For the 4th ASTIN colloquium in Trieste (1963) Robert E. Beard [1] had initiated the topic of extreme value theory and its application to actuarial problems. The impetus came from Gumbel's book [2], published in 1958, and the idea of applying extreme value theory ran about as follows:

Problem: Given: The size of the largest claim observed = XN

The number of observed claims = N.

Find: The excess of loss premium for a portfolio of independent risks identical with those under observation.

The idea was to use Gnedenko's limit distribution for the wide class of distribution functions with an unlimited tail and finite moments, the “exponential type” in Gumbel's terminology, and to calculate the excess of loss premium on the basis of this limit distribution.

The discussion in Trieste was very lively and culminated in Jan Jung's citation of Jan Jung [3]: “There is a natural law which states that you can never get more out of a mincing machine, than what you have put into it. That is, if the reinsurance people want actuarially sound premiums, they must get a decent information about the claim distribution”. In more mathematical language: The brilliant idea of Bobbie Beard is unfortunately leading to a non-robust procedure. A deviation in the true (but unknown) underlying distribution may lead to completely different results.

In any case the mincing machine argument seemed so powerful that we have had little reconsideration of the above problem at subsequent ASTIN colloquia. It may therefore be worthwhile to have another look at the problem now, i.e. more than ten years after the Trieste colloquium.

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.

This paper was inspired by comments by H. L. Seal in a series of lectures given to the Actuaries Club in New York and by a paper of his recently published in the Swiss Actuarial Journal (Seal, 1972 [6]). In his lectures he showed that the probability U(w, t) that a risk reserve at every epoch τ, where o < τ ≤ t will be non negative when the initial risk reserve is w is related to , the probability that the aggregate claim outgo through epoch t does not exceed by the relationship

where η is the security loading and f(x, t) = (∂/∂x) F(x, t).

It is assumed that the d.f. F(x, t) is differentiable eith regard to x with a possible exception at the point x = o.

Using an extension of the “ballot theorem” in Chapter III of Feller (1968 [4]) he showed that and observed that if numerical values of F(x, t) were available values of U(w, t) could be computed.

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)

The multiplicative ratemaking, model we have in mind is the following one. Within a certain branch of insurance we have, say for simplicity, two tarif arguments U and V. For example, in motor insurance we could think of U and V as being make of car and geographical district respectively. In fire insurance U could be class of construction for buildings and V could relate to fire defense capacities.

The arguments are of a qualitative nature and argument U has r levels, while argument V has k levels. To our disposal we have statistical experience of the business for a certain period of time, consisting of

—risk exposures nij (i = 1 … r, j = 1 … k).

Risk exposure nij thus corresponds to the ith U-level and the jth V-level. It could be e.g. number of policy years or sum insured during the period of observation for objects belonging simultaneously to U-level i and V-level j.

The nijS are known non-random quantities.

—(relative) risk measures pij(i = 1 … r, j = 1 …k).

Risk measure pij could be e.g. claims frequency, i.e. number of Claims divided by number of policy years, or claims cost per policy year or claims cost as a percentage of sum insured. In general pij is thus the observed number or the observed amount of claims belonging simultaneously to U-level i and V-level j, divided by the corresponding risk exposure nij.

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.

A diminishing number of Reinsurers still seem to work with percentage loadings which are applied with more or less sophistication to the pure risk premium rate.

A more sensible approach is to study the distribution of the total claims amount of the layer reinsured (standard deviation, variance, skewness).

In practice this distribution is not known and we have to work with estimates E* and σ*2 of the first moment which is the pure risk premium (E) and of the second central moment (σ2). Because of the positive skewness of the distribution, E* will fall below E in more than half of the cases and sometimes considerably below.

One security loading principle could be to add a proportion k of σ* to E* in such a way that

If ε is put equal to 0.25 this reflects our wish to ensure that our security loaded rates are too low only in 25% of the cases. A σ- loading here emerges as a result of the statistical uncertainty.

If we analyse the result fluctuations we are led to apply a variance loading (1), which can be interpreted as a price for capacity (2). (See also discussion contributions by B. Ajne, G. Benktander, G. Berger, H. Bühlmann; Transactions of the Congress, 5, Oslo 1972, p. 169 ff.)

