On the Concavity of Expected Shortfall

TL;DR

This paper demonstrates that Expected Shortfall, known as a convex risk measure, is actually concave with respect to probability distributions, showing it behaves differently than previously understood.

Contribution

It proves that Expected Shortfall is a concave risk measure with respect to probability distributions, contrasting its known convexity in risk positions.

Findings

Expected Shortfall is convex in risk positions

Expected Shortfall is concave in probability distributions

Implications for risk assessment methods

Abstract

It is well known that Expected Shortfall (also called Average Value-at-Risk) is a convex risk measure, i. e. Expected Shortfall of a convex linear combination of arbitrary risk positions is not greater than a convex linear combination with the same weights of Expected Shortfalls of the same risk positions. In this short paper we prove that Expected Shortfall is a concave risk measure with respect to probability distributions, i. e. Expected Shortfall of a finite mixture of arbitrary risk positions is not lower than the linear combination of Expected Shortfalls of the same risk positions (with the same weights as in the mixture).