Robust Distortion Risk Measures

50 Pages Posted: 23 Oct 2020 Last revised: 19 May 2022

See all articles by Carole Bernard

Carole Bernard

Grenoble Ecole de Management; Vrije Universiteit Brussel (VUB)

Silvana M. Pesenti

University of Toronto

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: May 18, 2022


The robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well-informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball - specified through the Wasserstein distance - around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterisation of sharp bounds on their value, and we obtain quasi-explicit bounds in the case of Value-at-Risk and Range-Value-at-Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimisation and to model risk assessment.

Keywords: Risk Bounds, Distortion Risk Measures, Wasserstein Distance, Distributional Robustness, Tail Value-at-Risk

JEL Classification: C52, C44, C58, G22, G23, G32

Suggested Citation

Bernard, Carole and Pesenti, Silvana M. and Vanduffel, Steven, Robust Distortion Risk Measures (May 18, 2022). Available at SSRN: or

Carole Bernard

Grenoble Ecole de Management ( email )

12, rue Pierre Sémard
Grenoble Cedex, 38003

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, 1050

Silvana M. Pesenti (Contact Author)

University of Toronto ( email )

100 St. George Street
Toronto, Ontario M5S 3G8

Steven Vanduffel

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050


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