On a relationship between distorted and spectral risk measures

Henryk, Gzyl and Silvia, Mayoral (2006): On a relationship between distorted and spectral risk measures.

Abstract

We study the relationship between two widely used risk measures, the spectral measures and the distortion risk measures. In both cases, the risk measure can be thought of as a re-weighting of some initial distribution. We prove that spectral risk measures are equivalent to distorted risk pricing measures, or equivalently, spectral risk functions are related to distortion functions. Besides that we prove that distorted measures are absolutely continuous with respect to the original measure.

• Item Type: MPRA Paper
• Language: English
• Keywords: Coherent risk measure; distortion function; Spectral measures; Risk Aversion Function
• Subjects: G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Acerbi, C. (2002) Spectral measures of risk: a coherent representation of subjective risk aversionJ.of Banking and Finance , 7, 1505--1518.

Acerbi, C., Simonetti, P. (2002) Portfolio Optimization with Spectral Measures of Risk, Working Paper, Abaxbank, avaible in www.gloriamundi.com.

Artzner, P.; Delbaen, F.; Eber, J.M. and Heath, D. (1999) Coherent measures of risk, Mathematical Finance 9 203-228.

Balbas, A., Garrido, J., Mayoral, S. (2006) Properties of distortion risk measures, Working Paper, University of Navarra.

Cherny, A.S. (2006) Pricing with coherent risk, available at: arcxiv.math.PR/0605049v1.

Cotter, J., Kevin D. (2006) Extreme Spectral Risk Measures: An Application to Futures Clearinghouse Margin Requirements. Forthcoming in Journal of Banking and Finance.

Denneberg D. (1997). Non--Additive Measure and Integral. Dordrecht-Boston- London: Kluwer Academic Publishers.

Denneberg, D. (2000) Non-additive measure and integral, basic concepts and their role for applications in Studies in Fuzzyness and Soft Computing, Grabisch, M. and Sugeno, M. (Eds.), Special volume of Fuzzy Measures and Integrals: Theory and Applications, 40, 42-69.

Delbaen, F.(2003) Coherent measures of risk on general probability spaces in Advances in Finance and Stochastics, Eds. K.Sandmann and J.P.Schonbucher, Springer--Verlag, Berlin, 1-37.

Embrechts, P.(1996) Actuarial versus financial pricing and hedgingRisk Finance, 1 (4), 17--26

Follmer, H., Shied A.(2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6(4), 429-447.

Gerber, H. and Shiu, E.(2002) Actuarial bridges to dynamic hedging and option pricing, Insurance: Mathematics and Economics, 18, 183-218.

Goovaerts, M.J. and Dhaene, J.(1997) On the characterization of Wang's clas of premium principles 26th International Congress of Actuaries, Birmingham (U.K.), 4, 121 - 134.

Goovaerts, M.J. and Laeven, R.J.A. (2006) Actuarial risk measures for financial derivative pricing, available at http://papers.ssrn.com

Hardy, M.R., Wirch, J.(2001) ``Distortion risk measures: coherence and stochastic dominance.'' Working Paper, avaible in www.gloriamundi.com.

Hurlimann, W. (2001) Distribution free comparison of pricing principles, Insurance: MAthematics and Economics, 17, 43-54.

Hurlimann, W. (2004) Distorsion measures and economic capital North American. Actuarial Journal, 8(1) 86–-95.

Kaas, R.; Goovaerts, M.J.; Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory Kluwer Academic Publishers, Dordrecht.

Kingman, J and Taylor, (1970) Measure Theory and Probability Cambridge Univ. Press.

Laurent, J.P., C. Gouriéroux and Scaillet O. (2000) Sensitivity Analysis of Values at Risk , Journal of Empirical Finance, 7 (3-4), 225--245.

Madan, D.B. and Unal, H. (2004) Risk-neutralizing statistical distributions with application to pricing reinsurance contracts on FDIC losses FDIC Center for Financial Research Working Paper No. 2004-01.

Reesor, R.M. and McLeish, D.L. (2003) Risk, entropy, and the transformation of distributions North American Actuarial Journal, 7(2), 128--144.

Rockafellar, R.T.; Uryasev, S. and Zabarankin, M.(2006) Generalized deviation measures in risk analysis Finance and Stochastics, 10, 51-74.

Schweitzer, M. (2001) "From Actuarial to Financial Valuation Principles" Insurance: Mathematics and Economics 28, 31-47

Wang, S.S. (1998) Aggregation of correlated risk portfolios: models and algorithms Proceedings of the Casualty Actuarial Society.

Wang, S., Young, V. and Panjer, (1996)Axiomatic characterization of insurance prices Insurance: Mathematics and Economics, 25, 337-347.