Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

In this paper we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures are asymptotically equivalent if the ratio of the worst-case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a non-coherent risk measure and a coherent risk measure, since the worst-case value of a non-coherent risk measure under dependence uncertainty is typically very difficult to obtain. The main contribution of this paper is that we establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice