In this paper we shall study the situation of two insurance companies which are negotiating with the view of concluding a reciprocal reinsurance treaty. We assume that the two companies are under no compulsion to reach an agreement. This means that if the companies conclude a treaty, the treaty must be such that both companies consider themselves better off than without any treaty. We futher assume that no third company can break into the negotiations. This means that the two companies either have to come to terms, or be without any reinsurance.

How the two parties reach an agreement in a situation like this, is one of the classical problems of theoretical economics. It is usually referred to as the “Bargaining Problem”. The problem appears very simple, but this is a deception. It has proved extremely difficult to formulate generally acceptable assumptions which give the problem a determinate solution. The “Theory of Games”, developed by von Neumann and Morgenstern (10), does not give a determinate solution, but it has greatly increased our understanding of such problems, and the present paper will draw heavily on that theory.

The situation which we propose to study, is very simple, may be too simple to have any bearing on reinsurance negotiations in real life. If there exists a reinsurance market, which also is a perfect market in the sense given to this term in economic theory, bartering between two companies does not make any sense. They could both do equally well or better by dealing in the market at the market price.

Section A of this paper uses the Canadian experience for private passenger automobiles to show (1) that merit rating is almost as effective as the class plan in separating the better risks from the poorer risks, (2) that both merit rating and class rating leave unanalyzed a considerable amount of variation among risks and (3) that certain available evidence supports the conclusion that annual mileage, which has long been felt to be an important measure of hazard, is a very significant cause of this unanalyzed variation among risks.

Section B presents a method for obtaining relativities among groups on which a multiple classification system has been imposed. The customary method of calculating class relativities uses the total experience for each class with all subdivisions within the classes added together. With the customary method it is difficult to make a completely accurate adjustment for different distributions by territory or merit rating, because any change in the class relativities disturbs the other sets of relativities and conversely. It is shown that even if such an adjustment were made, the customary method of calculating relativities one set at a time does not reflect the relative credibility of each subgroup and does not produce the best fit to the actual data. Moreover it produces differences between the actual data and the fitted values which are far too large to be caused by chance. In addition, for private passenger automobile insurance in Canada, it is shown that two sets of relativities which are multiplied together cannot produce the best fit to the actual data, and some of the consequences of trying to do so are explained. Some methods are advanced whereby all sets of relativities for classes, merit ratings, territories, and so forth, can be calculated simultaneously, which will overcome all the deficiencies in the customary method. These improved methods use the technique of minimizing a measure (technically known as the chi-square test) of the differences between the actual data and the fitted values. Some applications to other lines of insurance are mentioned.

Harald Cramér was nominated Professor in Actuarial Mathematics and Mathematical Statistics at the University of Stockholm in the late twenties. 1959 a book was “dedicated to Harald Cramér on his sixty-fifth birthday, in recognition of his pathbreaking work in probability and statistics”, with the title Probability and Statistics, The Harald Cramér Volume (Almqvist & Wiksell, Uppsala 1959; John Wiley & Sons, New York). The development during these three decades can be described in Cramér's own words in a recently published „Éloge de M. Paul Lévy.” “Pendant ces années” (the late twenties) “la théorie moderne des probabilités a commencé ce développement puissant qui, depuis cette époque, a continué sans cesse, et qui a fait de la théorie des probabilités une des branches les plus fécondes de l'analyse moderne, ayant les rapports intimes avec tous les domaines des mathématiques pures, et en même temps des applications pratiques dont l'étendue et l'importance vont toujours en croissant.”

A few historical notes will be given as a background for the review of The. Harald Cramér Volume, 1903-1934 Filip Lundberg formed the collective theory of risk. This theory is a particular part of the general theory of stochastic processes which latter was drafted, in its main lines, by Kolmogoroff, 1931 and 1933. (A stochastic process is constituted by a variable depending on chance which is simultaneously a function of one or more variates. For readers not used to this terminology such a variable may be exemplified with the total risk costs for a certain period paid for a group of policies issued by a fire insurance company. For simplicity, it is assumed that the group involves only agricultural risks, similar to each other and situated in the open country so that no conflagration can occur. Further we assume that during the period of observation no change occurs in the composition of the group neither through underwriting, nor through cancellation, nor through alteration.

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).

Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so that

If is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that