A Probability Metrics Approach to Financial Risk Measures

by Rachev, Svetlozar T.; Stoyanov, Stoyan V.; Fabozzi, Frank J.

• ISBN13: 9781405183697
• ISBN10: 1405183691
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-02-21
• Publisher: Wiley-Blackwell

Summary

A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies them to finance for the first time. Helps to answer the question: which risk measure is best for a given problem? Finds new relations between existing classes of risk measures Describes applications in finance and extends them where possible Presents the theory of probability metrics in a more accessible form which would be appropriate for non-specialists in the field Applications include optimal portfolio choice, risk theory, and numerical methods in finance Topics requiring more mathematical rigor and detail are included in technical appendices to chapters

Author Biography

Svetlozar T. Rachev is Chair-Professor in Statistics, Econometrics and Mathematical Finance at the University of Karlsruhe in the School of Economics and Business Engineering. Stoyan V. Stoyanov is a Professor of Finance at EDHEC Business School and Scientific Director for EDHEC-Risk Institute in Asia. Frank J. Fabozzi is Professor in the Practice of Finance in the School of Management at Yale University.

Table of Contents

Introduction

Probability metrics

Applications in finance

Probability distances and metrics

Introduction

Some examples of probability metrics

Engineer's metric

Uniform

Levy metric

Kantorovich metric

Lp-metrics between distribution functions

Ky Fan metrics

Lp-metric

Distance and semidistance spaces

Definitions of probability distances and metrics

Summary

Technical appendix

Universally measurable separable metric spaces

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

Choice under uncertainty

Introduction

Expected utility theory

St. Petersburg Paradox

The von Neumann-Morgenstern expected utility theory

Types of utility functions

Stochastic dominance

First-order stochastic dominance

Second-order stochastic dominance

Rothschild-Stiglitz stochastic dominance

Third-order stochastic dominance

Efficient sets and the portfolio choice problem

Return versus payoff

Probability metrics and stochastic dominance

Cumulative Prospect Theory

Summary

Technical appendix

The axioms of choice

Stochastic dominance relations of order n

Return versus payoff and stochastic dominance

Other stochastic dominance relations

A classification of probability distances

Introduction

Primary distances and primary metrics

Simple distances and metrics

Compound distances and moment functions

Ideal probability metrics

Interpretation and examples of ideal probability metrics

Conditions for boundedness of ideal probability metrics

Summary

Technical appendix

Examples of primary distances

Examples of simple distances

Examples of compound distances

Examples of moment functions

Risk and uncertainty

Introduction

Measures of dispersion

Standard deviation

Mean absolute deviation

Semi-standard deviation

Axiomatic description

Deviation measures

Probability metrics and dispersion measures

Measures of risk

Value-at-risk

Computing portfolio VaR in practice

Back-testing of VaR

Coherent risk measures

Risk measures and dispersion measures

Risk measures and stochastic orders

Summary

Technical appendix

Convex risk measures

Probability metrics and deviation measures

Deviation measures and probability quasi-metrics

Average value-at-risk

Introduction

Average value-at-risk

AVaR for stable distributions

AVaR estimation from a sample

Computing portfolio AVaR in practice

The multivariate normal assumption

The Historical Method

The Hybrid Method

The Monte Carlo Method

Kernel methods

Back-testing of AVaR

Spectral risk measures

Risk measures and probability metrics

Risk measures based on distortion functionals

Summary

Technical appendix

Characteristics of conditional loss distributions

Higher-order AVaR

The minimization formula for AVaR

ETL vs AVaR

Kernel-based estimation of AVaR

Remarks on spectral risk measures

Computing AVaR through Monte Carlo

Introduction

An illustration of Monte Carlo variability

Asymptotic distribution, classical conditions

Rate of convergence to the normal distribution

The effect of tail thickness

The effect of tail truncation

Infinite variance distributions

Asymptotic distribution, heavy-tailed returns

Rate of convergence, heavy-tailed returns

Stable Paretian distributions

Student's t distribution

On the choice of a distributional model

Tail behavior and return frequency

Practical implications

Summary

Technical appendix

Proof of the stable limit result

Stochastic dominance revisited

Introduction

Metrization of preference relations

The Hausdorff metric structure

Examples

The Levy quasi-semidistance and first-order stochastic dominance

Higher order stochastic dominance

The H-quasi-semidistance

AVaR generated stochastic orders

Compound quasi-semidistances

Utility-type representations

Almost stochastic orders and degree of violation

Summary

Technical appendix

Preference relations and topology

Quasi-semidistances and preference relations

Construction of quasi-semidistances on classes of investors

Investors with balanced views

Structural classification of probability distances

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