Mathematics and Statistics for Financial Risk Management

by Miller, Michael B.

• ISBN13: 9781118170625
• ISBN10: 1118170628
• Format: Hardcover
• Copyright: 2012-03-06
• Publisher: Wiley

Summary

In chapter 1, there is a review three math topics -- logarithms, combinatorics, and geometric series  and one financial topic, discount factors.  Emphasis will be given to the specific aspects of these topics that are most relevant to risk management. In chapter 2, the author explores the application of probabilities to risk management.  There is also an introduction to basic terminology and notations that will be used throughout the rest of the book. In chapter 3, Miller teaches how to describe a collection of data in precise statistical terms.  Many of the concepts will be familiar, but the notation and terminology might be new.  This notation and terminology will be used throughout the rest of the book. In chapter 4, some of the most common probability distributions will be pointed out, followed by a chapter on two closely related topics, confidence intervals and hypothesis testing.  For risk management, these are possibly the two most important concepts in statistics. Chapter 6 provides a basic introduction to linear regression models. At the end of the chapter, Miller explores two risk management applications, factor analysis and stress testing. The final chapter is on a class of estimators, which has become very popular in finance and risk management for analyzing historical data.  These models hint at the limitations of the type of analysis that we have been explores in previous chapters. This book has a lot of charts and equations.

Author Biography

Michael B. Miller studied economics at the American University of Paris and the University of Oxford before starting a career in finance. He has worked in risk management for more than ten years, most recently as the chief risk officer for a hedge fund in New York City.

Table of Contents

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