Risk measures with the CxLS property

TL;DR

This paper characterizes a class of law-invariant convex risk measures with convex level sets, linking them to generalized shortfall measures, and confirms expectiles as the unique elicitable coherent risk measures.

Contribution

It extends existing characterizations of risk measures by relaxing assumptions and identifies the class of generalized shortfall risk measures, also confirming the uniqueness of expectiles among elicitable coherent risk measures.

Findings

Law-invariant convex risk measures with convex level sets are characterized.

Expectiles are proven to be the only elicitable coherent risk measures.

A simple robustness criterion for convex risk measures is provided.

Abstract

In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (2014) and Bellini and Bignozzi (2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterization of robustness for convex risk measures in terms of a weak notion of mixture continuity.