This is the fourth, and, let it be hoped, the last, of a series of papers on mortality tables in their various aspects which the Institute has honoured me by accepting. The investigation had its origin many years ago in efforts to improve upon the then methods of constructing mortality tables from Census Returns; and, when the enquiry seemed to be nearing its conclusion, the fact became evident that the formulas which had presented themselves could be applied effectively in summation form to the graduation of tables derived from other sources. There was, however, a preliminary objection to be overcome. Summation Formulas of Graduation were under a cloud. Dr. Sprague had condemned them in unmeasured terms, and Mr. Todhunter, in the head-note to his paper (J.I.A., xxxii, 378), had accepted Dr. Sprague's authoritative judgment. Hence my first paper of the series, “On the Error introduced into Mortality Tables by Summation Formulas of Graduation”, (J.I.A., xli, 54), wherein it was attempted, and, I think, successfully, to show that that error is generally so minute as, with ordinary care, to be safely negligible in practice.

At the Congress of Actuaries in Berlin in 1906 I contributed a paper on Life Assurance in India. With certain modifications and additions, a considerable part of that paper is reproduced in that which I am now reading. My excuse for going over much of the same ground again is that the subject has not been before the Institute for many years past, and that the discussion here will no doubt elicit further information which will be useful to the profession.

At the present time India is one of the largest and most rapidly developing fields for insurance business in which the rates of mortality differ materially from those in temperate climates. Not only are there a greater number of Europeans from year to year engaging in the civil and military administration of the country and its mercantile pursuits who require the benefits of assurance, but the more Europeanized Natives themselves are also seeking the same protection.

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.

There are many references in the pages of the Journal to Old Age Pensions, and in view of the Act of last year the subject—from a financial and statistical aspect—may be discussed without transgressing on questions of State policy. I therefore venture to submit the following notes of some of the financial and statistical questions involved, stimulated in so doing by Mr. G. F. Hardy's statement in his Presidential Address, that he believed our most important work lies in the proper application of actuarial principles to the many practical questions which arise from time to time.

Old age pensions ranging from 1s. to 5s. a week according to the yearly means of the pensioners are granted to British subjects resident in the United Kingdom who have attained the age of seventy years, provided their yearly means do not exceed ￡31 10s., and provided they are not disqualified on the ground of Poor Law Relief, imprisonment, or the other reasons set forth in Section 3 of the Act.

The subject of Annuities, in relation to the transactions of the Life Offices of this country, has seldom received definite consideration at the meetings of our Institute. An examination of the contents of the Text-Book and of the Index to the Journal furnishes abundant proof of the attention which has been given to the theory, but in regard to the practical working of annuity business, little has of late years been written. The most important contribution in the Journal, from the latter point of view, is that by Mr. G. H. Ryan (read before the Institute seventeen years ago) “On a Method for determining the Gain or Loss from Mortality in an Annuity Company.” Therein reference was made to another contribution by the same author, read before the Actuarial Society of Edinburgh during the Session 1884-5, and published in the first volume of the Transactions of that Society, “On the Several Mortality Tables Employed in the Valuation of Annuity Contracts.” Changes of the utmost importance have taken place during the period which has since elapsed, and it is proposed herein to deal with the cause and effect of such changes, and to gather therefrom some guidance for the future.