1. Risks of a catastrophic nature, such as war or earthquake, can differ, in a statistical sense, from other ordinarily insured perils only in respect of the distribution of the probability function. For example, although the probability of a loss occurring may be small, its magnitude once it occurs may be very large. One reason for the latter feature is that the sum insured may itself be heavy. But still more important is the fact that there would be strong positive correlation between similar contiguously situated risks. This is the main reason for the loss taking on a catastrophic character.

2. The fact that the probability of a loss is ordinarily small, leads to the result that the expected loss and hence the pure premium for the risk is small. Equally, and for the same reason, the “loss strain at risk” is large. If the latter could be considerably reduced—which can only be done by charging impracticably high premiums—the risk becomes more easily insurable.

3. If there are n risks of this nature, 1, 2, 3 … n and νi(x) is the conditional frequency function of the ith risk we may write

While we may imagine this as a frequency curve, the complementary probability Ϸi of a loss not occurring can be represented as a vertical line at the origin—the different methods of representation of Ϸ and q are actually convenient. With this representation, the expansion of the binomial product

shows that the convolution of the n curves can be expressed as the sum of n + I functions

where

(ii) the term A1(x) comprises n separate curves of the form νi(x) each multiplied by its complementary probability, being of the form

(iii) the term A2(x) comprises terms each consisting of the convolution of the curves taken two at a time and multiplied by the complementary probability, being of the form

and so on.

In a note on the security loading of excess loss rates I am deducing a simple formula intended to replace some tedious calculations. In the beginning of that note I made the point that some authors recommend a loading proportional to the dispersion of the total claims amount of a treaty δ1 while others tend to favour .

I also stated that a loading proportional to δ1 or its estimate δ1* could be deduced from the statistical uncertainty in measuring the risk (section 4).

The question has been raised if and to what extent a loading system based on the dispersion is unduly punishing the smaller portfolios. This will be examined below.

The pricing concept will be analyzed from the point of view of a big dominating Reinsurer who wants to be fair in all directions. The conclusion of this study supports an affirmative answer to the question put above.

In a second part the loading is studied from a different angle bringing competition into the picture. The pricing or loading becomes a problem of operations research under the simplified assumption that profit is the only purpose of our activity. Not unexpectedly, the loading coming out from this aspect differs from those of part one.

Part two also deals with the question of how much of the loadings which we are aiming for, get lost in the competitive process. It is also shown that in most cases the harder the competition is, the higher loadings shall be used.

Part one and part two thus deal with the loading problem from different aspects, and illustrate the complexity of the problem. It is my hope that this note could stimulate further researches in this interesting and important area, also in a moment when some reinsurers are more concerned with the question of surviving than in fixing the loadings which should on the average and in the long run turn up as profits.

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.

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.

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.

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.