It is not intended in this paper to do more than go briefly again over part of the ground covered by Mr. Manly, in his admirable essay, printed in the 14th volume of the Journal of this Institute, on the Values of Policies as found by means of various Tables and Methods of Valuation.

The object of the following paper is to describe an investigation recently made, to determine, from observed facts, the risk incurred by persons making voyages to the Australian Colonies. This investigation was carried out under the superintendence of Mr. Sprague, at whose suggestion the following account of it was prepared.

The method proposed by the late Sir J. W. Lubbock ov approximating to the value ov a life annuity, appears hitherto to hav been very littl notist by mathematicians and not at all by practical men. It was demonstrated by him in the Cambridge Philosophical Transactions for the year 1829, in a paper entitld “On the Comparison of various Tables of Annuities”; and a short account ov it is also givn in the articl Probability, publisht by the Useful Knowledge Society, and very offen bound up with David Jones's work on Annuities: see § 56, p. 36. The former paper was reprinted in the Assurance Magazine (vol. v, p. 277). The original is disfigurd by numerous misprints, which ar all faithfully reproduced in the reprint; and when we add that very littl numerical illustration ov the application ov the method is givn, we hav probably said enuf to account for its attracting so littl attention in any quarter. Considerd as a means ov approximating to the value ov an annuity, the formula has been superseded by the simpler one givn by Mr. Woolhouse; but I believ it may still be practically useful in som cases.

The present paper is essentially elementary, and must therefore in a great measure be considered as addressed to the junior members of the Institute, although, perhaps, even to those more experienced persons to whom in the course of years these things have become familiar, a recapitulation, such as I now propose, may not be wholly uninstructive.

The rate of mortality experienced by residents in India will supply an answer to an enquiry of the highest interest in an actuarial sense. The enquiry may even be considered of national importance so long as our countrymen seek their fortunes in a region where the effects of the climate on their prospects of health and longevity continue to be regarded as uncertain and hazardous.

The subject of this paper has on two previous occasions been brought before the Institute; namely, by Mr. Jellicoe on 26 November 1855, and by Mr. Baden on 27 March 1871. The former of these gentlemen stated that, at the time his paper was written, a great diversity of opinion existed among actuaries, as to the proper method to be adopted in advising upon the proportions in which a settld fund should be divided by mutual consent between the life tenant and the reversioner; and so far as can be judged from the discussion that followd the reading of the latter paper, a similar diversity still existed then. One point, however, was clearly establisht by the discussion, namely, that Mr. Baden was quite mistaken in supposing that, because Mr. Jellicoe's conclusions had not been publicly calld in question, therefore they wer generally accepted by members of the profession.

Sir,—In compliance with your instructions, we have the honour to submit to you the following observations upon the Accounts and Statements submitted to the Board of Trade in pursuance of the Life Assurance Companies Act, 1870.

A “company” is defined for the purposes of the Act to mean “any person or persons, corporate or unincorporate, not being registered under the Acts relating to Friendly Societies, who issue, or are liable under policies of assurance upon human life within the United Kingdom, or who grant annuities upon human life within the United Kingdom.”

The determination of the probable effect, upon the increase of population, of the extinction of small pox, is a problem which has been discussed at some length by three of the great continental mathematicians of the last century, namely, D. Bernouilli, D'Alembert, and Laplace. I propose, first, to solve the problem by what may be termed the theory of the composition of decremental forces, and then to reproduce (by way of comparison) from Mr. Todhunter's History of the Theory of Probabilities, the solutions given by the three eminent mathematicians in question.

Many years having elapsed since the Eagle Insurance Company first undertook to issue policies of assurance upon unsound lives, and some doubts having arisen as to the way in which the funds of the Society had been affected by such transactions, the Directors decided that a thorough investigation should be made into the business connected therewith, with the view of obtaining not only a record of the past, but also, if possible, of establishing a guide for the future, and I now have the pleasure, with their permission, of making public the results of that investigation.

As a wish has been expressed for some further information as to the manner in which a base fee created by the conveyance of a tenant in tail may be converted into a fee simple, it is proposed to consider the question in the following pages.

It is well understood, that an estate in tail is an estate in fee inheritable by heirs special, namely by lineal descendants generally as heirs, or such descendants of a particular class, and that a tenant in tail may convert such an estate into a fee qualified to last so long as there shall be issue inheritable under the entail. Such a conversion may be made by any conveyance sufficient to pass a fee, and will in any such case bind the heir until entry by him,(a) but if enrolled in Chancery in the manner required by the Act for the Abolition of Fines and Recoveries will, notwithstanding the want of the consent of the protector of the settlement, wholly bar the issue in tail, leaving the estate absolute during their existence, and liable only to be defeated by the failure of issue, and then by the entry of the remainderman. When the protector consents in the manner prescribed by the Act, there is no base fee, but the estate tail is converted into a fee simple and the remainders are absolutely barred.

