Stochastic Bounds for Conditional Distributions Under Positive Dependence

Abstract

We provide stochastic bounds for conditional distributions of individual risks in a portfolio, given that the aggregate risk exceeds its value at risk. Expectations of these conditional distributions can be interpreted as marginal risk contributions to the aggregate risk as measured by the tail conditional expectation. We first provide general lower and upper stochastic bounds and then we obtain further improvements of the bounds in the case of a portfolio consisting of dependent risks. We also derive new characterizations of comonotonic random vectors.