Upper Bounds for Strictly Concave Distortion Risk Measures on Moment Spaces

25 Pages Posted: 2 Nov 2017 Last revised: 21 Jul 2018

Dries Cornilly

Vrije Universiteit Brussel (VUB); KU Leuven

Ludger Rüschendorf

University of Freiburg

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: January 28, 2018

Abstract

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

Keywords: Value-at-Risk (VaR), Coherent risk measure, Model uncertainty, Distortion function

JEL Classification: C02

Suggested Citation

Cornilly, Dries and Rüschendorf, Ludger and Vanduffel, Steven, Upper Bounds for Strictly Concave Distortion Risk Measures on Moment Spaces (January 28, 2018). Available at SSRN: https://ssrn.com/abstract=3063186 or http://dx.doi.org/10.2139/ssrn.3063186

Dries Cornilly

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brussels 1050
Belgium

KU Leuven

Celestijnenlaan 200B
Leuven, Vlaams-Brabant 3001
Belgium

Ludger Rüschendorf

University of Freiburg ( email )

Fahnenbergplatz
Freiburg, D-79085
Germany

Steven Vanduffel (Contact Author)

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

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