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’.

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 .

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.

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).