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Mathematics and Statistics for Financial Risk Management, 2nd Edition


"The first edition of this work was clear, comprehensive, and up-to-date. The Second Edition is all that and includes important new material on Bayesian and classical methods. Extensive examples and problems make clear how these concepts are used in the world's top financial institutions. The book is perfect for self-study or classroom use."
--Aaron Brown, author of Red-Blooded Risk, A World of Chance, and The Poker Face of Wall Street

"Michael B. Miller provides a very accessible ride across the daunting waters of mathematics for quantitative risk management."
--Attilio Meucci, founder, SYMMYS

"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."
--Seeking Alpha

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.

This incisive resource:

  • Covers basic statistical concepts--from standard deviation and correlation to regression analysis and hypothesis testing
  • Explores widely used risk models, including value at risk, factor analysis, Monte Carlo simulation, and stress testing
  • Contains numerous sample problems and end-of-chapter exercises (with answers)
  • Includes a companion website with Excel examples and templates
  • Features two new chapters, which cover multivariate distributions, copulas, and Bayesian analysis

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.

Preface ix

What’s New in the Second Edition xi

Acknowledgments xiii

Chapter 1 Some Basic Math 1

Logarithms 1

Log Returns 2

Compounding 3

Limited Liability 4

Graphing Log Returns 5

Continuously Compounded Returns 6

Combinatorics 8

Discount Factors 9

Geometric Series 9

Problems 14

Chapter 2 Probabilities 15

Discrete Random Variables 15

Continuous Random Variables 15

Mutually Exclusive Events 21

Independent Events 22

Probability Matrices 22

Conditional Probability 24

Problems 26

Chapter 3 Basic Statistics 29

Averages 29

Expectations 34

Variance and Standard Deviation 39

Standardized Variables 41

Covariance 42

Correlation 43

Application: Portfolio Variance and Hedging 44

Moments 47

Skewness 48

Kurtosis 51

Coskewness and Cokurtosis 53

Best Linear Unbiased Estimator (BLUE) 57

Problems 58

Chapter 4 Distributions 61

Parametric Distributions 61

Uniform Distribution 61

Bernoulli Distribution 63

Binomial Distribution 65

Poisson Distribution 68

Normal Distribution 69

Lognormal Distribution 72

Central Limit Theorem 73

Application: Monte Carlo Simulations

Part I: Creating Normal Random Variables 76

Chi-Squared Distribution 77

Student’s t Distribution 78

F-Distribution 79

Triangular Distribution 81

Beta Distribution 82

Mixture Distributions 83

Problems 86

Chapter 5 Multivariate Distributions and Copulas 89

Multivariate Distributions 89

Copulas 97

Problems 111

Chapter 6 Bayesian Analysis 113

Overview 113

Bayes’ Theorem 113

Bayes versus Frequentists 119

Many-State Problems 120

Continuous Distributions 124

Bayesian Networks 128

Bayesian Networks versus Correlation Matrices 130

Problems 132

Chapter 7 Hypothesis Testing and Confidence Intervals 135

Sample Mean Revisited 135

Sample Variance Revisited 137

Confidence Intervals 137

Hypothesis Testing 139

Chebyshev's Inequality 142

Application: VaR 142

Problems 152

Chapter 8 Matrix Algebra 155

Matrix Notation 155

Matrix Operations 156

Application: Transition Matrices 163

Application: Monte Carlo Simulations

Part II: Cholesky Decomposition 165

Problems 168

Chapter 9 Vector Spaces 169

Vectors Revisited 169

Orthogonality 172

Rotation 177

Principal Component Analysis 181

Application: The Dynamic Term Structure of Interest Rates 185

Application: The Structure of Global Equity Markets 191

Problems 193

Chapter 10 Linear Regression Analysis 195

Linear Regression (One Regressor) 195

Linear Regression (Multivariate) 203

Application: Factor Analysis 208

Application: Stress Testing 211

Problems 212

Chapter 11 Time Series Models 215

Random Walks 215

Drift-Diffusion Model 216

Autoregression 217

Variance and Autocorrelation 222

Stationarity 223

Moving Average 227

Continuous Models 228

Application: GARCH 230

Application: Jump-Diffusion Model 232

Application: Interest Rate Models 232

Problems 234

Chapter 12 Decay Factors 237

Mean 237

Variance 243

Weighted Least Squares 244

Other Possibilities 245

Application: Hybrid VaR 245

Problems 247

Appendix A Binary Numbers 249

Appendix B Taylor Expansions 251

Appendix C Vector Spaces 253

Appendix D Greek Alphabet 255

Appendix E Common Abbreviations 257

Appendix F Copulas 259

Answers 263

References 303

About the Author 305

About the Companion Website 307

Index 309