This paper deals with the Bonus-Malus system obtained when the claims frequency is submitted to trend. This system is specified in the two particular cases of Poisson-Gamma and Poisson-Inverse Gaussian distributions. The theoretical results are checked on data issued from automobile insurance policies observed during three years.

Excess claims lead to an unsatisfactory behavior of standard linear credibility estimators. We suggest in this paper to use robust methods in order to obtain better estimators. Our first proposal is the linear credibility estimator with the claims replaced by a robust M-estimator of scale calculed from the claims. This corresponds to a truncation of the claims with a truncation point depending on the data and different for each contract. We discuss the properties of the robust M-estimator and present several examples. In order to improve the performance for a very small number of years, we propose a second estimator, which incorporates information from other claims into the M-estimator.

Goovaerts and de Vylder (1983) provided a stable recursive algorithm for calculating the probability of ultimate ruin. Their algorithm yielded bounds for this probability. It is shown that in practice their method may be inherently unstable because it is based on the subtraction of nearly equal numbers. An alternative to this type of subtraction is provided. It is proved that their algorithm converges only at a linear rate to the true value. It is suggested that this slow rate of convergence be improved via an application of the Richardson extrapolation technique.

In this paper we discuss some properties of counting distributions whose discrete density satisfies a recursion in the form

with pn = 0 for n < 0 and present an algorithm for recursive evaluation of corresponding compound distributions.

The paper examines a type of insurance contract for which secondary markets do exist: default risk insurance is implicit in corporate bonds and other risky debts. It applies risk neutral martingale measure pricing to evaluate the option for a borrower with default risk, to prepay a fixed rate loan. A simple “matchbox” example is presented with a spreadsheet treatment.

In this paper, we will cover the bonus-malus system in automobile insurance. Bonus-malus systems are based on the distribution of the number of car accidents. Therefore, the modelling and fitting of that distribution are considered. Fitting of data is done using the Poisson inverse Gaussian distribution, which shows a good fit. Building the bonus system is done by minimizing the insurer's risk, according to Lemaire's (1985) bonus system.

The paper proposes a three parameter measure by which the risk of a portfolio can be assessed. The parameters are: the probability of ruin, the severity of ruin (i.e. the amount of the deficit when ruin occurs) and the time of ruin. This type of analysis does not lend itself to closed form solutions, but it can be easily carried out on a PC. The author presents some theoretical and practical examples.