The very title of this paper may cause some surprise, since economic theory so far has found virtually no application in insurance. Insurance is obviously an economic activity, and it is indeed strange that general economic theory should seem inapplicable to insurance.

This apparent paradox may to some extent be explained by the historic development. Actuarial mathematics and the essential scientific basis of insurance were developed into a self-contained and fairly complete theory long before economists could claim the name of science for their subject. Actuaries and other insurance people may from time to time have turned to economic theory for help on their problems. In most cases they must have turned away in disappointment, being convinced that actuarial mathematics was well ahead of general economic theory.

The last point is brought out clearly in a paper on the safety loading of insurance premiums which Tauber (19) presented at the Sixth International Congress of Actuaries in Vienna in 1909. In the introduction to this paper Tauber seems to consider risk bearing as a service and appears to assume that it like all other goods and services must have a price, determined by supply and demand in the market. Tauber did not develop this idea, apparently because the economic theory of his time was utterly unable to analyse the problem. That his approach was sound, and that the problem can be formulated and solved by modern economic theory, has been indicated in a previous paper (6).

During the last thirty years there has been an extremely rapid development in economic theory. The “General Theory” of Keynes (12) which appeared in 1936, is usually considered to have caused a revolution in economic theory. However, this “revolution” has not lead to any developments which seem to have an immediate application in insurance. Post- and pre-Keynesian economics are equally powerless when confronted with the problem which Tauber tried to formulate. It is therefore premature to conclude from the title of this paper that economic theory has caught up with actuarial science, and that the theory of insurance has now become a part of a general theory comprising all economic activities.

In his paper “Actuarial Activity in General Insurance in the Northern Countries of Europe” L. Wilhelmsen gives amongst other things an account of the work carried on by Centralstället för nordisk ömsesidig Brandförsäkringsstatistik (CNÖB, the Northern Central Office for Fire Insurance Statistics from Mutual Companies). In this he states that one part of the organisation's work is the carrying out of special investigations of current problems with material collected on each occasion for the purpose.

The object of this paper is to report investigations carried out in CNÖB on the connection between temperature and risk premium in fire insurance. The material used is made up exclusively of civil risks (buildings) and the material has been taken from Sweden and Norway. The background to the investigation consists in the fact that in Scandinavia the risk premiums for fire insurance show an apparent geographical variation, in that the amount clearly increases in the most northerly provinces.

As we know that the fire damage directly or indirectly caused by heating systems (chimney fires, cracked building blocks, embers from fireplaces or chimneys etc.) represents a large proportion of the damage in civil risks (40-50%, in some materials 60% or more), that the proportion is greatest in the northern parts and that in the Swedish material about 50% of the damage in any one area falls in the four months December to March, it is quite reasonable to trace the influence of the temperature factor behind the geographical variation; the colder the climate, the more lighting of fires and the more damage. This line of argument is connected exclusively to the frequency of damage; we shall return later to the mean degree of damage.

Le calcul de la prime de l'assurance Incendie des bâtiments se trouve compliqué par le fait qu'il doit tenir compte, non seulement du bâtiment assuré, mais également des immeubles voisins.

Si deux bâtiments sont suffisamment éloignés, le risque qu'un incendie se propage de l'un à l'autre est minime, mais le risque de contiguïté, et par conséquent d'incendie total, est d'autant plus grand que leur distance est plus faible.

La mesure dans laquelle le risque de contiguïté doit être retenu pour la tarification est un des problèmes les plus importants que l'Actuaire a à résoudre en assurance Incendie.

Pour qu'un incendie puisse s'étendre à d'autres bâtiments, le feu doit embrasser d'abord les éléments extérieurs du bâtiment assuré, c'est-à-dire les murs, la cheminée, la toiture ou les constructions en bois attenantes. Nous appellerons de tels incendies: incendies extérieurs par opposition à incendies intérieurs qui n'embrassent que le contenu du bâtiment et les aménagements intérieurs.

Par risque propre d'un bâtiment nous entendons l'indemnité moyenne annuelle de sinistre, sous la condition que ce bâtiment ne soit pas exposé au risque de contiguïté d'autres immeubles.

Le risque propre se divise en une fraction qui concerne les incendies intérieurs et une autre fraction concernant les incendies extérieurs,

Le deuxième terme Re peut être exprimé par le produit de la probabilité d'un incendie extérieur et de l'indemnité moyenne de cet incendie, soit

Les éléments de ce risque propre peuvent être observés pour différents groupes de risques, c'est-à-dire chaque groupe comprenant des bâtiments d'un même genre, par exemple en ce qui concerne leur construction, leur dimension ou leur destination.

