The relationships between actuarial and pure mathematics are curious. Actuaries have contributed to the development of mathematical theory: it is sufficient to mention, as examples, Fredholm of an earlier, and Cramér of a more recent generation. Scandinavian mathematicians, in particular, have been concerned with a very special type of stochastic process, reflected in the collective theory of risk, and the work of Philipson, Ammeter and others in this field is well known to readers of this Bulletin. However, the main stream of the theory of stochastic processes has little contact with actuarial applications.

On the other hand, many actuaries have studied and assimilated pure mathematics and have thrown light on actuarial matters by describing their own preoccupations in the terminology of modern, often abstract, mathematics. E. Franckx is one of their number.

The Instituto di Matematica Finanziaria of the University of Trieste (Faculty of Economics and Commerce) has published a booklet entitled

Essai d'une théorie opérationnelle des risques Markoviens which contains three lectures delivered by Professor Franckx in Trieste and a contribution which he presented to the 17th Congress of Actuaries, held in London in 1964.

The present referendum is only indirectly connected with the themes of the fifth Colloquium of ASTIN. It is more or less a memorandum about the actual work of Polish actuaries in the sector of the Insurance Theory and trying out a new conception and solution of social and economic problems of Property and Capital Insurances.

The principles and hypotheses of the cybernetic insurance theory posed in this referendum are a result of social situations in a planned economy. However, it is assumed that the cybernetic analysis of the economic role of insurance, and the methods of calculation resulting from this analysis, can also be applied to other social and economic situations.

In the judgment of the author, the problem under consideration would be of interest to the members of ASTIN and could possibly be included in the syllabus of one of the subsequent colloquia of ASTIN.

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

Discriminant analysis is an application of multivariate analysis, which may have its use in determining accident risk levels and premiums of industrial enterprises. This paper only aims to give some suggestions. The following questions will be considered.

1. Determining a discriminant function which makes it possible to discriminate between the risk levels of the industrial branches in an efficient way. The industrial branches comprise enterprises with comparable risk levels, hence they are to be considered as homogeneous groups. The function will at the same time serve as a means to classify seperate enterprises into one of these groups.

2. Fixing risk functions which enable us to rank the enterprises of an industrial branch to increasing risk on the ground of observations of a number of variates which characterize the risk situation.

3. Using these risk functions to calculate premiums. The classification-question mentioned under I was the reason to consider the technique of the discriminant analysis. By virtue of the Dutch Industrial Accidents Act every five years a tariffdecree is being published. This decree contains the premiums per wage-unit for the industrial branches. However, there are enterprises e.g. large compound enterprises which do not fall under these regulations. These enterprises ought to be classified according to their own experience. That means we need the knowledge of the risk levels of these particular enterprises in relation to the fixed risk levels of the industrial branches. As mentioned, this is a problem inherent to the typical Dutch situation. It seems, however, probable that such problems and the techniques we intend to sketch have a wider and more general meaning for the accident insurance.

In the present paper we study the question of how to allocate the reinsurance premium between the sub-portfolios when an excess of loss treaty is to be shared between several sub-portfolios. Several allocation schemes based on the expected value principle and the standard deviation principle are suggested. The calculations are relatively simple with unlimited free reinstatements. However, with limited and/or paid reinstatements the situation becomes rather tricky, and we therefore suggest a simulation scheme.

We present a dynamic pay-how-you-drive pricing scheme for motor insurance using telematics signals. More specifically, our approach allows the insurer to apply penalties to a baseline premium on the occurrence of events such as hard acceleration or braking. In addition, we incorporate a bonus-malus system (BMS) adapted for telematics data, providing a credibility component based on past telematics signals to the claim frequency predictions. We purposefully consider a weekly setting for our ratemaking approach to benefit from the signal’s high-frequency rate and to encourage safe driving via dynamic premium corrections. Moreover, we provide a detailed structure that allows our model to benefit from historical records and detailed telematics data collected weekly through an onboard device. We showcase our results numerically in a case study using data from an insurance company.

Motor insurers in the United Kingdom are not subject to Government control over the rating structures they use or the levels of premiums they charge. The market is highly competitive, and each insurer therefore needs to make the best estimate he can of the premium required for each category of risk, to produce a given level of profit. He also needs to estimate the extent to which departures from such premium levels can be justified, for example in order to have a simple rating structure or to meet competition.

The purpose of this note is to give some examples of the statistical tabulations being produced in one insurance office in the United Kingdom. The statistical system which has been developed covers many aspects of motor insurance management, but in this note we shall direct attention to just two, namely

(i) continuous review of the variations in the claims experience from one risk category to another, as a guide to the relative premiums required for the different categories; and

(ii) continuous review of the changing composition of the portfolio and of the movements in and out, to try to assess the extent to which gains and losses of business can be attributed to pricing differences, marketing strategies, etc.

There are two phases of difficulties in estimating a claims distribution. If we are going to estimate the claims distribution as accurately as possible, we should gather considerably long terms statistics. While economical and social environment will change. As a result the statistics gathered should be amended by a kind of trend value. One of difficulties here is the estimation of that trend value. Another difficulty is the estimation of claims distribution as being the stochastic distribution. In the case of considering claim size, the estimation becomes more difficult.

