Submodular risk measures

Summary. The manuscript claims to characterize submodularity for law-invariant convex risk measures. It asserts that expected losses are modular, that certainty equivalents are submodular exactly when the loss function is convex, and that law-invariant coherent risk measures are submodular if and only if they coincide with coherent distortion risk measures (including Expected Shortfall). It gives a complete characterization of submodular shortfall risk measures via the Arrow-Pratt measure of risk aversion, proves that optimized certainty equivalents are always submodular, and shows that adjusted Expected Shortfall is submodular only when it reduces to Expected Shortfall. Several deviation measures are also shown to be submodular. An empirical illustration on daily US equity returns reports no submodularity violations for ES, many for VaR, and relatively few for AES.

Significance. If the characterizations hold, the paper makes a useful contribution by linking submodularity directly to the class of coherent distortion risk measures through standard tools such as the Kusuoka representation and comonotonic additivity. The complete Arrow-Pratt characterization for shortfall risk measures and the precise reduction result for AES provide clear distinctions among risk measures that are relevant for applications. The empirical illustration, though secondary, supplies concrete evidence on violation rates in equity-return data. The work relies on established axiomatic properties without introducing free parameters or invented entities.