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.

We define axiomatically a concept of value for games without transferable utilities, without introducing the usual symmetry axiom. The model—a generalization of a previous paper [6] extending Nash's bargaining problem—attempts to take into account the affinities between the players, defined by an a priori set of “distances”. The general solution of all three- and four-person games is described, and various examples are discussed, like the classical “Me and my Aunt” and a reinsurance model.

Nous définissons de manière axiomatique un concept de valeur pour les jeux à utilités non-transférables, sans introduire l'axiome classique de symétrie. Le modèle — une généralisation d'un concept de valeur [6] étendant à plusieurs joueurs le problème de marchandage de Nash — tient compte des affinités entre les joueurs, données sous forme d'une matrice de “distances” a priori. Nous donnons la solution générale de tous les jeux à trois et quatre joueurs, et discutons plusieurs exemples classiques, dont le célèbre “Ma tante et moi” et le modèle de réassurance de Borch.

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.

The problem of rating the discount for an excess is considered under the condition where premium rates for excess-free policies are calculated according to a multiplicative (or equivalently, points rating) model.

Section 3 obtains some inequalities on the manner in which the discount should vary over the various risk-classes in the portfolio and compares the results with the current practical situation in Australia.

Section 4 derives a formula for rating the discount and gives some reasonable and practical approximations.

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.

Consider a claim size distribution with complementary d.f. H(.). Let E(x) denote the average claim payment under a policy subject to this claim size d.f. but with a deductible of x. That is

where E is the expectation operator and X is the random variable claim size before application of the excess.

It is well-known—see Benktander and Segerdahl (1960, p. 630)—that:

It was shown by them that E(x) is a constant for all x ≥ o if and only if the claim size d.f. is negative exponential:

This property of the negative exponential distribution is closely related to the fact that it is the only distribution with constant failure rate. See Kaufmann (1969, pp. 20-22).

The constancy of average excess claim size with varying deductible can be useful in practice. For example, if the distribution of motor vehicle (property damage) claim sizes can be assumed roughly exponential, which will often be reasonable, then a variation in the deductible will not induce any variation in the average size of claims paid by the insurer, i.e. after application of the deductible. This will be a particularly useful piece of information if for example one is examining trends in average claim size over a period during which a change in deductible occurred.

For the premium calculation the insurer will split up his collectivity of risks into risk groups which are homogeneous with respect to some directly observable risk factors. All risks of such a risk group will be charged the same base premium. But it is clear that by such an a priori classification not all determined factors can be taken into consideration, so that there will still remain accident proneness differentials within a risk group. Since these differentials will be reflected in the course of time by the claim experience of each risk, the insurer can come to a fair tarification by adjusting, each period, the base premium according to the individual claim experience of the risk. Such a system in which earlier neglected risk factors are taken into account a posteriori is an individual experience rating system. Our main interest goes to the following side-effect of experience rating: since an unfavourable claim experience results in a premium increase, an experience rated policyholder is stimulated to self-insure small damages. This phenomenon is well know in connection with bonus-malus systems in motor-car insurance, which explains why it is called “bonus-hunger”.

In the present paper a continuous time model for the bonus-malus system is set up which takes into account this hunger for bonus. An insured causing an accident will decide according to a certain decision rule whether to file a claim with his insurance company. The relevant information that he needs to make this decision is: his current risk class, the number of claims he has already filed during that period and the moment at which the decision is to be made.

The premium calculation principle is one of the main objectives of study for actuaries. There seems to be full agreement among the leading theoreticians in the field that the insurance premium should reflect both the expected claims and certain loadings. This is true for policy, risk or portfolio. There are three types of positive loadings: a) a loading to cover commissions, administrative costs and claim-settlement expenses; b) a loading to cover some profit (a cost-plus approach); and c) a loading for the risk taken by the insurer when underwriting the policy. The administrative costs can be considered a part of “expected gross claims”. Thus, the insurer's ratemaking decision depends on his ability to estimate expected claims (including costs) and on the selection of a fair risk loading.

