At the Lundberg Symposium, Stockholm 1968 Jung and Lundberg presented a report on similar problems as those treated in this note, and to the Astin colloquium, Berlin 1968 the present author presented a report with the same title as this note, where some of the results in the first-mentioned report were commented upon. Jung and Lundberg kindly discussed the topic here concerned with the present author some time after the colloquium. On account of this discussion, the present author withdrew his report from the publication in its original form. The following context is a revision and a completion of the author's report to the colloquium.

Let τ be a parameter measured on its original, absolute scale, and let be the same parameter measured on an operational scale with respect to the probability distribution (or the corresponding for t). The parameter will often be referred to as “time”, which does not imply a restriction of the theory to proper time parameters.

A random function X(s) is said to be distributed in a cPd i.w.s. (compound Poisson distribution in the wide sense), if the distribution function of X(s) for every fixed parameter point (s, τ) in a finite or infinite domaine of the parametric space as a function of τ can be written in the following general form

where the asterisk power m*, here and throughout this note, is taken to mean, for m ˃ O, the m times iterated convolution of the distribution function with itself, and, for m = o, unity. W(x, s) being the conditional distribution function of the size of one change in X(s) relative to the hypothesis that the change has occurred at s, here abbreviated to the change distribution. U(ν, τ) is a distribution function, called the structure function. In the general case V(x, s) and U(ν, τ) may depend on s and τ respectively. If, particularly, these functions are supposed to be independent of the parameter, they will be denoted V(x), U(ν) respectively. In the particular case, where V(x) = ε(ν—c1), c1 being an arbitrary but fixed constant and ε(ξ), here, and in the following context, the unity distribution equal to zero for negative values, and to unity for non-negative values of ξ, the cPd is said to be elementary and, in the opposite case, non-elementary. In the elementary case the distribution of X(s) is defined by the integral appearing in (Ia) with x = c1, so that W(x, s) = W(x) = I.

1. The claims experience of a motorcar insurance is assumed to give some indication about the risk (basic claim frequency) of that insurance. The experience rating systems in motor insurance are based on this supposition. In these systems the premium to be paid in some year is a function of the individual claims experience of preceding years. That premium can be considered as the difference between:

The bonus usually consists of at least the following components:

a. a component concerning the individual claim frequency.

b. an individual random factor.

c. a collective random factor.

Other factors, like the effect of the trend in the claim frequencies and the effect of the dependence of the claim frequency of age and experience of the driver are not taken into account in this paper.

2. The meaning of the three components mentioned in section I can be demonstrated by the following example.

The structure function U(χ) of the portfolio, being the distribution function of the individual basic claim frequencies χ is assumed to be defined by representing a gamma distribution with expected value q and variance qb. The probability Pi for an insurance to have i claims in a certain year is supposed to be

In this formula the factor y has the same value for all insurances in the portfolio in a certain year but varies from year to year according to the distribution function G(y) with expected value 1 and variance .

Although unnecessary assumptions are something we all try to avoid, advice on how to do so is much harder to come by than admonition. The most widely quoted dictum on the subject, often referred to by writers on philosophy as “Ockham's razor” and attributed generally to William of Ockham, states “Entia non sunt multiplicanda praeter necessitatem”. (Entities are not to be multiplied without necessity.) As pointed out in reference [I], however, the authenticity of this attribution is questionable.

The same reference mentions Newton's essentially similar statement in his Principia Mathematica of 1726. Hume [3] is credited by Tribus [2c] with pointing out in 1740 that the problem of statistical inference is to find an assignment of probabilities that “uses the available information and leaves the mind unbiased with respect to what is not known.” The difficulty is that often our data are incomplete and we do not know how to create an intelligible interpretation without filling in some gaps. Assumptions, like sin, are much more easily condemned than avoided.

In the author's opinion, important results have been achieved in recent years toward solving the problem of how best to utilize data that might heretofore have been regarded as inadequate. The approach taken and the relevance of this work to certain actuarial problems will now be discussed.

