In discussing the subject of assurance valuations and the most approved methods of making them, I have often had occasion to draw attention to the great importance of ascertaining with the utmost possible accuracy the rates of mortality and interest which have been found to prevail in any case under consideration, so as to determine what rates of premium are really required to provide, independently of other exigencies, for the sums assured, and how far the particular rates charged are sufficient or more than sufficient for that purpose. I have insisted the more strenuously on the necessity of this proceeding, from the conviction that in almost every Association some peculiarity in the prevailing mortality will be found to exist, distinguishing it from that of its neighbour, and arising from the different character of such influences as the condition of life of the persons whose lives are assured, the mode of selection, the greater or less predominance of the male sex and of assurances made by persons on their own lives over such as are effected on the lives of others, the constitution of the Company, and so on.

It will no doubt be in the recollection of many of our readers, that some few years ago attempts were made by the Council of the Institute to collect, from the Offices willing to contribute the results of their experience, such information connected with the extra risks undertaken by them as might afford the means of determining, if not exactly, at least with some approach to accuracy, the proper rates of premium to meet them. The Council made application to all the existing Companies; but did not succeed, unfortunately, in obtaining returns from more than fifty-two of them. These fifty-two Companies, however, were at the pains to supply as many as 6,154 cases—each of them of a special nature, as regarded unusual hazard to life arising from residence in foreign climates, or from particular pursuits; but so infinitely varied in their conditions and circumstances were they, that it became immediately obvious that considerable difficulty would arise as to grouping them into classes which should be sufficiently well defined to yield results of a useful and practical character. If, on the one hand, the conditions of the risk were kept within limits sufficiently restricted, the numbers in each class were all but insignificant; and if, on the other, attempts were made to remedy this last inconvenience, the groups comprised risks which were altogether heterogeneous.

Several writers, notably Messrs. Higham and Sprague, have treated of the Mortality Experience of Life Offices, basing their investigations upon various data; and it may be thought that there is no necessity to pursue the subject further. The question is, however, one of great interest and importance, both in its relation to abstract science, and in its bearing on the actuarial management of companies; and as it possesses collateral branches which have hitherto received but little attention, it occurred to me that my time would not be wasted in attempting to elicit new facts, and illustrate old ones, from the abundant materials now at our disposal.

Some apology is needed, perhaps, for proposing to rediscuss a subject which has been handled so often, and by so many distinguished actuaries. But I am probably addressing gentlemen who have not had occasion to become familiar with all that has been written or said on the subject, and others to whom it may usefully be presented from a fresh point of view; and there are some important practical questions too on which the profession seem scarcely yet agreed, which will admit of further consideration. I shall not attempt to notice the various papers which from time to time have been contributed by actuaries towards the solution of the questions I am to consider. For many of the views I am to express I am necessarily indebted to those who have preceded me; and the aim of this paper is less to introduce what is new, than to enforce, and illustrate if possible, sound and correct opinions. I am anxious to set the subject in a light which will make it intelligible to others than actuaries, and to discuss principles rather than formulas.

The current state of knowledge on the epidemiology of HIV infection and AIDS is reviewed, with extensive references to the medical literature. The particular focus is on those aspects of the epidemiology of HIV infection which are of interest in the formulation of models for projecting the future spread of the virus and its impact in terms of numbers of cases of AIDS and deaths from AIDS.

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.

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.

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.

The results of the investigation from 1945 to 1959 were presented in a paper to the Institute by R. E. Hayward and B. C. Lucena (J.I.A. 91, 286) and readers are referred to this for further detail on practical aspects of the investigation. This further report owes its existence to all those originally involved as well as those specifically involved in the follow-up. Professor J. M. Malins and Dr M. G. FitzGerald have continued to give us the help needed to obtain the required data from the hospital records. Those of our Society particularly involved in producing this follow-up report include B. C. Lucena, R. Betteridge, A. Roberts and M. Willett.

1.l. The aim of this paper is to examine a number of issues that arise in decision-making. In particular it seeks to explore the relevance to decision-making of the commonly made polarization as between the so-called objective and subjective approaches to probability. The paper is in three parts. The first concerns utility, and the second various aspects of probability. The third and final part looks at the interplay between these two concepts. Whilst a number of the comments and illustrations refer specifically to the insurance industry, the concepts discussed are broadly set in the wider context of business and commerce as a whole. It is hoped that the paper will thereby provide a basis for discussion on a topic that has acquired increasing importance in recent years.

The ordinary member of a pension fund, if asked whether there is any connection between the high cost of living at present oppressing him and the security of the pension to which he looks forward to support him in his old age many years hence, would probably fail to discover that the one is related to the other. The financial position of pension funds is, however, fundamentally affected by the sudden drop in the purchasing power of the sovereign during the last few years and more directly by the enhanced scales of salaries or wages which have resulted from the change in the value of money, and it is to be feared that in many funds a very serious position has to be faced. The actuary who is called upon to advise as to the situation and to formulate remedies by which equilibrium may be restored has not only difficult professional problems to grapple with, but, having made up his mind as to the course to be recommended, he has to carry with him the responsible officers of the fund, who in their turn have to secure the co-operation of the employer and also the assent of the " ordinary member " of the fund, to whom I have referred above, to the remedies put forward.

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.

The subject of Life Assurance Investments does not seem to have been discussed for a long time at the meetings of the Institute. Two valuable and instructive papers, bearing the honoured names of Mr. Samuel Brown and Mr. Arthur H. Bailey, were read to the Institute respectively on 25 January 1858 (J.I.A., vii, 241) and 24 February 1862 (J.I.A., x, 142), but they appear to be the only contributions of importance, dealing with the question in its entirety, to be found in the Journal. Notwithstanding the lapse of time which has taken place since the latter date, the principles of investment as defined by these two distinguished authorities are still very generally applicable to present needs; and the articles themselves have lost none of their charm for the student of life assurance finance, characterized as they are by soundness of judgment and by felicity of expression.

1. In the following experiments I simplify enormously, by assuming that the only benefit provided is a pension on retirement at age 65, that there are no benefits at all on death, not even a return of contributions, and that there are no ill-health retirements or early retirements. I also assume that all salaries, contributions and retirement pensions are paid annually in advance. This simplifies the calculations and hardly affects the results. I also assume complete certainty of inflation. salary increases and investment returns. The purpose of the experiment is to investigate the relationship between the amount of pension that may be deemed to have ‘accrued’ each year, and the amount of contribution (as a percentage of salary) needed to purchase the extra pension that has accrued in each year. under both a final salary defined benefit scheme and a revalued average salary scheme.

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.