Submodular risk measures

Abstract: We study submodularity for law-invariant functionals, with special attention to convex risk measures. Expected losses are modular, and certainty equivalents are submodular if and only if the underlying loss function is convex. Law-invariant coherent risk measures are submodular if and only if they are coherent distortion risk measures, which include the class of Expected Shortfall (ES). We proceed to consider four classes of convex risk measures with explicit formulas. For shortfall risk measures, we give a complete characterization through an inequality on the Arrow--Pratt measure of risk aversion. The optimized certainty equivalents are always submodular, whereas for the adjusted Expected Shortfall (AES) with a nonconvex penalty function, submodularity forces reduction to a standard ES. Within a subclass of monotone mean-deviation risk measures, submodularity can hold only in coherent distortion cases. In an empirical study of daily US equity returns using rolling historical estimation, no ES submodularity violations are observed, as expected from the exact ES structure of the estimator; VaR shows persistent violations linked to market stress, and AES shows a small percentage of violations.