More than ten years ago I wrote a paper with the title “The Economic Theory of Insurance” [6]. I was not particularly happy about this paper, and I do not think it contributed much to the development of a sastifactory theory. The paper did however make me—and I hope some readers—acutely aware of the difficulties and problems which must be overcome before a proper theory can be constructed. These problems are still unsolved, so I have on the present occassion chosen a more modest title for a paper on substantially the same subject.

Insurance is an economic activity of some importance, and there is an obvious need for a theory to explain and analyse the activity in the insurance sector of the economy. During the last decade many economists seem to have felt the need, and to have taken it as a challenge. The results have been a fair amount of research, and a number of publications, which I shall not try to review here. It may however be useful to refer to three very recent survey articles by Farny [10], Ferry [11] and Rosa [14], which give extensive bibliographies. The three articles seem to indicate that the economics of insurance is becoming a fashionable subject of research.

Stimulated by Karl Borch's paper [3] we have tried to analyze the paper written by K. Arrow [1] in 1953. Contrary to Borch's opinion we have some doubt whether this work contains a theory of insurance as a special case. Nevertheless, it has inspired us to this note, which tries to develop a somewhat more realistic model. As a matter of fact, our development is more in the spirit of another paper by Arrow [2]. We, however, have chosen a more general setup, and we believe that our treatment is also different.

The homogeneous (in time) model of credibility theory is defined by a sequence Θ, X1, X2, … of random variables, where for Θ = θ fixed, the variables X1, X2, … are independent and equidistributed. The structure variable Θ may be interpreted as the parameter of a contract chosen at random in a fixed portfolio, the variable Xk as the total cost (or number) of the claims of the kth year of that contract.

Bühlmann's linear credibility premium of the year t + 1 may be written in the form

where f is a linear function. In optimal semilinear credibility, we look for an optimal f, not necessarily linear, such that (1) is closest to Xt+1 in the least squares sense. In the first section we prove that this optimal f, denoted by f*, is solution of an integral equation of Fredholm type, which reduces to a system of linear equations in the case of a finite portfolio. That is a portfolio in which Θ and Xk can assume only a finite number of values.

In the second section we see that the structure of such a portfolio is closely connected with the decomposition of a quadratic form in a sum of squares of linear forms.

In the last section we calculate numerically the optimal premium for a concrete portfolio in automobile insurance. We limit ourselves to the consideration of the number of claims. The optimal premium is compared with the usual linear premium. The difference is far from negligible.

In various branches of applied mathematics the problem arises of making decisions to reconcile conflicting criteria. One example is the classical statistical problem, where a type 1 error cannot be arbitrarily reduced without increasing the probability for a type 2 error. Another example, quite familiar to actuaries, is graduation, where a compromise between smoothness and fit has to be reached. This motivates the concept of Pareto-optimal decisions, which is discussed in section 2. There is a simple method, maximizing a weighted average of the scores, to obtain certain Pareto-optimal decisions. In section 3 a condition is given, which is satisfied in most applications, that guarantees that all the Pareto-optimal decisions can be found by this method. This is applied in section 4, where the problem of risk exchange between n insurance companies is considered. The original model of Borch is generalized: it is assumed that some of the companies are not willing to contribute more than a certain fixed amount towards the aggregate loss of the other companies. The theorem in section 4 gives a characterization of all the Pareto-optimal risk exchanges. Because of the restrictions, these risk exchanges do not just depend on the combined surplus (which would amount to pooling) in general, and can be found by an algorithm. One benefit of this generalization of Borch's Theorem is that two seemingly unrelated results (optimality of a stop loss contract, and optimality of certain dividend formulas in group insurance) follow from it as special cases.

This paper presents a normative model for the sequential reinsurance and dividend-payment problem of the Insurance Company (I.C.). Optimal strategies are found in closed form for a class of utility functions. In some sense the model studied can be viewed as an adaptation of Hakansson's investment-consumption model of the individual [3] or a generalization of Frisque's model for the dynamic management of an I.C. [2].

In Section 2 the model is formulated as a discrete time dynamic programming problem. The objective of the I.C. is assumed to be maximization of the expected utility of the dividend streams paid to stock/policy-holders (s/p-holders). The initial reserves level is assumed to be known. The premiums to be collected in each period for selling policies are known in advance. The losses due to claims from policy-holders are random variables independent from period to period. In each period the I.C. must decide on the portion of the reserves to be paid as dividends and on the form and level of reinsurance with a reinsurer that quotes prices for any contract.

Optimal strategies in closed form are found in Section 3 when the utility function of the I.C. is given by the discounted sum of one-period utilities of dividends; and when the one-period utilities belong to the linear risk-tolerance class, which is given by: (la) u(x) = (ax + b)c+1/a(c + 1); ax + b > o, ac < o. (Ib) u(x) = log(ax + b); ax + b > o. (II) u(x) = — eγx; γ > o.

The effect of inflation of premium income and claims size distribution, but not of free reserves, on the probability of ruin of an insurer is studied.

An interesting similarity between this problem and the ruin problem in an experience-rated scheme is exhibited. This similarity allows the deduction of parallel results for the two problems in later sections.

It is shown that the probability of ruin is always increased when the (constant) inflation rate is increased.

