The question of bonus distributions has been much in the minds of actuaries during the post-war period. In October 1959 a paper was submitted by Benz to the Institute of Actuaries which he introduced with the following remarks: ‘Recent financial events have caused new problems to come before actuaries when advising on the finances of life offices and, in particular, on bonus distributions to with-profit policyholders. These problems may be said very briefly to centre around the question of what circumstances make it appropriate for an office to depart from the normal procedure for distributing bonuses.’

In a recent note (Bizley, 1960) the writer described a device which simplifies the solution of certain derangement problems. The present note explains and illustrates another device in combinatorial work (and hence in probability) having potentialities that do not seem to be well known. It is useful in tackling problems concerned with the choice of objects from an array when restrictions are placed upon the relative positions of those to be chosen, and it has applications to runs of consecutive events.

1. Consider the question: Given the group life assurance premium for death benefits, what extra should be charged for a premium rebate on the basis that the cost of the rebate must be covered by the extra? Assume no additional administrative costs are involved.

2. Let the rebate be a proportion s of the excess of a fraction a of the total office premium P′ over the total claim C in a year. The rebate is then s(αP′– C). Usually the benefit is more complicated, but as explained in §8, the extra premium for the more complex benefit is related in a simple manner to the extras for the benefit described. In order to avoid scale constants the average sum assured per life is taken as unity.

It is probably fair to say that during the last decade problems of internal administration have faded into insignificance besides the legal problems which now arise out of a life office's relationship with its policy-holders—in particular in regard to estate duty matters.

Before examining the position in detail I would like to make some general remarks regarding the burden of estate duty on life assurance policies today. Rates of duty are progressive, as the table on page 399 shows, and bear most heavily on the large estates. Therefore, not surprisingly, a good deal of interest and effort has centred in recent years in reducing this burden.