Stochastic bounds on sums of dependent risks

There is a growing concern in the actuarial literature for the effect of dependence between individual risks Xi on the distribution of the aggregate claim S=X1+⋯+Xn. Recent work by Dhaene and Goovaerts (Dhaene, J., Goovaerts, M.J., 1996. ASTIN Bulletin 26, 201–212; Dhaene, J., Goovaerts, M.J., 1997. Insurance: Mathematics and Economics 19, 243–253) and Müller (Müller, A., 1997a. Insurance: Mathematics and Economics 21, 219–223; Müller, A., 1997b. Advances in Applied Probability 29, 414–428) has led, among other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on E{φ(S)} for arbitrary non-decreasing, convex functions φ in situations of dependence between the Xi’s. This paper extends these results by showing how to compute bounds on P(S>s) and more generally on E{φ(S)} for monotone, but not necessarily convex functions φ. Special attention is paid to the numerical implementation of the results and examples of application are provided.