In preparing the notes on the subjects for discussion at the 4th ASTIN Colloquium at Trieste [1] I used some of the material which formed the basis of a talk given to the Scandinavian Actuarial Societies in September 1962. The papers presented in Trieste have established on a firm mathematical footing the formula for the excess loss premium derived in my note but the discussions also showed that some of the other 1962 material would be of interest.

It will be appreciated that the question originally proposed, i.e. to calculate an excess of loss reinsurance premium when the only information available is the largest claim experienced in each of a succession of periods, was deliberately phrased in this form as being the most troublesome case likely to arise in practice. Clearly if other information is available it would not be rejected in arriving at a premium, but this immediately extends the problem to one of finding the best methods of combining information of different kinds. For example if, say, the largest 5 claims in each period are known, what technique will make the maximum use of the data? Such extensions of the problem are not discussed in this note. Furthermore, I am not unmindful of the valuable comment by Jung, “that there is a natural law which states that you can never get more out of a mincing machine than what you have put into it” [2].

In his report to the XVIth International Congress of Actuaries in Brussels [5] in 1960 the author has given a survey of the German Motor Insurance Business and in this it was mentioned that a change might be expected from the flat-tariff prescribed by law into individual tariffs for every insurance company. The Federal Ministry of Economics decided this question at the end of 1959 by signing the price regulation PR 15/59 [I] The federal price-fixing treaty had thus come to an end after several decades.

Three steps are planned for the transitional period: For the years i960 and 1961 the flat-tariff remained in existence; free competition was introduced by providing for a return of premiums from technical surplus. At the beginning of 1962 the second phase of liberalisation began: Between 1962–1965 each insurance company has to compute its own tariff which has to be approved by the Federal Ministry of Economics. No deviations are permitted from the approved tariffs. Nobody can tell yet what the third phase in 1966 will bring.

At the time the price-fixing agreements were cancelled it was unanimously agreed that proper competition should be maintained and that ruinous price competition which would harm the interests of victims of traffic accidents must be avoided [4]. Trusts or tariff rings were not taken into consideration. However, some market regulations were set up by co-operation between the different parties.

The benefits (insurance conditions) are the same for all insurance enterprises. Also there are only few alternatives in the tariff conditions.

1. The comfound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value τ of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral

where U(ν, τ) for a fixed value of τ defines the distribution of ν. U(ν, τ) is called the risk distribution and is either τ-independent or, dependent on ν, τ.

2. The compound Poisson process in the narrow sense is defined as a process for which the probability distribution of the number of changes can be written in the form of (I) with a τ-independent risk distribution.

In their general form these processes have been analyzed by Ove Lundberg (1940). For such processes the following relation holds for the probability of i changes in the interval ο to τ, P̅i (τ) say

this relation does not hold for processes with τ-dependent risk distribution. Hofmann (1955) has introduced a sub-set of the processes concerned in this section for which the probability for non-occurrence of a change in the interval o to τ is defined as a solution of the differential equation

and ϰ ≥ o; the solutions may be written in the form where η is independent of and of two alternative forms one for ϰ = I and one for other values of ϰ. The probabilities for i changes in the interval o to τ in the processes defined by the solutions of Hofmann's equation are derived by Leibniz's formula, and are designated by and, in this paper, called Hofmann probabilities.

Il est peu de problèmes, dans l'industrie de l'assurance, dont la solution occasionne autant de difficultés que celui consistant à fixer dans l'assurance Automobile une prime équitable du point de vue technique.

On a tenté, dans plusieurs pays, de s'attaquer à ce problème par le moyen d'un bonus progressif dont bénéficierait le souscripteur n'annonçant pas de sinistre. Cependant, tant les théoriciens que les praticiens sont très partagés sur l'opportunité de cette solution.

Certains mathématiciens et juristes repoussent catégoriquement le système du bonus et qualifient l'idée de la ristourne d'une partie de la prime à l'assuré n'ayant pas eu de sinistre comme étant contraire à la notion même de l'assurance. Lors d'un des derniers congrès internationaux d'actuaires, ce procédé aurait — avec un peu d'exagération — été défini comme suit: Le bonus est, en matière d'assurance, une renonciation organisée de l'assurance. Cette opinion est principalement défendue par certains actuaires franҁais ).

