The precise wording of the heading is deliberate. The Canadian system is not a “bonus” nor is it properly described as “no claim”; it is a sub-classification of insureds according to whether they have been accident-free during a term of years ending at the inception of the insurance period. Specific rates are quoted in the Tariff for each sub-class, based broadly on the relative experiences; as a matter of arithmetical convenience, premium rates are calculated by applying discounts or surcharges to a “key” rate, but the percentage discounts or surcharges are not quoted in the Tariff, the Policy or the renewal papers.

In the early 1930's a “no claim bonus” on British lines was part of the Canadian automobile insurance system. A percentage reduction or “bonus” was allowed from the premium specified in the Policy, provided the insured had owned and operated a private passenger car, free of accident or revocation of driving license, for a period of up to three years before the inception of the Policy. At that time the Automobile Insurance Market was profitable and highly competitive, both as between Companies and as between Agents. Because of inter-Company competition, confirmation of entitlement to the bonus was rarely attempted. Often it was not possible, as most Companies did not maintain an alphabetical index of insureds; an effort to set up an inter-Company claim-recording bureau broke down through sheer weight of numbers of claims.

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.

In some recent papers ((1), (2) and (3)) about reinsurance problems I have made extensive use of utility concepts. It has been shown that if a company follows well defined objectives in its reinsurance policy, these objectives can be represented by a utility function which the company seeks to maximise. This formulation of the problem will in general make it possible to determine a unique reinsurance arrangement which is optimal when the company's objectives and external situation are given.

More than 50 years ago Guldberg (4) wrote (about the probability of ruin): “Wie hoch diese Wahrscheinlichkeit gegriffen werden soil, muss dent subjektiven Ermessen oder von Aussen kommenden Bedingungen überlassen bleiben”. This is the traditional approach to reinsurance problems. It does obviously not lead to a determinate solution. Most authors taking this approach conclude their studies by giving a mathematical relation between some measure of “stability”, such as the probability of ruin, and some parameter, for instance maximum retention, to which the company can give any value within a certain range. Such studies do usually not state which particular value the company should select for this parameter, i.e. what degree of stability it should settle for. This question is apparently considered as being outside the field of actuarial mathematics.

According to the provisions of the Swedish tables of rates the premium payable on a certain policy both for Third Party Liability and for Car Damage Insurance (without clause for particular self retention) is equal to the initial premium reduced with regard to points earned as related to years with no indemnities. One such point is earned for each year during which no indemnity has been paid on the policy; upon an indemnity one or two points are lost. The detailed rules for the reduction have been altered from time to time, the existing rules have been valid from the year 1952, these rules will here be called system II, while system I denotes the rules valid during the years immediately previous to 1952. The systems are described in the following table.

The probabilities for the occurrence of a claim and for the occurrence of a lapse for a policy can be considered functions of two variables, the points obtained in the bonus scale, b, and the “duration” i.e. the time elapsed since the policyholder signed his first policy, τ. For simplicity two kinds of models have been constructed, one assuming the probabilities to depend only on τ, the other assuming the probabilities to depend only on b.

By a general theorem the necessary and sufficient condition for a function φ0 (σ) being completely monotonic for σ lying in the right semi-plane, i.e. that the nth derivative with respect to σ has the sign of (— 1)n, is that the function may be represented by the Laplace-Stieltjes integral , where U (v) is a non-decreasing function of v, independent of σ and bounded in every finite interval, σ a real or complex variable represented in the right semi-plane, s a real constant ≤ the real part of σ. By the notation φn (σ) we designate , which for U (v) being independent of σ, as assumed above, is equal to .

Definition 1. A compound Poisson process (in the narrow sense) is a process for which the probability distribution of the number of changes in the random function Y (t), constituting the process, occurring while the parameter, which is represented on the positive real axis, is in the interval (o, t) for every value of t, is defined by the following relation

the function φn (σ) being defined by the integral given above and subject to the condition that φ0 (σ) tends to unity, when σ tends to s. The function U (v) in the integrand of φn (σ) is, then, a distribution function which defines the risk distribution, in this case said to be t-independent.

Although actuaries have been connected with life insurance business for more than 100 years, it is only more recently that actuarial activities in the non-life branches have taken place.

The problem of calculating level premiums for the steadily increasing risk in life insurance immediately calls for mathematical ability. The need for mathematically trained specialists is not so easily seen when the problem is to calculate premium rates for fire and casualty insurances. The premium rates in this field have to a great extent been fixed by persons with none or very little knowledge in mathematics and mathematical statistics. When the market conditions allow for considerable margins this arrangement works out very well. The statistical investigation needed later on for controlling the premium rates is not very complicated, and the premium rates can if necessary be changed on short notice. A mistake made in the premium stipulation, will generally not have a lasting effect. In life insurance on the other hand, the contracts often stretch over many years and a mistake in the premium stipulation cannot easily be corrected. These circumstances explain the reason why actuaries have been more often engaged in life insurance than in the fire and casualty branches.

But a closer study shows that from a mathematical point of view, the proper fixing of premium rates is often more complicated in the fire and casualty branches than in life. In the fire and casualty field a steady changing of market conditions seems to take place. The competition grows keener and keener, and the margins are cut down. Sooner or later we reach a point where it is of vital importance to know as exactly as possible where we stand. This means that we have to study the risk premium, the security loading and the expense loading separately, and to estimate these quantities with the highest efficiency.

