In the correspondence that has recently taken place respecting the origin of Gompertz's formula expressive of the law of mortality, it has been stated, much to my surprise, that the investigation first given by that gentleman in the Philosophical Transactions of the Royal Society for the year 1825 was defective and inaccurate. I do not here propose to reopen the controversy on the subject of the originality of discovery, or to enter upon a renewed discussion of certain other allegations that have been so efficiently disposed of by Professor De Morgan and Mr. Sprague; but, after the statement to which I have more particularly alluded, it is only right that I should avail myself of the present opportunity to communicate the circumstance that Mr. Gompertz kindly presented me with a copy of his valuable Memoir in January, 1839, in which the various misprints had, in the most legible manner, been previously corrected by himself, and that the processes so corrected are perfectly accurate in every respect. In order that any one may be enabled to judge of this for himself, I shall here give a faithful transcription of that portion of the Memoir which contains the mathematical investigation of the formula, as it will occupy but little space.

In AIDS Bulletins Nos. 1 to 4 the Institute of Actuaries AIDS Working Party presented projections of HIV infection and AIDS based on a model developed by Wilkie. This paper explores the sensitivity of those projections to the various different assumptions which have to be made and presents results on a number of alternative sets of assumptions.

The subject of Group Life Insurance schemes has been quite recently discussed by the Society (J.S.S. Vol. III. No. 3) and no useful purpose can be served by a repetition of remarks regarding such schemes.

However, there are two points which deserve further consideration. The first of these relates to what might be termed underwriting rules—or, in other words, regulations to which every contract must conform. Each individual contract should be written, so far as possible, on such a basis as will preclude the writing of unsatisfactory business and in this connection such points as the number of lives to be covered, maximum and minimum amounts of insurance, etc., fall to be considered.

The European Economic Community came into existence on 1 January 1958, following the ratification of the Treaty of Rome by the parliaments of the six member countries, Belgium, France, Germany, Italy, Luxembourg and the Netherlands. The long-term aim of ‘The Six’ in setting up the Community was to achieve a unified economic unit with a common economic policy, and whilst commonly referred to in the United Kingdom as the Common Market it should not be thought of merely as an advanced form of customs union.

In view of the long and honourable association of British life assurance institutions with life assurance in Canada, an association dating back prior to the formation of Canada's oldest company, and of the deep interest taken and aid rendered during the preparation of the Dominion Insurance Act, 1910, by the Canadian and Home officers of those British life institutions actively engaged in business in Canada, as well as by leadingindependent British actuaries, it would appear but appropriate that some reference should be made in the proceedings of the Institute relative to the new Act. Furthermore, in consequence of the increased interest taken in the affairs of Canada generally, and of the great development everywhere in evidence there, may it not be that a closer acquaintance with the principles underlying the new Canadian legislation may encourage those British life offices which though nominally having agencies in Canada are inactive in so far as new business is concerned, and other offices which have not yet ventured across the ocean, to seek aggressively for new business throughout the Dominion.

My intention is to give some simple illustrations of the difficulties which arise when the market rate of interest has sunk below the limit of solvency for parts of a company's stock of life contracts. In fact, the “partial liquidation” which at such a contingency would appear most natural to the unbiased mind of an actuary, does not seem to be seriously contemplated in insurance legislation. As regards Sweden, this mathematically correct reduction of the liabilities of some of the contracts could perhaps be realized within the legal frame in Mutual Offices. But it is impossible for other companies. It seems therefore necessary to attack the problem from the purely actuarial point of view. If we can find a technically satisfactory solution, it might then be possible to go farther with the question of giving the procedure legal shape.

Reference is made in the introductory remarks to the appointment of Investment Research Committees by the Faculty and Institute of Actuaries to undertake, in the first instance, the compilation of a comprehensive service of Index Numbers relating to British Stock Exchange Securities which will be suitable for Insurance Companies, Trust Companies, and so on. The Committees have been at work for some time, and it is hoped that publication of the results will be commenced at an early date. The Paper discusses the purposes of such indexes and the most suitable form of index numbers for the purposes decided upon. The difficulties arise from two sources ; the variety of uses for which the index must cater, and the requirement that the results must be comparable between any two dates in the series.

At every point in compilation the objects of the results must be carefully kept in view. These objects are divided under three headings : (a) uses directly connected with current investment activities ; (b) investigation of the effect of economic financial and political events on the level of security prices ; and (c) assistance in measuring approximately changes in the value of a portfolio of investments. It is suggested that securities should be divided into homogeneous classes, and a separate index calculated for each class. The wide groups “Fixed Interest” and “Variable Dividend” are not suitable for the purposes involved. The weighting system should approximate to the distribution likely to be found in an Insurance or Trust Company's investments.

Within each class companies' holdings will tend to approximate more nearly to equal amounts of money in each security than to any other distribution for which a system of varied weights would cater. Simple weights are accordingly advocated.

Various methods of averaging are discussed and reasons given for preferring the geometric average of price ratios to other methods. The dangers of adopting the arithmetic average—where the resulting figures are always dependent on the base prices—are illustrated in Table I. The geometric results are independent of base. Even using the geometric average great care must be exercised in applying the results to valuation purposes in groups where individual price movements show wide divergencies. The “error” involved by the geometric method, however, is always in the same direction. Capital appreciation is never overstated and depreciation never understated.

