Upper stop-loss bounds for sums of possibly dependent risks with given means and variances

Abstract

Consider non-negative random variables X 1,…, X n whose marginal means and variances are known. The purpose of this paper is to compare two different strategies for finding an upper bound on the stop-loss premium π(X 1+⋯+X n,d)=E{ max (0,X 1+⋯+X n−d)} that are valid for all retention amounts d⩾0 in the absence of information concerning the type or degree of dependence between the risks X i . One approach consists of maximizing the premium over all possible values ρ ij= corr(X i,X j), 1⩽i<j⩽n . As it turns out, however, a better solution exploits results of Dhaene et al. (Schweiz. Aktuarver. Mitt. (2000) 99) on the maximality of comonotonic risks in the stop-loss order. Explicit calculations and numerical illustrations of the proposed bounds are given.