For the premium calculation the insurer will split up his collectivity of risks into risk groups which are homogeneous with respect to some directly observable risk factors. All risks of such a risk group will be charged the same base premium. But it is clear that by such an a priori classification not all determined factors can be taken into consideration, so that there will still remain accident proneness differentials within a risk group. Since these differentials will be reflected in the course of time by the claim experience of each risk, the insurer can come to a fair tarification by adjusting, each period, the base premium according to the individual claim experience of the risk. Such a system in which earlier neglected risk factors are taken into account a posteriori is an individual experience rating system. Our main interest goes to the following side-effect of experience rating: since an unfavourable claim experience results in a premium increase, an experience rated policyholder is stimulated to self-insure small damages. This phenomenon is well know in connection with bonus-malus systems in motor-car insurance, which explains why it is called “bonus-hunger”.

In the present paper a continuous time model for the bonus-malus system is set up which takes into account this hunger for bonus. An insured causing an accident will decide according to a certain decision rule whether to file a claim with his insurance company. The relevant information that he needs to make this decision is: his current risk class, the number of claims he has already filed during that period and the moment at which the decision is to be made.