Contextual Areas Star-Shaped Risk Measures

In this paper, monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties are studied. The measures in this class, which arise when the subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and value-at-risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.

Automated summary

This paper characterizes and studies a type of monetary risk measures, called star-shaped risk measures, that possess positive superhomogeneity. These measures, which arise when the subadditivity property of coherent risk measures is disregarded and positive homogeneity is weakened, encompass all commonly used risk measures, including convex risk measures and value-at-risk. From a financial perspective, relaxing convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision-theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.