The method of valuation generally adopted in practice is that known as the Prospective Method; by it, the value of a policy is obtained by a consideration of the future, and is the discounted value of the payment to be made, less the discounted value of the payments (if any) to be received; discounted, that is, in each case, for both mortality and interest.

Some years ago, before the enactment of the Ten Hours Factory Bill, I had the honour to be engaged by the Earl of Shaftesbury (then Lord Ashley) on an investigation of the average distances traversed daily by children and others employed as “piecers” in the cotton mills about Manchester and the surrounding districts. After taking the various dimensions of the mules or spinningjennies, and making the requisite observations as to the number of strokes of each machine per minute, and the average number of breakages of the threads at each stroke, I found that the necessary calculation involved a preliminary discussion of a somewhat curious mathematical problem.

In the course of an investigation I have lately made, into the rate of mortality that has prevailed among the female lives insured in the Scottish Equitable Life Assurance Society, I have tried various methods of arranging the facts furnisht by the books of the office; and I think it may be useful to persons engaged in similar investigations, if I describe these methods, and state the conclusions at which I have arrived as to the best course of procedure.

I am always happy to be back where my Institute life started and I am grateful to your Committee for fitting me in at short notice.

When you start to write, ideas take on a being of their own. Life is a shy child which hides its face when you stand aloof, but is soon at your elbow to help when you kneel to build with its bricks. Then other bricks begin to appear. As it turns out, my helper is concerned about imagination so ‘imagination’ shall be our subtitle.

First, a brief reference to my 1972 paper: the winning themes were—in reverse order, as for beauty queens—third prize to the imminent collapse of our society, second prize to the essential unity of the universe, but an easy winner was double faults at tennis.

When interpolations are made between values of a function by any of the ordinary interpolation formulae, there are generally discontinuities in the slope and curvature of the successive interpolation curves, and when, as often happens in practical work, the given values are themselves derived from rough data, these discontinuities may be considerable. To remove these breaks of continuity T. B. Sprague in 1880 devised his osculatory interpolation formula, and in the present Paper is given a brief review of subsequent contributions to the subject. From the practical point of view, the most important of the earlier contributions were those of Mr. 6. King, who successfully applied the method to the construction and graduation of national and other life tables; but while two sets of national life tables have since been constructed on the lines proposed by him, there has, for the last fifteen years, been no further development in this country either in the theory itself or in its application.

In contrast to this has been the interest in the method shown by American actuaries, particularly in recent years. Their work is briefly reviewed, and special attention is directed to that of Mr. W. A. Jenkins, who, in 1926, put forward a method of greater generality than that underlying the earlier formulae.

The weak point of the osculatory method, regarded as a smoothing agent, rests on the fact that the graduated curve is required to pass through certain predetermined points. The curve will, in fact, be constrained to take a form similar to that assumed by a flexible steel wire which is clamped at fixed points, so that, while the curve is free from discontinuities, any departure of these points from the smooth curve will be reproduced with resulting undulations. To remove this tendency to waviness, Jenkins has devised his modified osculatory method, which, while requiring the successive interpolation curves to have the same slope and curvature at their common points at the ends of each interval, does not require the curves to pass through the points corresponding to the calculated values.

His fifth difference formula has been applied to regraduate the English Life Table No. 9, Males, and in connection therewith it is pointed out that the success of the method must depend to some extent on the provision of a good set of guiding values ; for, while the curve is not required to pass through the points corresponding to these values, the closer they lie to the smooth curve the better is the graduated curve likely to be. The method recommended for adoption is one based directly on summations in fiveyear groups of the unadjusted rates of mortality. By means of a formula derived from the modified osculatory formula quinquennial values are obtained. These are treated as first approximations, and the formula is again applied to them in order to produce rates which are treated as the graduated rates. The results are satisfactory both from the point of view of smoothness and of fidelity to the data, as measured by the agreement between actual and expected deaths.

An interesting point which emerges from the various graduations and from the comparisons of actual with expected deaths is that the crude rates of mortality appear always to exceed the graduated rates for ages centering about 24, 34 . . . and to fall below them for ages centering about 29, 39 . . . ; and it is suggested that this is due to some persistent tendency to misstatement of age, and that it is also largely responsible for the waviness of graduations of population data which is generally characteristic of the application of the ordinary osculatory method.

