For the 4th ASTIN colloquium in Trieste (1963) Robert E. Beard [1] had initiated the topic of extreme value theory and its application to actuarial problems. The impetus came from Gumbel's book [2], published in 1958, and the idea of applying extreme value theory ran about as follows:

Problem: Given: The size of the largest claim observed = XN

The number of observed claims = N.

Find: The excess of loss premium for a portfolio of independent risks identical with those under observation.

The idea was to use Gnedenko's limit distribution for the wide class of distribution functions with an unlimited tail and finite moments, the “exponential type” in Gumbel's terminology, and to calculate the excess of loss premium on the basis of this limit distribution.

The discussion in Trieste was very lively and culminated in Jan Jung's citation of Jan Jung [3]: “There is a natural law which states that you can never get more out of a mincing machine, than what you have put into it. That is, if the reinsurance people want actuarially sound premiums, they must get a decent information about the claim distribution”. In more mathematical language: The brilliant idea of Bobbie Beard is unfortunately leading to a non-robust procedure. A deviation in the true (but unknown) underlying distribution may lead to completely different results.

In any case the mincing machine argument seemed so powerful that we have had little reconsideration of the above problem at subsequent ASTIN colloquia. It may therefore be worthwhile to have another look at the problem now, i.e. more than ten years after the Trieste colloquium.

The multiplicative ratemaking, model we have in mind is the following one. Within a certain branch of insurance we have, say for simplicity, two tarif arguments U and V. For example, in motor insurance we could think of U and V as being make of car and geographical district respectively. In fire insurance U could be class of construction for buildings and V could relate to fire defense capacities.

The arguments are of a qualitative nature and argument U has r levels, while argument V has k levels. To our disposal we have statistical experience of the business for a certain period of time, consisting of

—risk exposures nij (i = 1 … r, j = 1 … k).

Risk exposure nij thus corresponds to the ith U-level and the jth V-level. It could be e.g. number of policy years or sum insured during the period of observation for objects belonging simultaneously to U-level i and V-level j.

The nijS are known non-random quantities.

—(relative) risk measures pij(i = 1 … r, j = 1 …k).

Risk measure pij could be e.g. claims frequency, i.e. number of Claims divided by number of policy years, or claims cost per policy year or claims cost as a percentage of sum insured. In general pij is thus the observed number or the observed amount of claims belonging simultaneously to U-level i and V-level j, divided by the corresponding risk exposure nij.

In the past the subject of optimal reinsurance has been dealt with by various authors such as Borch (1), Kahn (2) and Verbeek (3). Borch tries to look at the problem from the point of view of a company which acts as both Insurer and Reinsurer whereas the viewpoint of Kahn and Verbeek is that of a Ceding Company.

Let us study the simplest possible market situation of one Insurance Company C and one Reinsurer R. C writes a portfolio P and seeks reinsurance protection at R.

In the present note I will start with the viewpoint of optimizing simultaneously the situation of C and R.

For both C and R the variance is used as a measure of risk. The variance of P is V. C makes an effort to reduce V to a level Vc by way of a reinsurance treaty T. The corresponding variance of R is VR.

Out of the set of possible reinsurance treaty arrangements {T} we assume that the parties are first of all interested in such arrangements for which

In other words they will look for solutions which will lead to a substantial reduction of the variance.

It is possible to think of “nonsense” arrangements where no substantial reduction is achieved and in the extreme case even of arrangements for which

In a further paper we might deal with such a case.

In 1967, Bühlmann has shown that the credibility formula was the best linearized approximation to the exact Bayesian forecast.

His result for the credibility factor can be found back by means of some Bayesian inference techniques. Introducing a uniform prior probability density function for the credibility factor provides us with a method for estimating z, a correction term to the Bühlmann's result is obtained. It is shown how prior boundary conditions can be introduced.

The problem of the best fit to set ideal values under general inequality order restrictions is examined for asymmetric, quadratic, absolute, and Chebyshev norms. Special solution procedures are given in terms of network flow algorithms over a network associated with the given isotonic order relations, and the nature of the optimal solutions is characterized for the different norms.

The model is formulated in terms of finding an optimal insurance rate structure over given risk classes for which a desired pattern of tariffs can be specified. The suitability of different norms is considered in the context of corporate profitability, and the relationship to a simple rate relativities model is described.

