  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
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.
