In recent years, as a result of more concentrated research together with the ravages wrought upon some insurers by inflation, the fundamental significance of the so-called run-off triangle in the calculation of provisions for outstanding claims has been increasingly recognised. The run-off triangle, which is a two-way tabulation—according to year of origin and year of payment—of claims paid to date, has the following form, where Cij is the amount paid by the end of development year j in respect of claims whose year of origin is i, i.e. Cij is the total amount paid in year of origin i and the following j years.

The information relating to the area below this triangle is unknown since it represents the future development of the various cohorts of claims.

In this paper some ruin probabilities are calculated for an example of a lognormal claim distribution. For that purpose it is shown that the lognormal distribution function, Λ(y), may be written in the form

where V(x) is absolutely continuous and without being a distribution function preserves some useful properties of such a function.

An attempt is also made to give an approximant Λα(y) to Λ(y) such that Λα(y) is a linear combination of a low number of exponential distributions. For comparison, ruin probabilities are also calculated for two examples of Λα(y).

In the considered numerical cases it is assumed that the occurrence of claims follows a Poisson process.

The paper considers the problem of finding an upper bound for the Stop loss premium.

We will start with a brief sketch of the practical context in which this problem is relevant.

If it is reasonable to assume, that the accumulated claims variable of the underlying risk can be represented by a Compound Poisson Process, the following data are needed for fixing the Stop loss premium:

— the claims intensity,

— the distribution of the claimsizes (jump-size variable).

In practical situations it is usually possible to find a reasonable estimate for the claims intensity (expected number of claims in a given period).

Generally speaking, however, it is not so easy to get sufficient data on the claimsize distribution. Ordinarily only its mean is known. This deficiency in information can of course be offset by assuming the unknown distribution to be one of the familiar types, such as Exponential, Gamma or Pareto.

Stop loss premiums are however very sensitive to variations in the type of claimsize distribution and consequently it can make a lot of difference in the result what particular choice is made.

To gain some insight into the consequences of a specific assumption, it is useful to know within what range the premium can move for varying distributional suppositions. This means establishing an upper bound and a lower bound. The lower bound is trivially obtained if the mass of the claimsize distribution is solely concentrated at its mean. The upper bound on the other hand should correspond to the “worst” possible claimsize distribution. This means, that we have to look for a distribution which maximizes the Stop loss premium.

Many studies concerning the frequency of claims by size in health insurance are not generally known). A possible explanation of this circumstance could be the fact that in most countries this line of insurance has been brought entirely within the ambit of social insurance. Also from the side of the social insurance very few investigations have been published).

In this paper we will analyse the claim experience (relating, to the calendar year 1972) of a private health insurance business. The data have been subdivided according to three levels of coverage (in increasing order of benefits these are: class III, class IIb and class IIa). The claim payments comprise nursing costs, auxiliary costs and the fees for specialist treatment in and out of the hospital.

We will use the following notations:

Si: claim amount paid for the insured i in one year,

n: number of claims,

v: number of risks (policies insured).

In many instances the premium is simply determined as a level premium. In other words each insured pays the premium p, calculated as follows:

.

Actually we make the assumption that the claims are normally distributed, the parameters of which can be estimated as follows:

which permits the calculation of the premium according to:

.