Mr. Chairman, Honourable Colleagues!

Let me thank the organisers of this symposium for the honour they have done me by inviting me to present my report before this eminent audience. I would also like to express my thanks to all who are present here for the confidence shown to me as rapporteur.

The genesis and development of the theory of random processes is closely related to the problems of demography and insurance. Numerous problems stimulating the development of the theory of probability and the theory of random processes—the latter constituting one of the essential parts of the modern theory of probability—are connected with these two fields of research and activity.

The theory of random processes, being stimulated—similarly to other fields of mathematics—by concrete practical tasks, has created the general ideas and methods of research, achieving many valuable results. With the passage of time, it appeared that the theory of random processes can be usefully applied, by its conceptions, results and methods, to a number of scientific fields and practical activity. Its conceptions have found wide and important application to the earlier crystallized fields of human knowledge, as well as to the newly appearing ones.

These new fields of mathematics serve us as an example which once more corroborate the thesis, well known to us, according to which the course of the development of human knowledge starts from empirical observations, then passes up to abstract thinking and finally to practice.

Our discussions today will concern Subject A: Risk Theory, in particular the overall risk involved in operating an insurance concern. 7 papers have been handed in to the Colloquium on this subject.

To be quite candid, the referee would like to say that only one of these papers really treats the whole of the topic foreseen, and this is the paper by Mr. Colin M. Stewart (U.K.): The Assessment of Solvency. The other papers treat different special practical sides of the problem as well as general risk theoretical questions. One of the papers, that of Mr. Brichler, could have been attributed to Subject B as well as to Subject A, but it has been decided to take it up for treatment today.

Mr. Stewart delivers a lot of interesting points, which, I may say, take into account the solvency aspect as it looks to the supervisory authorities much more than has been done within the ASTIN discussions earlier. This, of course, is very related to the fact that regional as well as Common Market discussions have resulted in or will result in legislative rules for judging the solvency of insurance concerns, which rules will have great practical importance in the future. Mr. Stewart stresses the point that in case of non-life concerns it is the business on the books and its future contracted periods, short time business to be signed before judgment can be practically undertaken, claims reported, claims occurred but not yet reported, as compared with free reserves available that really decide the matter of sovency.

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. Le fait que le 8-e Colloque ASTIN a lieu en Pologne donne une bonne occasion pour informer les membres de l'ASTIN sur l'organisation actuelle des assurances dans ce pays qui est d'ailleurs pareille à celle des assurances dans les autres pays socialistes. Dans le présent rapport sera présenté en particulier le problème des assurances du patrimoine national (propriété de l'État) qui comprennent aussi les risques d'assurance importants et complexes.

2. Après la II-e Guerre Mondiale la Pologne a connu de profonds changements sociaux et économiques. La grande et moyenne industries ont été nationalisées. On a réalisé la réforme agraire. Le transport et l'échange des marchandises ont été confiés à des entreprises d'Etat et coopératives. Tout le système financier, y compris les banques et les assurances, a été nationalisé, etc. Actuellement en Pologne les activités d'assurance sont effectuées par deux établissements.

Panstwowy Zakład Ubezpieczeń — en abréviation PZU — (Assurance Nationale Polonaise);

Towarzystwo Ubezpieczeń i Reasekuracji „WARTA” — en abréviation „WARTA” (Compagnie d'Assurances et de Réassurances SA „WARTA”).

3. La PZU ne s'occupe que des assurances du pays, c.á.d. des assurances des objets situés en Pologne et des personnes domiciliées en Pologne.

La „WARTA” s'occupe de toutes assurances extérieures liées aux paiements en devises (contractées pour la plupart en monnaie étrangère), ainsi done avant tout des assurances des marchandises dans le transport maritime, des assurances des moyens de transport maritime et fluvial, des assurances d'aviation, des assurances des véhicules automobiles etc.

Effectively, a non-life insurance concern may be considered to be solvent if the supervisory authorities of the country or countries in which it operates allow it to continue operating. It is of no avail to claim that, by some other criterion, the concern may be considered to be solvent; it is by reference to the controls imposed by supervisory authorities that the concern must operate. For this reason, it is apposite to consider the principles of solvency assessment in the context of the financial statements and other documents generally available to supervisory authorities, as distinct from the theoretical mathematical concepts underlying insurance operation.

