During the last session of the Institute of Actuaries, I had the honour to submit a paper, the purpose of which was to devise a method of adjustment as even and correct as that of Mr. Woolhouse, and more easy in application. This was to be done by combining two summations of numbers increasing by third differences; the summations being so chosen as to introduce small factors only into the formula, and to admit of its being worked in columnar shape. (Journal, vol. xxiii, page 335.)

Thereupon, Mr. G. F. Hardy showed that Mr. Woolhouse's formula can be used in columnar shape; and Mr. Ackland, improving on Mr. Hardy's arrangement, contributed a table in which an adjustment of the HM decrement was so calculated. (Journal, xxiii, 354.)

This important and unexpected result of the reading of my paper rendered it of little value, except that it supplied a short formula which may be convenient for use where a rough graduation only is required. Any degree of smoothness can be obtained by repeating the process; but this is tedious, and I would now furnish a formula derived in the same manner, and yielding a satisfactory adjustment by a single operation.

The Arithmometer of M. Thomas (de Colmar) has been already brought under the notice of the readers of this Journal by General Hannyngton, in a remarkably lucid and suggestive paper, which will be found at p. 244, vol. xvi. General Hannyngton, in his paper, explains the manner of working the machine, and gives examples of some of its applications to the construction of actuarial tables, with hints as to others. These afford an idea of the very striking adaptation of the machine to the formation of such tables; and they cannot fail to have excited the interest of many of the readers.

In March 1944, while the second world war was still raging, the Royal Commission on Population was appointed ‘to examine the facts relating to the present population trends in Great Britain; to investigate the causes of these trends and to consider their probable consequences; to consider what measures, if any, should be taken in the national interest to influence the future trend of population; and to make recommendations’. This step was of twofold significance. First, it marked the recognition by the Government of the possible need to take policy decisions in the field of population—to translate the subject from the academic to the political plane. Secondly, it marked official recognition of the fact that despite a flood, during the immediate prewar years, of reports of grave foreboding by the demographers of the day (whose anxieties have since proved to have been exaggerated) the Government had insufficient information to decide whether or not there was a 'population problem' in Great Britain. That the problem does not now appear to be so pressing as was once thought does not abate in any way the necessity for observing the facts. The Royal Commission, indeed, emphasized the necessity for continuous study of the population problem which, they said, will always be changing.

Interpolation and other formulae involving advancing differences are frequently developed, or at least conveniently reproduced from memory, from the familiar operational identities

where

It is now nearly a decade since the author had the honour of submitting a paper on the mortality of assured lives in India, and the manner of its reception by the members of the Institute has emboldened him to make a second venture. Particular impetus was given to the writing of this paper by the fact that, in the discussion that followed the reading of the earlier paper, several speakers observed that they would be interested to know the results of an analysis of the mortality prevailing in the large subcontinent of India, subject to varying climates and other conditions of life, according to geographical situation, and an attempt has now been made to give this information as revealed by the statistics of the “Oriental”. The words, “the more investigations were made into mortality in various parts of the world and in various types of business the better”, with which Sir (then Mr) William Elderton closed his observations on the earlier paper, are typical of the attitude of the Institute towards papers on mortality investigations, particularly as applied to countries like India, which are becoming very important from a life assurance point of view. Information concerning the mortality to which the various parts of the country are subject is, however, still very meagre.

On the occasion of the bicentenary of the foundation of scientific life assurance, it was thought fitting that a work should be prepared in which the development of some aspect of the subject might be discussed. The authors undertook some research on the origins and disposal of surplus and the results are set out in a book entitled Surplus in British Life Assurance—Actuarial Control over its Emergence and Distribution during 200 Years. The following pages convey some of the impressions formed in the course of its preparation. Much of the later part of the paper consists, in fact, of extracts from the text of the book. The introductory sections present a short outline of the book as a whole.