The estimation of stop loss premiums can be based on some knowledge about the distribution function of the sum of all claims in a year (assuming that the stop loss insurance relates to a period of one calender year). Generally speaking there are two methods to obtain this knowledge about the distribution function.

1. The first method is to construct a distribution function from data concerning:

a. the distribution function of the number of claims per year, taking into account the variability of the parameter(s) of this distribution function.

b. the distribution function of the insured sums.

c. the distribution function of partial claims.

d. the correlation between the insured sum and the probability of occurring of a claim.

e. the probability of contagion.

2. The second method is to derive a distribution function from the year's totals of claims over a long series of years, expressed in e.g: units of the totals of insured sums in that years.

In practice it is often difficult to find a useful basis to apply one of these methods. Data concerning the distribution function of the number of claims per year, of the insured sums, and of partial claims are mostly available, but often nothing is known about the correlation between the insured sum and the probability of occurring of a claim.

The second method is mostly not applicable because, if the year's totals of claims over a long series of, by preference recent, years are available, these data often turn out to be heterogeneous or to be correlated with time. If, in that case, only the data of the most recent years are used, the number of these data is often a too small basis for the construction of a distribution function.

Risk Theory for Life Insurance is simplified by the fact that the distribution Ψ (x) of claim amounts x approximately coincides with the distribution of “Risk sums” (not exactly, owing to differaences in the claim frequency with age and actual state of health), ond this distribution is comparatively stable.—The dependence on the claim frequency is eliminated by the introduction of a new time variable, and the system reduced to a (stationary) Poisson Process, which should be valid at least for large risk systems and for the total Life branch for a moderate sequence of years.

In almost all non-life branches, partial claims will dominate and Ψ (x) can only be determined by risk statistics, leaving a certain space of indetermination, in particular for large claims and for mediumsized statistical risk groups.

In my previous analyses, in particular New York 1957, interest has been concentrated on traffic and motor car insurance, where the risk depends on cars insured and on the meeting traffic (including road conditions). In one year the same car can be involved in many accidents and double claims (=collisions) are rather frequent.—According to my experience, this system is best represented by a sequence of single and double risk situations in time (for individual cars or for risk groups).

Analysis is simplified for Fire Insurance (and many other non-life branches), because the risk system is composed of mostly independent insurances (or risk objects), which are best described by the ordinary Individual Risk Theory.

Assurance companies usually divide their portfolio into a series of tariff groups. This division is necessary for a comparison between the claims and the available premiums in the different tariff groups. By comparing these figures the company is able to verify if the tariff is appropriate or if a modification in line with the empirical data is to be recommended. Apart from this technical argument companies are sometimes compelled to make such a division because of their constitution (e.g. mutual companies) or because it is prescribed by the supervisory authorities. Companies underwriting in various countries are under the obligation, as a rule, to keep special accounts for different countries and currencies. All these reasons lead to a division of the whole business into more or less independent subdivisions which quite obviously have not as much capacity to equalize the risk fluctuations as the whole portfolio.

In every tariff group an exceptional fluctuation may occur either from a large individual claim or from claims under number of policies. This leads to a substantial loss in the relevant profit and loss statement at the end of the year. If this debit balance is carried forward to the new account every year, the account of such a tariff group may show a loss over a number of years. It is obvious that a company will endeavour to avoid such an awkward situation.

This sort of problem is usually dealt with by means of reinsurance and, to some extent this will be the right solution. However if every tariff group should be able to meet its own liabilities, a rather extended reinsurance is necessary. The capacity of the whole portfolio then would be of no importance, which suggests that a solution through reinsurance alone is not entirely satisfactory.

In the classical definition skewness is departure from symmetry. It was therefore natural to measure skewness by using a normalized third moment μ3/σ3. This condensed measure, however, is not refined enough to be used as an operational instrument for studying various functions which might be used to describe actual claim distributions. This is true especially when the interest is concentrated towards the higher values of the variate.

In their paper (1) Benktander-Segerdahl have suggested that the average excess claim m(x) as a function of the priority x should be used to reveal the characteristics of the tail of the distribution

where P(x) = 1 — H(x) denotes the distribution function.

This statistic is very apt when comparing actual claim distributions with possible theoretical models. It is also useful when classifying these models.

