Classical statistics deals with the following standard problem of estimation:

Given: random variables X1, X2 … Xn independent, identically distributed, and

observations x1, X2 … xn,

Estimate: parameter (or function thereof) of the distribution function common to all Xi.

It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namely

Given: risks R1R2 … Rn no contagion, homogeneous group,

Find: the proper (common) rate for all risks in the given class.

There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?

An insurance company aims to increase the volume of its business.

The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

pn = premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln = The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

We consider a set of risks B, which is divided into subsets Bi according to some property as profession, region etc. The subsets Bi are thought to be too small to permit a calculation of a premium from their own experience. On the other hand we assume that there are differences between the Bi. This paper tries to give a solution to the problem how the experience of the single Bi should be combined with that of the whole B.

The problem is treated only with respect to frequencies of claims. The amounts of the claims are not taken into consideration. I suppose that an extension of the theory to amounts is possible and highly desirable for practical applications. Lack of time did not permit me however to proceed to such an extension in this paper.

I think that the methods developed in the sequel might be applied to the following practical problems:

— rate making

— experience rating in the restrictive sense of the word, i.e. adjusting periodically the premium or granting premium refunds according to the experience of every single Bi

— judging the experience of a. Bi.

I do not give a list of references which would have to be very long. The basic idea was, as far as I know, first developed by Lundberg.

Les 5 contributions suivantes ont été présentées sur le thème no 3:

— par R. E. Beard:

2 contributions intitulées “Calculation of Reserves for Non-Life Insurance” (Le calcul des réserves pour les branches non-vie)

— par Karl Borch:)

“Control of a Portfolio of Insurance Contracts” (Le contrôle d'un portefeuille d'assurances)

—par B. H. Jongh:)

“The Insurer's Ruin” (La ruine de l'assureur)

— par T. Pentikäinen, Helsinki:

“On the Solvency of Insurance Companies” (De la solvabilité des compagnies d'assurance)

— par Erkki Pesonen, Helsinki

“Magnitude Control of Technical Reserves in Finland” (Le contrôle du montant des réserves techniques en Finlande)

Les questions traitées par ces auteurs peuvent être groupées comme suit:

1. Donnée la situation d'une compagnie d'assurance, quelles sont les affaires qu'elle devrait accepter?

2. Donnée la situation d'une compagnie d'assurance, dans quelle mesure doit-elle réassurer son portefeuille?

3. Le calcul des réserves techniques, c'est-á-dire de la réserve pour risques en cours et pour les sinistres non régiés.

4. La détermination des réserves de sécurité nécessaires en plus des réserves techniques proprement dites.

This report is a contribution to the discussion on the solvency problem, which has been taking place at ASTIN-meetings. In his report in Edinburgh 1964 Beard referred to many aspects which are closely connected with the problem. Such aspects are

1. the evaluation of liabilities;

2. the evaluation of assets;

3. the level of the premiums of long term policies and

4. reinsurance.

If all of these are not in order, there is no sense in speaking about solvency. E.g. a solvency margin defined as the difference between assets and the expected value of liabilities would not be a reliable measure of the financial state of an insurance company, if either of these—or maybe both—are not evaluated in a reliable way. The fixing of solvency margins is not an isolated problem, on the contrary it is only part of the security measures which must all be managed at the same time. The ultimate purpose of the security system prescribed by legislation must be to safeguard policyholders and claimants against losses.

This paper deals with some requirements put forward by the Finnish Supervisory Service on technical reserves mainly for solvency reasons. Consequently only one but nevertheless an important aspect of solvency problem is touched upon. The solvency question in its entirety is discussed in a paper of Dr. Pentikainen at this same colloquium.

According to the instructions of the Supervisory Service, given by virtue of the Insurance Companies Act, insurance companies are obliged to calculate a so-called equalisation reserve in a specific way. Simplifying slightly, this equalisation reserve is that part of the technical reserves which exceeds the premium reserve and the claims reserve calculated according to classical actuarial principles; that is to say, the present value of the expectation value of the net insurance liabilities. The Insurance Companies Act calls for this reserve because conventional technical reserves do not take into account the random feature of claims amount.

1.1.— Under Subject 4 at this Congress we have discussed the practical application of modern statistical techniques in different branches of insurance. During the last decades, there has been an almost explosive development in theoretical statistics and related branches of mathematics. I think it has been very useful to survey the techniques, which have been developed, and find out if they can be used in insurance.

1.2. — There may, however, be some danger in this approach. When new means become available, we should of course have an open mind, and examine these means in order to see if they can serve our ends. We should, however, not get so excited over the power of new techniques, that we distort our ends just for the sake of being able to apply the means.

Linear programming, to take an example, is a powerful tool, which has proved extremeiy useful in many, apparently very different fields. There is, however, little point in using this technique in insurance, unless we have problems which consist of determining the maximum of a linear expression, subject to linear restraints. If there are problems in insurance which can be cast in this form, with sufficient approximation, then linear programming is obviously useful. If, however, we lose something essential by reformulating our problems in this way, linear programming may become a dangerous temptation, which we should resist.

The relationships between actuarial and pure mathematics are curious. Actuaries have contributed to the development of mathematical theory: it is sufficient to mention, as examples, Fredholm of an earlier, and Cramér of a more recent generation. Scandinavian mathematicians, in particular, have been concerned with a very special type of stochastic process, reflected in the collective theory of risk, and the work of Philipson, Ammeter and others in this field is well known to readers of this Bulletin. However, the main stream of the theory of stochastic processes has little contact with actuarial applications.

On the other hand, many actuaries have studied and assimilated pure mathematics and have thrown light on actuarial matters by describing their own preoccupations in the terminology of modern, often abstract, mathematics. E. Franckx is one of their number.

The Instituto di Matematica Finanziaria of the University of Trieste (Faculty of Economics and Commerce) has published a booklet entitled

Essai d'une théorie opérationnelle des risques Markoviens which contains three lectures delivered by Professor Franckx in Trieste and a contribution which he presented to the 17th Congress of Actuaries, held in London in 1964.