The primary purpose of supervision is to make sure that a concern does not enter into obligations which it will be unable to fulfil. It is thus essentially a dynamic standard, not a static one having regard only to the business already accepted, but the first requirement is nevertheless to test the concern's finances in relation to the business already on the books. If this test is passed, then at least it is known that the concern will not be depending upon profits from new business and renewals to help to meet the cost of its present liabilities.

This raises the question of what controls might be applied to new business. The two main types of control possible are on the extent of selection of risks permitted, and on the premiums to be charged, but while it may be feasible for such controls to be applied to the home portfolio of a particular class, for example motor vehicle insurance, it is difficult to envisage the operation of such controls internationally in, for example, the marine, aviation and transit class.

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 several different countries there has lately been a very unfavourable development as far as fire losses are concerned. There have been many and very bad big fires which have especially affected industry and trade. The facts which can be considered to have caused this development are well-known. Bigger and bigger and more concentrated units of production, business enterprises, and storing form an accumulation of values which in itself implies increasing risks. On account of production-technical and business-economical conditions it has not always been possible to divide the buildings into sections desirable from a fire service point of view. The new production methods have sometimes caused new fire risks.

The Northern Central Office for Fire Insurance Statistics from Mutual Companies (CNÖB) has made an investigation in order to see to what extent the international trend towards increasing fire losses has any equivalence in the Northern countries (Denmark, Finland, Norway, and Sweden). I.a. there was made a graphical representation of how the observed rate of net risk premium of the portfolio has developed during the period 1946–1966, the result of which may be seen from fig. 1. No increasing trend can be observed from this. After having been decreasing during the immediate postwar time, the observed rate of net risk premium has been fairly constant from the beginning of the fifties, perhaps even somewhat decreasing except during the later years. The conclusion is that in so far facts as mentioned above have influenced the development of claims of the portfolios or parts of these in a deteriorating way, there must in that case have been circumstances which have worked as neutralizers.

On sait que la loi de Poisson simple représente mal la distribution des sinistres d'un groupe observé d'automobilistes du fait que tout groupe que l'on peut étudier en pratique, même s'il est composé d'assurés présentant des caractéristiques communes (même zone de circulation, même type de véhicule, même utilisation de ce véhicule, …) est hétérogéne quant aux autres caractéristiques et surtout quant au comportement personnel des assurés, élément dont l'influence sur les résultats du risque est prépondérante.

M. Delaporte a obtenu une représentation intéressante du phénomène en supposant que les sinistres d'un véhicule se répartis-sent suivant une loi de Poisson de moyenne donnée, et que les moyennes de chaque véhicule du groupe étudié se distribuent selon une loi de Pearson type III d'equation:

où Γ(b) est la fonction eulérienne de 21ème espèce:

a, b, et S0 étant des paramètres dont la valeur est calculée en égalant les expressions des 3 premiers moments théoriques aux moments correspondants observés.

Cette formule conduit à des calculs assez longs.

M. Depoid a proposé une formule plus simple.

On peut penser que la dimension des sinistres est la résultante d'un très grand nombre de causes indépendantes à effets positifs.

Soient

les variables aléatoires indépendantes qui représentent les facteurs élémentaires exerçant leur action dans l'ordre indiqué par les indices.

Si Xν est l'ampleur du sinistre dû aux facteurs ξ1, ξ2, …., ξν nous pouvons supposer que l'augmentation causée par ξν+1 soit proportionelle à ξν+1 et à une certaine fonction, g(Xν) de la dimension Xν.

C'est à dire, nous supposons qu'il y ait les relations:

Si chaque facteur n'apporte qu'un faible contribut à l'augmentation du sinistre, nous pouvons alors poser approximativement aussi:

Comme X varie de Xo à Xn (et que X = Xn désigne la dimension extrème du sinistre) le second membre de la (2) décrit une variable aléatoire, fonction de X.

Puisque par hypothèse ξ1,, ξ2, …., ξn sont des variables aléatoires indépendante, si n est suffisament grand, d'après le théorème de la limite des probabilités il s'insuit que la variable aléatoire au second membre de la (2), est, à la limite, distribuée suivant la loi normale. Si l'on pose g(t) = t − c, la variable aléatoire normale qui décrit la dimension du sinistre pourra être posée dans la forme In(X − c).

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.

