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A Probability Metrics Approach to Financial Risk Measures


Is the behavior of the stocks in our portfolio close to their behavior during the most recent crisis? How close is the strategy of hedge fund A to the strategy of hedge fund B? In which proportions do we invest in a given universe of stocks so that the resulting portfolio matches as much as possible the strategy of fund C?

All of these questions are essential to finance and they have one feature in common: measuring distances between random quantities. Problems of this kind have been explored for many years in areas other than finance. In A Probability Metrics Approach to Financial Risk Measures, the field of probability metrics and risk measures are related to one another and applied to finance for the first time, revealing groundbreaking new classes of risk measures, finding new relations between existing classes of risk measures, and providing answers to the question of which risk measure is best for a given problem. Applications include optimal portfolio choice, risk theory, and numerical methods in finance.

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).


About the Authors. Chapter 1 Introduction.

1.1 Probability Metrics.

1.2 Applications in Finance.

Chapter 2 Probability Distances and Metrics.

2.1 Introduction.

2.2 Some Examples of Probability Metrics.

2.2.1 Engineer's metric.

2.2.2 Uniform (or Kolmogorov) metric.

2.2.3 Levy metric.

2.2.4 Kantorovich metric.

2.2.5 Lp-metrics between distribution functions.

2.2.6 Ky Fan metrics.

2.2.7 Lp-metric.

2.3 Distance and Semidistance Spaces.

2.4 Definitions of Probability Distances and Metrics.

2.5 Summary.

2.6 Technical Appendix.

2.6.1 Universally measurable separable metric spaces.

2.6.2 The equivalence of the notions of p. (semi-)distance on P2 and on X.

Chapter 3 Choice Under Uncertainty.

3.1 Introduction.

3.2 Expected Utility Theory.

3.2.1 St. Petersburg Paradox.

3.2.2 The von Neumann-Morgenstern expected utility theory.

3.2.3 Types of utility functions.

3.3 Stochastic Dominance.

3.3.1 First-order stochastic dominance.

3.3.2 Second-order stochastic dominance.

3.3.3 Rothschild-Stiglitz stochastic dominance.

3.3.4 Third-order stochastic dominance.

3.3.5 Efficient sets and the portfolio choice problem.

3.3.6 Return versus payoff.

3.4 Probability Metrics and Stochastic Dominance.

3.5 Cumulative Prospect Theory.

3.6 Summary.

3.7 Technical Appendix.

3.7.1 The axioms of choice.

3.7.2 Stochastic dominance relations of order n.

3.7.3 Return versus payoff and stochastic dominance.

3.7.4 Other stochastic dominance relations.

Chapter 4 A Classification of Probability Distances.

4.1 Introduction.

4.2 Primary Distances and Primary Metrics.

4.3 Simple Distances and Metrics.

4.4 Compound Distances and Moment Functions.

4.5 Ideal Probability Metrics.

4.5.1 Interpretation and examples of ideal probability metrics.

4.5.2 Conditions for boundedness of ideal probability metrics.

4.6 Summary.

4.7 Technical Appendix.

4.7.1 Examples of primary distances.

4.7.2 Examples of simple distances.

4.7.3 Examples of compound distances.

4.7.4 Examples of moment functions.

Chapter 5 Risk and Uncertainty.

5.1 Introduction.

5.2 Measures of Dispersion.

5.2.1 Standard deviation.

5.2.2 Mean absolute deviation.

5.2.3 Semi-standard deviation.

5.2.4 Axiomatic description.

5.2.5 Deviation measures.

5.3 Probability Metrics and Dispersion Measures.

5.4 Measures of Risk.

5.4.1 Value-at-risk.

5.4.2 Computing portfolio VaR in practice.

5.4.3 Back-testing of VaR.

5.4.4 Coherent risk measures.

5.5 Risk Measures and Dispersion Measures.

5.6 Risk Measures and Stochastic Orders.

5.7 Summary.

5.8 Technical Appendix.

5.8.1 Convex risk measures.

5.8.2 Probability metrics and deviation measures.

5.8.3 Deviation measures and probability quasi-metrics.

Chapter 6 Average Value-at-Risk.

6.1 Introduction.

6.2 Average Value-at-Risk.

6.2.1 AVaR for stable distributions.

6.3 AVaR Estimation From a Sample.

6.4 Computing Portfolio AVaR in Practice.

6.4.1 The multivariate normal assumption.

6.4.2 The Historical Method.

6.4.3 The Hybrid Method.

6.4.4 The Monte Carlo Method.

6.4.5 Kernel methods.

6.5 Back-testing of AVaR.

6.6 Spectral Risk Measures.

6.7 Risk Measures and Probability Metrics.

6.8 Risk Measures Based on Distortion Functionals.

6.9 Summary.

6.10 Technical Appendix.

6.10.1 Characteristics of conditional loss distributions.

6.10.2 Higher-order AVaR.

6.10.3 The minimization formula for AVaR.

6.10.4 ETL vs AVaR.

6.10.5 Kernel-based estimation of AVaR.

6.10.6 Remarks on spectral risk measures.

Chapter 7 Computing AVaR through Monte Carlo.

7.1 Introduction.

7.2 An illustration of Monte Carlo Variability.

7.3 Asymptotic Distribution, Classical Conditions.

7.4 Rate of Convergence to the Normal Distribution.

7.4.1 The effect of tail thickness.

7.4.2 The effect of tail truncation.

7.4.3 Infinite variance distributions.

7.5 Asymptotic Distribution, Heavy-tailed Returns.

7.6 Rate of Convergence, Heavy-tailed Returns.

7.6.1 Stable Paretian distributions.

7.6.2 Student's t distribution.

7.7 On the choice of a distributional model.

7.7.1 Tail behavior and return frequency.

7.7.2 Practical implications.

7.8 Summary.

7.9 Technical Appendix.

7.9.1 Proof of the stable limit result.

Chapter 8 Stochastic Dominance Revisited.

8.1 Introduction.

8.2 Metrization of Preference Relations.

8.3 The Hausdorff Metric Structure.

8.4 Examples.

8.4.1 The Levy quasi-semidistance and first-order stochastic dominance.

8.4.2 Higher order stochastic dominance.

8.4.3 The H-quasi-semidistance.

8.4.4 AVaR generated stochastic orders.

8.4.5 Compound quasi-semidistances.

8.5 Utility-type Representations.

8.6 Almost Sstochastic Orders and Degree of Violation.

8.7 Summary.

8.8 Technical Appendix.

8.8.1 Preference relations and topology.

8.8.2 Quasi-semidistances and preference relations.

8.8.3 Construction of quasi-semidistances on classes of investors.

8.8.4 Investors with balanced views.

8.8.5 Structural classification of probability distances.