Individual automobile insurance claims are characterized by over-dispersion relative to the Poisson model. In addition, claim propensities vary among individuals in any insurance portfolio. This paper presents a model which takes account of both characteristics. The model employs the negative-binomial distribution as the distribution for individual-level claims and a Pareto distribution as the distribution for claim propensities within the portfolio. The paper shows that the resulting model is tractable and has a number of attractive properties which make it suitable for this application. The fit of the model to actual claim numbers for automobile third party liability insurance is examined and found acceptable. Bayes theorem is then applied to this model to calculate illustrative optimal premiums under the Bonus-Malus System (BMS).

For estimating the shape parameter of Paretian excess claims, certain Bayesian estimators, which are closely related to the Hill estimator, have been suggested in the insurance literature. It turns out that these estimators may have a poor performance – just as the Hill estimator – if a certain location parameter is unequal to zero in the Paretian modeling. In an alternative formulation this means that a scale parameter is unequal to 1. Thus, it suggests itself to add the scale parameter in the modeling and to deal with Bayesian estimators of the shape and scale parameters in a full Paretian model. These estimators will be applied to fire and motor reinsurance data. The performance of these estimators will be illustrated by means of Monte Carlo simulations.

The aim of this work is to present a method to compute the expected amount of annual claims in the health insurance. (A more detailed analysis of the method can be found in Kovářová (1998).) We will be especially concerned with the permanent health insurance and compare our numerical results with those computed and published for the permanent health insurance in C.M.I.R. (1991). At the same time we will give a general methodology applicable not only to the permanent health insurance but for other types of health insurance as well.

The methods used today can be divided into three groups: the method of the decrement tables, the Manchester Unity method and the multiple state model. We would like to describe a simple method to compute from the data of the multiple state model the bases for the Manchester United and for the decrement tables methods.

In Mack (1993), a formula for the standard error or chain ladder reserve estimates has been derived. In the present communication, a very intuitive and easily programmable recursive way of calculating the formula is given. Moreover, this recursive way shows how a tail factor can be implemented in the calculation of the standard error.