Bias and Prejudice

One type of unnecessary assumption lies in the supposition that a given estimator is unbiased when in fact it has a bias. We need not discuss this aspect of our subject at length here since what we might consider the scalar case of the general problem is well covered in textbooks and papers on sampling theory. Suffice it to say that an estimator is said to be biased if its expected value differs by an incalculable degree from the quantity being estimated. Such differences can arise either through faulty procedures of data collection or through use of biased mathematical formulas. It should be realized that biased formulas and procedures are not necessarily improper when their variance, when added to the bias, is sufficiently small as to yield a mean square error lower than the variance of an alternative, unbiased estimator.

This paper describes a rather simple application of credibility which was used to develop experience rating plans for Blue Cross and Blue Shield organizations in the United States. These are non-profit corporations which provide prepaid coverage for the cost of hospitalization and medical-surgical care. The benefits are in many instances full service benefits so that there is no additional cost to the subscribing member.

While coverage is being granted to individual applicants, the bulk of the unterwriting is composed of groups, that is employees of a given commercial entity. Since the plans are voluntary and, in many instances, the employer either acts only as collecting agent for his employees or else pays only a fraction of the premium, certain minimum participation percentages have been established to prevent anti-selection.

Because of the competion of Insurance Companies which have concentrated their efforts on larger and more profitable groups (especially those where the employer pays all of the premiums) it was early recognized that a uniform or “community” rate will result in a gradual loss of groups with good experience thereby requiring substantial increases in the average rates. The original program of experience rating which the author has developed for the Massachusetts Blue Cross in 1949 has been since adopted “mutatis mutandis” by a number of other Blue Cross and Blue Shield organizations.

The problem of estimating the final amount after settlement of Motor and General Third Party claims incurred during an underwriting year is of considerable importance for direct writing companies, as well as for reinsurers. While methods of considerable precision have been proposed for direct writing companies' business (ref. 1,2), a less exact but simpler method, which is applicable, i.a., to non-proportional reinsurance, could also be of interest.

The method uses the so-called “chain relatives” (ref. 3) computed from the figures for incurred losses similar to those quoted on p. 157, ref. 1. The year in which losses Si,j have occurred is designated by the subscript i, whilst the years in which the corresponding amount of losses paid plus outstanding has been ascertained (or estimated) are designated by the subscript j.

“Chain relatives” are defined as a ratio

If the available statistics embrace n years, ai,j can be computed for every j = 1, 2, 3, … n—1 but only for i ≤ n—j. To obtain estimates of ai,j for every i > n—j up to i = n we shall use E(ai,j)for fixed j.

The underlying hypothesis concerning ai,j is that every ai,j consists of a systematic part and of a random part. The random parts can always be assumed to have the mean = 0. For fixed j the systematic parts are a function of the index for the value of money and of several parameters defining the process of claims settlement. The index for the value of money obviously being constant for each fixed j, it is assumed that the other parameters are also constant for fixed j. This, of course, presupposes that the average proportion (over the years of occurrence) of settlement of the incurred losses is independent of their initial volume and of the year of occurrence.

Suppose that an automobile insurance plan is characterized by a double classification. The risks are thus divided into classes, i = 1, 2, …, p (e.g. defined by use of car and age of operator), and groups j = 1, 2, …, q (e.g. defined by licence and by accidents during the last three years). The experience of the company is described by the observed “relative loss ratios” rij and some measure of exposure nij. A general model, often used, is that the rij: s are observations of random variables with the expected values gij = g(αi, βj), where the relativities αi are parameters representing the classes i and the relativities βj represent the influence of the groups j. One of the ratemaker's problems is to find a realistic function g(α, β) and to obtain estimates ai of αi and bj of βj.

In their paper “Two Studies in Automobile Insurance Rate-making” (ASTIN Bulletin Vol. I, Part IV, page 192-217) Robert Bailey and LeRoy Simon have thoroughly analyzed this problem for private passenger automobiles in Canada. They have principally studied three different types of the function g(α β), namely g(α β) = αβ (Method 2), g(α β) = α + β (Method 3) and g(α β) = 3αβ— 2 (Method 4). The authors show in an appendix, that the variance of rij is approximately g(αi βj)/Knij where K ≅ 0.005 for the Canadian data. They estimate the relativities αi and βj; by making χ2 = K. Σnij(rij — gij)2/gij a minimum. For the Canadian material, the “method 4” agrees best with the observations. This method gives an observed χ2 value of about 8 for 11 degrees of freedom.

