The homogeneous (in time) model of credibility theory is defined by a sequence Θ, X1, X2, … of random variables, where for Θ = θ fixed, the variables X1, X2, … are independent and equidistributed. The structure variable Θ may be interpreted as the parameter of a contract chosen at random in a fixed portfolio, the variable Xk as the total cost (or number) of the claims of the kth year of that contract.

Bühlmann's linear credibility premium of the year t + 1 may be written in the form

where f is a linear function. In optimal semilinear credibility, we look for an optimal f, not necessarily linear, such that (1) is closest to Xt+1 in the least squares sense. In the first section we prove that this optimal f, denoted by f*, is solution of an integral equation of Fredholm type, which reduces to a system of linear equations in the case of a finite portfolio. That is a portfolio in which Θ and Xk can assume only a finite number of values.

In the second section we see that the structure of such a portfolio is closely connected with the decomposition of a quadratic form in a sum of squares of linear forms.

In the last section we calculate numerically the optimal premium for a concrete portfolio in automobile insurance. We limit ourselves to the consideration of the number of claims. The optimal premium is compared with the usual linear premium. The difference is far from negligible.

The effect of inflation of premium income and claims size distribution, but not of free reserves, on the probability of ruin of an insurer is studied.

An interesting similarity between this problem and the ruin problem in an experience-rated scheme is exhibited. This similarity allows the deduction of parallel results for the two problems in later sections.

It is shown that the probability of ruin is always increased when the (constant) inflation rate is increased.

The distribution of aggregate claims under inflationary conditions is described and used to calculate an upper bound on the ruin probability. Some numerical examples show that this bound is not always sharp enough to be practically useful. It is also shown, however, that this bound can be used to construct an approximation of the effect of inflation on ruin probability.

It is shown that if inflation occurs at a constant rate, then ruin is certain, irrespective of the smallness of that rate and of the largeness of initial free reserves and the safety margin in the premium. The corresponding result for experiencerated schemes is that a practical and “intuitively reasonable” experience-rating scheme leads eventually to certain ruin.

Finally, a simple modification of the techniques of the paper is made in order to bring investment income into account.

In many estimation problems, incomplete as well as complete samples are available for Bayesian prediction. After developing the theory for a special, but useful family of distributions, examples are given in life testing, renewal risk processes, life contingencies, and the problem of estimating a defective distribution.

De nombreux modèles, en actuariat, se basent sur l'existence d'une fonction d'utilité additive. Le but de cet article est de montrer que cette hypothèse enlève au problème traité son caractère dynamique et que la suppression de l'additivité conduit à une solution plus réaliste.

Stochastic-dynamic programming provides a technique for forecasting limits within which the insurance business will flow by a prefixed probability. The future development depends, among numerous other things, on management strategies, especially resources, which are planned for allocation in the acquisition of new business and for competition. This technique can be used to analyse different market situations. Various competitive measures and eventual counteractions by competitors can be assumed and simulated for the purpose. In this way the consequences of different strategies can be studied in order to find the most appropriate one. Our approach is similar to the well-known business games where teams play business in a simulated market. The idea of applying dynamic programming to business games was suggested by Esa Hovinen (discussion at the Astin Colloquium in Washington in 1977).

We define axiomatically a concept of value for games without transferable utilities, without introducing the usual symmetry axiom. The model—a generalization of a previous paper [6] extending Nash's bargaining problem—attempts to take into account the affinities between the players, defined by an a priori set of “distances”. The general solution of all three- and four-person games is described, and various examples are discussed, like the classical “Me and my Aunt” and a reinsurance model.

Nous définissons de manière axiomatique un concept de valeur pour les jeux à utilités non-transférables, sans introduire l'axiome classique de symétrie. Le modèle — une généralisation d'un concept de valeur [6] étendant à plusieurs joueurs le problème de marchandage de Nash — tient compte des affinités entre les joueurs, données sous forme d'une matrice de “distances” a priori. Nous donnons la solution générale de tous les jeux à trois et quatre joueurs, et discutons plusieurs exemples classiques, dont le célèbre “Ma tante et moi” et le modèle de réassurance de Borch.

For the premium calculation the insurer will split up his collectivity of risks into risk groups which are homogeneous with respect to some directly observable risk factors. All risks of such a risk group will be charged the same base premium. But it is clear that by such an a priori classification not all determined factors can be taken into consideration, so that there will still remain accident proneness differentials within a risk group. Since these differentials will be reflected in the course of time by the claim experience of each risk, the insurer can come to a fair tarification by adjusting, each period, the base premium according to the individual claim experience of the risk. Such a system in which earlier neglected risk factors are taken into account a posteriori is an individual experience rating system. Our main interest goes to the following side-effect of experience rating: since an unfavourable claim experience results in a premium increase, an experience rated policyholder is stimulated to self-insure small damages. This phenomenon is well know in connection with bonus-malus systems in motor-car insurance, which explains why it is called “bonus-hunger”.

In the present paper a continuous time model for the bonus-malus system is set up which takes into account this hunger for bonus. An insured causing an accident will decide according to a certain decision rule whether to file a claim with his insurance company. The relevant information that he needs to make this decision is: his current risk class, the number of claims he has already filed during that period and the moment at which the decision is to be made.

The premium calculation principle is one of the main objectives of study for actuaries. There seems to be full agreement among the leading theoreticians in the field that the insurance premium should reflect both the expected claims and certain loadings. This is true for policy, risk or portfolio. There are three types of positive loadings: a) a loading to cover commissions, administrative costs and claim-settlement expenses; b) a loading to cover some profit (a cost-plus approach); and c) a loading for the risk taken by the insurer when underwriting the policy. The administrative costs can be considered a part of “expected gross claims”. Thus, the insurer's ratemaking decision depends on his ability to estimate expected claims (including costs) and on the selection of a fair risk loading.

The main concern in the literature is the appropriate measurement of the risk and the exact loading formula. Bühlmann [1970, ch. 5] and others identified four possible principles of risk loading, namely, the expected value principle, the standard deviation loading, the variance loading, and the loading according to the principle of constant utility. Various studies point to the advantages and disadvantages of these principles and also examine some additional loading forms—semi-variance, skewness, etc. (e.g., Bühlmann [1970], Benktander [1971], Berger [1972], Burness [1972], Berliner [1974], Berliner and Benktander [1976], Bohman [1976], Cooper [1974], Gerber [1975] and others). Despite different preferences in choosing the appropriate loading calculation principle, all seem to agree that the risk loading must be positive, since, otherwise, the firm would just have to wait for its ruin, that is bound to come sooner or later, according to risk theory.

The purpose of this article is to re-examine the appropriate principle of premium calculation in light of the recent developments in the theory of finance and especially in the theory of capital market equilibrium. These developments may suggest a new point of view and raise a few questions regarding the loading rules.