A Theory of Insurance Premiums

Abstract

1.1. The buyer of an insurance contract buys security, and the seller accepts a risk. The premium charged by the seller must give him adequate compensation for the risk bearing service he provides, and of course be acceptable to the buyer. It is useful to see an insurance contract as a contingent claim. The buyer pays a premium in advance, and will get a random amount in return - as settlement of the claims he can make under the contract. Formally the transaction is of the same type as the purchase of a share in a risky business. The price of such shares is presumably determined by supply and demand in the market, and it is natural to assume that insurance premiums are determined in the same way. Economic theory has taken a long time to develop satisfactory models for the pricing of contingent claims. The breakthrough came just over thirty years ago, with the work of Arrow [1953]. In the following three decades a number of models have been developed, i.a. the so-called Capital Asset Price Model (CAPM) due to Sharpe [1964], Lintner [1965] and Mossin [1966], which has found extensive applications in practice. 1.2. Actuaries have of course been busy computing insurance premiums for more than a century. The more recent achievement in economics and finance do not seem to have had much influence on the work of actuaries. A recent survey volume of the theory of insurance premiums by Goovaerts et al. [1983] contains no references to highly relevant results in economic theory. One of the few exceptions is Buhlmann [1980] and [1984], which contain no references to literature. One explanation of why actuaries have ignored economics may be that they consider CAPM - a one-period model depending only on expectation and variance - as too primitive for their purposes. Instead of adapting CAPM and similar models, they have continued the development of the actuarial theory of risk, which places the focus on a class of stochastic process in continuous time. There is much pretty mathematics in this theory, but one inevitably feels that most contacts with economic reality have been lost. It is tempting to borrow a