The main concern in the literature is the appropriate measurement of the risk and the exact loading formula. Bühlmann [1970, ch. 5] and others identified four possible principles of risk loading, namely, the expected value principle, the standard deviation loading, the variance loading, and the loading according to the principle of constant utility. Various studies point to the advantages and disadvantages of these principles and also examine some additional loading forms—semi-variance, skewness, etc. (e.g., Bühlmann [1970], Benktander [1971], Berger [1972], Burness [1972], Berliner [1974], Berliner and Benktander [1976], Bohman [1976], Cooper [1974], Gerber [1975] and others). Despite different preferences in choosing the appropriate loading calculation principle, all seem to agree that the risk loading must be positive, since, otherwise, the firm would just have to wait for its ruin, that is bound to come sooner or later, according to risk theory.

The purpose of this article is to re-examine the appropriate principle of premium calculation in light of the recent developments in the theory of finance and especially in the theory of capital market equilibrium. These developments may suggest a new point of view and raise a few questions regarding the loading rules.

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 purpose of this statement is to provide general guidelines for the use of actuaries engaged in the establishment and review of loss and loss adjustment expense reserves. It is a statement of the Casualty Actuarial Society's Committee on loss Reserves. The statement consists of three parts.

1. Definitions

2. Considerations

3. Procedures

The balance sheet of the Fire and Casualty Annual Statement includes liabilities that are not subject to precise valuation. The reserves for unpaid losses (line 1, page 3 of the Annual Statement) and the reserve for unpaid loss adjustment expense (line 2, page 3 of the Annual Statement) cannot be precisely determined in advance. These reserves must be estimated. Because of their relative size and the difficulty in achieving accurate estimates of their values, these liabilities are vitally important balance sheet items. It is important that proper actuarial and statistical procedures be employed in order to improve the likelihood of reliable reserve estimates. Without reliable reserve estimates, an accurate evaluation of the financial condition of a fire and casualty insurer cannot be accomplished.

Loss reserving involves the current financial evaluation of costs associated with future contingent events, a matter of fundamental interest to actuaries. The contingencies involved are those factors which affect the cost of future payments on insured events which have already occurred. As ratemaking is another application of the same estimating process, the actuarial methodology is similar in many respects.

The definitions in the next section apply to both loss reserves and loss adjustment expense reserves.

For the purpose of this statement the terms “loss” and “claim” will be used interchangeably.

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.

A numerical procedure is described to evaluate the stop-loss premium in case the risk process is a compound Poisson process. The method is mainly based on an algorithm of R. Piessens and M. Branders for the numerical evaluation of Fourier transforms.

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.

Gerber (1974) has given a characterization of the exponential utility function by proving the fact that a premium calculation principle of zero utility is iterative iff the underlying utility function is linear or exponential. In the present note we prove the more general result that the premium calculation principle introduced by Bühlmann, Gagliardi, Gerber and Straub (1977) is iterative iff the underlying function v is linear or exponential or when the principle is a mean value principle.

It has been known for generations that the fire risk rate increases with the size of the insured object in a similar way as the death rate increases with the age.

Professor d'Addario and other Italian mathematicians have shown that statistical data often can be graduated by the formulas where S denotes the sum insured:

In 1940, d'Addario) in a practical case found the values α = 0.78, β = 0.44. In 1956, Blandin and Depoid have used the same formulas. Although these formulas in practical cases often lead to good approximations, one can hardly say that a proper mathematical or physical model describing the behaviour of fires lies behind.

About 1950, I worked with the statistics of a group of Danish fire insurance companies, in particular covering farm buildings. This investigation was organized by Gunnar Benktander. Our data confirmed the increase of the fire risk rate with the size of the buildings.

In one special group: Farm houses with thatched roofs, this increase was so important that the risk premium was approximately proportionate to the square of the sum insured. The statistics fully justified the tariffing and we tried to construct a model describing and explaining the observed facts.

The risk group in question was characterized by the overwhelming importance of total or practically total losses. Only a few per cent of the damages went to minor fires. When a fire breaks out in such a building, and reaches a certain slight extent, then it is not possible to save the building from total destruction. With sufficient approximation, we may say that only total losses occur (as in life assurance).