If, however, emphasis mainly is laid on classifying distributions according to their skewness, another statistic might be preferable. Let μ(x)dx denote the probability that a stochastic variable which is known to be at least equal to x, does not exceed x+dx. In other words, μ(x)dx represents the probability that a claim or the corresponding stochastic variable, which, when observed from the bottom, is “alive” at x, “dies” in the interval (x, x + dx). The lower this claims rate of mortality, the skewer and more dangerous is the claim distribution.

Il a été créé en France, en 1948, dans le cadre de l'Association Générale des Sociétés d'Assurances contre les Accidents, un pool dénommé „Section de Réassurance Mutuelle” ayant pour objet la réassurance en mutualité, en excédent de sinistres, des risques de Responsabilité Civile Automobile, de Responsabilité Civile Générale et d'Accidents du Travail.

Le seuil au dela duquel les sinistres ont été répartis en mutualité a été:

Etaient réassurés par la Section tous les sinistres survenus en France, Afrique du Nord, Union Française et Belgique, mais la part relative aux risques assurés hors de la France métropolitaine était faible.

Nous n'avons tenu compte dans la présente étude que des sinistres de Responsabilité Civile Automobile supérieurs à:

L'adhésion à ce pool était libre. A titre d'indication le tableau ci-après donne par année les encaissements globaux en France et Algérie de toutes les Sociétés d'Assurances pratiquant l'Assurance Automobile. (Rapport annuel de la Direction des Assurances) et le montant des primes Auto et R.C. diverses ayant servi de base au calcul de la cotisation des Sociétés adhérentes à la Section de Réassurance Mutuelle.

La question posée au colloque constitue un cas particulier du problème de la chance et de la malchance. Il appartient à une catégorie plus large de problèmes qui touche aussi bien à la statistique qu'à la recherche opérationnelle.

Pour une catégorie de risque déterminée, il faut définir deux éléments essentiels.

a) pour une période de temps de référence conventionnelle — la semaine par exemple, il faut disposer de la loi de survenance des sinistres c'est-à-dire que pour la période de temps considérée, il faut disposer de la loi qui constate qu'il se présente

la fonction génératrice de cette loi de survenance, toujours définie pour o ≤ s ≤ 1.

b) pour la catégorie de risques, nous considérons d'autre part la fonction de distribution des sinistres survenus.

S désignant la variable aléatoire du montant du sinistre, m la somme payée.

Ces deux éléments essentiels peuvent être donnés soit sur leur forme mathématique (fonctions spécifiées) sont sur leur forme expérimentale (image statistique d'un échantillon).

a) On peut caractériser la situation du portefeuille d'une société d'assurance par:

le montant de ses réserves libres: Z

le coefficient de sécurité attaché aux primes pures P encaissées: λ

la fonction de fréquence des montants de sinistre: ν(x)

le bénéfice ou gain moyen par prime pure unitaire: δλ

Aux trois premiers élements correspond une probabilité de ruine ε; nous définissons cette dernière comme la probabilité d'extinction d'un fonds de valeur initiale Z, alimenté par des primes (I + λ) P et débité des montants de sinistre x liquidés ou mis en réserve, soit pendant une période infinie ou finie (processus de ruine continu), soit à des époques fixées d'avance, (processus de ruine discontinu).

Cette probabilité de ruine peut, dans des hypothèses assez générates, s'exprimer par:

où C est une fonction, dont la valeur est inférieure à l'unité, de λ et éventuellement de Z et de la longueur μ de la période considérée.

In the discussion in the ASTIN Colloquium 1962, which followed my lecture on the numerical evaluation of the distribution functions defining some compound Poisson processes, a remark was made, which drew my attention to the paper quoted here below under reference number (1), On composed Poisson distributions I.

As I have the impression that this remark may induce some confusion of the terms “compound” and “composed”, the more as the same French word (composé) is used for the two terms, a comparison between the two kinds of processes shall be made.

1. The most general propositions of (1) are a theorem which concerns a general homogeneous Markoff process (1.c. § 2) and a theorem for the family {P(k,p)} of distribution functions of positive integervalued variables with mean p, where p runs over all non-negative numbers, and where the convolution of P(k,p1) and P(k,p2) is equal to P(k,p1 + p2) (l.c. § 3). By these propositions the characteristic functions corresponding in the 1st case to the distribution functions defining the homogeneous Markoff process, and in the 2nd case to the distribution functions belonging to the family {P(k,p)} can all be written in the same form, namely

where u is an entirely imaginary variable, p is the parameter of the process, respectively of P(k,p) and c1, c2 … non-negative constants such that converges. If, in the 1st case, converges, and as, in the 2nd case, this series converges, the distributions defined by these characteristic functions are called composed Poisson distributions, which define homogeneous composed Poisson processes.