The total costs of damage caused by passenger car traffic accidents per distance interval of an individual motorist can be split into two components: the number of accidents which occur to the motorist, and the amount of damage associated with these accidents. In our study we present a probability model for the number of accidents of an individual motorist. We also discuss models, on empirical results, for the amount of damage. The integration of both models results in a general model for the costs of damage. From this general model we derive a model for the costs of indemnity to be paid by car insurance companies. The models are applied to a sample of Dutch car insurance data.