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.

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.

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

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.

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.

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.

The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.

The calculations were based on the following assumptions.

Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula

and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula

As to the distribution function F(x, t) it is assumed that

where

Pn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,

V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, and

Vn*(x) is the nth convolution of the function V(x) with itself.

V(x) is supposed to be normalized so that the mean = I.

Let F(x) be the distribution function of the total amount ξ of claims of an insurance company. It is assumed that this company reduces its liability by means of one or several reinsurance arrangements. Let the remaining part of claims amount be equal to η, which is another random variable. Reinsurance arrangements are supposed to fulfil the following consistency condition: if ξ = x, then almost certainly o ≤ η ≤ x. Presumably all reinsurance arrangements occuring in practice can be supposed to fulfil this requirement.

The problem is to find an optimal reinsurance arrangement, i.e. an optimal random variable η, in the sense that, the net reinsurance premium being given, the variance of η reaches its minimum. In other words, a variable η is looked for, which gives

E{η} = P = const.; V{η} = minimum

In the sequel we use conditional expectations in the sense defined by DOOB ([I]).

Let η be an arbitrary random variable satisfying the consistency condition. If

R(x) = E{η|ξ =x},

then evidently ≤ R(x) ≤ x. Further we have

For the variance the inequality

holds true, since

Clearly the arrangement E {η|ξ} gives also to the reinsurer a smaller variance than the original arrangement.

Reinsurance forms can roughly be classified into proportional and non-proportional. The authors of this paper had planned to investigate the “efficiency” of two different reinsurance forms, one from each of these categories. Efficiency is here understood as reduction in the variance of the annual results of the risk business achieved per unit of ceded reinsurance risk premium. This investigation may be carried out in full later.

This note will only deal with the interplay between surplus and excess of loss reinsurance; more specifically the effect of changes in the volume of surplus cessions on the excess of loss risk premium.

The study came out of a practical Fire Reinsurance rating problem and will be carried through under very simplified assumptions. Thus we will ignore the conflagration hazard and the possibility of a wrongly taxed PML. This means that if amounts above a PML of M are ceded on a surplus basis the highest loss per event will be M, and an excess cover above a priority m will never pay more than M-m per event.

The following notations will be used

R(M) ceded risk premium volume on surplus basis, the PML retention being M.

πM(m) excess of loss risk premium if priority is m and surplus cessions are made above a PML of M.

1. In an article on such processes in ASTIN Bull. (II-3, pp. 450–451) the author has derived a relation between cPp i.n.s. (compound Poisson processes in the narrow sense) and composed Poisson processes, which is quite general. In this note this relation shall be simplified under the rather weak condition that Po(t), Pn(t) being the distribution of the number of changes occurring in the interval (o, t), for every fixed value of t, tends to a positive limit less than unity when t tends to infinity.

2. By the theory expounded by Thyrion (Bull, de l'Assoc. Act. Beiges, 1959 and 1963) the characteristic function of the number of events in the interval (o, t) can for each fixed value of t in this case be written

where η is a real variable and i the imaginary unity.

It is easily seen that (1) is equal to the following expression

which has the form of a characteristic function of the number of changes in a composed Poisson process.