The question which it is proposed to consider in the present paper is the correct mode of valuation of current policies, or true measure of proof to which claimants thereon are entitled in the liquidation of a Life Office. The subject is of especial interest at the present time, because its solution is even now waiting settlement by the Court of Chancery; and the interest is heightened by the fact, that conflicting judgments have been delivered by two eminent equity Judges, namely, by Lord Justice James, in Bell's case, when sitting as Vice-Chancellor in the Court of Chancery, and by Lord Cairns, in Lancaster's case, when sitting as Arbitrator in the Albert Arbitration. It does not appear that among actuaries there is a complete agreement. Nevertheless, the case ought not to present any great difficulty, and whether among actuaries or in a Court of Equity must be decided by argument alone. It is therefore proposed to consider the question, in the first instance, upon principle; and secondly, with reference to the doctrines laid down in the cases to which we have referred.

With the exception perhaps of extra premium, there is, it seems to me, no actuarial point in which the practice of assurance offices is so empirical as that of the allowance to be given for the surrender of a policy. The usual custom, so far as I am aware, is to find the value of the policy (generally by the table of mortality and interest used in calculating the reserve) and from that value to deduct a certain proportion, handing over the remainder to the policyholder. The amount of this deduction is purely arbitrary, and were it not for the equalizing effects of competition, would show very startling diversities; since, even with this counteracting effect, the differences in practice can hardly help striking anyone who has examined the amounts given as surrender value by the various offices interested in large reassurance cases which from any reason do not follow the rule of the parent company. In fact, making any return whatever under discontinued policies is the result of that struggle for existence called competition; and like all things growing from this root, the practice has been moulded much more by the pressure of circumstances than by theoretical considerations. Consequent on this method of growth we find extraordinary and indefensible anomalies imperatively calling for rectification. Further it seems to me, looking at the matter as a purely practical question and not one of actuarial fitness at all, there is no point more needful of examination. As things are at present, nobody can say positively whether as its proportion of the value the assurance office retains too little or too much. If the former, the sooner a practice resulting in loss is reformed, the better.

A paper on Extra Premium may treat of the subject in one or other of two ways. It may be, on the one hand, an attempt to deduce tables of mortality for the various classes of under-average risks occurring in a Life Office's transactions. Or, on the other hand, this part of the subject may be ignored and the paper be simply an enquiry into the effects resulting from the various methods employed in charging extra—an analysis of the actuarial points raised in consequence of a Society admitting not merely minimum risk lives, but others not eligible on similar terms. In short, the subject of Extra Premium may be considered either in respect to the amount to be imposed or in regard to the manner and effect of its imposition. The first question must be determined mainly by actual mortality observations—the second does not require in the same degree for its examination the light of experience. It is this latter question which I have taken up in the following paper; but as both branches of the subject are intimately connected, it may be well to make a few general remarks on the former also.

Gompertz's Function for expressing the probability of a person surviving anygiven period, x, is . For thepurpose of adapting this expression to the entire period of adult life Ihave added the factor ∈−ax. The two expressionsbecome identical in form when the factor∈−δx representing the effect of discount is introduced, and both may therefore becorrectly designated by the name of “Gompertz's Function.”

Of all the securities proposed to insurance companies, reversionary life interests are the most troublesome to deal with in the ordinary way of mortgage; and the objections to so dealing with them have been considered by many actuaries so serious, that they have laid down the rule that the only safe way of making an advance on the security of a reversionary life interest is by way of reversionary charge. The transaction, they think, should be in the nature of a sale rather than a mortgage, a portion of the reversionary life interest being sold in consideration of the present advance.

The desirableness of obtaining trustworthy statistics as to the mortality prevailing in Infancy and Childhood has been frequently insisted upon by Actuaries; no apology therefore is needed for placing before the Members of the Institute the results of some original investigations into the subject. Whilst regretting that the materials at my disposal are not very extensive, I trust that this contribution to the subject of Vital Statistics will, nevertheless, be found to possess some degree of value and interest.

