Most large insurance companies have today electronic computers that enable not only efficient actuarial statistics, but also research in applied risk theory. An important task in this latter field is the developing of an information system for control of the business as to the statistical balance between premiums and claims. The entire system can be separated into two parts, one descriptive and one analytic part. The descriptive part, that also may be called statistics production, is the base of the whole system and must be constructed in a general way to make it possible to apply mathematical tools in risk analysis. For the analytic part and its applications for computers there is a growing interest among actuaries as can be noticed from the reports in actuarial journals. The classical models of collective risk theory have recently been extensively illustrated by numerical calculations performed by the convolution committee in Sweden.

When starting to construct the analytical part of the information system one finds that in spite of programming for the computer there is firstly a hard work to find realistic mathematical models, especially mathematical expressions for claim distributions, and secondly to estimate their parameters. According to the mathematical theory of risk, or the risk process, it is necessary to assume and test specific mathematical forms of two distributions, the number of claims and claims amount.

We suppose that a risk business issues a single type of contract under which, in return for a unit “premium”, it will pay a “sum insured” m (m being an integer) on the occurrence of a contingency of probability q ˂ ½. The expected gain on each contract is i—mq and is assumed to be positive. This type of enterprise is conveniently denominated a simple risk business. Two “real life” situations it simulates are those of a group life policy for a uniform amount covering a number of young lives (e.g., university students), and a roulette casino where the stakes are uniform and the bets are limited to the single numbers o to 36.

The risk business is supposed to commence its operations with a “risk reserve” of K units. Each premium is added to this reserve as it is received and all claims by contract holders are paid there-from. We say that the business is “ruined” as soon as the risk reserve becomes zero or negative (though it could be argued that it would be unethical to accept a premium once the risk reserve is less than m — 1). On the other hand, if the reserve reaches an amount M units no further premiums are paid into it until a claim occurs to reduce it below M (de Finetti, 1957). The risk business intends to continue its operations for a long, but finite, period unless ruined in the meantime.

We consider two probabilities: (i) υx, the chance of eventual ruin given that the risk reserve is now x, and (ii) υx, n, the probability that ruin occurs as a result of the nth contract (simultaneous contracts being ranked in a prearranged, e.g., alphabetical, order). Clearly

and we will write I — νx=μx

In this paper an attempt is made to find an answer to the question, “What is the most advantageous size for the retention limit of a risk portfolio, given the fact that a certain stability requirement is to be satisfied?”

This problem will be approached from the viewpoint of an insurer who wishes to obtain a certain degree of stability at lowest cost.

It is assumed that in his choice of reinsurance methods, the insurer restricts himself to either a surplus treaty, a stop loss treaty or a combination of both these types.

Moreover it is assumed that “stability” can be adequately measured by the variance of the risks retained for own account.

We start to consider a reinsurance policy based on the surplus system where the amount of risk in excess of a retention limit u is ceded.

By thus limiting the potential loss on each risk individually, the variance is kept at a certain level, but at the expense of an amount of premium payable to a reinsurer.

The insurer could, of course, reduce the reinsurance cost by increasing his retention but he then is bound to incur a higher variance in his portfolio, which would mean a loss of stability.

One might ask, however, whether a suitably chosen stop loss coverage could bring the variance down again to the proper level at lesser cost than the profit obtained by taking a higher retention. A reduction in reinsurance cost would then have been effected.

The question leads to an optimization problem, which in a more general setting, has been discussed by K. Borch.

It is often found even today in Europe that for certain statistical investigations the conclusion is drawn that the extent of the available statistical data is not sufficient. Going to the root of this pretention, however, we notice that there is a want of clear conception about the extent that is in fact necessary in order that a valid conclusion may with greater probability be arrived at. This, for instance, is the case when obvious tariff reductions are shirked from by entrenching oneself behind the law of large numbers, which by its very nature can in actual practice be never accomplished in its inherent sense.

