These notes are not intended to be a history of the Second World War. They are no more than a survey of the effect of that war on the financial institutions, namely, the Life Insurance Companies, with which British actuaries are particularly concerned. It is hoped that the survey will be of value to those members of the Students' Society who have been out of touch—through no fault of their own—with what has been happening in the field in which their peace-time activities mainly centre.

The Companies operate under conditions of ‘Freedom with Publicity’ in furtherance of which there was published, pre-war, the yearly volume of Assurance Companies' Returns, which, with the summaries appended to it, offered a convenient source of facts and figures. The publication of the Returns ceased during the war years, the last figures available relating to the operations of the year 1937; for most purposes, therefore, 1937 has been used as the ‘pre-war basic year’.

Those of us who have had occasion to study the Friendly Societies: Act of 1896 will be aware of the provisions contained therein relating to the dissolution of friendly societies and of the fact that the rules of a registered society must incorporate such provisions.

The chance that many of our members will meet a case of dissolution in practice is perhaps small, but, nevertheless, the study of the problems involved in assessing the shares of the members in the funds of a society is well worth while, since it involves the consideration of the basic principles of friendly society valuation.

After a short discussion of the legal provisions, this paper introduces some of the more important problems that can arise in connexion with the dissolution of societies of the most common type granting both sickness and death benefits.

The considerable public discussion of the past two years on the subject of our social insurance schemes and their post-war reconstruction has culminated in the publication of the Government's plans as a White Paper (Cmd. 6550–1). At present, social insurance in this country (now to be called National Insurance) is divided into a number of independent sections, each with its own statistical arrangements covering a large part of the population. At the same time some aspects of the whole population are regularly surveyed by the Registrars-General by means of an independent system of recording. The Government's plan greatly expands the field of insurance to cover all classes and groups. Before this unified plan can begin to function, fresh statistical machinery will have to be prepared, adequate to the task of keeping it efficiently in operation, and there will in consequence be a great opportunity for coordinating, simplifying and generally improving the national population records as a whole.

Two persons turn up cards alternately, each starting with a similar, but differently shuffled, pack of n cards. What is the probability that before all the cards have been dealt two identical cards are showing at the same time?

If the two players deal simultaneously the answer is unity minus the probability that none of the n pairs of cards is an identical pair. The problem is thus reduced to the calculation of the number of ways in which n letters can be put into n addressed envelopes without any letter going to the correct addressee. This is the classical example of derangements, and the solution can be found in Chap, IV of Whitworth's Choice and Chance, viz.

Statistics forms a not unimportant part of the actuary's studies, and is understood to be attracting greater attention than was the case, say, fifty years ago. To some extent this is undoubtedly due to an increasing recognition of its importance in this field, but we must also recognize that it is a subject in which big advances have been made on the theoretical side during the present century. Now there must always be of necessity a great gulf fixed between the papers of an Aitken and those of the Institute's examinations. It does not follow, however, that the elements of the subject must continue to be taught in the same old way. The enigma facing the student when extensions to knowledge are being continually made can be resolved, and very often is resolved, by a complete rewriting of the elementary text-books. To give an example, we have all been taught at one time or another how to test the difference between two sample means by using the standard error, on the large sample theory.

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.

Let f(x), f″(x), f″(x) denote a function of x, its first and second derivatives respectively. If f(x), f″(x) are one-valued, finite and continuous, and f″(x) exists in the neighbourhood of the value a of x, then

where o < θ < I. Suppose the value a is an approximation to a root of the equation f(x) = o, the true value of which is a + h. Equation (I) becomes

and supposing that f″(a) ≠ o

that is, a + h differs from .

Ever since the abrupt announcement that Lend Lease was to be discontinued, the attention of the British public has been focused on Anglo-American economic problems. So furious has been the spate of articles (well informed and otherwise) launched by the Press, and so vociferous have been the supporters of wholly divergent views, that most people must by this time be heartily sick of the whole subject. I must therefore apologize for inflicting upon the Society the subsequent remarks on such a well-worn theme, but in self-defence would explain that the subject was chosen before the recent clamour and shouting arose.

This note is a direct outcome of the discussion provoked by the recent report of the Committee of the Students' Society on the Future of the Profession. One of the points mentioned in the report was the high average age at qualification for the Fellowship. This age depends, inter alia, on

(1) the age of starting,

(2) the frequency of holding the examinations,

(3) the number of parts that may be taken at a time,

(4) the total number of parts to be passed,

(5) the stringency of the examinations.

This note deals only with the last two of these factors.

In normal circumstances, a student desiring to become a Fellow of the Institute of Actuaries finds himself faced with the prospect of having to pass seven examinations, viz. the Preliminary and Parts 1–6. The total numbers who sat for each part and the numbers and percentages of passes for each year from 1928 onwards are given in a table at the end of this note.

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.

There is a high chance against your picking up any recent number of the Bulletin de l'Association des Actuaires suisses which does not contain an article dealing with the theory of the changing population ‘Le renouvellement, quelques problèmes connexes et les équations intégrates du cycle fermé (Féraud, vol. XLI, 1941, p. 81) and ‘Über die Integralgleichung der Bevölkerungstheorie’ (Hadwiger, vol. XXXVIII, 1939, p. I) are two titles of papers belonging to this category. The English names for the relevant notions can be found in many publications by Lotka (e.g. ‘The theory of industrial replacement’, Skand, Aktuar Tidskr. 1940; ‘On an integral equation in population analysis’, Ann. math. Statist. 1939; ‘The structure of a growing population’, Human Biology, vol. III) and one can nearly always see that a paper deals with this subject by noticing an equation of the following form:

from which the function G (t) is to be determined. (For a purely mathematical investigation of this ‘integral equation of the second type’ see Feller, ‘On the integral equation of renewal theory’, Ann. math. Statist. Vol. XII, 1941, and compare also C. D. Rich, J.I.A. Vol. LXV).

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.