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

Consider non-negative random variables X1,...,Xn 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 [pi](X1+...+Xn,d)=E{max (0,X1+...+Xn-d)} that are valid for all retention amounts d[greater-or-equal, slanted]0 in the absence of information concerning the type or degree of dependence between the risks Xi. One approach consists of maximizing the premium over all possible values [rho]ij=corr(Xi,Xj), 1[less-than-or-equals, slant]i<j[less-than-or-equals, slant]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.