The foregoing remarks refer to price indexes. To obtain maximum utility from the results yield statistics also are required. The very different nature of the problem of yields is stated and an arithmetic average of individual yields in each group is advocated. While the price index should be related to 100 at the starting-point it is obviously preferable to publish actual yield averages.

Towards the end of Mr. Brown's term of office as President I submitted to him the Table which appears in the appendix to this paper asking whether he thought it of sufficient interest for publication in our Transactions. Mr. Brown replied by inviting me to go further and write a paper for the Faculty on the subject of Investments using the Table as an illustration of past history. Later our present President supported this idea and the notes which I now have the honour to submit are the outcome. Apart from the fact that I dealt with the history of Life Offices' Investments at some length when addressing the Students' Society a few years ago, it seemed to me that something more than a historical survey was desirable. There are few papers in our Transactions dealing with Investment Policy and this was the subject on which I decided. I think that the correct prelude to a discussion of Investment Policy is its own history, and so I give in Part I of this paper a very short general survey of the years from 1871 to 1935. The Table in the appendix will give information additional to what is contained in my remarks to those who wish to go further.

The history of the actuarial profession in South Africa starts, as far as is known, during the last decade of the nineteenth century when three actuaries arrived in the Cape Colony from the United Kingdom. Two of these took up positions with the two South African life offices which were then in existence and the third became Government Actuary to the Cape Colony. By the time the Union of South Africa was established in 1910, four more actuaries had arrived from the United Kingdom, and a further four had arrived by 1920. Some of these gentlemen established permanent homes in this country, while others returned overseas after varying periods of time. It was no doubt due to these early beginnings that the actuarial homes of South African actuaries are today London and Edinburgh.

The first South African-born actuary qualified as a Fellow of the Faculty in 1921, and since that time increasing numbers of South Africans have become qualified, mostly as Fellows of the Faculty.

The rate of mortality experienced by residents in India will supply an answer to an enquiry of the highest interest in an actuarial sense. The enquiry may even be considered of national importance so long as our countrymen seek their fortunes in a region where the effects of the climate on their prospects of health and longevity continue to be regarded as uncertain and hazardous.

1. Kennedy & Howroyd (1956) have discussed the application of the Lagrangian multiplier technique to actuarial problems. This technique permits the analytical solution of constrained optimization problems, where both the function to be optimized and the constraints must satisfy stringent analytical conditions. In particular, the first derivatives of all functions must exist.

When constraints comprise inequality as well as equation restrictions, as is the case in linear and non-linear programming, then the conditions required for the Lagrangian multiplier technique do not hold. It was therefore found necessary to develop a new body of techniques, known as mathematical programming techniques, for the solution of constrained optimization problems involving inequality constraints.

The subject of Insurance, theoretical and practical, has been so copiously and minutely discussed of late, especially in Lectures and Essays, that serious perplexity awaited me when I began to meditate upon my Introductory Address. I felt, under these circumstances, that a Paper constructed on the usual lines must either weary you with tedious repetition or discredit myself with the reproach of platitudes. I accordingly determined to see whether I could not,—at all events, partially,—succeed in adding some novelty and interest to our proceedings by an attempt to ascertain whether the vast Doctrine of Evolution applied in any valid degree to the business in which we are engaged. Such an investigation possesses at least the merit,—however I may fail in my exposition,—of appealing to the sympathy and thought of all concerned in the several Departments of our common work, for results which may be found to bear upon Life Assurance will also, mutatis mutandis, apply to other Branches. I confess, moreover, that the subject strongly attracted my attention on the ground that an enquiry of this interesting character had not previously been attempted.

The object of the actuary, in applying the science of vital statistics to pecuniary transactions, being to determine the values of payments dependent upon conditions of human survivorship, and therefore necessarily deferred for periods of longer or shorter duration, it is obvious that the rate of interest at which money may be increased is an element of his calculations, nearly if not quite as important as the probable risk of mortality that may affect the lives involved. The labours of actuaries have been generally directed to determining with precision the laws of mortality, rather than towards the present more humble branch of inquiry–the usefulness of which, however, must always become apparent with the earliest professional experience. Believing that an extensive and well digested collection of facts relating to the rate of interest would lead to some valuable and important conclusions as to future fluctuations, I trust that the present attempt to contribute to such a result will not be without interest for the members of the Institute, although I make no pretension to give a complete view of the whole subject, still less to discuss the important and intricate questions arising out of it; my object being merely to offer a general historical sketch, leaving the full prosecution of the inquiry to others, possessing more leisure and better qualifications for the task. Apart from its professional, the subject is one of historical interest. Men's pecuniary affairs afford unfailing indications of their characters, habits, and inclinations; and it has been truly remarked by a philosophical writer, that “a right measure and manner in getting, saving, spending, giving, taking, lending, borrowing and bequeathing, would almost argue a perfect man.” What is true in this respect of individuals is true to nearly the same extent of communities; and hence the conviction, becoming every day more general, of the importance of financial, for the illustration of general history.