  In this paper monetary risk measures that are positively superhomogeneous,
called star-shaped risk measures, are characterized and their properties
studied. The measures in this class, which arise when the controversial
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.