In venturing to bring this subject before the Institute to-night, I am conscious of inviting attention to a matter which has been frequently treated before and by far abler hands than mine. I at once confess that I have little original matter to bring forward; indeed, the main object of this paper is to endeavour to bring into one focus and to a common standard of criticism the various formulas for ascertaining the rate of mortality, which have already been so often discussed in this room. The expression “rate of mortality” is advisedly used here instead of “exposed to risk”, because it would seem that attention has hitherto been unduly centred in the latter.

The remainder form of Newton's formula of interpolation with divided differences has been known for at least 37 years; yet actuaries have taken hardly any notice of the complete formula which ought to be one of their principal tools. It will, therefore, not be superfluous to begin by reproducing the very elementary proof from first principles.

One of the most striking features in the history of Life Assurance business during the past few years is the marked increase—both absolute and proportional—which has been shown in the number and amount of policies effected under the Endowment Assurance plan. There is every reason to expect that this increase will continue, and that ere long Endowment Assurances will divide honours with whole life policies as regards importance in the periodical valuations. It is consequently very desirable so to improve the methods of valuing such policies as to minimize the amount of labour involved, and it will become more and more necessary to abandon methods which answered very well so long as Endowment Assurances were looked upon as “Special Policies”, but which are extremely cumbersome when applied to a large mass of contracts. The shorter methods that have hitherto been suggested, though admirably adapted for use in test or check valuations, involve an error which many actuaries will consider too appreciable to be neglected, more especially as the approximate methods nearly always bring out a reserve which is smaller than the true reserve, and it accordingly becomes desirable to seek some process that will be free from this objection.

It is quite common for stocks (usually loan stocks) to be issued by companies and purchased by investors, which include among their terms an option to convert at some future date into other stocks (usually ordinary shares) on specific terms. Such options obviously have value, but I am unaware of any simple published method for estimating the value of such options in practical cases. The purpose of this note is to suggest a formula, based upon easily understood estimates of the future market prices of the stocks in question, which may be applied readily in practice to estimate the value of such options.

Gentlemen, on the present occasion I propose to submit to you a few remarks upon the business aspects of our work, and some of the circumstances on which the prosperity of a life insurance office depends. In the first instance, it may be useful to examine what is the real importance of life insurance to the country; and for this purpose I will take some of the figures which are now available by means of the returns made in pursuance of the Act of 1870, and compare them with other figures relating to the position and progress of the country. It appears from the Board of Trade returns that the income and assets of the life offices are steadily, not to say rapidly, increasing; the total income of the British life offices having increast from nearly 14½ millions sterling in 1871 to over 22 millions in 1884. The total life funds of the same companies now amount to about 140 millions, the increase of the funds since 1871 having been over 52 millions. These are very large figures, and they prove that the business in which we are engaged is a very important one. If, however, we confine our attention to them, we shall perhaps conceive an exaggerated idea of the importance of our business; and I will therefore compare them with the figures relating to some other industries. Thus I find it stated in the Statesman's Year-Book that the value of the total imports into the United Kingdom in the year 1883, was nearly 427 millions sterling; and the value of the exports of British produce about 240 millions; the largest articles of import being corn and flour of the value of about 67½ millions, and the largest item of export being cotton manufactures of the value of about 76½ millions.

British life offices transacting life assurance outside the United Kingdom have until now confined themselves almost entirely to the countries of the British Commonwealth—because among the countries with stable currencies the members of the Commonwealth have in the main been the ones with which we have had the largest numbers of other business contacts. Another influence has been the absence of a language barrier. Before the 1914–18 War some offices transacted business on the Continent of Europe but thereafter most of them pulled out; today as far as I know only one British office transacts life assurance in France, one in Germany and none in Italy.

One might be forgiven therefore for thinking that actuaries handling life office affairs need not concern themselves with life business on the Continent of Europe—particularly as, at the beginning of last year, the negotiations for Britain's entry into the European Common Market broke down.