Most persons who have dealt with mortality experiences have, no doubt, observed the variation in the average rates deduced according to the grouping of the exposures, but a few observations thereon, for the benefit of students of life contingencies, may, perhaps, be acceptable.

The first process in the investigation of the rate of mortality among lives assured in an office, or a special class, as annuitants, is to obtain the number exposed to risk of mortality, and the deaths which have actually occurred, at each age, during a certain period of time, as, say, the past five years, or since the company was established. The method to be used for deducing the numbers exposed to risk will depend on the special circumstances of the experience under observation, and need not here be discussed; but, assuming the exposures and deaths to have been ascertained, the mortality experience of the company is deduced by dividing the number of deaths at each age by the corresponding exposures.

Having had occasion, in connection with the recent Indian Census, to investigate the question of native mortality as prevailing in the various Indian Provinces, a short statement of the methods employed and the results obtained may perhaps be of interest to the members of the Institute of Actuaries. The problem presented and methods employed belong distinctly to the work of the Actuary, although the results may, perhaps, have more interest for the Statistician.

There is no subject of greater importance in the conduct of the business of Life Assurance than that of the valuation of policies. It is a matter which comes before the actuary from day to day and from year to year, whether in the form of fixing the amount to be returned to an individual policyholder on the cancelment of his contract, or in that of estimating what reserve should be made in the aggregate for a mass of policies existing at the close of a valuation period. Both of these questions have been discussed before the Institute on many previous occasions, and it is not necessary to enter into them now. The task I have set before myself is the humbler one of dealing with results already arrived at. I do not propose to enquire what are the reasons for preferring one mortality table or rate of interest to another, or on what principle of calculation the surrender-value of a policy should be based. I accept the results brought out by the use of any mortality table or rate of interest agreed on, and I propose to consider how these results may be represented by means of diagrams and models, and whether by this means we can arrive at any clearer understanding of the relation to one another of the reserves brought out by different standards of valuation.

Gentlemen, on the present occasion I propose to submit to you a few remarks upon the business aspects of our work, and some of the circumstances on which the prosperity of a life insurance office depends. In the first instance, it may be useful to examine what is the real importance of life insurance to the country; and for this purpose I will take some of the figures which are now available by means of the returns made in pursuance of the Act of 1870, and compare them with other figures relating to the position and progress of the country. It appears from the Board of Trade returns that the income and assets of the life offices are steadily, not to say rapidly, increasing; the total income of the British life offices having increast from nearly 14½ millions sterling in 1871 to over 22 millions in 1884. The total life funds of the same companies now amount to about 140 millions, the increase of the funds since 1871 having been over 52 millions. These are very large figures, and they prove that the business in which we are engaged is a very important one. If, however, we confine our attention to them, we shall perhaps conceive an exaggerated idea of the importance of our business; and I will therefore compare them with the figures relating to some other industries. Thus I find it stated in the Statesman's Year-Book that the value of the total imports into the United Kingdom in the year 1883, was nearly 427 millions sterling; and the value of the exports of British produce about 240 millions; the largest articles of import being corn and flour of the value of about 67½ millions, and the largest item of export being cotton manufactures of the value of about 76½ millions.

In December 1879 I read before the Institute a paper (J.I.A. xxii, 117) on the probability that there will hereafter be issue of a marriage hitherto childless. The statistics on which I based my conclusions, were those given by Dr. Matthews Duncan in his work, Fecundity, Fertility, and Sterility; and by Mr. Charles Ansell, Jr., in his Statistics of Families of the Upper and Professional Classes. But I was not altogether satisfied with the results I obtained; for, as I pointed out, the two sets of statistics differ from each other in some material respects, and I was unable to give any satisfactory explanation of the difference, or to say which set was the more trustworthy. For the purpose I had in view both sets had one great defect, namely, that they gave no information as to the number of childless marriages which corresponded to those that became fruitful at different intervals as stated in the tables. I have now, by means of a careful examination of the records of the British Peerage families, obtained a fresh set of statistics; and I purpose on the present occasion applying them to the solution of the same problem as I discussed in 1879.

