ONE‐PARAMETER FAMILIES OF DISTORTION RISK MEASURES

This paper introduces parametric families of distortion risk measures, investigates their properties, and discusses their use in risk management. Their derivation is based on Kusuoka's representation theorem of law invariant and comonotonically additive coherent risk measures. Our approach is to narrow down a tractable class of risk measures by requiring their comparability with the traditional expected shortfall. We make numerical comparison among them and propose a method of estimating the value of the distortion risk measures based on data. Their use and interpretation in risk management will also be discussed.

[1]  C. Robertson,et al.  Generating Ordered Families of Lorenz Curves by Strongly Unimodal Distributions , 1987 .

[2]  Shaun S. Wang,et al.  Axiomatic characterization of insurance prices , 1997 .

[3]  Dilip B. Madan,et al.  New Measures for Performance Evaluation , 2007 .

[4]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[5]  Shaun S. Wang.  Premium Calculation by Transforming the Layer Premium Density , 1996, ASTIN Bulletin.

[6]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[7]  D. Cox.  Regression Models and Life-Tables , 1972 .

[8]  P. McCullagh.  Regression Models for Ordinal Data , 1980 .

[9]  Kjell A. Doksum,et al.  Rank estimates in a class of semiparametric two-sample models , 1989, Annals of the Institute of Statistical Mathematics.

[10]  Abaxbank,et al.  Spectral Measures of Risk : a Coherent Representation of Subjective Risk Aversion , 2002 .

[11]  Shaun S. Wang,et al.  Insurance pricing and increased limits ratemaking by proportional hazards transforms , 1995 .

[12]  G. Choquet.  Theory of capacities , 1954 .

[13]  D. Tasche,et al.  Expected shortfall and beyond , 2002, cond-mat/0203558.

[14]  Dieter Denneberg,et al.  Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation , 1990, ASTIN Bulletin.

[15]  Shaun S. Wang.  A CLASS OF DISTORTION OPERATORS FOR PRICING FINANCIAL AND INSURANCE RISKS , 2000 .

[16]  Erich L. Lehmann,et al.  The Power of Rank Tests , 1953 .

[17]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[18]  Jan Dhaene,et al.  The Concept of Comonotonicity in Actuarial Science and Finance: Theory , 2002, Insurance: Mathematics and Economics.

[19]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[20]  Eric R. Ziegel,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2004, Technometrics.

[21]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[22]  D. Tasche,et al.  On the coherence of expected shortfall , 2001, cond-mat/0104295.

[23]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[24]  G. P. Szegö,et al.  Risk measures for the 21st century , 2004 .

[25]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[26]  E. Jouini,et al.  Law Invariant Risk Measures Have the Fatou Property , 2005 .

[27]  A. S. Cherny,et al.  Weighted V@R and its Properties , 2006, Finance Stochastics.