We are using the following terminology—essentially following Feller:

a) Compound Poisson Variable

This is a random variable

where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variable

hence

(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functions

b) Weighted Compound Poisson Variable

This is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)

hence

or in the language of characteristic functions

Let [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads then

It is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

1. There is a general rule applicable to all insurance and reinsurance fields according to which the level of the so-called technical minimum premium should be fixed such that a certain stability criterion is satisfied for the portfolio under consideration. The two bestknown such criteria are

(i) the probability that there is a technical loss in any of the future years should be less than a given percentage

(ii) the probability that the company gets “ruined” i.e. initial reserves plus accumulated premiums minus accumulated claims becomes negative at any time of a given period in the future should be less than a tolerated percentage.

Confining ourselves to criterion (i) in the present paper we may then say that the problem of calculating technical minimum premiums is broadly spoken equivalent with the problem of estimating loss probabilities. Since an exact calculation of such probabilities is only possible for a few very simple and therefore mostly unrealistic risk models and since e.g. Esscher's method is not always very easy to apply in practice it might be worthwhile to describe in the following an alternative approach using results and techniques from Reliability Theory in order to establish bounds for unknown loss probabilities.

It would have been impossible for me to write this paper without having had the opportunity of numerous discussions with the Reliability experts R. Barlow and F. Proschan while I was at Stanford University. In particular I was told the elegant proof of theorem 3 given below by R. Barlow recently.

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.

E. Sparre Andersen [1]) presented to the XVth International Congress of Actuaries, New York, 1957, a model of a collective risk process with a positive gross risk premium where the epochs of claims formed a renewal process. Let Ψ(u) (where u denotes the original risk reserve) denote the ruin probability in this model. Generalizing the classical result Sparre Andersen deduced the inequality

where R is a suitable positive number depending on the distribution function (continuous to the right), P(y), — ∞ < y < ∞, P(o) < 1, for the amounts of claims in case a claim occurs and also depending on the distribution function, K(t), t > o, K(o) = o, for the times between the epochs of successive claims, (The times between the epochs of successive claims, the inter-occurrence times, are assumed to be independent and identically distributed random variables. The time between the starting point and the epoch of the first claim is assumed to be independent of and to have the same distribution function as the inter-occurrence times. The amounts of claims are assumed to be independent of each other and of the epochs of claims and to be identically distributed.)

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.

As is well-known, in the early 60's a Swedish committee set to work at the numerical calculation of the distribution function of the total amount of claims and of the related stop loss premiums in the Poisson and Polya cases (Bohman and Esscher [6]). Since the characteristic function for the said distribution function was easily available in terms of the characteristic function for the distribution function of an individual claim, the committee chose to base the numerical calculations on the C-method by H. Bohman (Bohman [5]). The calculation of the ruin probability for a finite or infinite period was not considered by the committee.

The last-mentioned problem has now been taken up by a new committee formed by the Swedish Council for Actuarial Science and Insurance Statistics. The committee—consisting of H. Bohman, J. Jung, N. Wikstad and the present author—has to consider several aspects of the practical applicability of the collective risk theory. However, without possibilities of calculating—at least approximately—the ruin probability for a finite period the applicability of the existing ruin theories seems to be rather limited, so the committee has looked around for such possibilities. At the present stage the committee is considering the classical Poisson theory and Sparre Andersen's generalization of this theory [2]. It is the hope of the committee that, at a later stage, also the Polya theory and the theory recently presented by Segerdahl [11] combining the Sparre Andersen theory and the Polya theory may be treated.

In this paper we are going to study some properties of a stochastic process, which has been proposed by Cramér (1968) as a model of the claims arising in an insurance company. This process has been studied by Cox in a different context. A few elementary results, concerning moments, are given by Cox and Lewis (1966). The present paper will be a survey of some results derived by the author (1970:1) and (1970:2). For detailed proofs we refer to these papers.

Let λ(t) be a real-valued stochastic process, such that P{λ(t) < o} = o. We further assume that Eλ(t) = 1 and that Eλ2(t) < ∞ for every fixed value of t. We denote the covariance

The process λ(t) will play the role of an intensity function. That means, that for every fixed realization of the process, the probability of

and that the number of events in disjoint intervals are independent.

We now define a point process N(t), where N(t) is the number of events which have occurred in (o, t]). With this definition we get

where

The integral is assumed to exist almost surely. This process will be called the N-process.