Apart from a proper understanding of the limits within which a set of statistical data may subject to certain assumptions be ascribed full measure of credence, there is further a lack of the necessary tools that would permit, on the basis of ascertainable values alone, far-reaching conclusions to be drawn or a maximum of useful information to be gathered from an investigation of which the scope is evidently not sufficient.

Credibility Theory, of which Prof. Mowbray [4] may be regarded as the initiator, was evolved in the U.S.A. about 50 years ago to fill this lacunae. The development of Credibility Theory may be considered as one of the most significant contributions of American actuarial science, and it is frankly astonishing that apart from certain specific realisations in the collective risk theory which are fairly closely related to Credibility Theory, it is only in recent years that this interesting topic has met with the required attention in Europe.

Hans Bühlmann traite dans son ouvrage “Optimale Prämien-stufensysteme” (1964) principalement le problème de trouver l'estimation optimale de la prime de la n-ième période à partir des observations de sinistres au cours des périodes précédentes.

Max Gürtler a traité dans le “Bonus ou Malus” du Astin Bulletin vol. III part I deux systèmes différents pour la détermination de la prime individuelle pour l'assurance automobile, c'est-à-dire

a) système “Absence de sinistre”

b) système “Nombre de sinistre”.

A l'aide de simples modèles (examples page 56) pour le groupement de portefeuilles de risques bons et lourds Gürtler étudie l'effet des deux systèmes et en tire des conclusions générales sur l'applicabilité des systèmes. Gürtler souligne d'abord que le système b) utilise considérablement mieux l'information, parce que tous les sinistres y sont considéréd et non pas, comme dans le système a), seulement le premier sinistre étant registré pour chaque assurance.

Marcel Derron traite dans “A theoretical study of the no-claim bonus problem” du même cahier de la publication la quote-part du ER (Error ratio) qu'il définit comme une quote-part égale au montant absolut de la différence entre la prime payée et la vraie prime divisée par la prime payée. Il calcule la quote-part du ER pour le modële de Gürtler et pour des modèles alternatifs et cherche le système de bonification qui rend la quote-part du ER aussi petite que possible pour le modèle choisi.

1.1. In this paper we shall consider a given portfolio of insurance contracts, and we shall study the following two problems:

(i) How should this portfolio be reinsured?

(ii) What reserves should the company maintain to pay claims which will be made under the contracts in the portfolio?

This means that we shall ignore all questions as to how the company acquired the portfolio, i.e. some of the most important questions concerning management control of insurance companies, such as rating policy, underwriting control etc.

1.2. Even the two simple problems, which are singled out for study in this paper, cannot be solved unless we specify the objectives and the external circumstances of the company.

It is obvious that the reinsurance problem cannot be solved unless we know something about the company's “attitude to risk”, and about the cost of obtaining cover for various kinds of risk in the reinsurance market.

It is also obvious that we cannot solve the reserve problem unless we specify the safety requirements, which the company has to satisfy. We shall see in the following that we may also have to specify the portfolios, which the company expects to underwrite in the future.

This paper was written in connection with the preparation of Marché Commun regulations in the insurance sector and has been submitted to the Commission Technique pour l'étude d'un indice de solvabilité relatif aux entreprises d'assurances contre les dommages. It aims at explaining the scope of the problem to non-mathematicians and for that reason emphasizes its logical in contradistinction to its computational aspects.

The probability of the insurer's ruin has two aspects. The occurrence of the event ruin may be considered with respect to a fixed period but also with respect to a period of undetermined length. In both cases the period starts at a moment at which the insurer's capital (patrimoine) is known and it is intuitively clear that in both cases the probability of ruin will be the higher as the insurer's capital is smaller and the risk to which he is exposed heavier.

For some purposes more precise conclusions are required. This requirement gives rise to problems which will be considered here with respect to the probability of the occurrence of ruin in a period of undetermined length. It will be taken for granted that besides the artificial events occurring in games of chance there are other classes of uncertain events to which numerical probabilities can be assigned and that, as far as claims are concerned, the insurer's losses belong to one of the said classes. On this understanding it makes sense to consider a numerical probability of ruin depending on a fixed initial capital, random losses to which numerical probabilities are assigned, and other profits and losses.