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
Date Written: October 19, 2012
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
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