In this paper we will investigate the following reinsurance problem: An insurer, whose total claims for a certain period may be regarded as a random variable x with expected value Ex = m, wishes to cede part of his business to a reinsurer. A reinsurance treaty will consist of rule for the division of x between the two parties. For any observed value of x it should define uniquely what amount should be borne by the ceding insurer. The amount borne by the reinsurer is then simply the remaining part of x.

We shall assume that the insurer has already decided how much of his business he wishes to cede, in the sense that he wants to retain a part of the total risk with expected value m — c, where c is a fixed constant, o < c < m.

Using the terminology introduced by Kahn in (2) we will describe a reinsurance contract by a transformation (or function) T that for a given x yields the amount Tx borne by the cedent. The random variable x is thus divided into two parts

and the properties of the reinsurance contract described by T are summarized in the distributions of the two random variables Tx and (1 — T)x = — Tx.

The motivation for reinsurance is generally held to be a desire for stability, in other words the cedent wishes to choose a T such that the random fluctuations in Tx are in some sense smaller than those of x. This choice will in our case be performed under the restriction that ETx = m — c.

It is clear that we can never talk about an optimal choice of T without defining exactly what criterion we shall use when comparing two transformations, T1 and T2. According to the above, the criterion should refer to the properties of the distributions of T1x and T2x, so that if one distribution is “more concentrated” around some central value, the corresponding transformation is deemed preferable to the other. However, we still have to define what we mean by “more concentrated”.