As a further test of the possibilities of the method, and, in particular, of its smoothing powers, it has been applied to graduate the data of the Government Female Annuitant Experience 1900-1920. These data had proved very intractable to curve-fitting methods, but had been found to show an element of periodicity up to about age 90 with an apparent change in the form of the curve after that age. In connection therewith, formulae are given for applying the method to the data at the ends of the experience where the main method is inapplicable. The result is to produce a curve which is almost perfectly smooth and which closely follows the original data.

The method is essentially a graphic method in which the curves are drawn in accordance with mathematical formulae instead of being drawn by hand, and the work is arranged in such a way that it can be carried out practically from beginning to end by purely mechanical processes.

In the Introduction to his famous Théorie Analytique des Probabilités, Laplace, after an eloquent summary of the progress of Astronomy from the days when every rare celestial phenomenon was regarded as a portent of human misery, to the verification in April 1759 of the perihelion passage of Halley's Comet, proceeds to say :—

“The regularity which Astronomy shows us in the movement of Comets exists, beyond doubt, in every phenomenon of nature. The curve described by a simple molecule of air or of vapour is regulated in a manner as certain as the planetary orbits: there is no difference between the two cases save our ignorance.

In submitting a revised edition of the paper which I read before the Actuarial Society of Edinburgh more than six years ago, I wish to express the pleasure I feel that this, my first effort, should not only have proved of benefit to students, but that the Faculty of Actuaries consider it worthy of a place in their Transactions. The subject was taken up originally to condense the best actuarial opinions and bring together some of those practical questions which had been dealt with by actuaries of ripe experience. Premiums can be profitably discussed at almost any time, because an exchange of views is sure to shed light on some obscure point of more or less importance. The present time is peculiarly opportune for further discussion, because of the recent publication of the New Mortality Experience, and the knowledge of such statistics which we have all been gradually absorbing.

Written concurrently with the first item in C.M.I.R.3, the Faculty of Actuaries Mortality Research Group's paper determines a range within which mortality rates of Life Office Pensioners may be expected to change in the foreseeable future. Comparisons are made between the observed changes in pensioner mortality rates and those observed for the population of England and Wales, and reference is also made to the trends of mortality rates assumed in recent British population projections. From these considerations two forecasts are made, based upon “optimistic” and “pessimistic” future mortality assumptions, between which it is expected the actual future rates of mortality change will lie.

In the second part of the paper the financial effects of the range of forecasts are set out, when used to project the graduated pensioner Mortality Experience 1967-70 (C.M.I.R., 2, 57). The implications are illustrated in the context of two model funds, one based upon life offices' data, and the other based upon a non-insured pension scheme for which the contribution rates vary in accordance with the levels of future expected pensioner mortality.

In the course of some recent work (cf. reference 2) it proved necessary to construct a life table from a set of graduated values of mx, the central death rate at exact age x. This meant that values of qx had to be derived in order to produce the life table from a given radix. Since the values of mx were known only for integer values of x, it was necessary to use an approximate method to derive the corresponding values of qx at each age. In this note we discuss briefly the approximations which we adopted. Our results are necessarily of an elementary nature and the principal reason for setting them down is to provide a reference for possible future use. It is perhaps worth pointing out that in a situation where the values of mx are given by an explicit mathematical formula values of qx may be obtained exactly using the results of reference 1.

Mr. President and gentlemen, I should like to express my pleasure at being back in the Faculty Hall, where I was privileged to listen to your interesting discussion last year on “The Growth of Pension Rights”. I am glad to find that the number of occasions on which economists and actuaries are not only on speaking terms but able to take counsel of one another is increasing, for I am sure that there are many problems, of which the future of pensions is only one, that can only be satisfactorily resolved through our joint efforts and deliberations. This conviction rests partly on my experience as a member of the Phillips Committee, which was heavily dependent both on the actuaries who served on it and on the members of the profession who, in one way or another, helped it along; but it is a conviction that is just as forcibly brought home to me when I look across the Atlantic to the inquiry that has been in progress since 1950 into the long-term trends in savings and investment in the United States—an inquiry carried out by economists but sponsored and largely financed by the Life Assurance Association of America.

The Paper discusses the general problem of appreciation and depreciation, with special reference to the relation between maturing assets and maturing liabilities, and describes a method of calculating the bonus-earning power of different classes of policy. This method is used to test the effect on bonus-earning power of two types of changes in interest rates:–

1. (a) a change, such as that caused by the War Loan conversion of 1932, which takes effect gradually; and

2. (b) a subsequent change, due to a rise in the rate of income tax, which has an immediate effect on the income from existing as well as investments.