Let me begin with some practical examples of experience rating.

a) Swiss Automobile Tariff 1963

— Within each tariff-position there are 22 grades:

— The new owner of a car starts at grade 9

— The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car

— The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades.

b) Sliding Scale Premiums in Reinsurance

Excess of Loss Contracts often stipulate that:

The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows:

subject to a minimum of 0,04

and a maximum of 0,08

c) Participation in Mortality Profit in Group Life Insurance

A group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formula

refund = x% gross premiums — y% claims (where x < y)

All these examples fall under the heading “Experience Rating”. What do they have in common?

Definition: A system by which the premium of the individual risk depends upon the claims experience of this same individual risk.

1. Risks of a catastrophic nature, such as war or earthquake, can differ, in a statistical sense, from other ordinarily insured perils only in respect of the distribution of the probability function. For example, although the probability of a loss occurring may be small, its magnitude once it occurs may be very large. One reason for the latter feature is that the sum insured may itself be heavy. But still more important is the fact that there would be strong positive correlation between similar contiguously situated risks. This is the main reason for the loss taking on a catastrophic character.

2. The fact that the probability of a loss is ordinarily small, leads to the result that the expected loss and hence the pure premium for the risk is small. Equally, and for the same reason, the “loss strain at risk” is large. If the latter could be considerably reduced—which can only be done by charging impracticably high premiums—the risk becomes more easily insurable.

3. If there are n risks of this nature, 1, 2, 3 … n and νi(x) is the conditional frequency function of the ith risk we may write

While we may imagine this as a frequency curve, the complementary probability Ϸi of a loss not occurring can be represented as a vertical line at the origin—the different methods of representation of Ϸ and q are actually convenient. With this representation, the expansion of the binomial product

shows that the convolution of the n curves can be expressed as the sum of n + I functions

where

(ii) the term A1(x) comprises n separate curves of the form νi(x) each multiplied by its complementary probability, being of the form

(iii) the term A2(x) comprises terms each consisting of the convolution of the curves taken two at a time and multiplied by the complementary probability, being of the form

and so on.

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”.

As reports on work done on subjects discussed at earlier colloquia and on new lines of investigation 6 papers have been laid down on my table (To be exact, 5 on my table and the last one on my chair at the beginning of this colloquium on Wednesday morning).

Four of them deal with problems concerning the estimation of reserves. These are the papers by Beard, Molinaro, Benedikt and Hovinen.

The other two papers, each of which has two authors, namely Kauppi and Ojantakanen and van Klinken and Groenenberg deal with mathematical questions of more general interest.

I can repeat what my colleagues have said this week:

Although the number of the papers is rather small, the importance and the quality of them seem to me to be very great.

The paper of R. E. Beard concerns the studies he has made on the data provided by the 6th Conference of European Insurance Supervisory Services. Suitable statistics of motor insurance on which to experiment have been made available relating to 28 European insurance companies. Beard has made certain preliminary studies of these statistics. His paper has been written against the background of these studies. Although only the broadest indications of numerical values are given, since the detailed statistics are confidential, nevertheless the comments made are based on the studies of the figures. Therefore these comments have a considerable practical utility.

As a basic principle on which to build calculations of technical reserves it is assumed that the claim frequency rate will be constant and that the distribution of claims by amount will be stable, both apart from random fluctuations. Although it is hardly possible to prove the validity of these assumptions, there could be no basis for calculation if these principles were not true.

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.

In a note on the security loading of excess loss rates I am deducing a simple formula intended to replace some tedious calculations. In the beginning of that note I made the point that some authors recommend a loading proportional to the dispersion of the total claims amount of a treaty δ1 while others tend to favour .

I also stated that a loading proportional to δ1 or its estimate δ1* could be deduced from the statistical uncertainty in measuring the risk (section 4).

The question has been raised if and to what extent a loading system based on the dispersion is unduly punishing the smaller portfolios. This will be examined below.

The pricing concept will be analyzed from the point of view of a big dominating Reinsurer who wants to be fair in all directions. The conclusion of this study supports an affirmative answer to the question put above.