The distribution of aggregate claims under inflationary conditions is described and used to calculate an upper bound on the ruin probability. Some numerical examples show that this bound is not always sharp enough to be practically useful. It is also shown, however, that this bound can be used to construct an approximation of the effect of inflation on ruin probability.

It is shown that if inflation occurs at a constant rate, then ruin is certain, irrespective of the smallness of that rate and of the largeness of initial free reserves and the safety margin in the premium. The corresponding result for experiencerated schemes is that a practical and “intuitively reasonable” experience-rating scheme leads eventually to certain ruin.

Finally, a simple modification of the techniques of the paper is made in order to bring investment income into account.

The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves.

To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.

When the distribution of the number of claims in an interval of time of length t is mixed Poisson and the moments of the independent distribution of individual claim amounts are known, the moments of the distribution of aggregate claims through epoch t can be calculated (O. Lundberg, 1940, ch. VI). Several approximations to the corresponding distribution function, F(·, t), are available (see, e.g., Seal, 1969, ch. 2) and, in particular, a simple gamma (Pearson Type III) based on the first three moments has proved definitely superior to the widely accepted “Normal Power” approximation (Seal, 1976). Briefly,

where the P-notation for the incomplete gamma ratio is now standard and α, a function of t, is to be found from

the kappas being the cumulants of F(·, t). An excellent table of the incomplete gamma ratio is that of Khamis (1965).

The problem that is solved in this paper is the production of an approximation to U(w, t), the probability of non-ruin in an interval of time of length t, by using the above mentioned gamma approximation to F(·, t).

This paper concerns the Borch model of a reinsurance market seen as a model of an economy under uncertainty.

In a market of this type the goods traded are unit coverings contingent to a particular state of nature (n-tuple of claims).

Our idea is to regard the probability of a state of nature as a sort of intrinsic value of the related contingent covering. From this point of view we examine the role of the reinsurance market in modifying values in market equilibrium prices and other questions, related to this classical economic problem, in the particular case of a quadratic utility function for all companies.

In many estimation problems, incomplete as well as complete samples are available for Bayesian prediction. After developing the theory for a special, but useful family of distributions, examples are given in life testing, renewal risk processes, life contingencies, and the problem of estimating a defective distribution.

We compute a merit-rating system for automobile third party liability insurance by two different ways, both with the help of an exponential utility function.

(i) We apply the principle of zero utility to exponential utilities.

(ii) We break the symmetry between the overcharges and the undercharges by weighting them differently through the introduction of a utility function, in order to penalize the overcharges.

The results are applied to the portfolio of a Belgian company and compared to the premium system provided by the expected value principle.

Deux méthodes différentes, basées sur l'emploi de fonctions d'utilité exponentielles nous permettent de définir un système bonus-malus en assurance automobile:

(i) le principe de l'utilité nulle;

(ii) la pénalisation des injustices de la compagnie, obtenue en pondérant les erreurs de prime au moyen d'une fonction d'utilité de manière à briser la symétrie entre les primes trop élevées et les primes trop basses.

Les résultats théoriques sont appliqués au portefeuille d'une compagnie belge et comparés aux primes fournies par le principe de l'espérance mathématique.

De nombreux modèles, en actuariat, se basent sur l'existence d'une fonction d'utilité additive. Le but de cet article est de montrer que cette hypothèse enlève au problème traité son caractère dynamique et que la suppression de l'additivité conduit à une solution plus réaliste.

Let us denote by E(x) the pure risk premium of an unlimited excess cover with the retention x and by H(x) and m(x) the corresponding expected frequency and severity.

We thus have E(x) = H(x) · m(x).

H(x) is a non-increasing function of x and for practical purposes we can assume that it is decreasing; H′(x) < o. The equation H(x) = n has then only one solution xn, where n is a fixed integer.

Let En denote the risk premium for a reinsurance covering the n largest claims from the bottom.

Let us define Intuitively we feel that is a good approximation for En.

We shall first show that when the claims size distribution is Pareto and the number of claims is Poisson distributed, is a good approximation for En, being slightly on the safe side. We further include a proof given by G. Ottaviani that the inequality always holds.

In the Pareto case we have

where the Poisson parameter t stands for the expected number of claims in excess of 1 (equal to a suitably chosen monetary unit) and

.

Stochastic-dynamic programming provides a technique for forecasting limits within which the insurance business will flow by a prefixed probability. The future development depends, among numerous other things, on management strategies, especially resources, which are planned for allocation in the acquisition of new business and for competition. This technique can be used to analyse different market situations. Various competitive measures and eventual counteractions by competitors can be assumed and simulated for the purpose. In this way the consequences of different strategies can be studied in order to find the most appropriate one. Our approach is similar to the well-known business games where teams play business in a simulated market. The idea of applying dynamic programming to business games was suggested by Esa Hovinen (discussion at the Astin Colloquium in Washington in 1977).

The concept efficiency of a bonus-mains system was defined, apparently in a totally different way, consecutively by Loimaranta (1972) and Lemaire (1975, 1976). In this paper we start with a more general model that leads us to a definition of efficiency that contains both earlier ones as special cases. Further we introduce the definition of efficiency over a finite planning horizon and consider the efficiency not only for a single risk but also for the entire risk group. As a consequence of our approach we can also generalize the concepts excess premium and central value as they were introduced by Loimaranta.