Let me begin with some practical examples of experience rating.

a) Swiss Automobile Tariff 1963

— Within each tariff-position there are 22 grades:

— The new owner of a car starts at grade 9

— The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car

— The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades.

b) Sliding Scale Premiums in Reinsurance

Excess of Loss Contracts often stipulate that:

The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows:

subject to a minimum of 0,04

and a maximum of 0,08

c) Participation in Mortality Profit in Group Life Insurance

A group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formula

refund = x% gross premiums — y% claims (where x < y)

All these examples fall under the heading “Experience Rating”. What do they have in common?

Definition: A system by which the premium of the individual risk depends upon the claims experience of this same individual risk.

As reports on work done on subjects discussed at earlier colloquia and on new lines of investigation 6 papers have been laid down on my table (To be exact, 5 on my table and the last one on my chair at the beginning of this colloquium on Wednesday morning).

Four of them deal with problems concerning the estimation of reserves. These are the papers by Beard, Molinaro, Benedikt and Hovinen.

The other two papers, each of which has two authors, namely Kauppi and Ojantakanen and van Klinken and Groenenberg deal with mathematical questions of more general interest.

I can repeat what my colleagues have said this week:

Although the number of the papers is rather small, the importance and the quality of them seem to me to be very great.

The paper of R. E. Beard concerns the studies he has made on the data provided by the 6th Conference of European Insurance Supervisory Services. Suitable statistics of motor insurance on which to experiment have been made available relating to 28 European insurance companies. Beard has made certain preliminary studies of these statistics. His paper has been written against the background of these studies. Although only the broadest indications of numerical values are given, since the detailed statistics are confidential, nevertheless the comments made are based on the studies of the figures. Therefore these comments have a considerable practical utility.

As a basic principle on which to build calculations of technical reserves it is assumed that the claim frequency rate will be constant and that the distribution of claims by amount will be stable, both apart from random fluctuations. Although it is hardly possible to prove the validity of these assumptions, there could be no basis for calculation if these principles were not true.

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.

I) Parmi les différentes questions qui se posent dans l'étude de l'assurance automobile, celle de l'évaluation des réserves techniques est sans doute une des plus importantes. Dès que l'augmentation progressive du parc des véhicules a entraîné une hausse sensible de la chiffre des affaires et que le besoin de sauvegarder l'indemnisation des sinistres de circulation a transformé la base même du rapport contractuel, le problème des garanties, indissociable de celui de la solvabilité de l'entreprise, s'est imposé à l'attention générale.

En raison de cette évolution et de la nécessité d'un contrôle sur les engagements des compagnies, la fixation du montant exact de la réserve sinistres est fondamentale soit par rapport à la validité du tarif adopté (congruité des primes), soit à la possibilité de faire face aux engagements techniques moyennant des activités correspondantes. Deux aspects done d'un même problème, dont l'importance est augmentée dès que les risques se sont alourdis, les résultats de la branche défavorables, la situation économique et monétaire de plusieurs pays est devenue instable.

2) Le problème du calcul de la réserve pour sinistres en cours avait été posé, il y a plus que trente ans, par M. d'Addario: dans ses études il avait mis en relief l'existence d'une relation dans l'échelonnement des paiements valable pour une période de stabilité monétaire et d'une loi de distribution par rapport à l'âge du portefeuille sinistres (génération sinistres)). A cette oeuvre, disons prioritaire, ont suivi les remarquables travaux de M. M. Thépaut, Burlot, Henry, Pellegrin, Depoid: à la lumière des données expérimentales du marché français ils ont eu le grand mérite de pousser les études sur le coût moyen des sinistres et sur la cadence des règlements.

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.

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.

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.

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.

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.

1.1. In the different versions of the “Theory of Risk” it is almost universally assumed that ruin or bankruptcy marks the end of the game. The earlier versions of the theory tried to estimate the probability of this event, and studied the steps which an insurance company could take to bring probability of ruin down to an acceptable level. The more modern versions of the theory of risk tend to formulate the problem in economic terms, and study the cost of postponing or avoiding ruin.

In a recent discussion of a paper [4] surveying the development of the theory of risk, Professor Bather suggested that ruin may not necessarily be the end. If an otherwise sound insurance company gets into difficulties, so that ruin looms large, it is very likely that steps will be taken to rescue the company, for instance by refinancing, or in more extreme cases, by a merger.

1.2. To practical insurance men the simple suggestion of Professor Bather may seem next to trivial. Insurance companies get into difficulties fairly regularly, and rescue operations are considered in the insurance world, if not daily, at least annualy. The suggestion has, however, far-reaching implications for the theory of risk, and these do not seem to have been fully realised. If ruin does not mean the end of the game, but only the necessity of raising additional money, the current theories of risk may have to be radically revised. In this paper we shall discuss some of these implications.

1. Some years ago I discussed optimal reinsurance treaties, without trying to give a precise definition of this term [1]. I suggested that a reinsurance contract could be called “most efficient” if it, for a given net premium, maximized the reduction of the variance in the claim distribution of the ceding company. I proved under fairly restricted conditions that the Stop Loss contract was most efficient in this respect.

I do not consider this a particularly interesting result. I pointed out at the time that there are two parties to a reinsurance contract, and that an arrangement which is very attractive to one party, may be quite unacceptable to the other.

2. In spite of my own reservations, it seems that this result —which I did not think deserved to be called a theorem—has caused some interest. Kahn [4] has proved that the result is valid under far more general conditions, and recently Ohlin [5] has proved that the result holds for a much more general class of measures of dispersion.

In view of these generalizations it might be useful to state once more, why I think the original result has relatively little interest. In doing so, it is by no means my purpose to reduce the value of the mathematical generalizations of Kahn and Ohlin. Such work has a value in itself, whether the results are immediately useful or not. I merely want to point out that there are other lines of research, which appear more promising, if our purpose is to develop a realistic theory of insurance.

1. In a paper presented to the fifth Astin Colloquium (Lucerne, 1965) Bühlmann has given some propositions with regard to Experience Rating understood as a sequence of estimates of the expectation with respect to the distribution function of θ, H(θ) say, of μ(θ) which is the mean for fixed θ with respect to the distribution function G(x; θ) of the variables for each value of ν = 1, 2 … n. In such problems the estimator function for μ(θ) is generally chosen to be the conditional mean of μ(θ) for a given set of observed values of This is generally justified by the principle of least square deviation. According to Bühlmann this justification is not sufficient. Therefore, he bases the choice of this estimator function upon a postulate of equilibrium, described in the following lines.

Let X′ be a subset of X, and C(X′) a cylinder with the base X′ ⊂ X in the product space X × Θ, where X is the set of all possible and Θ the set of all possible θ, then the postulate of equilibrium implies the equality between the expectations of μ(θ) and of a function of on each cylinder C(X′). This is exactly Kolmogoroff's definition of the conditional expectation:

Bühlmann states, further, that the best linear estimate of (1) based on the arithmetic mean of n sample values xν of ν = 1, 2 … n can be written in the form of the Credibility Formula:

where the symbol E[·] denotes the expectation over the product space X × Θ and

If the are independent and identically distributed, this leads to where k = E [σ2 (θ)] / Var [ν(θ)], and σ2(θ) is the variance for a fixed θ of G(x; θ). This case of the credibility formula is, generally, applied in American practice. (The proposition has later been proved for more general conditions).