An underwriter who is confronted with two portfolios having the same first and second moments should, if conditions are the same, prefer the one with the lower third moment. A special loading for skewness is thus indicated.

In the past the subject of optimal reinsurance has been dealt with by various authors such as Borch (1), Kahn (2) and Verbeek (3). Borch tries to look at the problem from the point of view of a company which acts as both Insurer and Reinsurer whereas the viewpoint of Kahn and Verbeek is that of a Ceding Company.

Let us study the simplest possible market situation of one Insurance Company C and one Reinsurer R. C writes a portfolio P and seeks reinsurance protection at R.

In the present note I will start with the viewpoint of optimizing simultaneously the situation of C and R.

For both C and R the variance is used as a measure of risk. The variance of P is V. C makes an effort to reduce V to a level Vc by way of a reinsurance treaty T. The corresponding variance of R is VR.

Out of the set of possible reinsurance treaty arrangements {T} we assume that the parties are first of all interested in such arrangements for which

In other words they will look for solutions which will lead to a substantial reduction of the variance.

It is possible to think of “nonsense” arrangements where no substantial reduction is achieved and in the extreme case even of arrangements for which

In a further paper we might deal with such a case.

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 the classical risk theory the interdependence between the security loading and the initial risk reserve is studied. It could be said that the purpose of these studies are to state how large the security loading must be in order to avoid ruin of the insurance business. It has often been said that the classical approach with an infinite planning horizon is unrealistic. The main reason for this attitude is that if the security loading is equal to zero then ruin is certain. Since in practice it is often difficult to estimate the true size of the security loading the whole problem of ruin or non-ruin seems to rest on a rather shaky foundation. This attitude to the problem is reflected in studies in risk theory performed in recent years. The infinite planning horizon is then often replaced by a finite time period. Since the probability of ruin during a short period of time depends to a minor extent on the size of the security loading these studies are concentrated mainly on the shape of the claim distribution, while the security loading is of minor interest.

Let us think of a gambling-house, where coin-tossing is practised. Let us assume that the gambling-house for reason of fairness decides to pay two dollars to each winner who has staked one dollar. Probability theory tells us that however rich the gambling-house may be, it will be ruined in the long run. This simple example reminds us of the trivial fact that insurance business without a sufficient security loading in the premium is commercially impossible.

In 1967, Bühlmann has shown that the credibility formula was the best linearized approximation to the exact Bayesian forecast.

His result for the credibility factor can be found back by means of some Bayesian inference techniques. Introducing a uniform prior probability density function for the credibility factor provides us with a method for estimating z, a correction term to the Bühlmann's result is obtained. It is shown how prior boundary conditions can be introduced.

The problem of the best fit to set ideal values under general inequality order restrictions is examined for asymmetric, quadratic, absolute, and Chebyshev norms. Special solution procedures are given in terms of network flow algorithms over a network associated with the given isotonic order relations, and the nature of the optimal solutions is characterized for the different norms.

The model is formulated in terms of finding an optimal insurance rate structure over given risk classes for which a desired pattern of tariffs can be specified. The suitability of different norms is considered in the context of corporate profitability, and the relationship to a simple rate relativities model is described.

In a recent paper on the theory of demand for insurance Arrow [1] has proved that the optimal policy for an insurance buyer is one which gives complete coverage, beyond a fixed deductible. The result is proved under very general assumptions, but its content can be illustrated by the following simple example.

Assume that a person is exposed to a risk which can cause him a loss x, represented by a stochastic variable with the distribution F(x). Assume further that he by paying the premium P(y) can obtain an insurance contract which will guarantee him a compensation y(x), if his loss amounts to x. The problem of our person is to find the optimal insurance contract, i.e. the optimal function y(x), when the price is given by the functional P(y).

In order to give an operational formulation to the problem we have outlined, we shall assume that the person's attitude to risk can be represented by a Bernoulli utility function u(x), and we shall write S for his “initial wealth”. His problem will then be to maximize

when the functional P(y) is given, and y(x) є Y. The set Y can be interpreted as the set of insurance policies available in the market. It is, natural to assume that o ≤ y(x) ≤ x, but beyond this there is no need for assuming additional restrictions on the set Y.

