Stochastic bounds on sums of dependent risks

There is a growing concern in the actuarial literature for the effect of de pendence between individual risks Xi on the distribution of the aggregate c laim S = X-1 + . . . + X-n. 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 Muller ( Muller, A., 1997a. Insurance: Mathematics and Economics 21, 219-223; Muller , A., 1997b. Advances in Applied Probability 29, 414-428) has led, among ot her things, to the identification of the portfolio yielding the smallest an d largest stop-loss premiums and hence to bounds on E{phi(S)} for arbitrary non-decreasing, convex functions phi in situations of dependence between t he X-i's. This paper extends these results by showing how to compute bounds on P(S > s) and more generally on E{phi(S)} for monotone, but not necessar ily convex functions phi. Special attention is paid to the numerical implem entation of the results and examples of application are provided. (C) 1999 Elsevier Science B.V. All rights reserved. JEL classification: IM11; IM12; IM30.