Plusieurs communications sont présentées au Colloque sur les Mathématiques de l'Assurance Automobile. Afin de pouvoir mieux situer chacune d'elles dans ce vaste sujet, il peut être bon de schématiser l'ensemble de la question. Il comprend:

1. la survenance des accidents

2. le tarif et la mesure de son efficacité par rapport à l'estimation optimale.

Groupons dans une classe les propriétaires de véhicules ayant un même ensemble de caractéristiques extérieures de risques: même modèle de voiture, même zone de garage habituel, même usage ou profession du propriétaire, etc. Les risques s de survenance d'accident ne sont pas identiques pour tous les propriétaires de la classe, mais sont distribués selon une fonction de répartition F(s).

Désignons par P(s′ ∣ s) la probabilité de survenance de s′ sinistres pour un assuré dont le risque individuel est s (des études antérieures ont montré que P(s′ ∣ s) est représentable par une loi de probabilité de Poisson). La probabilité qu'un assuré, pris au hasard dans la classe, ait s′ sinistres est:

s0 étant le risque minimum de la classe.

Désignons par g(X) la densité de probabilité de la distribution des couts X des sinistres. On peut admettre qu'à l'intérieur de la classe, au moins en première approximation, g(X) est indépendant de s, d'où la probabilité de survenance d'un sinistre de coût X est:

1. If you introduce a franchise or raise its amount, the number of claims will decrease. One reason why you will operate a franchise in car damage insurance is to avoid the troublesome and expensive handling of the large number of very small claims.

Therefore, the management could use a mathematical model describing the number of claims as related to the size of the franchise. By varying this figure one may find the franchise that provides for the optimal business conditions.

2. The mathematical model to be described in this paper was found by studying the distribution of the claims in two portfolios. The general conditions of the two groups were as equal as possible, there was no-claim bonus in neither group, but group B had a franchise of 250 DKr. (about US $ 35) while group A covered the smaller claims also. Each policy had been unaltered for at least one year. The number of claims during 12 successive months was registered as follows:

The distribution could be described by a negative binomial distribution

where the parameters α and γ were estimated by

These estimates are not maximum likelihood estimates but it was not regarded necessary to improve the estimation as the χ2 values of the incomplete χ2-test were found as

The parameters were estimated as follows:

3. Special interest is attached to the variations of the γ-parameter. It is well known, that if γ → ∞ and α → ∞ under the condition then the negative binomial distribution will converge towards a Poisson distribution with the parameter λ.

In this paper motor liability insurance is considered from the viewpoint of an excess of loss reinsurer.

The reinsurer pays portions of claim amounts in excess of a contractually agreed limit, further referred to as excess point.

Aspects of practical interest are the rating of new contracts and cost-projections on claims occurred in past years which have not yet been reported.

What causes a major difficulty to a rating approach is the timelag between the occurrence of a claim and its settlement.

During this timelag, which may extend to ten years or even longer, the size of a claim will usually increase considerably. This is due to such factors as inflation and more victim oriented legal procedures.

As a result of this phenomenon, a claim, initially not involving the reinsurer, may confront him many years later, when its size overtakes the excess point. Hence, each risk year produces a generation of claims stretching far into the future.

The problem facing the reinsurer trying to compute premiums is that he must look, say ten years into the future, on the basis of incomplete data.

The table below is a schematic presentation of the time history of a contract that started k years ago.

The experience is assumed to be expressed in numbers of claims exceeding a constant excess point.

Successive rows show the generations of excess claims down to the last completed risk year k.

In the pages that follow we will attempt to project expected numbers of excess claims into the future, using information as schematized in this table.

Cette note ne risque pas de prêter le flanc à une critique souvent adressée à ASTIN, l'excès de spéculations mathématiques. Elle s'expose plutôt au reproche contraire, car elle a été, autant que possible, délibérément dépouillée de considérations théoriques, pour s'en tenir à quelques constatations arithmétiques fort simples sur la variable „Nombre de sinistres”.

Ces constatations sont orientées vers la pratique, particulièrement vers la mise en œuvre d'un système de bonus-malus.

This paper deals with bonus systems used in Denmark, Finland, Norway, Sweden, Switzerland and West Germany. These systems are studied by methods given by Mr. Loimaranta). The bonus rules of Denmark and Sweden have been modified because they contradict to one of the assumptions of the theory.

