Balancing an insurance portfolio by class of business

Abstract

The paper considers the benefits to be gained from diversification of an insurance operation into a number of classes of business. This is formulated mathematically. It is assumed that the insurer's objective is to maximize increase in utility of wealth over the period up to his planning horizon (Section 2). Several forms of solution of this problem are considered (Section 3). In particular, it is found that there is no benefit in diversification if the parameters characterizing the stochastic properties of the claims of each class of business are known with certainty. In these circumstances, utility is maximized by underwriting all premium in that class of business projected to produce the greatest profitability. It is also seen that, unless the concept of the insurer's risk aversion is incorporated somehow, for example by means of a utility function, no benefits from diversification appear (Section 5.2). The benefits of diversification arise from uncertainty concerning the basic parameters of the claims process in the various classes of business. A parallel with the concepts of diversifiable and undiversifiable risk from Modern Portfolio Theory is pointed out (Section 4). A measure of the balance achieved within a portfolio is constructed. A somewhat simplified version of the problem is considered in Section 5, and an explicit algebraic solution obtained. Of the total budgeted premium to be underwritten in a particular year, the solution prescribes the percentages to be underwritten in the various classes of business. The solution depends upon the projected profitability and the parameters underlying the claims process in each of the classes. Some verbal interpretation of the algebraic solution is provided. That solution is applied to a realistic numerical example (Section 6). The apportionment of total budget premium by class of business arrived at in the numerical example is compared with the actual budget of a particular insurer. A number of qualifying remarks concerning the application of the results of the paper are made in Section 7.