When an insurance company accepts new insurances or when the premiums of earlier accepted insurances have to be changed on renewal the company has to

— search for the factors that influence the premium and

— calculate the premium according to the values of these factors.

In order to calculate the premiums the company gathers data consisting of factors eventually influencing the amount of claims. On the basis of these data the company calculates the tariff which has to fulfill the following general principles:

1. The tariff has to be as correct as possible in relation to different risk groups.

2. The structure of the tariff has to be such that the calculation of the insurance premium is quite straightforward. With this in mind the factors influencing the tariff have to be few enough and the structure of the tariff has to be the simple (e.g. linear or multiplicative) function of the factors or it should be rather easy to put them into tabular form.

These principles are partly contradictory. If the premium is correct, the structure of the tariff is not usually simple.

In his paper to the Tenth ASTIN Colloquium the author presented generalised extreme value techniques for making use of all large losses that are available for analysis and not merely the largest. In this paper the problem of assessing the relative contributions of various factors to fire losses is investigated. A model concerned with multiple regression with extreme observations of given rank is developed. It takes into consideration the biases due to the use of extremes and the differences between categories of risks in regard to the frequency of fires (or claims). By way of illustration the model was applied to the largest and second largest losses in the textile industries in the United Kingdom during the six-year period 1965 to 1970. The presence or absence of sprinklers, whether the buildings were single-storey or multi-storey, and total floor area were the independent variables included in this preliminary investigation. Judged from extreme losses sprinklers appear to reduce considerably the expected damage in all fires.

The technique enables different estimates to be obtained for each regression parameter for different ranks. It is desirable to have a single overall estimate for each parameter; and for this purpose a second model is developed for performing a regression analysis combining observations pertaining to a number of ranks. Covariances of the residual errors are also taken into account in this model.

This paper is concerned with certain problems which arise in the course of rate fixing for a portfolio comprising a large number of direct risks.

This is, of course, the traditional territory of the actuary and there is no lack of papers discussing specific aspects of the subject, such as the analysis of claim frequency, or the fitting of distributions to claim amount data. Equally there is a literature which discusses in general terms the philosophy which it is suggested should be followed by the non-life insurance ratemaker. It is when one tries in practice to follow a logical and consistent path through from portfolio records to rates that gaps in the literature become apparent and it is with covering these gaps, or at least some of them, that the present paper is concerned.

It is not the concern of the author in this paper to attempt to describe the entire ratemaking process. That there is no reference to the analysis of office expenses, commissions, or to the application of “judgment” or “experience” should not be taken to imply that these aspects are not susceptible to the kind of approach advocated here. Attention here is focused upon what might in a sense be regarded as the most basic aspect: the claim process or experience.

In many insurance operations the position which seems to be developing is that there is no lack of necessary computer hardware, and in principle, of data: the difficulties arising relate more to the form in which data should be retained and extracted, and the manner in which it should be used to assist in rate-making.

By intuition, the subdivision of an insurance portfolio into a number of classes is said to be good if it reflects the heterogeneity of the portfolio in an efficient way. To illustrate this rather vague statement we take the following very simple example:

The portfolio consists of 20 independent risks, 10 of them producing an expected loss ratio of say 30% each (type A risks) and 80% each (type B risks) respectively.

This “natural” subdivision is certainly better than, for instance, no subdivision at all;

or, the finest possible subdivision with 20 classes consisting of only 1 risk each (because there is no point in differentiating between risks of the same type);

or, 5 classes each containing two A- and two B-risks (here the number of classes is unnecessarily high and the heterogeneity has been completely wiped out; statistics based on this subdivision would even make us believe that the portfolio is totally homogeneous).

As a matter of fact, the above “natural” subdivision is—of course!—the best of all subdivisions, it is the optimal subdivision in this case.

In practice, however, as we all know, it is not easy to find the optimal subdivision. For one thing, the inherent structure or “natural” subdivision is not known a priori and secondly, for many different reasons, we can only choose from a limited number of subdivisions and not from all theoretically possible solutions. Note that even with only 20 risks there are 58.1012 possibilities of subdividing the portfolio. Thus, in practice, there is only a relatively small number of admissible subdivisions and the optimal one may not be among them, but we still need some sort of statistical criterion to choose the best one from these admissible subdivisions.