Improved Hoeffding–Fréchet bounds and applications to VaR estimates

Abstract

The classical Fréchet bounds determine upper and lower bounds for the distribution function F of a random vector X, when the marginal df’s F _i are fixed. As consequence these bounds imply also upper and lower bounds for the expectation E _φ(X) of a certain class of functions φ(X). The classical examples are the Hoeffding bounds for the expectation of the product EX ₁ X ₂ of two random variables. In this paper we review and partially elaborate on several developments of improved Hoeffding–Fréchet bounds which assume some restriction on the dependence structure additional to the information on the marginals. We describe applications of the results to obtain improved VaR bounds for the joint portfolio of risk vectors. We consider in particular improved VaR bounds in the case where information of the joint distribution function resp. on the copula is available on some subsets and the case where higher order marginal information is available.