Textbook
Mathematics and Statistics for Financial Risk ManagementISBN: 9781118170625
304 pages
March 2012, ©2012

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 easytoread 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 ValueatRisk, factor analysis, Monte Carlo simulations, and stress testing are also explored. A chapter on time series analysis introduces interest rate modeling, GARCH, and jumpdiffusion 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.
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
ChiSquared Distribution 94
Student’s t Distribution 95
FDistribution 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
DriftDiffusion 199
Autoregression 200
Variance and Autocorrelation 205
Stationarity 206
Moving Average 212
Continuous Models 212
Application: GARCH 215
Application: JumpDiffusion 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
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.
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Chapter 3 Coskewness Excel example from Chapter 3 
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Chapter 3. Kurtosis Excel example from Chapter 3 
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Chapter 4. Asian Option Pricing Excel example from Chapter 4 
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Chapter 4. Normal Distribution Excel example from Chapter 4 
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Chapter 4. Uniform to Bernoulli Excel example from Chapter 4 
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Chapter 5. Sample Mean Convergence Excel example from Chapter 5 
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Chapter 6. Cholesky Excel example from Chapter 6 
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Chapter 7. PCA Equity Indexes Excel example from Chapter 7 
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Chapter 7. PCA Interest Rates Excel example from Chapter 7 
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Chapter 8. Univariate Regression Excel example from Chapter 8 
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Chapter 9. AR(1) Monte Carlo Excel example from Chapter 9 
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Chapter 9. ARCH Excel example from Chapter 9 
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Chapter 9. Jump Diffusion Excel example from Chapter 9 
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Chapter 9. Spurious Correlation Excel example from Chapter 9 
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Chapter 9. Spurious Regression Excel example from Chapter 9 
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