Classical statistics deals with the following standard problem of estimation:

Given: random variables X1, X2 … Xn independent, identically distributed, and

observations x1, X2 … xn,

Estimate: parameter (or function thereof) of the distribution function common to all Xi.

It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namely

Given: risks R1R2 … Rn no contagion, homogeneous group,

Find: the proper (common) rate for all risks in the given class.

There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?

Trois traits caractérisent, me semble-t-il, le développement récent de la théorie du risque et influencent done l'exposé dont notre Président m'a chargé, avec une amabilité tenement pressante que je n'ai pu m'y dérober.

Le temps n'est plus où l'on pouvait déplorer un manque d'intérêt des actuaires pour la théorie du risque: tel est le trait le plus immédiatement évident. ASTIN y a certainement contribué. Le cercle restreint d'adeptes — dirais-je d'initiés — a éclaté. Les idées traversent les frontières et même l'océan. Ce faisant, elles s'enrichissent au passage, se généralisent, ou bien encore rencontrent une idée-soeur qui leur ressemble parfois comme une jumelle mais porte un autre nom. On voit les idées “reςues” se heurter à des conceptions rajeunissantes. On voit aussi les nécessités pratiques infléchir la théorie. Bref il se produit un véritable bouillonnement de recherches et d'applications, des plus heureux pour la théorie du risque, mais des plus pénibles pour le conférencier qui voudrait en cerner convenablement l'évolution. Pour ce faire, l'ampleur et la diversité des publications exigeraient un professionnel en la matière, disposant du temps voulu pour s'informer complètement. Ce n'est pas mon cas et cela explique — et excusera, je l'espère — le cadre délibérément restreint de mon exposé. Car je n'ai pas l'intention de suivre la théorie dans ses multiples démarches réeentes et dans tous ses perfectionnements les plus subtils. Je vous propose plutôt de jeter quelques regards sur le développement récent de la théorie du risque et ainsi de tenter de dégager et d'illustrer quelques idées essentielles sur cette évolution et d'essayer d'entrevoir son déroulement futur. Telles sont les limites de mon objectif.

An insurance company aims to increase the volume of its business.

The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

pn = premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln = The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

The response to the invitation for the 5th Colloquium has been, especially for subject 1, very gratifying; a large number of meritorious papers have been sent in. Fortunately, it took not much time in this case to find a good guide for all this work. R. E. Beard presents in his article “Some Actuarial Aspects of Non-Life Insurance Company Management”) a broad cross-section of the many problems that confront the non-life insurer. It was therefore regarded as not quite unwise to follow his steps and to discuss the other papers at suitable points.

In the first half of his work R. E. Beard deals with a practical example from motor insurance, which is fully calculated with the help of numerical data. The question relates to the estimation of the value of outstanding claims, a quantity which is of decisive importance for the evaluation of the results of a portfolio. Apart from chance fluctuations which might affect the statistics from year to year, there are three major factors that render the task of estimation difficult:

(a) the natural growth of the portfolio

(b) the fact that bigger claims require longer time for settlement than the smaller ones

(c) the change in the value of money with time.

We consider a set of risks B, which is divided into subsets Bi according to some property as profession, region etc. The subsets Bi are thought to be too small to permit a calculation of a premium from their own experience. On the other hand we assume that there are differences between the Bi. This paper tries to give a solution to the problem how the experience of the single Bi should be combined with that of the whole B.

The problem is treated only with respect to frequencies of claims. The amounts of the claims are not taken into consideration. I suppose that an extension of the theory to amounts is possible and highly desirable for practical applications. Lack of time did not permit me however to proceed to such an extension in this paper.

I think that the methods developed in the sequel might be applied to the following practical problems:

— rate making

— experience rating in the restrictive sense of the word, i.e. adjusting periodically the premium or granting premium refunds according to the experience of every single Bi

— judging the experience of a. Bi.

I do not give a list of references which would have to be very long. The basic idea was, as far as I know, first developed by Lundberg.

