The Ministry of Social Affairs and Health, being the Supervising Office of Insurance in Finland, has established a special working group to investigate the problems involved with the solvency of insurers. A report will be compiled in a near future. The capacity of risk carriers is one of the problems dealt with, and it will be preliminarily reviewed in this paper.

The problem was treated by the working group parallelly by means of

1. an empirical approach observing actual fluctuations in underwriting gains of insurers, and

2. a theoretical approach, constructing a stochastic-dynamic model and studying its behaviour, especially its sensitivity to numerous background factors.

First the methods of investigation are described and their application is then demonstrated using some numerical data. Because a comprehensive report will be published by the working group separately, only the main schedule is given. For the same reason the consideration is limited here to stochastic risks, omitting the fact that the solvency of an insurer is also jeopardized by numerous “non-stochastic” risks such as failure in investments, political interference of the authorities, mismanagement of the company, or misappropriation of its property.

Compound distributions such as the compound Poisson and the compound negative binomial are used extensively in the theory of risk to model the distribution of the total claims incurred in a fixed period of time. The usual method of evaluating the distribution function requires the computation of many convolutions of the conditional distribution of the amount of a claim given that a claim has occurred. When the expected number of claims is large, the computation can become unwieldy even with modern large scale electronic computers.

In this paper, a recursive definition of the distribution of total claims is developed for a family of claim number distributions and arbitrary claim amount distributions. When the claim amount is discrete, the recursive definition can be used to compute the distribution of total claims without the use of convolutions. This can reduce the number of required computations by several orders of magnitude for sufficiently large portfolios.

Results for some specific distributions have been previously obtained using generating functions and Laplace transforms (see Panjer (1980) including discussion). The simple algebraic proof of this paper yields all the previous results as special cases.

To some extent, this paper provides theoretical back-up for Taylor's (1980b) see-saw method of analysis of outstanding claims.

Section 3 deals with the payments per claim finalized (PPCF) method. This assumes PPCF to be a function of development year only.

Section 4 carries out a theoretical investigation of the validity of this assumption. It is found that the assumption is not justified in general; that PPCF can (and usually will) be sensitive to changes in speed of finalization of claims; and that this sensitivity will depend on the convexity properties of PPCF as a function of development year. The analysis is made on the basis of the so-called hypothesis of invariant order.

Section 5 develops the see-saw method from this theoretical basis, although Section 7 is at pains to point to situations in which that method remains applicable despite violation of the hypothesis of invariant order.

Section 6 provides a numerical example.

This paper uses the same notations and some of the results of Baton and Lemaire (1981). The reader is referred to that work for more details about the classical risk exchange model, which will not be recalled here. The main result of that former paper was to characterize the core of the market in the case of exponential utilities, and to show that it is never empty. Since the core always exists, and since it is such an intuitive notion, one might wonder why we introduce here a much more complicated concept. The reason is that the core is presently subject to a heavy fire of criticisms–both experimental and theoretical–from leading researchers in game theory; they claim that the core is much too static, that it does not take into account the real dynamics of the bargaining process, that it does not introduce the full spectrum of negotiation threats of the traders. Indeed, experimental data consistently produce final payoffs that lie outside the core, but within the bargaining set (abbreviated: b.s.). We shall attempt to illustrate those criticisms in 4. We shall define the b.s. in a general non transferable game in 2, and characterize it in the special case of a 3-company reinsurance market in 3, but first of all we would like to explain intuitively the basic mechanisms of the b.s. by means of a simple example (with transferable utilities).

The basic difficulty in modelling a negotiation process is to express what is the purpose of the game. Certainly, the objective is not just to get the maximal amount of profits, because if everyone demands the highest payoff he can obtain in a coalition, no agreement will be reached; the goal of the process is to attain some state of stability, to which the bargainers should agree if they want any agreement to be enforced. This stability should reflect in some way the power of each player, which results from the rules of the game, his initial situation, his attitude towards risk, …

A bargaining process is a multi-criteria situation, in which the players certainly attempt to maximize their payoffs, but also try to enter into a “safe” or “stable” coalition.

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjer's result.

Dans trois études récentes (1977a, 1977b, 1979), J. Lemaire a appliqué à un ensemble d'observations du risque automobile quelques méthodes de sélection fréquemment utilisées en analyse de la régression. Les variables explicatives (traitées comme variables indépendantes) sont les variables décrivant le risque. Les variables étudiées (traitées comme variables dépendantes d'un modèle de régression) sont les deux variables généralement prises en considération dans ce contexte: le nombre des sinistres et leur montant cumulé.

