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.

This paper is concerned with certain problems which arise in the course of rate fixing for a portfolio comprising a large number of direct risks.

This is, of course, the traditional territory of the actuary and there is no lack of papers discussing specific aspects of the subject, such as the analysis of claim frequency, or the fitting of distributions to claim amount data. Equally there is a literature which discusses in general terms the philosophy which it is suggested should be followed by the non-life insurance ratemaker. It is when one tries in practice to follow a logical and consistent path through from portfolio records to rates that gaps in the literature become apparent and it is with covering these gaps, or at least some of them, that the present paper is concerned.

It is not the concern of the author in this paper to attempt to describe the entire ratemaking process. That there is no reference to the analysis of office expenses, commissions, or to the application of “judgment” or “experience” should not be taken to imply that these aspects are not susceptible to the kind of approach advocated here. Attention here is focused upon what might in a sense be regarded as the most basic aspect: the claim process or experience.

In many insurance operations the position which seems to be developing is that there is no lack of necessary computer hardware, and in principle, of data: the difficulties arising relate more to the form in which data should be retained and extracted, and the manner in which it should be used to assist in rate-making.

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?

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.

By intuition, the subdivision of an insurance portfolio into a number of classes is said to be good if it reflects the heterogeneity of the portfolio in an efficient way. To illustrate this rather vague statement we take the following very simple example:

The portfolio consists of 20 independent risks, 10 of them producing an expected loss ratio of say 30% each (type A risks) and 80% each (type B risks) respectively.

This “natural” subdivision is certainly better than, for instance, no subdivision at all;

or, the finest possible subdivision with 20 classes consisting of only 1 risk each (because there is no point in differentiating between risks of the same type);

or, 5 classes each containing two A- and two B-risks (here the number of classes is unnecessarily high and the heterogeneity has been completely wiped out; statistics based on this subdivision would even make us believe that the portfolio is totally homogeneous).

As a matter of fact, the above “natural” subdivision is—of course!—the best of all subdivisions, it is the optimal subdivision in this case.

In practice, however, as we all know, it is not easy to find the optimal subdivision. For one thing, the inherent structure or “natural” subdivision is not known a priori and secondly, for many different reasons, we can only choose from a limited number of subdivisions and not from all theoretically possible solutions. Note that even with only 20 risks there are 58.1012 possibilities of subdividing the portfolio. Thus, in practice, there is only a relatively small number of admissible subdivisions and the optimal one may not be among them, but we still need some sort of statistical criterion to choose the best one from these admissible subdivisions.

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.