Probability Metrics Approach to Financial Risk Measures

Svetlozar (Zari) T. Rachev is Chair-Professor in Statistics, Econometrics and Mathematical Finance at the University of Karlsruhe in the School of Economics and Business Engineering. He is also Professor Emeritus at the University of California, Santa Barbara in the Department of Statistics and Applied Probability. He has published seven monographs, eight handbooks and special-edited volumes, and over 300 research articles. His recently coauthored books published by Wiley in mathematical finance and financial econometrics include Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing (2005), Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008).  He is cofounder of Bravo Group, now FinAnalytica, specializing in  financial risk-management software, for which he serves as Chief Scientist. Stoyan V. Stoyanov, Ph.D. is the Head of Quantitative Research at FinAnalytica specializing in financial risk management software. He is author and co-author of numerous papers some of which have recently appeared in Economics Letters, Journal of Banking and Finance, Applied Mathematical Finance, Applied Financial Economics, and International Journal of Theoretical and Applied Finance. He is a coauthor of the mathematical finance book Advanced Stochastic Models, Risk Assessment and Portfolio Optimization: the Ideal Risk, Uncertainty and Performance Measures (2008) published by Wiley. Dr. Stoyanov has years of experience in applying optimal portfolio theory and market risk estimation methods when solving practical problems of clients of FinAnalytica.Frank J. Fabozzi is Professor in the Practice of Finance in the School of Management at Yale University. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in the Sloan School at MIT. Professor Fabozzi is a Fellow of the International Center for Finance at Yale University and on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University. He is the editor of the Journal of Portfolio Management. His recently coauthored books published by Wiley in mathematical finance and financial econometrics include The Mathematics of Financial Modeling and Investment  Management (2004), Financial Modeling of the Equity Market: From CAPM to Cointegration (2006), Robust Portfolio Optimization and Management (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008).

• Preface xiiiAbout the Authors xv1 Introduction 11.1 Probability Metrics 11.2 Applications in Finance 22 Probability Distances and Metrics 72.1 Introduction 92.2 Some Examples of Probability Metrics 92.2.1 Engineer’s metric 102.2.2 Uniform (or Kolmogorov) metric 102.2.3 Lévy metric 112.2.4 Kantorovich metric 142.2.5 Lp-metrics between distribution functions 152.2.6 Ky Fan metrics 162.2.7 Lp-metric 172.3 Distance and Semidistance Spaces 192.4 Definitions of Probability Distances and Metrics 242.5 Summary 282.6 Technical Appendix 282.6.1 Universally measurable separable metric spaces 292.6.2 The equivalence of the notions of p. (semi-)distance on P2 and on X 353 Choice under Uncertainty 403.1 Introduction 413.2 Expected Utility Theory 443.2.1 St Petersburg Paradox 443.2.2 The von Neumann–Morgenstern expected utility theory 463.2.3 Types of utility functions 483.3 Stochastic Dominance 513.3.1 First-order stochastic dominance 523.3.2 Second-order stochastic dominance 533.3.3 Rothschild–Stiglitz stochastic dominance 553.3.4 Third-order stochastic dominance 563.3.5 Efficient sets and the portfolio choice problem 583.3.6 Return versus payoff 593.4 Probability Metrics and Stochastic Dominance 633.5 Cumulative Prospect Theory 663.6 Summary 703.7 Technical Appendix 703.7.1 The axioms of choice 713.7.2 Stochastic dominance relations of order n 723.7.3 Return versus payoff and stochastic dominance 743.7.4 Other stochastic dominance relations 764 A Classification of Probability Distances 834.1 Introduction 864.2 Primary Distances and Primary Metrics 864.3 Simple Distances and Metrics 904.4 Compound Distances and Moment Functions 994.5 Ideal Probability Metrics 1054.5.1 Interpretation and examples of ideal probability metrics 1074.5.2 Conditions for boundedness of ideal probability metrics 1124.6 Summary 1144.7 Technical Appendix 1144.7.1 Examples of primary distances 1144.7.2 Examples of simple distances 1184.7.3 Examples of compound distances 1314.7.4 Examples of moment functions 1355 Risk and Uncertainty 1465.1 Introduction 1475.2 Measures of Dispersion 1505.2.1 Standard deviation 1515.2.2 Mean absolute deviation 1535.2.3 Semi-standard deviation 1545.2.4 Axiomatic description 1555.2.5 Deviation measures 1565.3 Probability Metrics and Dispersion Measures 1585.4 Measures of Risk 1595.4.1 Value-at-risk 1605.4.2 Computing portfolio VaR in practice 1655.4.3 Back-testing of VaR 1725.4.4 Coherent risk measures 1755.5 Risk Measures and Dispersion Measures 1795.6 Risk Measures and Stochastic Orders 1815.7 Summary 1825.8 Technical Appendix 1835.8.1 Convex risk measures 1835.8.2 Probability metrics and deviation measures 1845.8.3 Deviation measures and probability quasi-metrics 1876 Average Value-at-Risk 1916.1 Introduction 1926.2 Average Value-at-Risk 1936.2.1 AVaR for stable distributions 2006.3 AVaR Estimation from a Sample 2046.4 Computing Portfolio AVaR in Practice 2076.4.1 The multivariate normal assumption 2076.4.2 The historical method 2086.4.3 The hybrid method 2086.4.4 The Monte Carlo method 2096.4.5 Kernel methods 2116.5 Back-testing of AVaR 2186.6 Spectral Risk Measures 2206.7 Risk Measures and Probability Metrics 2236.8 Risk Measures Based on Distortion Functionals 2266.9 Summary 2276.10 Technical Appendix 2286.10.1 Characteristics of conditional loss distributions 2286.10.2 Higher-order AVaR 2326.10.3 The minimization formula for AVaR 2346.10.4 ETL vs AVaR 2376.10.5 Kernel-based estimation of AVaR 2426.10.6 Remarks on spectral risk measures 2457 Computing AVaR through Monte Carlo 2527.1 Introduction 2537.2 An Illustration of Monte Carlo Variability 2567.3 Asymptotic Distribution, Classical Conditions 2597.4 Rate of Convergence to the Normal Distribution 2627.4.1 The effect of tail thickness 2637.4.2 The effect of tail truncation 2687.4.3 Infinite variance distributions 2717.5 Asymptotic Distribution, Heavy-tailed Returns 2777.6 Rate of Convergence, Heavy-tailed Returns 2837.6.1 Stable Paretian distributions 2837.6.2 Student’s t distribution 2867.7 On the Choice of a Distributional Model 2907.7.1 Tail behavior and return frequency 2907.7.2 Practical implications 2957.8 Summary 2977.9 Technical Appendix 2987.9.1 Proof of the stable limit result 2988 Stochastic Dominance Revisited 3048.1 Introduction 3068.2 Metrization of Preference Relations 3088.3 The Hausdorff Metric Structure 3108.4 Examples 3148.4.1 The L´evy quasi-semidistance and first-order stochastic dominance 3158.4.2 Higher-order stochastic dominance 3178.4.3 The H-quasi-semidistance 3208.4.4 AVaR generated stochastic orders 3228.4.5 Compound quasi-semidistances 3248.5 Utility-type Representations 3258.6 Almost Stochastic Orders and Degree of Violation 3288.7 Summary 3308.8 Technical Appendix 3328.8.1 Preference relations and topology 3328.8.2 Quasi-semidistances and preference relations 3348.8.3 Construction of quasi-semidistances on classes of investors 3358.8.4 Investors with balanced views 3388.8.5 Structural classification of probability distances 339Index 357