Some Results on Optimum Premium Payment Plans
Any insurance plan consists of a sequence of payments every year (or some other fixed time interval) in return for certain death benefits. The benefits may take the form of a wide variety of insurances or annuities. For simplicity, we will assume that premiums and benefits are paid annually. In this paper, we investigate the appropriateness of this type of plan. Naturally, appropriateness of any plan cannot be measured without an optimality criteria. Three such criteria, which are statistical in nature, are introduced in this paper. For the principal "safety" criterion which we use, the optimal premium are those which minimize a certain "profit variance" subject to a familiar profit constraint. We also develop a "profitability" criterion and then solve an associated optimality problem. Our main results state that if the sequence of present values of total benefits is nonincreasing, then the profit variance is minimum when the insured pays a net single premium at once, and if this cannot be done, the insured should pay off the policy as early as possible.