Drs. H. Bohman and F. Esscher have reported in a recent paper) an extensive research performed in Sweden on the different methods of calculation of the distribution function of the total amount of claims. In the present paper certain methods are discussed in so far as they are different from those presented in the above quoted paper. The consideration is restricted to the generalised Poisson function even though some results can be easily extended. The author has already commented on some of the results represented in the sequel at a special meeting of the 17th International Congress of Actuaries in Edinburgh.

I. Lemma. Let be the generalised Poisson function under investigation. If aiSi(x), where Σ ai = 1 (the functions Si need not be distribution functions, neither must the constants ai be real numbers of interval [0,1]), then

F(x; n, S) = F(.; a1n, S1) * …*F(.; arn, Sr) (x),

as is easily verified by the use of characteristic functions. This component representation is repeatedly used in the sequel.

2. A Modified Esscher Method. The Esscher method is based on an observation that the well-known Edgeworth expansion is more advantageously applicable to a conveniently modified distribution function instead of the original generalised Poisson function. Let us assume that the value of F(x) is required at a point

Les 5 contributions suivantes ont été présentées sur le thème no 3:

— par R. E. Beard:

2 contributions intitulées “Calculation of Reserves for Non-Life Insurance” (Le calcul des réserves pour les branches non-vie)

— par Karl Borch:)

“Control of a Portfolio of Insurance Contracts” (Le contrôle d'un portefeuille d'assurances)

—par B. H. Jongh:)

“The Insurer's Ruin” (La ruine de l'assureur)

— par T. Pentikäinen, Helsinki:

“On the Solvency of Insurance Companies” (De la solvabilité des compagnies d'assurance)

— par Erkki Pesonen, Helsinki

“Magnitude Control of Technical Reserves in Finland” (Le contrôle du montant des réserves techniques en Finlande)

Les questions traitées par ces auteurs peuvent être groupées comme suit:

1. Donnée la situation d'une compagnie d'assurance, quelles sont les affaires qu'elle devrait accepter?

2. Donnée la situation d'une compagnie d'assurance, dans quelle mesure doit-elle réassurer son portefeuille?

3. Le calcul des réserves techniques, c'est-á-dire de la réserve pour risques en cours et pour les sinistres non régiés.

4. La détermination des réserves de sécurité nécessaires en plus des réserves techniques proprement dites.

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.

The purpose of this paper is to describe a technical procedure, which enables one to compute values of the generalised Poisson distribution function, with an accuracy considered sufficient for insurance companies and with satisfactory speed. The procedure requires a fast medium sized computer.

The computation of values of the generalised Poisson distribution function has become a timely problem in Finland, because of the introduction by the Supervisory Service of more stringent requirements in determining limits of the so called equalisation reserves, which have their theoretical basis in the random fluctuations of claims amounts. The question has also been discussed in papers submitted by Dr. Pentikainen [3] and Dr. Pesonen [4] to this Colloquium. Because the practical computation is a further problem, the Federation of Finnish Insurance Companies set up a committee in 1962 to gather and work up the necessary statistics from various branches of insurance and to develop the computational methods ready for use in practice. The committee has almost completed its work, and one of the results, a procedure to compute values of the generalised Poisson function with a mixed method, is described below. The method is referred to briefly in [4]. The programming and further planning work has been done by Dr. Loimaranta and M. Sc. Porn.

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

Modern life is characterized by risks of different kind: some threatening all persons and some restricted to the owners of property, motor cars, etc., while still others are typical for some individuals or for special occupations. The corresponding accidents, losses or claims will occur suddenly and unexpectedly and may involve considerable financial loss. It is quite evident that modern life is a fit subject for risk theory, and that some results in the pure mathematic theory might have applications in the study of problems in real life.

In practice, however, we can identify risk theory with insurance risk theory or with the application of the theory of probability on insurance risk problems. This general definition has the advantage, that it covers a wide field of different risks and risk problems as specified in the insurance texts—and a great collection of risk situations ₌ claims occurred (with corresponding loss amounts) is available in the claims acts. In fact, I believe that any actuary or mathematician, starting his researches in risk theory or in risk statistics, should begin his studies by a series of actual claims acts.