Insurance contracts portfolios with heterogenous insured ages

In this paper we consider two portfolios: one of m endowment insurance contracts and one of m whole life insurance contracts. We introduce the majorization order and Schur functions. We assume that the owners of the portfolios are of different ages at issue time and are exposed to a common life-distribution. We study the effect of the aging heterogeneity on the premiums and on the death benefits of the insurance contracts. We show that the premiums paid in both contracts and the death benefit awarded in the whole life contract are Schur functions. We provide upper and lower bounds for the premiums and for the death benefit, and compute the bounds for some distribution functions used frequently in the actuarial sciences.