The intention of this paper is to propose an actual way of estimating stochastic claims distribution considering various kind of claim size by the use of a computer. Regarding the problem of amending claims distribution by a trend Value I will discuss at another time.

It is rather a pity that we received only three papers especially devoted to subject 2, “applications of methods of operations research and modern economic theory”. Maybe the time for real applications of operations research techniques in the insurance field has still to come. However, I am convinced that sooner or later such techniques—in connection with the use of the computor—will play a more or less important part in the management of insurance business. An impulse to this of course will be the appearance of —let me say—“textbooks” on this subject; such as the book written by Wolff, Methoden der Unternehmensforschung im Versicherungswesen, Springerverlag 1966, a copy of which I received a few days ago.

Meanwhile, the fact that only a few papers have been sent in gives me the opportunity to report on them in some detail and to make a few side remarks.

The studies presented are:

Borch. Dynamic decision problems in an insurance company, de Leve and Weeda. Driving with Markov-programming.

d'Hooge, Franckx and Gennart. Utilisation pratique de la méthode de simulation dans l'assurance “non-life”.

The paper Prof. Borch submitted, deals essentially with some problems related with random walk theory. Partly it can be considered to be the counterpart of the classical discontinuous ruin theory as its subject is the expected duration of life of the company and maximisation of the discounted value of dividend payments.

The question of calculating the expected duration of life and more specially the means the company has to influence its future lifetime involves reinsurance and utility theory to which topic in recent years Prof. Borch has devoted a number of very interesting publications.

Modern life is characterized by risks of different kind: some threatening all persons and some restricted to the owners of property, motor cars, etc., while still others are typical for some individuals or for special occupations. The corresponding accidents, losses or claims will occur suddenly and unexpectedly and may involve considerable financial loss. It is quite evident that modern life is a fit subject for risk theory, and that some results in the pure mathematic theory might have applications in the study of problems in real life.

In practice, however, we can identify risk theory with insurance risk theory or with the application of the theory of probability on insurance risk problems. This general definition has the advantage, that it covers a wide field of different risks and risk problems as specified in the insurance texts—and a great collection of risk situations ₌ claims occurred (with corresponding loss amounts) is available in the claims acts. In fact, I believe that any actuary or mathematician, starting his researches in risk theory or in risk statistics, should begin his studies by a series of actual claims acts.

La question est controversée de savoir si l'on doit tenir compte du revenu des placements — et éventuellement des bénéfices sur réalisations de valeurs — pour apprécier le résultat technique de la branche Automobiles.

Rappelons qu'en France, réglementairement, dans les états B3 qui font apparaître le résultat technique de la catégorie, on tient compte des frais généraux, ce qui est normal, mais non des intérêts.

En réalité, il semble bien qu'il soit difficile de faire abstraction d'un élément non négligeable de profits et pertes qui dérive de l'exercice même de l'industrie de l'Assurance, mais la question demeure de savoir dans quelle mesure il faut en tenir compte.

Nous allons montrer que la question est intimement liée à la conception même que l'on se fait du montant des réserves techniques à inscrire au passif du bilan.

On ne peut avoir une idée saine tant des résultats techniques que des revenus de placements que si la comptabilité est établie par exercice de souscription. C'est l'hypothèse que nous retiendrons dans ce qui va suivre).

Un tel exercice donne lieu à encaissement d'un certain montant de primes, au paiement de commissions et de frais généraux et enfin au paiement de sinistres s'échelonnant dans le temps, au delà de la clôture calendrier de l'exercice.

Pour plus de rigueur même — et l'on peut se référer au mode de raisonnement appliqué en Assurance sur la Vie — il faut considérer séparément les différentes échéances de primes, chacune se comportant comme une sorte d'exercice isolé: ensemble des assurances à échéance du Ier janvier, ensemble des assurances à échéance du 2 janvier, etc… jusqu'à celles à échéance du 31 décembre. (Il est indifférent de raisonner sur un ensemble ou „en valeur probable” selon le principe actuariel classique).

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.

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.

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.

In order to make this report clear to those without experience in insurance matters, we first present some basic facts about the insurance business. In so doing we intentionally omit certain facts irrelevant to the present study. The most important omission of this kind is our assumption that the insurance business operates without administrative expenses and without sales costs. We also assume that the insurance business is run in such a way that no profit is made. It will be evident to readers already directly connected with insurance problems what further omissions we have made and why we have made them.

Insurance is the establishment of a contract between the insurance company and the insured person This contract is by tradition called the “insurance policy” and the insured person is called the “policyholder”. By agreeing to the insurance policy, the policyholder commits himself to paying certain premiums to the insurance company, and the insurance company commits itself to paying certain amounts to the policyholders. The conditions under which such payments are to be made are of many different kinds: the policy-holder dies, the policyholder becomes ill, a homeowner's house is burnt down, or the policyholder has a collision in his car. The circumstances under which a payment is to be made to the policy-holder are described in detail in the insurance policy. The amount to be paid is either fixed or variable: in life insurance the amount to be paid is always fixed and stated in the insurance policy, but in most other forms of insurance the intention is that the insurance compensate the policyholder for the losses he might incur as a consequence of the events covered by his policy.