In a second part the loading is studied from a different angle bringing competition into the picture. The pricing or loading becomes a problem of operations research under the simplified assumption that profit is the only purpose of our activity. Not unexpectedly, the loading coming out from this aspect differs from those of part one.

Part two also deals with the question of how much of the loadings which we are aiming for, get lost in the competitive process. It is also shown that in most cases the harder the competition is, the higher loadings shall be used.

Part one and part two thus deal with the loading problem from different aspects, and illustrate the complexity of the problem. It is my hope that this note could stimulate further researches in this interesting and important area, also in a moment when some reinsurers are more concerned with the question of surviving than in fixing the loadings which should on the average and in the long run turn up as profits.

At the ASTIN Colloquium held in Lucerne I reported on the request from the 6th Conference of European Insurance Supervisors for co-operation in the studies being made into the subject of technical reserves in non-life insurance. Copies of notes prepared for the 6th Conference were distributed to members and an advisory committee was formed to assist in dealing with the request. Two reports have been sent to all members of this committee. It has not been practicable to prepare further reports and the present paper sets out the current position of the work being done.

2. It will be recalled that in a number of papers I have represented that research into methods of determining technical reserves has been severely restricted because of the absence of suitable statisticson which to experiment. This point has been well taken by the European Insurance Supervisors and recently suitable statistics of motor insurance on which to experiment have been made available on a confidential basis relating to 6 Swedish, 7 German, 5 French, 6 Italian and 4 British companies. These returns, whilst not compiled on a uniform basis, provide an immense amount of data not previously obtainable, and it will be appreciated that the analysis of the data is a task of considerable magnitude.

3. I have made certain preliminary studies as a result of which a number of points of detail have arisen which have been referred back for clarification or explanation. Nevertheless these studies have provided a very useful background to the whole problem and the present paper is written against this background and of my earlier papers on the topic. Since the detailed statistics are confidential and agreement for publication would mean some delay, only the broadest indications of numerical values are given. Nevertheless, the comments made are based on the studies of the figures for the 28 companies and thus have a considerable practical utility.

At the Lundberg Symposium, Stockholm 1968 Jung and Lundberg presented a report on similar problems as those treated in this note, and to the Astin colloquium, Berlin 1968 the present author presented a report with the same title as this note, where some of the results in the first-mentioned report were commented upon. Jung and Lundberg kindly discussed the topic here concerned with the present author some time after the colloquium. On account of this discussion, the present author withdrew his report from the publication in its original form. The following context is a revision and a completion of the author's report to the colloquium.

Let τ be a parameter measured on its original, absolute scale, and let be the same parameter measured on an operational scale with respect to the probability distribution (or the corresponding for t). The parameter will often be referred to as “time”, which does not imply a restriction of the theory to proper time parameters.

A random function X(s) is said to be distributed in a cPd i.w.s. (compound Poisson distribution in the wide sense), if the distribution function of X(s) for every fixed parameter point (s, τ) in a finite or infinite domaine of the parametric space as a function of τ can be written in the following general form

where the asterisk power m*, here and throughout this note, is taken to mean, for m ˃ O, the m times iterated convolution of the distribution function with itself, and, for m = o, unity. W(x, s) being the conditional distribution function of the size of one change in X(s) relative to the hypothesis that the change has occurred at s, here abbreviated to the change distribution. U(ν, τ) is a distribution function, called the structure function. In the general case V(x, s) and U(ν, τ) may depend on s and τ respectively. If, particularly, these functions are supposed to be independent of the parameter, they will be denoted V(x), U(ν) respectively. In the particular case, where V(x) = ε(ν—c1), c1 being an arbitrary but fixed constant and ε(ξ), here, and in the following context, the unity distribution equal to zero for negative values, and to unity for non-negative values of ξ, the cPd is said to be elementary and, in the opposite case, non-elementary. In the elementary case the distribution of X(s) is defined by the integral appearing in (Ia) with x = c1, so that W(x, s) = W(x) = I.