Nous avons déjà souligné (Hallin, 1977) combien les hypothèses qui se trouvent à la base de l'analyse de la régression — normalité, homoscédasticité et linéarité de la régression — sont loin d'être remplies dans le contexte de l'assurance automobile. Le montant cumulé annuel prend la valeur zéro avec une probabilité proche de 0.9! Le nombre annuel de sinistres vaut 0, 1, 2, rarement plus! Même très approximativement, de telles variables peuvent difficilement être considérées comme normales. En outre, la plupart des variables explicatives sont de type nominal ou ordinal, ce qui rend délicate l'utilisation de modèles linéaires, les interactions de tous ordres étant très importantes, ainsi qu'on pourra le constater. Ces réserves sont d'ailleurs prévues par Jean Lemaire lui-même, qui ne propose ses conclusions qu'à titre de première approximation. Les mêmes données ont encore été soumises (Masure, 1978) aux méthodes de l'analyse discriminante, et les mêmes réserves peuvent être faites en ce qui concerne l‘utilisation des méthodes et l'interprétation des résultats (combinaisons linéaires, etc.).

Nous avons proposé dans Hallin (1977a, b) un ensemble de méthodes qui, selon les hypothèses distributionnelles pouvant être faites (et qui vont des hypothèses classiques de l'analyse de la variance à celles, beaucoup moins restrictives, des méthodes de rangs), constituent des généralisations de celles qui sont utilisées par Jean Lemaire. En particulier, celle que nous appliquons ici est entièrement “distribution free”. Ces méthodes sont également une extension de celle qu'a proposée Pitkänen (1975, 1976).

We develop Hachemeister's regression model in credibility theory (without proofs) and indicate how the involved structural parameters can be estimated from the observable variables (with proofs for the simple results and those not yet published).

Large families of unbiased estimators are available. From the practical viewpoint this is rather a handicap because it creates the problem to decide what estimators actually to use. In order to fix optimal estimators, we adopt the small-sample criterion of minimum-variance. But in the research for general solutions three kinds of difficulties arise.

(i) The calculations become too lengthy.

(ii) The optimal estimators depend on some of the parameters to be estimated. (Then we call them pseudo-estimators).

(iii) The optimal estimators depend on new structural parameters defined in terms of fourth-order moments.

Only a compromise allows to cope with this reality. Situation (iii) creates new estimation problems. They can only be avoided at the cost of the introduction of special assumptions or approximations. Then problem (i) is more or less automatically solved. By an obvious method of successive approximations pseudo-estimators can serve as true estimators. Thus (ii) is no real problem.

In a series of celebrated papers, K. Borch characterized the set of the Pareto-optimal risk exchange treaties in a reinsurance market. However, the Pareto-optimality and the individual rationality conditions, considered by Borch, do not preclude the possibility that a coalition of companies might be better off by seceding from the whole group. In this paper, we introduce this collective rationality condition and characterize the core of this game without transferable utilities in the important special case of exponential utilities. The mathematical conditions we obtain can be interpreted in terms of insurance premiums, calculated by means of the zero-utility premium calculation principle. We then show that the core is always non-void and conclude by an example.

The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.

The result is a generalization of Lundberg (1940).

Consider the distribution of

where X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution function

and N is an integer valued random variable with probability function

Then the distribution function of S is given by

If MX(t), MN(t) and MS(t) are the associated moment generating functions, then

Zehnwirth (1981) contains some flaws. If

is the Esscher premium for a risk X, the loading is H(X) — E(X) and not h as Zehnwirth states. The first and third formulas on page 78 are wrong, since o(h) is a quantity such that

A correct statement would have been that

or simply that H(X) is a continuous function of the parameter h. However, this continuity is not uniform in all risks, which is illustrated by (3). No matter how small h is, there is always an X such that the difference between H(X) and E(X) is substantial. In view of this what is the meaning of a statement like “… the Esscher premium is a small perturbation of the linearized credibility premium”?

The notion of ordering and danger of claim size distributions is extended to claim frequency distributions.

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).

Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.

The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).

Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given by

That is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss function

where a is the estimate of Y, and of course the loading h is greater than zero.

Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.