Cash-subadditive risk measures without quasi-convexity
Xia Han, Qiuqi Wang, Ruodu Wang, Jianming Xia
TL;DR
This paper studies cash-subadditive risk measures without assuming quasi-convexity, motivated by uncertainty in interest rates and defaultable claims. It proves that any cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex cash-subadditive risk measures, and introduces quasi-star-shapedness and quasi-normalization as natural properties to obtain refined representations (notably for $\Lambda\mathrm{VaR}$). It further develops representations for risk measures with additional properties, including normalization and SSD-consistency, linking them to families of quasi-convex or law-invariant measures and to familiar constructs like Value-at-Risk and Expected Shortfall. The results broaden the analytical framework for risk measures beyond convexity, enabling robust portfolio optimization and decision-theoretic analysis under non-quasi-convex cash-subadditive settings.
Abstract
In the literature on risk measures, cash subadditivity was proposed to replace cash additivity, motivated by the presence of stochastic or ambiguous interest rates and defaultable contingent claims. Cash subadditivity has been traditionally studied together with quasi-convexity, in a way similar to cash additivity with convexity. In this paper, we study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced, and we obtain a corresponding representation result via the lower envelope of normalized, quasi-convex and cash-subadditive risk measures.