In the year 1865, the late Rev. John Hodgson published a pamphlet entitled “Observations in reference to the Duration of Life amongst the Clergy of England and Wales;” to which is added, in the form ofa supplement, a letter from Mr. Samuel Brown, containing various tables,having the observations of Mr. Hodgson for their basis, applicable toquestions connected with Life Assurance, and the Valuation of NextPresentations to Livings and Advowsons.

The influence that occupation has upon health is a subject that has hitherto attracted but little attention, for though in the works of one or two writers on vital statistics it has been slightly alluded to, no work has yet appeared which, while founded on direct observation, has been devoted to the treatment of this subject.

There are few things so discouraging to an agent, after successfully canvassing for an assurance, as to find that the medical examiner is unable to recommend the case (which has, perhaps, been secured by a serious expenditure of time and trouble), without an addition to the ordinary rate of premium. Knowing the difficulty he has experienced in persuading the proposer to take the decisive step, the agent feels that, in nine cases out of ten, the demand for an extra premium is tantamount to the rejection of the proposal—for, somewhat unreasonably, perhaps, instead of impressing more strongly upon the party concerned the importance of life assurance, the effect is generally to irritate and annoy—or even to create in his mind a suspicion that he is treated unfairly in being required to pay a higher rate than other persons of the same age.

We have now, in conclusion, to deduce from the foregoing investigations the proper formulas for deducing a mortality table from observations arranged as in § 17. The preceding notation will not be strictly adhered to.

The Arithmometer of M. Thomas (de Colmar) has been already brought under the notice of the readers of this Journal by General Hannyngton, in a remarkably lucid and suggestive paper, which will be found at p. 244, vol. xvi. General Hannyngton, in his paper, explains the manner of working the machine, and gives examples of some of its applications to the construction of actuarial tables, with hints as to others. These afford an idea of the very striking adaptation of the machine to the formation of such tables; and they cannot fail to have excited the interest of many of the readers.

In recent years, the writer who has done the most good service in bringingscientific methods to bear on the collection and comparison of statistics,is M. Quetelet, the Director of the Royal Observatory in Bruxelles,President of the Central Commission of Statistics, and the PerpetualSecretary of the Academy of Sciences. Nearly all the learned Academies ofEurope claim the credit of enrolling him as a member. He was amongst thesmall number of illustrious men, including Professor Babbage, Whewell,Malthus, Drinkwater, Jones, Col. Sykes, &c., who at the third meeting ofthe British Association for the advancement of Science, at Cambridge in1833, succeeded in establishing the special Section F on Economic Scienceand Statistics, and which afterwards led to the formation of the StatisticalSociety of London and various provincial Societies. He was also theoriginator of the International Statistical Congresses, which have been souseful in improving the methods of collection and publication of facts inall branches of statistical enquiry by every Government in Europe.

The discussion of the methods of dealing with the mortuary statistics of a life insurance company has been hitherto almost entirely neglected by American actuarial writers. This is not a little singular, considering the attention bestowed on other subjects of not greater difficulty or importance. Viewing the large and rapidly-growing body of data already accumulated by American companies, the problem is not of less concern to us than to our English and German brethren, to whom, notwithstanding, we are indebted for the principal contributions to this branch of actuarial science.

Statistics, generally considered, is a term of very comprehensive import, and is to be understood as having reference to an important collection of facts properly arranged and systematized in the form of numerical tables, for the purpose of conveying such information or data as may best assist in the investigation and discussion of particular subjects of inquiry. The general principles applicable to these investigations are, for the most part, intimately allied with the mathematical theory of probabilities, and constitute the true science of statistics. It will, moreover, be found, on examination, that the same identical principles lie at the foundation of all the physical and inductive sciences so far as they originally and necessarily depend upon experiment and observation.