For a number of reasons it is important for an insurance company to estimate the claims costs of a year within the different branches of non-life insurance as soon as possible after the end of the year. The claims cost of a year is hereby defined as the total cost, before taking reinsurance into account, of all claims generated by events that have occurred during the year. When the estimation has to be done, part of these claims will be reported and closed, others will be reported and still open, and the remaining ones will be incurred but not yet reported. The total cost of the claims is defined as the sum of all payments that have been made or will be made on account of the claims. Thus, in this definition no regard is paid to interest, i.e. no discount factors are applied to payments to be made in the future.

Instead of considering a year, we could consider an arbitrary period of twelve consecutive months. The estimation problem is the same, and estimates of the claims costs of consecutive twelve months periods will allow a closer following up of trends and yield predictions for the present year.

For estimates to be available quickly, it is necessary that the estimation procedure be founded on data that are available immediately at the end of the year or the latest twelve months period. This means a.o. that for the bulk of the open claims, individual estimates of reserves by claims adjusters are out of the question. In other words, the estimation procedure has to be basically of a statistical character. In addition, for continuous estimates to be produced it has to be well adapted to electronic data processing. Indata to the procedure have to be stored in the memories of the computer.

This paper contains little which can be considered as new. It gives a survey of results which have been presented over the last 10-15 years. At one time these results seemed very promising, but in retrospect it is doubtful if they have fulfilled the expectations they raised. In this situation it may be useful to retrace one's steps and see if problems can be reformulated or if new approaches can be found.

Mathematical models have been used in insurance for a long time. One of the first was the Gompertz mortality law; a more recent model, which has been intensively studied is the Compound Poisson Distribution in Lundberg's risk theory.

When a model is introduced, one usually proceeds by stages. The first step is to see if the model appears acceptable on a priori reasons. If it does, the second step is to examine the implications of the model, to see if any of these are in obvious contradiction with observations. If the result of this examination is satisfactory, the third step is usually a statistical analysis to find out how well the model approximates the situation in real life, which one wants to analyse. If the model passes this second examination, the next and final step may be to estimate the parameters of the model, and use it in practice, i.e. to make decisions in the real world.

The advantage of working with a model is that it gives an overall purpose to the collection and analysis of data. A good model should tell us which data we need, and why.

Stimulated by the paper of W. S. Jewell) on multidimensional credibility I should like to show that at level 2 assumptions (Jewell's terminology) one can obtain explicit formulae for forecasting total losses in the future based upon total losses and number of claims observed in the past.

L'assurance est un jeu. Le rôle principal de l'actuaire est d'aider l'assureur à „trouver son adversaire”. En assurance-vie, le risque est défini par la table de mortalité et l'assuré est trouvé quant il déclare son âge et son état de santé. En assurance non life, la question n'est pas aussi simple. Mais la pratique commerciale a introduit la notion de classe du tarif. Les risques qui appartiennent à une classe de tarif n'appartiennent pas à une classe homogène, mais les conséquences d'un sinistre sont comparables au sens du calcul des probalités. La valeur moyenne d'un sinistre résultant de la classe de tarif est statistiquement stable. Ce qui fondamentalement a garanti le fonctionnement de l'assurance non life est l'application de la loi des grands nombres. La régularité statistique sera toujours l'élément de classement et de décision. Nous considérons donc la classe de tarif comme étant l'élément primaire du risk-business, même si la valeur moyenne d'un sinistre varie dans le temps.

Etablir un modèle est toujours une approximation plus au moins bien réussie de la réalité. Le problème de la représentation d'un risque couvert par un contrat d'assurance ou un ensemble de contrats (réassurance) est un problème théorique. Autre chose est le problème de l'efficacité c'est-à-dire la possibilité d'effectuer des calculs pratiques. Le modèle gaussien de la loi des erreurs de mesure n'est pas théoriquement exact, mais il permet les calculs de compensation. Le modèle de Dodson en assurance-vie est une première approximation de la réalité mais il permet à l'actuaire d'effectuer ses tarifications.

A premium calculation principle is a general rule that assigns a premium P to any given risk S. Intuitively, P is what the insurance carrier charges (apart from an expense allowance) for taking over the risk S (see [3], p. 85-87). Mathematically, S is a random variable, and P depends on S through its distribution function. The value of P may be finite or infinite; in the latter case we speak of an uninsurable risk.

A premium calculation principle is called additive, if the premium assigned to the sum of two independent risks is the sum of the premiums that are assigned to the two risks individually. For example, the variance principle, P = E[S] + β Var [S] (β > o), is additive, because the variance of the sum of independent random variables equals the sum of the variances. Additivity is a very desirable property, from a theoretical as well as from a practical point of view (as pointed out by Borch [2], p. 429).

The variance principle is not entirely satisfactory for various reasons. For one thing it does not take account of the skewness of S (a risk whose distribution is skewed to the right seems to be more dangerous than one with a symmetrical distribution). Furthermore, it produces in some cases a premium P that exceeds S with probability one (example: β = .3, S = o or 10, each with probability 1/2).

Le dépouillement statistique d'une classe de tarif hétérogène permet d'observer les fréquences relatives qi des contrats qui pendant une période de référence ont été frappés de (i — 1) sinistres.

On a:

et

Ces fréquences déterminent dans l'espace Rm un vecteur à m composantes, que nous notons:

Une base de décomposition est constituée par un éventail fini de η classes homogènes. Chaque classe homogène Ck est définie par une variable aléatoire, dont la loi de probabilité (λk) dépend du paramètre λk. Théoriquement ce paramètre peut être tout à fait général (un triplet réel p.e.); pour les applications pratiques traitées au paragraphe 4, il suffit pourtant de considérer λk comme un nombre réel.

