I hope that you will allow me first of all to congratulate Dr. Buhlmann on the excellent lecture he has given us and I am sure that you can all agree with me that after his general survey on experience rating and credibility, there is no need at all for me to go into the general aspects of the subject. Therefore I can start immediately to report on the contributions delivered, and I am very happy to state that, though the number of papers is rather small, namely six, their contents are qualitatively as high as has become usual at our ASTIN-Colloquia.

As regards the number of contributions I just mentioned, I like to draw your attention to the fact that because the present subject was on the programme of our last colloquium in Luzern too, in some cases we had to decide whether a contribution should be considered as relevant to subject I, or as a report on further work done on subjects from previous meetings. If perhaps in some cases our choice is not in accordance with the intention of the contributor I would like to apologize for that in advance. Moreover I wish to remind you that on Friday after the summaries of the discussions, Mr. Jansen will report on those papers submitted, which were not regarded as contributions to subject 1 or 2.

I would like to deal first with the papers of a more theoretical nature, afterwards going on to the contributions on special applications. Among the papers of general interest I ask your attention for the paper of Franckx entitled “Le comportement de l'assureur”. I consider this contribution as a very important attempt to compare the method of credibility theory, where as you know credibility is used as a function of the volume, with a method in which decisions are made in advance on the amount of certain levels of significance, in this way fixing what Franckx calls “le comportement a priori”.

La théorie américaine de la crédibilité est considérée par nos collégues américains comme une pièce-maîtresse de la théorie actuarielle „non life” (Longley Cook: An Introduction to Credibility Theory). Elle a re¸u la consécration de l'expérience sur le marché le plus large du monde. Dirigeants d'entreprises et autorités de contrôle se sont accordéd sur son emploi. Ce sont la des résultats positifs qui ne peuvent que faire réfléchir tous les actuaires. ASTIN a mis le probléme á l'ordre du jour du colloque de Lucerne en 1965. Cette année, la réunion d'Arnhem reprend le même sujet, á juste titre.

Mais il faut le constater, si les méthodes d'adaptation américaines donnent des résultats pratiques indiscutables, elles ont été établies par des méthodes pragmatiques, d'abord basées sur certains cas particuliers et admises, ex cathedra, en toute généralité. Cette situation laisse un sentiment de malaise et souléve de nombreuses critiques. (Derron: Kredibilität und Versicherungwesen) (Mayerson: A Bayesan view of Credibility). La théorie de la crédibilité impose avant tout la recherche d'une base théorique plus valable.

La question est délicate. Elle touche directement au probléme de l'inférence statistique qui constitue entre probabilistes et statisticiens un domaine trés controversé. La présente note apporte une contribution trés subjective á la théorie américaine. Elle tend á élargir le débat et conduit á la conclusion que l'établissement d'une théorie du comportement de l'assureur serait souhaitable. Dans une telle théorie, les mathématiques ont leur importance, mais elles présupposent l'admission á priori de certaines régies d'inférence.

1. The claims experience of a motorcar insurance is assumed to give some indication about the risk (basic claim frequency) of that insurance. The experience rating systems in motor insurance are based on this supposition. In these systems the premium to be paid in some year is a function of the individual claims experience of preceding years. That premium can be considered as the difference between:

The bonus usually consists of at least the following components:

a. a component concerning the individual claim frequency.

b. an individual random factor.

c. a collective random factor.

Other factors, like the effect of the trend in the claim frequencies and the effect of the dependence of the claim frequency of age and experience of the driver are not taken into account in this paper.

2. The meaning of the three components mentioned in section I can be demonstrated by the following example.

The structure function U(χ) of the portfolio, being the distribution function of the individual basic claim frequencies χ is assumed to be defined by representing a gamma distribution with expected value q and variance qb. The probability Pi for an insurance to have i claims in a certain year is supposed to be

In this formula the factor y has the same value for all insurances in the portfolio in a certain year but varies from year to year according to the distribution function G(y) with expected value 1 and variance .

This paper describes a rather simple application of credibility which was used to develop experience rating plans for Blue Cross and Blue Shield organizations in the United States. These are non-profit corporations which provide prepaid coverage for the cost of hospitalization and medical-surgical care. The benefits are in many instances full service benefits so that there is no additional cost to the subscribing member.

While coverage is being granted to individual applicants, the bulk of the unterwriting is composed of groups, that is employees of a given commercial entity. Since the plans are voluntary and, in many instances, the employer either acts only as collecting agent for his employees or else pays only a fraction of the premium, certain minimum participation percentages have been established to prevent anti-selection.

Because of the competion of Insurance Companies which have concentrated their efforts on larger and more profitable groups (especially those where the employer pays all of the premiums) it was early recognized that a uniform or “community” rate will result in a gradual loss of groups with good experience thereby requiring substantial increases in the average rates. The original program of experience rating which the author has developed for the Massachusetts Blue Cross in 1949 has been since adopted “mutatis mutandis” by a number of other Blue Cross and Blue Shield organizations.

Suppose that an automobile insurance plan is characterized by a double classification. The risks are thus divided into classes, i = 1, 2, …, p (e.g. defined by use of car and age of operator), and groups j = 1, 2, …, q (e.g. defined by licence and by accidents during the last three years). The experience of the company is described by the observed “relative loss ratios” rij and some measure of exposure nij. A general model, often used, is that the rij: s are observations of random variables with the expected values gij = g(αi, βj), where the relativities αi are parameters representing the classes i and the relativities βj represent the influence of the groups j. One of the ratemaker's problems is to find a realistic function g(α, β) and to obtain estimates ai of αi and bj of βj.