The purpose of this paper is to describe a possible application on the rating of excess of loss covers of Mr. Bühlmann's work on Experience Rating and Credibility). One of the most important problems in connection with the rating of such treaties consists in estimating the number of excess claims and the average excess claim amount. Especially with cases where claims data are scarce there is a temptation to estimate these two quantities by means of the credibility theory. This approach leads, on the one hand, to a relatively complicated formula when considering the average excess claim amount, and, on the other, to a rather simple one for the credibility factor of the number of excess claims.

One of the basic functions of risk theory is the so-called generalised Poisson function F(x), which gives the probability that the total amount of claims ξ does not exceed some given limit x during a year (or during some other fixed time period). For F(x) is obtained the well known expansion

where n is the expected number of claims during this time period and Sk*(x) is the k:th convolution of the distribution function S(z) of the size of one claim. The formula (1) is, however, much too inconvenient for numerical computations and for most other applications. One of the main problems of risk theory, which is still partly open, is to find suitable methods to compute, or at least to approximate, the generalised Poisson function.

A frequently used approximation is to replace F(x) by the normal distribution function having the same mean and standard deviation as F as follows:

where α1 and α2 are the first zero-moments of S(z):

SM(Z) is here again the distribution function of the size of one claim. To obtain more general results a reinsurance arrangement is assumed under which the maximum net retention is M. Hence the portfolio on the company's own retention is considered. If the reinsurance is of Excess of Loss type, then

where S(z) is the distribution function of the size of one total claim.

This report aims at describing the procedures and statistics prepared for use in calculation of the limits of the equalisation reserves of Finnish insurance companies. The report is based on work done by a committee which the Federation of Finnish Insurance Companies set up in 1962. Its purpose was to collect and work up the necessary statistics and to develop computational methods to be used in practice for the computation of equalisation reserves. In one respect the work was initiated by the Supervisory Service, which prepared new and more precise regulations concerning the determination of limits of the above reserves.

Because solvency problems of insurance companies have had great attention in recent years internationally, and the Finnish equalisation reserves are closely related to them, the theme may be sufficiently interesting to be presented once more setting out practical results. The solvency problem among other questions has been dealt with by Drs. Pesonen and Pentikäinen in former colloquia. This report can be considered as a continuation of Dr. Pesonen's paper “Magnitude Control of Technical Reserves in Finland” submitted to the Lucerne Colloquium and my paper “A Procedure to Compute Values of the Generalised Poisson Function” to the same colloquium.

The principles and theoretical formulae to be used in computation of the limits for the equalisation reserves have already been stated in the papers of Drs. Pesonen and Pentikainen. Other technical reserves are not dealt with here.

The equalisation reserve is roughly defined as that part of the technical reserves which exceeds the conventional premium reserve and the claims reserve and which secures the company's solvability against random fluctuations of claims and fluctuation of the basic probabilities of the claims. The equalisation reserve E must be greater than a minimum Emin from solvability reasons and, because of taxation, should not exceed a maximum amount Emax.

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.

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.

This note deals with a way to determine upper and lower bounds for the coefficient of variation of the total claim costs in a year in excess of a certain limit value. In many a publication attention has been paid to the question of calculating, in addition to the mean of the excess costs, the variance of these costs. In this connection we only mention here the study of Vajda: Minimum Variance Reinsurance, ASTIN-Bulletin, September 1962. It is evident that in studying such problems of “statistics of large claims” the coefficient of variation, defined as the ratio of the standard deviation and the mean, may be a useful tool. If we exclude special “dangerous” claim distributions and also distributions with a very “irregular” tail, it appears possible to derive bounds for this coefficient of variation and to indicate its asymptotic behaviour. Hereby good use can be made of Jensen's inequality for convex functions. Jensen's inequality has frequently been applied to problems in the field of life insurance mathematics, but as far as the authors of this note know, not to the question mentioned here.