The first part of this paper was devoted to an investigation of the actual rates charged by the different companies, with a view of showing how important a bearing they have upon a valuation. Before commencing the second part of my subject, which deals with different modes of distributing profits when placed side by side with the rates charged, I would recapitulate the conclusions drawn from the first part of my paper, and, in addition, refer to some of the observations which were made thereon when the paper was read before the Institute last session.

Prior to the commencement of the present century, no direct method had been adopted to ascertain the number of the population in England. Various estimates, founded upon Domesday Books, Subsidy Rolls, and payments of Hearth and Poll taxes furnish, with more or less exactness, the numbers at previous periods. Three such calculations relating to the population towards the close of the 17th century are mentioned by Macaulay as being entitled to peculiar attention. “Of these computations one was made in the year 1696 by Gregory King, Lancaster Herald, a political arithmetician of great acuteness and judgment. The basis of his calculations was the number of houses returned in 1690 by the officers who made the last collection of hearth money. The conclusion at which he arrived was that the population of England was nearly five millions and a half. About the same time, King William the Third was desirous to ascertain the comparative strength of the religious sects into which the community was divided. According to the reports laid before him from all the dioceses of the realm, the number of his English subjects must have been about 5,200,000. Lastly, in our own days, Mr. Finlaison, an actuary of eminent still, subjected the parochial registers to all the tests which the modern improvements in statistical science enabled him to apply. His opinion was, that, at the close of the seventeenth century, the population of England was a little under 5,200,000 souls. … We may, therefore, with confidence pronounce that, when James the Second reigned, England contained between five million and five million five hundred thousand inhabitants.”

Mr. Sprague, in his report on Mr. Finlaison's Tables of Mortality among Government Life Annuitants, states as the result of his study of the subject of graduation that the most satisfactory system of adjustment is the graphic method; and he supports this opinion by referring to the irregularities which he regards as a consequence of the use of Mr. Woolhouse's method of adjustment. I do not doubt that Mr. Sprague would conceive and correctly draw a curve to satisfy all the conditions of good graduation; but in my own hands the graphic method would be a failure, and there may be others who doubt if they have skill to draw the curve and patience to amend it by the successive trials which Mr. Sprague himself finds necessary.

In an article on Life Assurance that appeared in the Economist some time ago, the following passage occurs:—“We should be “all the more sanguine of future success if the purely actuarial “element were given somewhat less prominence and power in the “direction of the offices. The fact that they (i.e., the actuaries) “are immersed in mathematical abstractions is calculated to “impair their efficiency as men of business. They are not “brought sufficiently in contact with the public to know and “appreciate their wants, or to discover the best means of satisfying them. They are men of the study rather than of the “market place.”

It has been suggested that a formula given at page 17 of the present volume should be demonstrated, and its use in graduation more clearly explained.

A series of numbers increasing by third differences is summed continuously p at a time, the results are summed q at a time, these results again r at a time, and so on, until all is collected into one term, S.

In introducing a consideration of this subject to the Institute, I do not purpose putting forward any new theories; many papers have been written upon the different methods of valuation and the modes of distributing profits, which have been most valuable contributions for determining the best theoretical basis to be adopted. I need only refer to the papers of Messrs. Jellicoe, Sprague, Bailey, Meikle, Manly, Valentine, King, and others, all of which are to be found in the Journal of the Institute, in order to remind you how fully these subjects have been considered at different times. The present paper, however, is intended to investigate the practical effect of applying the respective theories, not upon any theoretical assumption, but as they have to be applied in actual practice, and to see how far they are influenced by the different rates of premium actually charged by the different companies.

Life assurance is the compound growth, first, of our commercial necessities, aided largely by a love of speculation, and later, of our progressive civilization. For the former, rough and ready means of estimation were resorted to; for the latter, a long and elaborate course of progressive investigation was needed.

The development of the business has extended over some three or four centuries, perhaps more. It has passed through three distinct phases, as (1). The Experimental Period; (2). The Speculative or Transitional Period; (3). The Period of Scientific Exactitude. These periods, of course, more or less overlap each other, but they each possess very marked distinctions.

By the assessment of life-risks, I mean the gathering together all the scattered influences that bear on the duration of life of particular individuals; the connoting in the mind the weight to be attached to the several factors that make up such influences; and, finally, the fixing the price of assurance for each individual life which approximately puts all upon a level.