1.1. — In this paper we shall consider some of the decisions which have to be made in the normal course of business in an insurance company. We shall see that the “right” decisions can be found only when the problems are analysed in their proper dynamic context.

As examples of the decision problems which we shall study, we can mention the following:

(i) What premium rates should be quoted on the insurance contracts, which the company offers to the public?

(ii) How much should the company spend to promote the sale of its policies?

(iii) When should the company refuse to underwrite a proposed insurance contract?

(iv) How shall the company reinsure its portfolio of insurance contracts?

(v) What reserve funds should an insurance company keep?

(vi) How shall the company's funds be invested?

Any actuary will be familiar with such problems, and he will probably feel that these problems cannot be satisfactorily solved with the methods offered by the classical actuarial theory.

1.2. — In some earlier papers [I] and [2] it has been argued that such problems can best be solved in the frame work of utility theory. As an illustration we shall take Problem (iii) in the preceding paragraph, and consider an insurance company in the following situation:

(i) The company has a capital S.

(ii) The company holds a portfolio of insurance contracts which will lead to a total payment of x to settle claims. F1(x) is the distribution of the variate x.