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.

Tht method, according to Mr. Altenburger (J.I.A., xxxv, 332), was devised by Dr. Karup in 1878, and communicated by him to the Austrian Government in 1882, and to the Swiss Government in 1884; but, unless perhaps in Germany, of which I have no knowledge, it does not seem to have been published until comparatively recently, and no doubt it has been discovered independently by other actuaries. So far as I am aware, the first account of it in the English language is that given by the late Wm. D. Whiting, in 1894, in the Transactions of the Actuarial Society of America, vol. iii, p. 427. Mr. Whiting wrote at the suggestion of an unnamed member of the Council of the American Society, and he does not say to whom the authorship of the formula should be attributed. The first reference in the Institute Journal is to be found in a letter of Mr. Julius Altenburger, in J.I.A., xxxiv, 152, and therefore the formula sometimes goes by his name, but he himself says that the name of Dr. Karup should be attached to it.

In the present note, attention is directed mainly to Endowment Assurances, and Policies for the Whole of Life by Limited Payments; but the method is available for many other classes, and its very general applicability is elaborately worked out by Mr. Frederick Bell in his paper (J.I.A., xxxix, 17) read before the Institute, 19 December 1904. The principles of the method are therefore well known, but, in so far as they have yet been developed, there remains a certain amount of rigidity which it is the object of this note to remove. Elasticity may be introduced, which will render the formula more convenient under the presentday varying conditions of British companies.

Mr. Lidstone has kindly communicated the following notes of corrections to his paper “On the Rationale of Formulæ for Graduation by Summation”, in the present volume of the Journal—

(i) (P. 108), 2nd line of par. 28, for “rates”, read “ratios.”

MR. T. G. ACKLAND, in the course of the discussion upon Mr. Manly's paper “On the Valuation of Staff Pension Funds”, presented the following problem for solution by the members of the Institute—” In the case of a superannuation fund, established—“ in connection with a municipality or corporation, to which the “employees contribute at a fixed rate, what will be the probable “annual charge upon the rates of the City during the next ten “years, during the following ten years, and so on for fifty years, “supposing that the present staff of the Corporation is adequate “in number for the needs of the Corporation, and that new “entrants come in only in replacement of those who die, “withdraw, or retire” ? (J.I.A., xxxvi, p . 283).

It is sometimes the case that a commercial undertaking merely guarantees the stability of its staff pension fund, without making an adequate annual contribution from which to build up the reserve necessary to meet the actuarial liability, or, more usually, it is the practice of a corporation to meet the current claims under its pension scheme out of the current income from its employees' contributions and to use any excess of income in relief of the rates, and to charge any deficiency, when arising, upon the rates. In such cases the ordinary actuarial valuation is of little practical use, but the main question is that put by Mr. Ackland—“ What will be the charge upon the employer, or the rates, at a given date, or over a given period”?

The question whether the net premium method of valuation, now almost universally adopted by offices distributing their profits in the form of reversionary bonuses, should be regarded as a final and scientific method of ascertaining divisible profits or as merely a temporary expedient which must, in the words of Mr. Sorley, “inevitably disappear before the advance of true actuarial science”, is one which has frequently engaged the attention of actuaries. The importance of the subject is so great to all connected with the administration of Life offices that in the hope of provoking an interesting and useful discussion on the general question, I venture to submit the following notes on an alternative method of valuation, based on a suggestion by Mr. Manly (in the discussion on Mr. Warner's paper on the Net Premium Method of Valuation reported in the 37th volume of the Journal, p. 57), that in the valuation balance sheet the present value of the bonus at the rate to be declared at a given valuation should be treated as a liability for the future existence of the policies.

As the alternative method is, to some extent, based on the original principles of the net premium method, I propose, as briefly as possible, to summarize the modifications which have been introduced in the last 30 years in the principles and application of the net premium method.

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.

“Before proceeding to examine the different methods pursued “in the valuation of the liabilities of an office under its policies, “I will briefly state what appears to me to be the true principle “upon which the valuation should be based. On the establish-“ment of an office it is assumed, or ought to be, that a certain “table will exactly correspond with the mortality likely to prevail “among the persons whom they will insure; and that a certain “rate of interest (generally 3 per-cent) is what they may with “certainty depend upon realizing. Under these circumstances, “premiums calculated upon such data, increased by a small “addition sufficient to cover all expenses likely to be incurred, “are charged. We will leave out of consideration the addition “ made for bonus, as it is only charged to be returned again. “Suppose now that all these conditions are realized, and that “after a few years it is found that the mortality experienced “corresponds exactly with the table adopted; that the rate of “interest realized is exactly 3 per-cent; and that the additional “charge, technically called the loading, has exactly covered all “the expenses. How much ought the office to have in hand? “The answer naturally will be,—the total net premiums received “in each year, less the claims paid in that year, accumulated at “3 per-cent compound interest.

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.