Next to England, Germany is the country in which the principles of life assurance have had the most successful development. It is true the more recent French Companies, La Caisse Paternelle, L'Equitable, La Caisse des Ecoles et des Families, which also call the assurances granted by them assurances sur la vie, show a greater extension in a shorter time; but what they guarantee to the public are not life assurances in our sense of the word. They are neither assurances granted for previously specified sums, nor is the payment of the claim dependent on the death of the assured. The more modern French life assurances are rather indefinite reversions obtained by paying in any sum the assured pleases, the interest of which (as in the Tontines), according to the mortality found to prevail in the different classes, may prove sometimes greater, sometimes less, and is only divided amongst those who attain the term of life which may have been previously agreed on. The contributions of those who die early, together with the interests thereon, go to augment the dividends to the surviving members. Whilst life assurance amongst the English and Germans is therefore calculated to be on the death of the assured a source of provision for their families, the more recent mode of life assurance amongst the French principally aims at granting to the assured himself, in his lifetime, an annuity or sum of money, which, increasing according as the computations are made for an earlier or later period, affords him the means of extending his business, of completing his education, of securing a dowry on marriage, or a provision in old age. In consequence, partly of the prospect of selfenjoyment of the reserve thus made, partly the possibility of obtaining a high profit for a small investment through a greater mortality in the classes than the tables assumed, this kind of assurance became very much in vogue amongst the French. Recently, however, a clearer understanding of the matter, and the exposure of some deceptions which a few of the Companies or their managers have been guilty of, have in some degree cooled the public favour.

In making a valuation of the ordinary whole-life assurance contracts of a company according to the methods in general use at the present day, there are two separate sets of questions to be considered.

In the first place, the basis of valuation has to be chosen; that is to say, we have to select the table of mortality and the rate of interest which are to be employed. The questions that arise in this connection are of great importance, and for their settlement we have the assistance of the powerful instrument devised by Mr. King—the Model Office and its Valuations by various Tables of Mortality and at various Rates of Interest.

Perhaps the most important lesson many of you can learn—one that is frequently lost sight of by persons who are solely or mainly engaged in theoretical studies—is that the researches which you are pursuing, by the aid of a variety of complicated symbols, are, after all, not merely elaborate mathematical puzzles to be solved principally for the purpose of awakening or sharpening the intellect, but have, for you at least, an eminently practical aim as well, and are insisted on as part of the education of an Actuary, with a view to their serving as the framework on which, when afterwards called upon, he must be prepared to base actual business transactions involving large sums of money, or, it may be, to lay down principles for the regulation in all its various details of a life assurance office.

The investigation of the experience, and valuation of the liabilities, of what are known as Clerks' Associations, present some features of special interest; and I have thought that it might he useful to discuss and explain some of the methods that have been employed in analyzing the data, and in deducing therefrom the tables of money values appropriate for the valuation of the risks.

In a paper read a short time ago before the Institute, I endeavoured to show the inexpediency of using, for the purposes of valuation, tables constructed on the principle which characterizes the Northampton table. I did not intend to pursue this particular branch of the subject further; but, finding that more difference of opinion exists in reference to it than I was aware of, and considering that the practice of making valuations by the Northampton table, and by tables framed upon similar principles, is still carried on to a great extent, I am led to offer a few more observations upon the matter previously to entering upon the consideration of some other questions, the correct solution of which materially depends upon a right understanding of this one.

The presentation in November 1905, of three papers on “methods of valuation with allowance for selection” (J.I.A., vol. xl, pp. 1, 15, 42) might well be considered to have left room for no further word on the subject. Any attempt to produce a new method merely to achieve an equal degree of accuracy would undoubtedly be presumptuous, for Mr. King's method affords absolute accuracy, Mr. Diver follows with a microscopic error of £9 on a reserve of more than £3,000,000, and Mr. Ackland closes the series with a larger but still relatively insignificant error of £1,183 on a total reserve of upwards of £13,500,000. It is also shown that in both Messrs. Ackland's and Diver's methods, the error may be expected to be always less than ·01 per-cent.

The practical application of all three methods involves, however, a departure from the familiar system of classifying whole-life assurances according to attained ages, with the loss of all the advantages attaching to that system. This will be viewed by many with regret, and was especially referred to by Mr. Lidstone and Mr. R. P. Hardy in the subsequent discussion. Their remarks, together with Mr. Ackland's cordial expression of goodwill to “other workers in this field” led me to consider the subject from the standpoint of a grouping according to attained ages without any sub-grouping according to duration.

