Loss Models: From Data to Decisions, 4th Edition

18.4 The credibility premium

In Section 18.3, a systematic approach is suggested for treatment of the past data of a particular policyholder. Ideally, rather than the pure premium μ_{n + 1} = E(X_n+1), one would like to charge the individual premium (or hypothetical mean) μ_n+1 (θ), where θ is the (hypothetical) parameter associated with the policyholder. Because θ is unknown, the hypothetical mean is impossible to determine, but we could instead condition on x, the past data from the policyholder. This leads to the Bayesian premium E(X_n+1|x).

The major challenge with this approach is that it may be difficult to evaluate the Bayesian premium. Of course, in simple examples such as those in Section 18.3, the Bayesian premium is not difficult to evaluate numerically. But these examples can hardly be expected to capture the essential features of a realistic insurance scenario. More realistic models may well introduce analytic difficulties with respect to evaluation of E(X_n+1|x). whether one uses (18.12) or (18.13). Often, numerical integration may be required. There are exceptions, such as Examples 18.11 and 18.12.

We now present an alternative suggested by Bühlmann [17] in 1967. Recall the basic problem: We wish to use the conditional distribution or the hypothetical mean μ_n+1 (θ) for estimation of next year’s claims. Because we have observed x, one suggestion is to approximate μ_n+1(θ) ...