The authors have studied the combined data on claims in fire insurance of dwelling houses reported 1958-1969 by Swedish fire insurance companies The claims were cleared of deductibles and adjusted according to a suitable index Only losses above the largest deductible (in real value) applied during the observation period were included.

The material contains four different classes according to the fire resistibillty of the building construction For international comparisons, the pure classes B1 (“stone” dwellings) and B4 (wooden houses) are of interest The distribution of the claims could be well approximated by the log-normal distribution in B1and by the Pareto distribution in B4 An equally good or better fit was obtained by assuming the original loss, reported or not, being distributed according to these distributions and applying the distributions, conditioned by the loss being larger than the deductible in both cases the distribution parameters are functions of the insurance amount in such a way, that the mean value of the loss is described as a power of this amount.

The authors refrain from any theoretical arguments for the general applicability of the distributions used They observe, however, the good approximation by wellknown parametric distributions which facilitates many actuarial taks, such as the determination of first loss premiums, deductible premium factors, excess-of-loss premiums etc The agreement between model and reality make these functions fit for use in the models underlying the general risk theory and in the more comprehensive models of the non-life insurance business.

A few years ago Scurfield (JSS 18) indicated a method used inone U.K. non-Life Office for estimating claims liability for the Motor class of business. Since that paper was written a considerable amount of development has taken place and it is now used in the office as an effective continuous method, using computer techniques, for estimating claims liability. It has also been used for projecting expected cash flow for claims arising in the past.

Whilst it is accepted that the method has limitations, it has been found that an automatic method is required especially in the production of financial models in the non-Life field, and its limitations are outweighed by the fact that there are, in our opinion, no easier methods of producing a satisfactory working model.

In the following paper it is the intention to set out briefly the details of the method, then concentrating on the known limitations and the methods so far devised to combat these limitations. In the appendices we have reproduced some computer printout and figures which will provide practical information on the method.

The principle behind the method is that the “Run-off” of claims payments for any “year of claim”, or similar cohort of claims, follows a particular pattern which experience has shown to be reasonably stable. We can thus study the pattern of claims payments at each stage of “run-off”, say each month and thus ascertain the average and range of values that the historic pattern shows (Fig. 1).

Fig. 1.

Period of delay since beginning of claims Period (Years)

Our purpose is to introduce some models of inference for risk processes. The bayesian viewpoint is adopted and for our treatment the concepts of exchangeability and partial exchangeability (due to B. de Finetti, [6], [7]) are essential.

We recall the definitions:

The random variables of a sequence (X1, X2 …) are exchangeable if, for every n, the joint distribution of n r.v. of the sequence is always the same, whatever the n r.v. are and however they are permuted.

From a structural point of view an exchangeable process X1, X2 … can be intended as a sequence of r.v. equally distributed among which a “stochastic dependence due to uncertainty” exists. More precisely the Xi are independent conditionally on any of a given set (finite or not) of exhaustive and exclusive hypothesis. These hypotheses may concern, for instance, the values of a parameter (number or vector) on which the common distribution, of known functional form, of Xi depends. We shall restrict ourselves to this case. Therefore, we shall assume that, conditionally on each possible value θ of a parameter Θ, the Xi are independent with F(x/θ) as known distribution function. According to the bayesian approach, a probability distribution on Θ must be assigned.

In this paper, we are interested in a model related to a number of periods of Company's activity.

The Company calculates an amount s which then could be given as a supplementary interest to the shareholders. This calculated amount is taken from the risk reserve. Let us assume the risk reserve = S at the beginning of an operating period. If the Company gives an amount s to the shareholders, then the risk reserve is S — s. (Borch 1972; Seal 1969).

We must determine the best policy of dividends, that is a rule which determines the payments to be made each year to the shareholders of the Company, maximising a definite criterion.

The problem of dividends must be approached in the “dynamic programming manner”. Indeed, the payments of dividends have an effect upon further gains of the Company and its capacity to pay dividends in the future.

The objective of the Company is, for example, maximising the average utility of the dividend payments, which is calculated according to the distribution of claims. (Borch 1964a; Wolff 1966).

In 1935 the New York Insurance Department introduced the concept of special contingency funds for certain types of insurance. Such requirements had first been introduced in 1925 for mutual workmen's compensation companies. Clear, consistent principles for these funds were not stated at the time, but their purpose seems to be to provide a cushion that may be used in time of serious financial difficulty.

In group life insurance, this fund is a “special contingency reserve” and is carried at the suggestion of the New York Department). For mutual casualty, nonprofit hospitalization and medical indemnity plans, and for reciprocal insurers, the fund is treated as “special contingent surplus”, and is a mandated substitute for the minimum capital required of stock insurers). The U.S. federal Life Insurance Company Income Tax Act of 1959 recognizes special credits of a similar nature for health, non-participating life and group life and health insurance coverages.

In all of these cases, the accumulations or credits are defined by a designated percentage of premiums). For the New York Department group life reserve and the Income Tax health and group life credit, a maximum is defined by a second percentage of the same annual premiums base. For the nonprofit plans' surplus, a similar maximum in terms of premium is used. However, in the case of mutual casualty companies, and of reciprocal insurers, the “special contingency surplus” must be built up to a defined absolute amount equal to the amount that would be required as capital of a stock company; there are no statutory provisions for withdrawing any part of such a fund once it is accumulated.

The credibility formula used in casualty insurance experience rating is known to be exact for certain prior-likelihood distributions, and is the minimum least-squares unbiased estimator for all others. We show that credibility is, in fact, exact for all simple exponential families where the mean is the sufficient statistic, and is also exact in an extended sense for all regular distributions with their natural conjugate priors where there is a fixed-dimensional sufficient statistic.

The distribution of the sum of n mutuality independent random variables with a common distribution f(x) plays an important role in many insurance problems. The paper presents alternative methods of deriving the distribution of the sum of n random variables when f(x) is a mixed density and mass function. The methods are illustrated and compared.

The Sparre Andersen model assumes that the interclaim times and the amounts of claims are independent random variables, the former identically distributed according to a distribution function K(t), t ≥ o, K(o) = o, the latter identically distributed according to a distribution function P(y) — ∞ < y < ∞ As is well known, the Poisson risk process corresponds to the particular case K(t) = 1 — eβt. In the present paper it is pointed out that another particular case, viz. K(t) = є(t — h), corresponding to a fixed (and thus — strictly speaking—nonrandom) mterclaim time, h, has interesting applications. Thus, the ruin problem considered by Giezendanner, Straub and Wettenschwiler in a paper to the 1972 International Congress of Actuaries in Oslo can be formulated by means of this particular case. The same can be said about the earlier model brought forward by Ammeter in his 1948 paper in Skandinavisk Aktuarietidskrift.

About the contents of the paper the following further information may be given. The general Sparre Andersen model is first presented and then the ruin formulas are given for the case with a positive gross risk premium. Thereafter, a modified and more direct method for deriving certain necessary auxiliary functions is illustrated by examples including 1 a the Giezendanner—Straub—Wettenschwiler model. The rest of the paper contains a discussion from the point of view of the Sparre Andersen model of (1) the discrete (equidistant) inspection of a Poisson process for ruin, (11) the Ammeter model and analogous models, and (111) the Giezendanner—Straub—Wettenschwiler model.