207,997 1871 Very good water supply.

Large number

55

177,089 1871

JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

On the Mortality amongst Assured Lives, and the requisite Reserves of Life Offices. By GEORGE KING, of the Alliance Assurance Company, Fellow of the Institute of Actuaries, Member of the Physical Society, &c.

[Read before the Institute, 24 April 1876.]

PART I.-Statistical.

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.

Mr. Sprague's essay, in vol. 15 of the Journal, on the "Rate of Mortality prevailing among Assured Lives" is very exhaustive; and had the discovery of the law of their mortality been my object, it would have been presumption in me to follow him; but Mr. Higham, in the years 1850 and 1851, read two papers before the Institute, in which he analyzed the experiences of the Equitable Society and of the Seventeen Offices, deducing annuity values

VOL. XIX.

2 D

therefrom; and it seemed to me that the time had now arrived when the more ample statistics in these days available might be similarly dealt with. I also thought it would be well to go further, and inquire how policy-values and office reserves are affected by the peculiar mortality obtaining among the lives assured, and to prepare for the purpose, from the most recent figures, a model office somewhat on the plan adopted by Mr. Manly, in his Essay in vol. 14 of the Journal, and value it by my own new tables, and by standard tables for comparison.

My paper divides itself naturally into two parts, the first briefly styled "statistical", and the second "financial". In the former are sorted the raw materials found in the volume of Mortality Experience; and the latter will include monetary tables based upon the results of the first part, and the valuations of the new model office above referred to. The first part I have now the honour to submit to the Institute. The calculations for the second are already well advanced, and I hope before long to offer it also. On the very threshold we are met by an objection which has been strongly urged by some. The Mortality Experience includes every policy of the twenty companies where the duration of human life was involved; and yet, for the purposes of our investigation, we assume that all the business was for the whole of life. We have principally in view to measure the effects of the "selection against a company "after its policies have been issued; but the force, and its consequent action upon the mortality, must vary according to the description of policy. If, therefore, there were any large admixture of other kinds with whole-life policies, and especially of such kinds as might follow a very different law of discontinuance, it would be useless to take the trouble which a laborious analysis involves. It will be well, before proceeding further, to consider this point, and see how far any conclusions we may form are vitiated.

We are unable to examine the actual transactions contained in the experience, and a perfectly satisfactory answer is not possible, but the Parliamentary Blue-books supply us with information of which we can avail ourselves, with probably an error of insignificant magnitude. We shall, moreover, err most likely on the safe side, by exaggerating in appearance the untrustworthiness of the materials; for the tendency of companies seems to be to invent new forms of life assurance, and so increase the proportion of the special to the total business.

The following table, A, gives an analysis of the business on

the books of 19 out of the 20 companies, when last they valued. The remaining company is omitted, it not being an independent institution, and its schedules not appearing separately in the Bluebooks; but, had its returns been accessible, it is not likely that the percentages would have been sensibly different.

TABLE A.-Analysis of the Business of the 19 Companies.

The upper portion of the table gives the gross sums assured, the sums reassured, and the percentage of reassurance for each of the eight broad classes into which I have found it possible to distribute the transactions. From the variety of nomenclature adopted by the different companies, the classification has not been so easy as would be imagined, and a few minor inconsistencies may have crept in; but they will not affect the general results. The lower portion gives the amounts assured in each class, less reassurances-deducted in order to eliminate, as far as possible, duplicate policies on one life-and the percentages these amounts bear to the total business. The most striking feature is the enormous preponderance of assurances for the whole of life with

uniform premiums, over all the other business taken together, they being 90.33 per-cent of the whole. It should also be observed that this excess would have been still more marked, had we been able to present lives instead of amounts assured; for it is evident, from the much larger percentages of reassurance, that in the miscellaneous groups the policies must be for heavier sums, and consequently comparatively fewer in number than those in the first line. Thus it is clear that, even if policies of every other quality exercised an unfavourable influence on our deductions, yet the ordinary whole-life cases are so numerous as nearly to obliterate it. But many of the special whole-life cases, and the joint-life and endowment assurances, probably follow a very similar law of discontinuance; and the remainder, not much more than two percent of the whole, may safely be neglected.

Having thus cleared the ground of preliminary obstacles, we may proceed with confidence to our task. As stated above, in 1850 Mr. Higham read a paper on "The Value of Selection among Assured Lives, and its effect upon the Adjustment of a Scale of Premiums as between persons assuring at different ages." The paper was not published in the Journal, which at that early period did not exist; and it has long since been out of print. This is unfortunate, as very interesting results were ably worked out by Mr. Higham, the usefulness of which has in great measure been lost through their not having been accessible. It was through the kindness of Mr. Higham that they were placed in my hands, and I was enabled to profit by his labours. In this paper Mr. Higham analyzed the published experience of the Equitable Society, preparing a number of mortality tables for various ages at entry, and giving the expectations of life deduced from these tables, and the corresponding annuities and premiums. The returns of a single company were hardly sufficient to afford trustworthy data for such an extended investigation, the observations which Mr. Higham could utilize embracing only 12,947 lives; and in 1851 he read another paper printed in vol. 1 of the Journal, on "The Value of Selection as exercised by the Policyholder against the Company", in which he analyzed in a similar way the greater mass of material supplied by the seventeen offices, embracing for ages at entry 20 to 65, 55,506 lives. As this is so well known it is needless for me to do more than paper refer to it, in order to state that that portion of the experience of the twenty companies which I have been able to utilize, consists of 127,471 entrants, as against Mr. Higham's 55,506.