Si Pi(λk) représente la probabilité à priori d'avoir exactement (i — 1) sinistres dans la classe Ck, nous pouvons noter (λk) par:

(p1(λk), p2(λk), … Pm(λk))

On a:

et

Nous supposons que les vecteurs (λk) sont linéairement indépendants.

Résoudre l'hétérogénéité d'une classe de tarif consiste tout d'abord à choisir une base de décomposition, à représentation analytique connue, et puis à décomposer la classe de tarif par l'éventail des classes homogènes Ck de cette base.

Credibility theory is concerned with the problem of forecasting the mean performance (claim frequency, total losses, etc.) of an individual risk, selected from a collective of heterogeneous risks, based upon the statistics of the collective, and upon the individual's experience data. The classic results, derived by American actuaries in the 1920's, were further strengthened by Bailey and Mayerson in 1950 and 1965, who showed that these results were exact Bayesian for certain risk distributions. Bühlmann, in 1967, then showed that the credibility formulae were the best least-squares linearized approximation to the exact Bayesian forecast, for general risk distributions.

This paper extends credibility theory to the problem of forecasting the distribution of individual risk, based upon collective statistics and individual experience data. Although the problem is, in principle, solved by finding a Bayesian conditional distribution, this approach requires a detailed knowledge of collective structure. The credible distribution, on the other hand, requires fewer prior statistics, and is also a best least-squares linearized approximation to the exact Bayesian distribution.

Following the theoretical development, detailed computational results are given.

In the following, consequential losses after fires are considered. A generalisation to consequential losses after other risks is most likely rather trivial.

The purpose of this paper is rather asking questions and putting a problem before the readers than presenting solutions. According to our experience, little has been done to seek logical systems or to construct mathematical or physical models in order to study these problems. The volume of business is rapidly growing but it could grow much more if a more reasonable tariffing was made. Many insured with fire insurances hesitate to take out an additional consequential coverage. They can see the need of the coverage, but they will not accept—and rightly—the present market rates. Actuarial colleagues are invited to join in building up suitable models to guide the practical underwriter.

The office premium today is most often composed of the following parameters:

a. the premium rate of the corresponding fire insurance

b. the insurance sum, defined e.g. as the expected turnover less variable costs

c. the coverage period, an upper limit of the period in which indemnity can be given. In Denmark, for a period of 12 months the premium of a consequential loss' insurance is normally defined as the insurance sum multiplied by the fire premium + 25 per cent.

We shall regard various risk situations in order to study the relationship between the fire risk and the consequential loss. The most simple situation is an undertaking where practically spoken only total fire losses will occur. One may think of a fireworks factory or a telephone exchange building. The fireworks factory will burn out physically. After a fire the contents of the telephone building, even if the majority of the materials is at hand, will be worthless because of the corrosion caused by the PVC contents of the wires. In the cases mentioned, an outbreaking fire might, of course, be extinguished so early that practically no damage will occur. If the fire, however, exceeds a certain extent, a total loss is likely to follow.

The increasing difficulties facing the insurer when he measures certain risks and fundamentaly when that measurement has to be carried out “a priori” are not easily overcome. Thus, the greater part of the insurance companies still use in numerous branches subjective methods of assessing a risk, which unfortunately do not lead to the best results. Especially in certain risks with undefined characteristics, the problem becomes more accentuated to the point of being practically impossible to class risks whose behaviour would be homogeneous.

For example we can analyse the numerous problems of this nature arising in the motor insurance in all countries in assessing the combined risk motor-car and driver.

Numerous solutions have been proposed and amongst them a system of premiums adapted to the actual risk and very especially the credibility premiums represent a fundamental step in the evaluation of these types of risks, for they have an advantage of at least as experience increases assuming an authentic measurement “a posteriori” of the true degree of risk which they cover.

The statistical theory of extreme values well described by Gumbel [1] has been fruitfully applied in many fields, but only in recent times has it been suggested in connection with fire insurance problems. The idea originally stemmed from a consideration of the ECOMOR reinsurance treaty proposed by Thepaut [2]. Thereafter, a few papers appeared investigating the usefulness of the theory in the calculation of an excess of loss premium. Among these, Beard [3, 4], d'Hooge [5] and Jung [6] have made contributions which are worth studying. They have considered, however, only the largest claims during a succession of periods. In this paper, generalized techniques are presented which enable use to be made of all large losses that are available for analysis and not merely the largest. These methods would be particularly useful in situations where data are available only for large losses.

Quadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.

Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.

Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.

The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.

A common procedure for experience rating is to use Whitney's credibility formula with the manual premium per risk unit estimated by the observed average claim amount per risk unit. As pointed out by Whitney himself, this observed average also needs to be subjected to credibility adjustment. This suggestion is pursued in the paper and an experience rating procedure developed in which there are two stages:

(i) revision of the prior expected manual premium in accordance with the observed average claim amount per risk unit;

(ii) experience rating of the individual risk classes by the usual methods but using the manual premium obtained in stage (i).

Both stages of this procedure involve credibility indices, for which formulae are developed.

Conditions under which the “common procedure” referred to above is fairly reliable are found and it is seen, somewhat surprisingly, that the procedure is often more valid than one expects on intuitive grounds. Care is necessary, however, when there are wide differences in size between different risk classes.