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.

When an insurance company accepts new insurances or when the premiums of earlier accepted insurances have to be changed on renewal the company has to

— search for the factors that influence the premium and

— calculate the premium according to the values of these factors.

In order to calculate the premiums the company gathers data consisting of factors eventually influencing the amount of claims. On the basis of these data the company calculates the tariff which has to fulfill the following general principles:

1. The tariff has to be as correct as possible in relation to different risk groups.

2. The structure of the tariff has to be such that the calculation of the insurance premium is quite straightforward. With this in mind the factors influencing the tariff have to be few enough and the structure of the tariff has to be the simple (e.g. linear or multiplicative) function of the factors or it should be rather easy to put them into tabular form.

These principles are partly contradictory. If the premium is correct, the structure of the tariff is not usually simple.

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.

On 12 February 1974, a limited no-fault insurance system was introduced to cover persons killed or injured in road accidents in Victoria, Australia. This system supplements the existing compulsory third party insurance system, under which payments are made only if an insured driver can be shown to be negligent.

The costs of the no-fault benefits are met by compulsory third party insurers, who are allowed to charge not more than the premium fixed by the Government. This paper examines the adequacy of the no-fault component of the premium.

A theoretical model has been developed, covering all the components of compulsory third party and no-fault insurance. This model incorporates the limited available accident and insurance statistics, together with a number of assumptions. As in many practical situations, the limited data available has severely restricted the factors considered, and has reduced the reliability of the estimates made from the model.

The theoretical model does however:

—indicate approximately the premium required for no-fault benefits

—suggest areas where statistical investigation would be valuable

—provide estimates for different types of benefits to different groups of persons, which can be checked as more data becomes available

—provide useful data when considering other forms of no-fault insurance, such as the national rehabilitation and compensation scheme currently being considered by the Australian Government.

In his paper to the Tenth ASTIN Colloquium the author presented generalised extreme value techniques for making use of all large losses that are available for analysis and not merely the largest. In this paper the problem of assessing the relative contributions of various factors to fire losses is investigated. A model concerned with multiple regression with extreme observations of given rank is developed. It takes into consideration the biases due to the use of extremes and the differences between categories of risks in regard to the frequency of fires (or claims). By way of illustration the model was applied to the largest and second largest losses in the textile industries in the United Kingdom during the six-year period 1965 to 1970. The presence or absence of sprinklers, whether the buildings were single-storey or multi-storey, and total floor area were the independent variables included in this preliminary investigation. Judged from extreme losses sprinklers appear to reduce considerably the expected damage in all fires.

The technique enables different estimates to be obtained for each regression parameter for different ranks. It is desirable to have a single overall estimate for each parameter; and for this purpose a second model is developed for performing a regression analysis combining observations pertaining to a number of ranks. Covariances of the residual errors are also taken into account in this model.

The concept of utility is twofold. One may think of utility:

1) as a tool to describe a “fair game”

2) as a quantity that ought to be maximized.

The first line of thought was initiated by Daniel Bernoulli in connection with the St. Petersburg Paradox. In recent decades, actuaries, economists, operations researchers and statisticians (this order is alphabetical) have been concerned mostly with optimization problems, which belong to the second category. Most of the actuarial models can be found in a paper by Borch [4] as well as in the texts by Beard, Pesonen and Pentikainen [3], Bühlmann [6], Seal [15], and Wolff [17].

We shall adopt the first variant and stipulate the existence of a utility function such that the surplus process of an insurance company is a fair game in terms of utility. This condition is naturally satisfied under the following procedure: a) a utility function is selected, possibly resulting from a compromise between an insurance company and supervising authorities, b) whenever the company makes a decision that affects the surplus, it should not affect the expected utility of the surplus.

Mathematically, this simply means that the utility of the surplus is a martingale. Therefore martingale theory (that was initiated by Doob) is the natural framework in which we shall study the model. We shall utilize one of the most powerful tools provided by this theory, the Martingale Convergence Theorem.