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 = 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 other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on E{phi(S)} for arbitrary non-decreasing, convex functions phi in situations of dependence between the 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 necessarily convex functions phi. Special attention is paid to the numerical implementation of the results and examples of application are provided. (C) 1999 Elsevier Science B.V. All rights reserved. JEL classification: IM11; IM12; IM30.