Submodular risk measures

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.