In forming mortality tables from the experience of assurance or annuity societies or analogous institutions, we deal with individual lives, tracing each throughout his recorded history, from the moment he comes under observation, until he is removed either by death or withdrawal,—and here the word “withdrawal” includes not only lapse or surrender, but the termination of the contract in any other way, as also the closing of the observations at some determined point of time. We therefore have something approaching to accuracy, and the deaths take place among the lives actually observed, there being close relationship between those who are exposed to risk and those who die. With tables of this class the mortality function which first presents itself is the rate of mortality, qx.

With mortality tables formed from census returns and death registers the case is different. We know nothing of individuals, but deal solely with fluctuating groups, and take the lives in the aggregate; and many of the recorded deaths are not those of persons included in the enumerations. For instance, the English Life Table No. 2 was based on the population of England and Wales enumerated at the census of 6 June 1841, and the deaths recorded in the seven years 1838 to 1844; so that many of the deaths took place before the date of the census, and could not have been deaths among the population counted.

The principles of investments proper to Life Offices were laid down many years ago by Mr. S. Brown and Mr. A. H. Bailey, and Mr. A. G. Mackenzie summarized them conveniently at the beginning of his paper “On the Practice and Powers of Assurance” Companies in regard to the Investment of their Life Assurance “Funds” in 1891 (J.I.A., xxix, 185). These principles may be said to be the laws which govern our practice; but as in all such matters, it is the application of the law which is of importance, it may serve a useful purpose to-night to review, in the first place, recent changes in the investments of the Offices and ascertain their tendency; secondly,to consider the nature and causes of the fluctuations in values which have recently been strikingly prominent, and finally to make some rough forecast of the future.

It is at once apparent, from the nature of securities easily convertible at short notice, that though suitable for bankers, they are not specially adapted for insurance companies, whilst another great class of investors, namely, trustees and executors, are usually limited to particularly well-secured investments, either by law, or by special provisions incorporated in the settlements or the wills under which they act. The resulting competition in these classes of gilt-edged securities is keen, large sums constantly coming on the market for investment, so that the rates of interest yielded are comparatively low, a fact which tends to drive investors not so limited in power further afield. It is not surprising, therefore, to find that many insurance companies have of late years enlarged their power of investment, as it is now recognized that to give ample discretionary power is the wisest policy.

In venturing to submit a paper on the subject of bonuses to the Institute, I feel that a word of explanation is necessary as to the reasons that led me to take up this investigation, and as a preliminary to this, I would first call attention to some results presented by previous writers.

In 1887 the late Mr. Sunderland read a paper (J.I.A., vol. xxvi) dealing with Whole-Life Assurances, and he followed this up by a further paper (J.I.A., vol. xxviii) dealing with Endowment Assurances. A valuable set of tables was given with each paper, based on a HM 3 per-cent valuation, showing, among other things, the bonus arising from surplus interest margins of ½, 1, 1½, and 2 per-cent for whole-life, and 1 and 1½ per-cent for endowment assurances.

He also gave the bonuses arising from an annual surplus margin of premium of 10s. at the various experience rates of interest, and showed that bonus from surplus premium in any year is proportional to the amount of such surplus premium. Dealing with mortality, he showed the results of experiencing Sprague's Select Mortality under whole life policies, and gave an illustration of the effect of experiencing a favourable mortality under a 25-year endowment assurance. The papers were practically independent, and did not aim at comparing bonuses under the two classes, although so far as interest and loading profits were concerned, they gave the means of doing so.

The new method described in Mr. King's recent paper, “On the Construction of Mortality Tables from Census Returns and Records of Deaths”, marks such a great advance on that employed in the construction of the official English Life Table, that it occurred to me that it might be worth examining whether the numerical application could not be further simplified by the use of central differences. Everett's formula (J.I.A., xxxv, 452) lends itself admirably to the construction of tables by subdivision of intervals. It involves only even central differences of each of the two middle terms of the series between which the interpolation has to be made, and, as was pointed out by the author in communicating his formula to the Journal, “each sum of three terms does double duty, serving both for “the preceding and the succeeding interval. In an extended “computation, the number of ‘sums of three terms’ to be “calculated is accordingly practically identical with the number “of intervals, and the labour of calculation is only about half “what it appears to be on the face of the formula.”

The problem before us is that of fitting between consecutive pairs of a series of points a series of partial interpolation curves, which shall have the same slope and curvature at their points of junction. In what follows, the central difference notation is that introduced by Dr. W. F. Sheppard in his paper on “Central Difference Formulæ” (Proceedings of the London Mathematical Society, xxxi, 449).

The case of Kettlewell v. Refuge Assurance Company, which deals with the effect of unauthorized representations by assurance agents, and has, by reason of its importance, attracted considerable attention, has already been referred to in these Notes in connection with its appearance before the Divisional Court (J.I.A., xli, 574). The case has since been heard on appeal by the Court of Appeal, and it may be convenient briefly to recapitulate the facts. The plaintiff effected a policy on the life of one James Kettlewell, with the defendants. After paying the premiums for a year, she proposed to allow the policy to lapse, but on the representations of the defendants' district superintendent and agent, that if she continued to pay premiums until the expiration of five years from the date of policy she would be entitled to a free policy, she continued to pay the premiums until the end of these five years, when she claimed the free policy. The representations so made by the defendants' agents were untrue to their knowledge, and were made without the authority of the defendants, and in these circumstances the latter refused to grant the free policy. The plaintiff then brought an action to recover back the premiums paid by her on the faith of these representations. It was given in evidence for the assurance company that their agents were employed only for the purpose of procuring proposals for policies, and that they had no authority to make any contract for the defendants. The County Court judge entered judgment for the plaintiff for the amount of premiums paid since the date when the false representations were made, and on appeal, the Divisional Court upheld this decision, holding that the fact that the defendants had been at risk during the four years covered by these premiums did not disentitle the plaintiff to rescind the contract of insurance and recover back the premiums paid.