Besides, it may have some interest to point out a connection between the question of estimating the excess of total claim costs and the theory of life times. We may translate “the limit value” as “the age already reached” and “the mean of the excess of loss” as “the expectation of life”. The estimation of the coefficient of variation ot the life time now leads to some well-known biometric formulae on life times and at the same time to an interesting observation as regards human life tables.

Let us consider the following problem.

A motorist has decided to effect an accident insurance under the following conditions. The insurance runs for one year. The premium for the first year amounts E0. If no damages have been claimed during i successive years, i = 1, 2 or 3 the premium is reduced to Ei. After four years of damagefree driving no further premium reduction is granted, so the premium remains E3. The premium is due on the first day of the year. The own risk amounts a0.

The number of accidents of our motorist during a time period T is assumed to be Poisson distributed with parameter λT. The extent of the damage has distribution function F(s) with finite mean and variance.

The problem of our motorist will be to decide whether to claim a damage or not. He will have to develop a strategy that specifies his decisions in every possible situation. His strategy will be called optimal if it minimizes the expected costs in the long run.

We may expect that in view of the premium reduction, it will be unprofitable to claim damages which are not much larger than a0. Once a damage is claimed it will be profitable to claim all following damages that exceed a0 during the remaining part of the year.

Hence his decisions will also depend on the time of the year and the premium paid at the beginning of that year. So we distinguish between four types of years, for each premium one.

Les colloques successifs d'Astin ent eu le mérite de clarifier certaines idées théoriques importantes. En invitant au colloque d'Arnhem les actuaires á penser aux méthodes de Monte Carlo, les organisateurs ont forcément orienté les recherches vers le domaine des applications. Car simuler, c'est á. dire établir des résultats conformes á un processus aléatoire bien déterminé, suppose á priori que les données numériques d'introduction (ou de définition) ont été fixées.

Nous ne reprendrons pas les principes de la méthode de simulation, ils sont donnés de maniére magistrale dans “Notes on operations research 1959” du centre de recherche opérationnelle du Massachusetts Institute of Technology. Le chapitre II, portant titre „Simulation of Random processes” rédigé par Herbert P. Galliher donne un exposé complet du probléme. Nous renvoyons á cette référence.

Parmi les idées que l'on doit retenir, car elle est d'application dans tous les domaines, nous citons „Never simulate a single numerical problem;always simulate several of a class of problems”.

Dans le domaine non life, cet aspect est fondamental. Ce qui y fait défaut, ce sont des algorithmes généraux de résolution numériques. Une des difficultés majeures rencontrées par les actuaires a été et est encore le calcul effectif des convolutions. Ceux qui ont établi cette note se sont posé la question de savoir si la méthode de Monte Carlo ne pouvait fournir un procédé:

— évitant les convolutions.

— suffisamment général pour être applicable à une classe de risques arbitraire.

— donnant comme sous-produit des éléments intéressant aussi bien les théoriciens que les praticiens.

In this paper we will investigate the following reinsurance problem: An insurer, whose total claims for a certain period may be regarded as a random variable x with expected value Ex = m, wishes to cede part of his business to a reinsurer. A reinsurance treaty will consist of rule for the division of x between the two parties. For any observed value of x it should define uniquely what amount should be borne by the ceding insurer. The amount borne by the reinsurer is then simply the remaining part of x.

We shall assume that the insurer has already decided how much of his business he wishes to cede, in the sense that he wants to retain a part of the total risk with expected value m — c, where c is a fixed constant, o < c < m.

Using the terminology introduced by Kahn in (2) we will describe a reinsurance contract by a transformation (or function) T that for a given x yields the amount Tx borne by the cedent. The random variable x is thus divided into two parts

and the properties of the reinsurance contract described by T are summarized in the distributions of the two random variables Tx and (1 — T)x = — Tx.

The motivation for reinsurance is generally held to be a desire for stability, in other words the cedent wishes to choose a T such that the random fluctuations in Tx are in some sense smaller than those of x. This choice will in our case be performed under the restriction that ETx = m — c.