In their paper “Two Studies in Automobile Insurance Rate-making” (ASTIN Bulletin Vol. I, Part IV, page 192-217) Robert Bailey and LeRoy Simon have thoroughly analyzed this problem for private passenger automobiles in Canada. They have principally studied three different types of the function g(α β), namely g(α β) = αβ (Method 2), g(α β) = α + β (Method 3) and g(α β) = 3αβ— 2 (Method 4). The authors show in an appendix, that the variance of rij is approximately g(αi βj)/Knij where K ≅ 0.005 for the Canadian data. They estimate the relativities αi and βj; by making χ2 = K. Σnij(rij — gij)2/gij a minimum. For the Canadian material, the “method 4” agrees best with the observations. This method gives an observed χ2 value of about 8 for 11 degrees of freedom.

It is rather a pity that we received only three papers especially devoted to subject 2, “applications of methods of operations research and modern economic theory”. Maybe the time for real applications of operations research techniques in the insurance field has still to come. However, I am convinced that sooner or later such techniques—in connection with the use of the computor—will play a more or less important part in the management of insurance business. An impulse to this of course will be the appearance of —let me say—“textbooks” on this subject; such as the book written by Wolff, Methoden der Unternehmensforschung im Versicherungswesen, Springerverlag 1966, a copy of which I received a few days ago.

Meanwhile, the fact that only a few papers have been sent in gives me the opportunity to report on them in some detail and to make a few side remarks.

The studies presented are:

Borch. Dynamic decision problems in an insurance company, de Leve and Weeda. Driving with Markov-programming.

d'Hooge, Franckx and Gennart. Utilisation pratique de la méthode de simulation dans l'assurance “non-life”.

The paper Prof. Borch submitted, deals essentially with some problems related with random walk theory. Partly it can be considered to be the counterpart of the classical discontinuous ruin theory as its subject is the expected duration of life of the company and maximisation of the discounted value of dividend payments.

The question of calculating the expected duration of life and more specially the means the company has to influence its future lifetime involves reinsurance and utility theory to which topic in recent years Prof. Borch has devoted a number of very interesting publications.

Let us consider the following problem.

A motorist has decided to effect an accident insurance under the following conditions. The insurance runs for one year. The premium for the first year amounts E0. If no damages have been claimed during i successive years, i = 1, 2 or 3 the premium is reduced to Ei. After four years of damagefree driving no further premium reduction is granted, so the premium remains E3. The premium is due on the first day of the year. The own risk amounts a0.

The number of accidents of our motorist during a time period T is assumed to be Poisson distributed with parameter λT. The extent of the damage has distribution function F(s) with finite mean and variance.

The problem of our motorist will be to decide whether to claim a damage or not. He will have to develop a strategy that specifies his decisions in every possible situation. His strategy will be called optimal if it minimizes the expected costs in the long run.

We may expect that in view of the premium reduction, it will be unprofitable to claim damages which are not much larger than a0. Once a damage is claimed it will be profitable to claim all following damages that exceed a0 during the remaining part of the year.

Hence his decisions will also depend on the time of the year and the premium paid at the beginning of that year. So we distinguish between four types of years, for each premium one.

Les colloques successifs d'Astin ent eu le mérite de clarifier certaines idées théoriques importantes. En invitant au colloque d'Arnhem les actuaires á penser aux méthodes de Monte Carlo, les organisateurs ont forcément orienté les recherches vers le domaine des applications. Car simuler, c'est á. dire établir des résultats conformes á un processus aléatoire bien déterminé, suppose á priori que les données numériques d'introduction (ou de définition) ont été fixées.

Nous ne reprendrons pas les principes de la méthode de simulation, ils sont donnés de maniére magistrale dans “Notes on operations research 1959” du centre de recherche opérationnelle du Massachusetts Institute of Technology. Le chapitre II, portant titre „Simulation of Random processes” rédigé par Herbert P. Galliher donne un exposé complet du probléme. Nous renvoyons á cette référence.

Parmi les idées que l'on doit retenir, car elle est d'application dans tous les domaines, nous citons „Never simulate a single numerical problem;always simulate several of a class of problems”.

Dans le domaine non life, cet aspect est fondamental. Ce qui y fait défaut, ce sont des algorithmes généraux de résolution numériques. Une des difficultés majeures rencontrées par les actuaires a été et est encore le calcul effectif des convolutions. Ceux qui ont établi cette note se sont posé la question de savoir si la méthode de Monte Carlo ne pouvait fournir un procédé:

— évitant les convolutions.

— suffisamment général pour être applicable à une classe de risques arbitraire.

— donnant comme sous-produit des éléments intéressant aussi bien les théoriciens que les praticiens.

1.1. — In this paper we shall consider some of the decisions which have to be made in the normal course of business in an insurance company. We shall see that the “right” decisions can be found only when the problems are analysed in their proper dynamic context.

As examples of the decision problems which we shall study, we can mention the following:

(i) What premium rates should be quoted on the insurance contracts, which the company offers to the public?

(ii) How much should the company spend to promote the sale of its policies?

(iii) When should the company refuse to underwrite a proposed insurance contract?

(iv) How shall the company reinsure its portfolio of insurance contracts?

(v) What reserve funds should an insurance company keep?

(vi) How shall the company's funds be invested?

Any actuary will be familiar with such problems, and he will probably feel that these problems cannot be satisfactorily solved with the methods offered by the classical actuarial theory.

1.2. — In some earlier papers [I] and [2] it has been argued that such problems can best be solved in the frame work of utility theory. As an illustration we shall take Problem (iii) in the preceding paragraph, and consider an insurance company in the following situation:

(i) The company has a capital S.

(ii) The company holds a portfolio of insurance contracts which will lead to a total payment of x to settle claims. F1(x) is the distribution of the variate x.