Remarks on Quantiles and Distortion Risk Measures

European Actuarial Journal, 2(2), 319-328

12 Pages Posted: 20 Oct 2012 Last revised: 7 Dec 2012

See all articles by Jan Dhaene

Jan Dhaene

Katholieke Universiteit Leuven

Alexander Kukush

Catholic University of Leuven (KUL)

Daniël Linders

University of Illinois

Qihe Tang

University of Amsterdam - Amsterdam School of Economics (ASE)

Date Written: October 19, 2012

Abstract

Distorted expectations can be expressed as weighted averages of quantiles. In this note, we show that this statement is true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of quantiles.

Keywords: comonotonicity, distorted expectation, distortion risk measure, TVaR, quantile

Suggested Citation

Dhaene, Jan and Kukush, Alexander and Linders, Daniël and Tang, Qihe, Remarks on Quantiles and Distortion Risk Measures (October 19, 2012). European Actuarial Journal, 2(2), 319-328. Available at SSRN: https://ssrn.com/abstract=2164229

Jan Dhaene

Katholieke Universiteit Leuven ( email )

Naamsestraat 69
Leuven, 3000
Belgium

Alexander Kukush

Catholic University of Leuven (KUL) ( email )

Leuven, B-3000
Belgium

Daniël Linders (Contact Author)

University of Illinois ( email )

306 Altgeld Hall,
1409 West Green Street
Champaign, IL 61822
United States

Qihe Tang

University of Amsterdam - Amsterdam School of Economics (ASE) ( email )

Roetersstraat 11
Amsterdam, North Holland 1018 WB
Netherlands

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