This paper is mainly concerned with the division of a trust fund between the beneficiaries, when an actuary is asked for a solution which, in his opinion, is fair to all parties concerned. It may be that the beneficiaries are all sui juris and form a closed class, in which case they approach the actuary as umpire, and adopt a passive role, at least until they have considered the implications of his recommendations. Alternatively, the actuary's report may be required to assist the Court to approve a scheme involving beneficiaries, e.g. infants, whom it is the Court's object to protect. In that case the actuary's role—as discussed later in the paper—is rather different, but it may still be necessary for him to state what in his judgment would be an impartial solution.

I. Equity-linked contracts have been the subject of discussions at the Faculty of Actuaries (J. B. Maclean, T.F.A.23, 327), the Institute of Actuaries Students' Society (R. H. Blunt and L. H. Lane, J.S.S.14, 259) and the Institute of Actuaries (J.I.A.84, 37). The papers delivered and the discussions devoted considerable space to the question of the desirability, from the point of view of both the industry and the public, of the issue of such contracts.

Since these discussions, several offices have entered the market both for assurances and for ‘self-employed’ annuity contracts, and this paper, recognizing the de facto position, is concerned with practical points arising from the issue of the contracts.

In a series of trials with constant probability p of ‘success’ (S) and q ( = I–p) of ‘failure’ (F), the problems arise of determining the expected number of trials required to obtain a specified pattern of results (e.g. SSFSFFSSSFF), and of calculating the probability that such a pattern will appear in a given number of trials. We are here concerned essentially with ‘general’ patterns which may exhibit no clear regularity; many methods are available, and well known, for dealing with regular patterns (e.g. SSSSSS or SFSFSF), which do not apply to irregular ones. Feller, has shown how to solve the problems for general patterns, using an ingenious definition of the event corresponding to the appearance of the pattern whereby it becomes a recurrent event; his powerful general theory of recurrent events then applies and yields the required information by means of generating functions. A different method is given on page 171 of Bizley for finding the expected number of trials required to obtain a general pattern; this uses only the simplest mathematical tools but involves rather a lot of work for long patterns. Under each of these methods, however, every given pattern has to be treated individually and a separate calculation performed.