Textbook

# Mathematics and Statistics for Financial Risk Management

ISBN: 978-1-118-17062-5
Hardcover
291 pages
US \$85.00

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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