1. If you introduce a franchise or raise its amount, the number of claims will decrease. One reason why you will operate a franchise in car damage insurance is to avoid the troublesome and expensive handling of the large number of very small claims.

Therefore, the management could use a mathematical model describing the number of claims as related to the size of the franchise. By varying this figure one may find the franchise that provides for the optimal business conditions.

2. The mathematical model to be described in this paper was found by studying the distribution of the claims in two portfolios. The general conditions of the two groups were as equal as possible, there was no-claim bonus in neither group, but group B had a franchise of 250 DKr. (about US $ 35) while group A covered the smaller claims also. Each policy had been unaltered for at least one year. The number of claims during 12 successive months was registered as follows:

The distribution could be described by a negative binomial distribution

where the parameters α and γ were estimated by

These estimates are not maximum likelihood estimates but it was not regarded necessary to improve the estimation as the χ2 values of the incomplete χ2-test were found as

The parameters were estimated as follows:

3. Special interest is attached to the variations of the γ-parameter. It is well known, that if γ → ∞ and α → ∞ under the condition then the negative binomial distribution will converge towards a Poisson distribution with the parameter λ.