Bounds and Approximations for Sums of Dependent Log-Elliptical Random Variables

Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a,b) have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed form approximations. We also present numerical examples to show the accuracy of the proposed approximations.