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.

La sélection des risques est un des premiers principes fondamentaux que la théorie de l'assurance a dégagés. Il se traduit essentiellement par la répartition des risques en classes de tarif homogènes et il prémunit l'assureur contre le danger de voir „le mauvais risque chasser le bon”.

Pour créer des classes de tarif homogènes, il faut théoriquement repérer tous les facteurs influençant le risque et en chiffrer l'effet. Si cela est fait, la fluctuation des résultats individuels autour de la moyenne n'est que l'effet accidentel du hasard et ne peut donner lieu a posteriori à rectification de la prime: il n'y a rien d'inéquitable à ce que les titulaires des contrats non sinistrés paient pour les autres puisque tous sont égaux devant le risque.

Comme dans toute science appliquée, la mise en oeuvre des principes se heurte parfois à de grandes difficultés d'exécution. Dans la pratique il n'est généralement pas question de repérer tous les facteurs du risque, mais seulement les principaux, ne fût-ce qu'en raison de l'impossibilité de conduire des études statistiques valables sur une multitude de groupes peu étoffés. On ne peut donc faire grief au praticien d'estimer qu'il a créé des classes homogènes lorsque celles-ci tiennent compte des principaux facteurs de risque.

Mais l'optique change si, pour l'une ou l'autre raison qu'il importe peu de préciser, le tarif néglige un ou des facteurs dont le bon sens et l'expérience — à défaut du calcul — prouvent l'influence appréciable.

Thank you, Mr. Masterson, Sir George and friends. Mr. Masterson told you that Mr. Longley-Cook was going to give this report originally. Since he can't, he has turned it over to me, and I shall deliver it practically the way he has written it.

The profession of actuary originating in the need for a scientific approach to the problems of early life insurance companies, has broadened its scope over the years, and many actuaries now provide valuable contributions to the scientific study of problems in insurance, commerce and industry which are completely unrelated to life insurance. Because a large majority of actuaries are concerned, however, with problems of life insurance and pension plans, those outside these fields have felt the need for an association which is concerned with their special interests.

I feel most honoured to take part in the birth of Astin, an association for the study of insurance problems outside of the life insurance field. It was most wise that Astin has been formed as a section of the International Congress and not as an independent organization because, although our problems may be different, we will gain much in our mutual association.

A specialist should not, in his devotion to his particular line of study, lose touch with the broader and more general developments of science. Cross-fertilization of ideas can, on many occasions, prove invaluable. It was over 48 years ago that Mr. Stanley Otis and others suggested that a society of actuaries and statisticians of liability companies be formed. Some five years later, the Casualty Actuarial and Statistical Society of America—later to be known by the shorter title of Casualty Actuarial Society—was inaugurated.

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.

The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such that

where c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

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

The following comments on the papers written in English, French and German under Subject IVA and published in Trans. XVth. International Congress of Actuaries 1957, formed the basis of an address by Mr. C. Philipson, Sweden, on Thursday 17th October, to the members of Astin in New York.

Let sn be the maximum value of si and tn the minimum value of the angular coefficient of the straight line connecting the point (si, Ai) with point (1, 1), when i takes the values 1, 2 … n and the point (si, Ai) is taken to mean the vertex of the convex curve representing the generating function of the ith favourable risk. In a portfolio of m + n independent risks—where m are unfavourable and n favourable—Franckx finds that the superior limit of the probability that the aggregate amount of claims shall not exceed k can be written

, where Im is taken to mean the product of the generating functions for the m unfavourable risks.

Speaking of the theory of risk as first given by Filip Lundberg and developed by Cramér and the Scandinavian School he writes:

“Cela n'empêche qu'il y a encore un monde entre la théorie et la pratique et que les actuaires n'auront la partie gagnée que dans la mesure ou leurs travaux seront suffissamment simples que pour qu'ils soient et compréhensibles et directement utilisables”.

La Caisse nationale suisse d'assurance en cas d'accidents à Lucerne, chargée du service de l'assurance obligatoire en cas d'accidents en Suisse, a publié il y a quelque temps les „Résultats de la statistique des accidents de la huitième période quinquennale 1953-1957”. Cette publication, qui est la huitième, paraît tous les cinq ans en français et en allemand). Ces différents rapports donnent un aperçu intéressant de la situation de l'assurance-accidents obligatoire en Suisse. Ils constituent dans leur ensemble une documentation précieuse touchant l'un des aspects du développement de la sécurité sociale en Suisse depuis 1918. C'est en effet à cette date que l'assurance obligatoire en cas d'accidents est entrée en vigueur et que la Caisse nationale suisse d'assurance en cas d'accidents, établissement de droit public autonome, a commencé son activité.

Le dernier rapport sera aussi consulté avec fruit par quiconque s'intéresse aux problèmes de l'assurance sociale. Cette publication d'environ 180 pages, illustrée par de nombreux tableaux, graphiques et illustrations, fournit une quantité d'indications très utiles présentées de façon claire. Les premiers chapitres sont surtout consacrés aux statistiques tirées du règlement des sinistres, tandis que les derniers chapitres traitent de quelques problèmes d'assurance spéciaux qui sont particulièrement actuels.