At a meeting of London Actuaries held in 1848 to discuss the formation of a professional organization, the following resolution was carried: ‘That this meeting considers it desirable to establish a scientific and practical association amongst the actuaries, secretaries and managers of the Life Assurance Societies of Great Britain.’ It is clear therefore that at the very beginning the prospective members regarded themselves as in some sense scientific.

The subject of Analysis of Surplus is often approached by the student with unwarranted misgivings, but it is hoped to show in this paper that the main requirement is a clear understanding of the valuation processes and assumptions.

Every valuation is based on a series of estimates and assumptions and it is a natural sequence that, at the end of a valuation period, these should be tested to see to what extent their adoption has been justified by experience. It can therefore be said that Analysis is the corollary of Valuation.

In the theory of Life Contingencies there arises the problem of determining the probability that exactly r lives shall survive a stated period of time out of n lives of different ages. This is of course a special case of a general question in Probability which may be stated thus: If there are n independent events whose individual probabilities of occurring at a single trial are p1,p2…pn, what in terms of p1,p2…pn is the probability that at a single trial of each exactly r of them shall occur?

The developments given in the text-books of King and Spurgeon are familiar to all, but these are not the only methods that have been evolved. Different proofs have appeared from time to time in various publications, most of them in foreign languages, and it is the object of the present paper to collect these together so that they may be easily accessible for reference and comparison. This would appear to be desirable because the question cited above is fundamental in Life Contingencies and in Probability, and also because some of the most satisfactory proofs are to be found only in foreign technical works and have not hitherto been published in English.

Mr A. F. Murray's paper published in the July issue of this Journal covers such a wide and complex field that it is obviously impossible to discuss it adequately in three to four pages. Within this limited space I have selected for comment a few points which not only constitute the main props in Mr Murray's arguments, but are also, largely for accidental reasons, of rather special interest. Their interest derives from the fact that in the last article written by the regretted Lord Keynes, these points are discussed in great detail and—most important—entirely different conclusions are reached.

If I understand him rightly Mr Murray's argument is roughly this. Multilateral international trade can function properly only if the following two conditions are satisfied:

(i) each of the participating nations must pursue a policy of full employment at home;

(ii) each nation must pursue abroad an economic policy such that its balance of payments over a long term is of manageable dimensions.

The distribution discussed by Mayhew and Vajda (J.S.S. Vol. vi, 1946, pp. 67–75 (70)), is the binomial with negative index (having origin, i.e. first term, at n instead of o), about which a good deal has been written, and their problem is the one discussed by Yule in 1910 (J.R. Statist. Soc. Vol. LXXIII, p. 26). The authors base their proof on an idea of Laplace, but the full power of the characteristic function derivation can be seen if we go further than they did. The results are of some interest, particularly as Kendall (Advanćed Theory of Statistics, Vol. 1, §5.13) does not go beyond deriving mean and variance by the elementary method in this case.

I have been asked to give you some idea of the activities which occupied me for four years of the war, namely, my duties with the Admiralty. For obvious reasons it is not possible for me to give in detail many of my impressions or the problems that arose in the work which was mainly directed to the end of securing the right number of the right type of Naval Aircraft at the right place at the right time. In practice the emergency of the problem precluded the use of sophisticated methods, and recourse had to be made to practical measures designed to provide a fair approximation to the answer sought.

As an instance of the actuarial nature of the problem involved it may be noted that the life of an aircraft is made up of three components:

(a) a period when, although serviceable, the aircraft is ineffective through being in transit, etc.,

(b) a period when it is effective and in use (including that in ‘ready reserve’), and

(c) a period when the aircraft is ineffective while undergoing repair, etc.