrix Operations 129

Application: Transition Matrices 136

Application: Monte Carlo Simulations Part II: Cholesky Decomposition 138

Problems 141

CHAPTER 7 Vector Spaces 143

Vectors Revisited 143

Orthogonality 146

Rotation 152

Principal Component Analysis 157

Application: The Dynamic Term Structure of Interest Rates 162

Application: The Structure of Global Equity Markets 167

Problems 171

CHAPTER 8 Linear Regression Analysis 173

Linear Regression (One Regressor) 173

Linear Regression (Multivariate) 183

Application: Factor Analysis 188

Application: Stress Testing 192

Problems 194

CHAPTER 9 Time Series Models 197

Random Walks 197

Drift-Diffusion 199

Autoregression 200

Variance and Autocorrelation 205

Stationarity 206

Moving Average 212

Continuous Models 212

Application: GARCH 215

Application: Jump-Diffusion 217

Application: Interest Rate Models 218

Problems 220

CHAPTER 10 Decay Factors 223

Mean 223

Variance 230

Weighted Least Squares 231

Other Possibilities 232

Application: Hybrid VaR 233

Problems 234

APPENDIX A Binary Numbers 237

APPENDIX B Taylor Expansions 239

APPENDIX C Vector Spaces 241

APPENDIX D Greek Alphabet 242

APPENDIX E Common Abbreviations 243

Answers 245

References 283

About the Author 285

Index 287

"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." (SeekingAlpha, April 2012)
Chapter 3 Coskewness
Excel example from Chapter 3
Chapter 3. Kurtosis
Excel example from Chapter 3
Chapter 4. Asian Option Pricing
Excel example from Chapter 4
Chapter 4. Normal Distribution
Excel example from Chapter 4
Chapter 4. Uniform to Bernoulli
Excel example from Chapter 4
Chapter 5. Sample Mean Convergence
Excel example from Chapter 5
Chapter 6. Cholesky
Excel example from Chapter 6
Chapter 7. PCA Equity Indexes
Excel example from Chapter 7
Chapter 7. PCA Interest Rates
Excel example from Chapter 7
Chapter 8. Univariate Regression
Excel example from Chapter 8
Chapter 9. AR(1) Monte Carlo
Excel example from Chapter 9
Chapter 9. ARCH
Excel example from Chapter 9
Chapter 9. Jump Diffusion
Excel example from Chapter 9
Chapter 9. Spurious Correlation
Excel example from Chapter 9
Chapter 9. Spurious Regression
Excel example from Chapter 9