It is clear that we can never talk about an optimal choice of T without defining exactly what criterion we shall use when comparing two transformations, T1 and T2. According to the above, the criterion should refer to the properties of the distributions of T1x and T2x, so that if one distribution is “more concentrated” around some central value, the corresponding transformation is deemed preferable to the other. However, we still have to define what we mean by “more concentrated”.

The aim of this study is to find suitable methods for utilizing available statistical information on the risk development of terminated insurances. This information may comprise not only data respecting sex, age, insurance tariff, etc., but also the points of time when certain events—disability, injuries or damages—have occurred as well as their durability and cause. The method study here presented is based on statistical material gathered from long-term disability insurances which have ceased to be valid, either in consequence of the expiration of the term of insurance, or the insured's death. We endeavour to study the point of time when the first event (disability) occurred and the relation between subsequent events, searching suitable methods for assessing risks after the occurrence of the primary event.

The influence of passed events upon the future risk process may be explained by either a direct dependence between actual events—actual damage or disability may have an impairment for the future of the risk—or by heterogeneity a priori. In the former case, we may speak of a contagious process (Polya-Eggenberger), and, in the latter case, the occurrence or non-occurrence of an event may, according to Lexis and Newbold, act as a risk differentiating factor providing a theoretical basis for a technique of experience rating.

It is difficult to decide whether the relationship noted in respect of a risk process is of the direct or the indirect kind. A direct dependence between one event and the following ones has no relevance to a study of the incidence of the first event and the time of its occurrence. We avoid in such a study the obstacles created by formal rules and regulations with regard to the problem of whether two or more disability periods occurring at short intervals should be considered as a single disability period or not. In order to measure the frequency of the first event we have—as in the case of measuring mortality—to limit the period exposed to risk until the occurrence of the first event.

1.1. — In this paper we shall consider some of the decisions which have to be made in the normal course of business in an insurance company. We shall see that the “right” decisions can be found only when the problems are analysed in their proper dynamic context.

As examples of the decision problems which we shall study, we can mention the following:

(i) What premium rates should be quoted on the insurance contracts, which the company offers to the public?

(ii) How much should the company spend to promote the sale of its policies?

(iii) When should the company refuse to underwrite a proposed insurance contract?

(iv) How shall the company reinsure its portfolio of insurance contracts?

(v) What reserve funds should an insurance company keep?

(vi) How shall the company's funds be invested?

Any actuary will be familiar with such problems, and he will probably feel that these problems cannot be satisfactorily solved with the methods offered by the classical actuarial theory.

1.2. — In some earlier papers [I] and [2] it has been argued that such problems can best be solved in the frame work of utility theory. As an illustration we shall take Problem (iii) in the preceding paragraph, and consider an insurance company in the following situation:

(i) The company has a capital S.

(ii) The company holds a portfolio of insurance contracts which will lead to a total payment of x to settle claims. F1(x) is the distribution of the variate x.

Mr. Benedikt uses “chain relatives” based on the incurred claim totals included in Part 5 of Schedule “P” of the annual statement required for fire and casualty companies in the United States. Each total is for the losses as developed to end of calendar year (j) for claims incurred because of accidents in calendar year (i). Each total is the sum of the actual payments made before the end of year (j) plus the reserve for estimated payments to be made after the end of year (j) for claims incurred in year (i). The “chain relatives” are ratios. The “chain relative” ai,j is the ratio of developed losses to end of (j + 1) to the developed losses at the end of year (j).

Each total of Part 5 of Schedule “P” equals the sum of the total payments to date plus the total reserves for future payments for the corresponding classification of claims. Separate totals for these amounts are given in Part 1 of Schedule “P”. The totals of Part 5 are not secured directly from Part 1 because Part 1 gives totals by policy year of issue only and Part 5 separates the totals by policy of issue by calendar year in which claims are incurred. The two parts are prepared from the same basic claim information and agree in total.

The accumulated total paid losses for most casualty lines increase with passage of time. This accumulated total for paid losses can be reduced only if there are recoveries for losses previously paid, such as can occur for auto collision. For auto collision, the insurance company for this insurance can pay the insured for the damage to his car and then later recover from the insurance company that provided the liability insurance for another car involved in the same accident. Such substantial recoveries normally do not occur for auto liability insurance for bodily injury and property damage.