The President of the Institute of Actuaries was in the Chair. The opener prefaced his formal remarks as follows:

I would first of all like to say how pleased and honoured I feel to have been asked once again to take a part in the proceedings of this Society, particularly as this must be regarded as a special occasion since we are honoured by the presence in the chair of you, Sir, the President of the Institute, and also because this meeting is in the nature of an experiment.

It may be within your recollection, Sir, that on the first occasion, some thirty years ago, when I was asked to take part it was also an experiment. We met at the Old Bell and my part was in what has come to be known as Heath's play. It was therefore not so much a surprise to me, as it might have been to some, when three or four months ago the Chairman of the Society telephoned me and asked me to take part in another experiment, mentioning a symposium. Being one of those who dodged compulsory Greek in my undergraduate days I temporized. I consulted a Pocket Oxford Dictionary and found that my memory had not let me down: ‘symposium = a Greek drinking party’. Pursuing the matter further, I consulted the larger edition: ‘symposium = a drinking party, a convivial meeting for drinking and conversation’, and the only example it gives of the use of the word, ‘our symposium at the King's Head broke up, 1787’. This seemed conclusive and, remembering the symposia I had enjoyed with Alan Gardner in the past, I agreed to come along. I was a little disturbed when I received the notice of meeting which mentions a review, but I concluded this was just a printer's error for revue, although I thought the title unusual. My doubts were quite set at rest on Monday when you yourself announced in this Hall that the Society would meet tonight for a symposium.

The title of this lecture was chosen over a year ago before I had formed a clear picture of its scope. It is, however, adequately descriptive of my purpose which is to develop consistent general principles, as opposed to detailed methods, which can serve as a guide to the individual computer—primarily one working alone with an ordinary calculating machine. I shall endeavour to illustrate the principles by actual examples, but it will be outside my present compass to concentrate upon those methods which rely upon brilliance of technique rather than on soundness of principle.

It is clearly desirable to say precisely what ‘computation’ embraces, and I therefore put forward the following definitions of ‘calculation’ and ‘computation’—in each case it being understood that the word is prefixed by ‘numerical’:

Calculation is the application of the four rules of arithmetic to the numerical evaluation of an algebraic expression.

The paper concerns the situation in which an evaluation of outstanding claims is discounted, in anticipation of investment return earned by the funds supporting that liability. Factors bearing upon the choice of an appropriate rate of return to be assumed in this evaluation are considered.

The ‘standard approach’ to this problem is criticized in two main respects:

(i) there is usually no statement (indeed, no consideration) of whether the discounted value of outstanding claims is to be associated with assets at book, market, or some other value;

(ii) the discounting of outstanding claims is often performed by means of an assumed ‘inflation gap’, i.e. an assumed difference between future rates of inflation and investment return.

Various other matters germane to the determination of an appropriate rate of investment return are listed in Section 2.

Sections 3 and 5 deal particularly with points (i) and (ii) raised above. It is suggested that, in certain circumstances, an evaluation of outstanding claims which fails to address (i) or is carried out on an ‘inflation gap’ basis will be virtually meaningless.

Section 4 deals with the case of an insurance fund in which assets are matched with technical liabilities by amount and term. The considerations which would cause the rate of investment return projected in these circumstances, to differ from that projected in the case of an absence of matching are discussed.

Section 6 deals with the issue of exactly which assets are to be regarded as supporting the liability for outstanding claims. It is apparent that the identification of such assets will affect the rate of return to be regarded as referable to outstanding claims. Particular matters considered include:

(i) assets to be associated with acquisition of new business (and hence not with outstanding claims);

(ii) debtor and creditor items. particularly outstanding premiums and inwards loans.

Section 7 gives a numerical example of the projection of future rates of investment return. This is done by means of a computer projection of various items of a hypothetical insurance fund, taking into account projected future:

(i) interest rates;

(ii) rate of growth of new premium;

(iii) the required distribution of assets by sector;

(iv) profitability of business underwritten.

Conclusions on the various matters considered are dotted through the paper. For convenience, they are collected together in Section 8.

ALL the methods of group valuation submitted to the Institute in the last decade require the calculation of the second sum of the quantities valued, whether in the form of multiplications by —4½, —3½, … 3½, 4½, i.e., the second sum about the centre, or the more convenient second sum about the end. The object of this paper is to show that the α, β method can be transformed so as to avoid this difficulty.