A class of claim frequency distributions discussed by Sundt and Jewell (1981) is completely enumerated. Computational techniques for the associated compound total claims distribution in the presence of policy modifications are then derived.

It is shown how the stationary distributions of a bonus–malus system can be computed recursively. It is further shown that there is an intrinsic relationship between such a stationary distribution and the probability of ruin in the risk-theoretical model. The recursive algorithm is applied to the Swiss bonus–malus system for automobile third-party liability and can be used to evaluate ruin probabilities.

The Pareto-optimal design for profit-sharing is derived under general assumptions as to the utility function of both the insured and the insurer. This generalizes the result of Jones and Gerber and explains commonly used dividend formulas in terms of risk aversion.

A model for the claim number process is considered. The claim number process is assumed to be a weighted Poisson process with a three-parameter gamma distribution as the structure function. Fitting of this model to several data encountered in the literature is considered, and the model is compared with the two-parameter gamma model giving the negative binomial distribution. Some credibility theory formulae are also presented.

For a general class of reinsurance treaties the author gives an upper bound for the net premium. This result can be seen as the counterpart to a premium bound for the classical stop-loss reinsurance cover (see Bowers, 1969). For some special cases some preliminary work can be found in Kremer (1983).

We consider a general credibility model for the prediction of IBNR-claims which allows for random fluctuations in the underlying delay distribution. Such fluctuations always bring about decreasing credibility. It is shown that even negative credibility is achieved for more substantial fluctuations in the delay distribution. Special attention is paid to the mixed Poisson case for claim numbers including the discussion of parameter estimation.

It is shown how the upper bounds for stop-loss premiums (and approximations to tail probabilities) obtained by replacing the individual model for a portfolio of risks by the collective model can be improved upon at the cost of only slightly more computer time. The method used is simply to keep a restricted number of large risks as they are instead of approximating them by a compound Poisson distribution. In a real-life example, the relative error in the stop-loss premium is shown to be reduced drastically by keeping only 10 out of 743 risks unchanged.

The paper describes how a study group, appointed by the Belgian Professional Union of Insurance Companies, designed a new tariff structure in motor third party liability. Particular emphasis was given to the construction of a more efficient bonus–malus system.