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 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 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 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.

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.

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.

In practical statistical work one frequently meets certain problems. For instance, we may have the following data about loss ratios in certain insurance companies and corresponding, numbers of insurance in force:

I assume further that we have no reason to believe that the companies, their loss ratios and their structure of insurances in force differ in any other way than by the size of companies. The problem is how to get quick estimates of mean losses and their variances in different companies?

A straitforward way to estimate the mean loss ratio would be to compute the usual mean of numbers pi, (Σpi/6) = 14; its standard deviation is 6,5. As this procedure of the “first statistician” seems to be too simple and naive, a “more cautious” statistician would compute the weighted mean loss ratio

The “more cautious” statistician would arque that his result is much better than the other result 14. But what would be the variance of the estimate 7,1, and what is the variance in the different companies?

Conditions under which the natural conjugate prior is not zero on its boundary are given, correcting an argument about conditions for exact credibility given in another paper.

A simple risky situation is studied in the framework of consumption theory. Saving is shown to be a substitute to insurance. Two new concepts, riskbearing budget and effective risk coverage, are introduced in order to give a more accurate insight into the optimal risk-bearing decision. The effect of a variation in current consumption and in wealth upon the optimal insurance coverage is analysed.

It is likely that in the future applications of actuarial methods to the decision making in non-life companies will more and more relate to the utility concept as was proposed by K. Borch [1] about fifteen years ago. In this connection it will be important to have workable numerical methods. The calculation of the distribution function of the profit is an unavoidable problem from a practical point of view. Even if it is possible to compute this function today accurately with computers by using the ingenious technique developed by H. Bohman [2], integrals become very laborious when applied to the decision making procedure based on utility concepts. This paper intends to show that the NP-technique,—proposed for the first time by L. Kauppi and P. Ojantakanen in actuarial science [3]—, is particularly suitable in integrals needed for utility calculations.

LetF(x) be the distribution function of the total amount of claims and let its mean, standard deviation, skewness and kurtosis be respectively m, σ, γ1 and γ2. The NP-technique uses the system of equations

where Φ(y) is the standardized normal distribution function. If the parameters m, σ, γ1 and γ2 and F(x) are known, y is directly found from the tables of the normal distribution function, and thereafter the second equation directly gives the value of x. If, vice versa, x and the above parameters are known, F(x) is obtained by solving y from the second equation (1), or, more practically, by using the converted NP-expansion instead of (i), i.e. [4]:

where

Sometimes it is sufficient to use short forms of the formulae (1) and (2), obtained by omitting the terms in the brackets. If these rougher approximations are used, the estimation of the kurtosis γ2 remains unnecessary.

In order to motivate the series of Monte Carlo simulations we have carried out in the following article we would like readers to imagine that a small rural casualty insurance company, the Farm Fire and Flood Damage Ins. Co. (FFFDIC), is to be bought by an entrepreneur (whom we shall designate by EP) provided his consulting actuary (the author of this article) can satisfy his requirements which are as follow:

(i) A 15-year investment is foreseen at the end of which time EP wishes to be able to sell, hopefully without loss.

(ii) The risk-capital is to be invested and (although some of it must be in easily liquidable securities) should yield a rate of return comparable with that obtainable on the same amount of capital invested in the market.

(iii) The premiums will not have risk-loadings, as such, but will be loaded for profit by 15%.

(iv) The risk-capital should, on the average, be returnable at the end of the 15-year investment.

In recent times, the subject of deductibles in Industrial Fire insurance has gained significance to an increasing extent. In fact, up to a short time ago it was by no means common to apply deductibles in Industrial Fire insurance in Europe. The situation is entirely different in the USA where deductibles are the usual thing and are even obligatory with individual risk categories (e.g. petroleum refineries), hazards (e.g. explosion), or special types of companies (e.g. factory mutuals).

Today, however, there is a definite increase in the demand for deductibles in Industrial Fire insurance in Europe, too.

This trend may be explained by the fact that in order to reorganize their business successfully, insurers have been forced to increase their premiums by a considerable amount. It seems to be an unwritten law that when premiums in general are increased considerably, the point will some time or other be reached when the insured do not accept any further premium increases. Thus, even if such increases are completely justified in a view of claims experience, the insured will demand that premiums are reduced either by the insurer granting rebates for appropriate loss prevention measures or by the introduction of deductibles. In the USA, for example, such a situation arose around 1960 when considerable premium increases very quickly led to the introduction of substantial deductibles in Industrial business. In some European countries we are currently experiencing a similar development.