D'autres mathématiciens, cependant, sont d'avis que le bonus est une précieuse disposition permettant, sur la base des sinistres survenus, d'évaluer chaque risque individuel plus rigoureusement que si I'on renonҁait à cette pratique). Et, à leur avis, il est inopportun de ne pas faire usage de cet élément d'information.

2. “Absence de sinistres” et “Nombre de sinistres”

Normalement, les primes d'assurance dépendent d'un certain nombre de facteurs caractéristiques du risque, à l'exclusion cependant des sinistres concernant un contrat particulier.

The no-claim bonus problem has given rise to a considerable amount of discussion throughout the whole world. There is quite a difference of opinion among the actuaries and other experts concerned in this field and several exchanges of view have taken place the last few years. The ASTIN section of the Permanent Committee has been well aware of this fact and it has devoted one Colloquium to this subject and discussed it at others.

In 1959 the major part of the La Baule meeting was dedicated to this subject and attention was focussed on this problem once again at Rattvik in 1961. Nevertheless controversies on this subject still continue. Almost every conference where the bonus problem is discussed is marked by a widespread difference of opinion.

As is well known, the claim frequencies under insurance policies show a considerable heterogeneity, especially in the early years. It is not possible to get homogeneous sub-groups by means of a continuous subdivision; what may be gained in homogeneity, is lost in credibility. It seems therefore that a subsequent adjustment of premiums according to the past claim record may well be a suitable way of obtaining a fair premium.

Those who are in favour of a rating procedure granting a bonus at a careful driver will stress that criticism is useless as long as no better solution is available, whereas actuaries who reject such a rating system argue that the unfairness of a flat rate is not at all eliminated by means of a bonus.

Une Compagnie d'assurances délivre par l'intérmediaire d'Agences de Voyage des contrats d'assurances individuelles garantissant les personnes prenant place dans des autocars contre les décès par accident. Le capital garanti est uniforme. Il sera pris égal à I par simplification.

Cette Compagnie prend à sa charge le paiement de i décès au maximum dans chaque accident et demande à un Réassureur de payer, aux termes d'un traité en excédent de sinistre, les autres décès; autrement dit, si un accident cause d décès, la Compagnie paie I, le Réassureur en excess paie d — I.

Il s'agit de calculer la prime de ce traité en excédent de sinistre?

La solution dépend de trois paramètres:

n nombre de passagers par autocar, égal au maximum au nombre de places; on supposera, pour fixer les idées, n = 502).

d nombre de décès survenus dans un même accident. C'est une variable stochastique comprise entre 0 et n.

Il n'y a pas lieu de faire intervenir la fréquence d'accidents (entraînant des morts) q, car celle-ci est un des éléments constitutifs du tarif appliqué par la Cédante et la prime de réassurance est exprimée comme une fraction de la prime originale. Si celle-ci est incorrecte, trop basse ou trop élevée, l'Assureur réalise une perte ou un bénéfice et il est normal que le Réassureur suive en cela la fortune de sa Cédante. La seule question est celle d'un partage équitable de la prime perçue, bonne ou mauvaise, entre la Cédante et le Réassureur.

The classical treatment of stochastic models in non-life insurance is to first derive the well-known Poisson distribution by considering the question of how many claims take place during a definite period t. In deriving this distribution the following three assumptions are made:

( I) The population studied is homogeneous.

( II) The occurrence of a claim is a rare event, viz. in an infinite-simal time interval [t, t + Δt], the probability of more than one occurrence must be of the order of magnitude o(Δt).

(III) The occurrence of any later claim is not influenced by previous ones (no contagion).

In my purely theoretical study [5]), the consequences of discarding one or more of the above assumptions were considered. By so generalizing the Poisson distribution, a great many stochastic models can be built, although the results were not always successful. The following study concentrates on some considerations based on assumption (II). The theoretical formulation of the model will be dealt with briefly and the author would first make reference to the instructive article of Thyrion [7] which was unfortunately unknown to him when he was preparing his already mentioned paper. Ammeter [2] and Arfwedson [3] have also considered special cases of this generalization. With the help of the statistics over traffic accidents in the city of Zurich, I hope to throw some more light on the practical aspects of the problem. To the Statistics Office of this city I would express my thanks for kindly placing all documents at my disposal.