Submodular risk measures

TL;DR

This paper characterizes submodularity in law-invariant risk measures, especially convex ones, and provides empirical insights into their violations in US equity returns.

Contribution

It offers a complete characterization of submodular shortfall risk measures and analyzes submodularity properties of various risk measures including ES and VaR.

Findings

Expected Shortfall (ES) is submodular only when it reduces to AES.

Many Value-at-Risk (VaR) violations of submodularity are observed.

Empirical analysis shows no ES violations but some VaR violations in US equity data.

Abstract

We study submodularity for law-invariant functionals, with particular attention to convex risk measures. Expected losses are modular, and certainty equivalents are submodular exactly when the loss function is convex. Law-invariant coherent risk measures are submodular exactly when they are coherent distortion risk measures, including Expected Shortfall (ES), and several deviation measures are also submodular. Beyond positive homogeneity, submodularity is restrictive for convex risk measures. We give a complete characterization for shortfall risk measures via the Arrow--Pratt measure of risk aversion, show that optimized certainty equivalents are always submodular, and prove that adjusted Expected Shortfall (AES) is submodular only when it reduces to ES. An empirical illustration for daily US equity returns finds no ES submodularity violations, many Value-at-Risk (VaR) violations, and relatively few AES violations.