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.

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.

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).

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

.

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.

Recently two types of insurance policy covering automobile liability have become available. With this kind of policy, the insured party receives a prepaid discount on the annual premium; however, he must make an additional payment to the insurer on first report of an accident.

We shall examine only one of these two types of policy, since the second is very similar to the first.

On stipulation of the contract, the insured party pays a premium equal to 78 percent of that currently in force for complete coverage, depending on the various limits. However, at the same time at which the insured reports his first accident in the course of the year (and only in this case), he must make an additional payment equal to 35 percent of the premium due in the case of complete coverage. This additional payment, considered as a deposit, is repaid to the insured party if the accident has no follow up within four months; however, it may be requested again by the insurer if the case is subsequently reopened and leads to payment of damages. This additional payment will thus become a part of the premium only in the case of payment of damages for the first accident reported.

In this way, the premium paid by the insured is equal to 78 percent or to 113 percent of the current premium, depending on whether or not the insured reports accidents.

By the term actual run-off triangle we shall mean the two-way tabulation—according to year of origin and year of payment—of claims paid to date, which has the following form:

where Cij is the amount paid during development year j in respect of claims whose year of origin is i.

The information relating to the area below and/or to the right of this triangle is unknown since it represents the future development of various cohorts of claims.

Now in seeking to use this triangle as a basis for projection of claims in future development years for each of the years of origin 0, 1, 2, etc., we must recognise that the entries Cij in the above triangle, being random variables, contain random deviations from their expected values uij. It is the corresponding triangle of these expected values in which we are interested, and which shall be called the expected run-off triangle.

The paper discusses the problem of estimating the mean of a long-tailed claim size distribution when the investigator's knowledge of the distribution is only vague.

One method of dealing with this problem, the method developed by Johnson and Hey, is examined and found to produce strongly biased estimators.

The situation in which a sufficient statistic (but nothing else) for the claim size distribution is known is examined, and an approximately unbiased estimator developed. This estimator is substantially more efficient than the arithmetic mean in some cases. It appears to be quite successful when the sufficient statistic is real-valued. It is of limited use when the sufficient statistic is vector-valued.

The problem of distribution-free parameter estimation in recent credibility theory is discussed in the papers [1], [3] and [4] of the bibliography. Here, we consider a multiclass model with regression assumption. In that case, already treated by Ch. Hachemeister, [3], this author obtains an unsymmetrical matrix as an estimator of a covariance matrix. Of course, for practical use, this matrix is symmetrized in the obvious way. We show that this procedure can be avoided and that a lot of symmetrical unbiased estimators can be obtained at once.

By particularisations to the 1-rank model, we find the estimators given by Bühlmann and Straub, [1], [4].

In the multirank case, a generalization of the minimumvariance principle (minimization of the trace of the covariance matrix) leads to an optimal estimator of the mean regression vector. A noteworthy conclusion of our discussion is that there is no difference at all between the various credibility formulae (the inhomogenous formula, the homogeneous formula, the meanfree formula) if the mean regression vector is estimated optimally.

Finally we show that it must not be hoped to find, in a large set of unbiased estimators of the covariance matrix, one estimator furnishing, always, a semidefinite positive estimate.

After having a look at the results obtained by adjustment of automobile claims amounts distribution, we research how the number and the time-configuration of past claims condition the claims law of probability.

We have statistics about a group of 471 000 cars which was followed for three years: 1970, 1971 and 1972. We use mathematical technics and among multi-dimensional analysis, we use factorial analysis of correspondance (A.F.C.). A.F.C. permits us to show the link which exists between the claim amount of the third year and the number of claims during the two years before. A quantitative analysis of the corporal claims shows that, of the frequency of corporal claims during the third year growths up in function of the number of past claims, the expected corporal claims amount of the third year decreases as the square of the material claims number during the two first years.