It is assumed that the number of claims in a year follows the Poisson distribution with a mean λ. Further it is assumed that the value of λ is independent of time.

In each case bonus rules are given in form of transformations Tk defined in ref. 1., i.e. Tk(i) = j when a policy moves from class i to class j after k claims. The class where a new policy starts from is called the initial class. Bonus scales, the vectors B, are normed so that the premium of the initial class is 100.

For each bonus system the efficiency η of the system and the discrimination power d of the bonus rules as a function of the mean claim frequency λ have been calculated. The graphs of these functions are presented in the figures 2 to 4 respectively. In the following pages, different bonus rules are described in detail and some simple analysis based on the curves on pp. 211-214 has been made. The calculations were performed quite recently and any deeper analysis of the results has not been possible because of lack of time.

In order to be able to apply the theory of Markov chains to a bonus system we must require among others that the transition to a certain class depends only on the number of claims occurred during last period ignoring possible former claims. The Danish system has originally four bonus classes labeled from 0 to 3.

Motor insurers in the United Kingdom are not subject to Government control over the rating structures they use or the levels of premiums they charge. The market is highly competitive, and each insurer therefore needs to make the best estimate he can of the premium required for each category of risk, to produce a given level of profit. He also needs to estimate the extent to which departures from such premium levels can be justified, for example in order to have a simple rating structure or to meet competition.

The purpose of this note is to give some examples of the statistical tabulations being produced in one insurance office in the United Kingdom. The statistical system which has been developed covers many aspects of motor insurance management, but in this note we shall direct attention to just two, namely

(i) continuous review of the variations in the claims experience from one risk category to another, as a guide to the relative premiums required for the different categories; and

(ii) continuous review of the changing composition of the portfolio and of the movements in and out, to try to assess the extent to which gains and losses of business can be attributed to pricing differences, marketing strategies, etc.

In motorcar insurance—at least in Europe—is widely used a merit rating system called bonus (or malus) system, characterized among others by the fact that only the number of claims occurred does modify the premium. A great variety of different bonus systems is in use. Each country—if not each company—seems to prefer a different bonus system and there is a lack of agreement whether one of them is better than the other. Efforts to construct a theoretically optimal bonus system have not been able to change the situation. General methods to evaluate and compare different existing or proposed bonus systems are therefore wanted.

In the present paper the theory of Markow chains is used to develope formulas for some asymptotic properties of bonus systems. Quantities: efficiency of a bonus system, discrimination power of bonus rules and minimum variance bonus scale are introduced. The last of them gives an asymptotical solution for the problem to find locally “best” bonus scales for given bonus rules. Finally the Danish bonus system is used as an illustration.

We call a merit rating system bonus system if the following assumptions are valid:

i. All policies of a given risk group can be devided into a finite number of classes so that the premium of a policy for a given period depends solely on the class for that period.

ii. The actual class is uniquely defined by the class for the previous period, and the number of claims occurred (regulated) during the period.

E. Franckx [1] has established the distribution function of the largest individual claim of a portfolio. By assuming the number of claims to be Poisson distributed, H. Ammeter was able to develop the distribution function of the total loss excluding the largest individual claim [2] as well as the distribution function of the nth largest claim [3].

Of course, the nth largest claim is dependent on the largest claim, second largest claim and so on, down to the (nth — 1) largest claim. If we assume the number of claims to be Poisson distributed and the amount of the individual claim to be Pareto distributed, the correlation between the mth largest and the nth largest claim can be expressed by an analytical formula which is susceptible to numerical computation.

With this knowledge we shall be able to compute the variance of the sum of the n largest claims and moreover the correlation between the sum of the n largest claims and the total loss amount. Although an excess of loss reinsurance treaty and a treaty reinsuring the n largest claims are very different in their construction, this paper will show that from a practical point of view there exists a similarity between the two treaties. The correlation coefficient between the sum of the n largest claims and the sum of all claims exceeding a certain limit enables us to assess the degree of similarity.

The correlation coefficient and thus the degree of similarity will prove to be high even in case of the reinsurance of only a small number of largest claims.

Finally, the knowledge of the two first moments of the sum of the n largest claims allows us to compute the premium and the security or variance loading for the reinsurance of the n largest claims.