It is now some seven years since Mr. Makeham pointed out that if our mortality tables were graduated by his formula, , “all summations connected with questions of population, the expectation of life, as well as the values of annuities, would be reduced” to the form . Since that time Mr. Makeham's formula, though not representing the law of mortality so closely as some others which have been or can be presented, has nevertheless, on account of its comparative simplicity and its great practical advantages, made manifest progress in the favor of the profession. Messrs. Woolhouse and Brown, among others, have constructed life tables in accordance with it, and it is understood that many actuaries favor its general adoption.

Problem V.—To construct a complete table of the Values of Temporary Annuities; that is, a table which shall comprise all ages and all durations.

This benefit, being a function of the ages at commencement and cessation, say x and w (to the former of which ages also its value is referred), it will be here for convenience denoted, in accordance with the usual functional notation, by ϕ(x, w); and its duration, consequently, will be w–x=n

Since the passing of the Life Assurance Companies Act, 1870, the accounts and valuation statements lodged by the various companies with the Board of Trade, in accordance with the requirements of the Act, have been printed and published annually. The valuation returns afford, in the majority of cases, sufficient information to enable an actuary to form a tolerably correct opinion of the financial strength or weakness of the companies to which they refer. In some cases, however, owing to the special character of the facts brought out in the schedules, it is important that the actuary should have before him, when testing the sufficiency of the reserve, some facts, relating to the growth of the business, and the purposes to which the expenditure has been applied, which are not elicited by the Act. For instance, he should know the annual amount of new business which has been transacted, and the annual amount of assurances surrendered and dropped, as well as the claims by death, in order to enable him to form an opinion as to the average duration of the existing policies. He should also have a division of the expenses, showing those applicable to new business and those applicable to the management of business already acquired. An approximation to these particulars may generally be made by the actuary after a study of the reports and other voluntary publications of the companies; so that, on the whole, it may be said that the Act affords to the public the means of judging, through the medium of the actuarial profession, of the relative positions of the various offices as regards stability and profitableness.

It may perhaps be thought by some persons, that the subject upon which I am to address you tonight, is one that requires no urging on your attention. It is admitted on all hands, I may be told, that a good actuary must necessarily go through a course of mathematical training, and nothing further need be said upon the point. But if an apology for the choice of my subject is required, I would remind you that a mathematician takes nothing for granted. Whereas many people will say that it is self-evident that “two and two make four”, and that this neither requires nor admits of proof, the mathematician knows that it is a proposition, which is not in the true sense of the word self-evident, but can be strictly proved. While we deal with simple matters of this kind, little harm is likely to result from assuming as self-evident, propositions which admit of proof. But the case is very different when we come to more complicated and obscure questions; and not the least advantage to be derived from a mathematical training, is the gradual forming of the habit of mind which requires strict proof to be furnished of many important propositions which are usually taken for granted. I purpose, then, instead of taking for granted the use of mathematical studies, to examine exactly why and how they are of service to the actuary; and I hope that the conclusions I shall arrive at, may be of some assistance to those among you who wish to extend their mathematical reading beyond the point that is by common consent considered indispensable to the actuary.

The Northampton Table has played such an important part in the life insurance transactions of this country, and is still used to such a considerable extent, that a short explanation of the method by which it was constructed, may not even at the present day be without interest to the actuarial profession. In many ways we are greatly indebted to Dr. Price, and in the writer's opinion anything emanating from him is well worthy of our consideration. It is to be hoped, therefore, that no further apology need be made for the following remarks on one of his most important works.

If the death take place in the (t+1)th interval in the year of death, then the present value of the t instalments of each paid in that year previously to the death, is .

The manner in which current policies should be valued for proof upon the insolvency of a Life Office has been much discussed in our pages, and under cover of the conflicting decisions in Bell's and Lancaster's cases has given rise to much diversity of opinion. The interest of the question can scarcely be said to be exhausted. We are, therefore, glad to be able to present to our readers an elaborate judgment on the subject delivered by Lord Romilly, the late Master of the Rolls, which is not only interesting from the eminence of its author, but important as carrying the weight of a judicial decision. It will be best apprehended by perusal, and we will therefore only state that it unhesitatingly decides in accordance with the rule laid down by Lord Justice James in Bell's case, and against that of Lord Cairns in Lancaster's case. In order, however, to bring the whole matter down to the present time, we must remind our readers that it has been dealt with in the Life Assurance Companies Amendment Act, 1872, and again by the determination of Lord Westbury in the European Arbitration.

The business of Life Assurance is founded on the principle that the number of deaths which will occur among a large number of persons in a given time is not a matter dependent entirely on what is called chance; but is subject to a law of average so uniform in its operations, and so trustworthy as to its results, as to be capable of forming the basis of calculations on which the shareholder may stake his capital, and the assured the welfare of those for whom it is his duty to provide. Proceeding on this principle, tables of mortality have been constructed,—from data collected at different places and under a variety of circumstances, and tabulated and adjusted with various degrees of accuracy,—which show, out of a given number of persons born, the number who survive each year of age, and the number who die at each age from year to year. From such tables, companies transacting the business of assurance calculate the rates of contribution to be required from their clients; and on the sufficiency of the rate of mortality founded on in any instance depends, of course, the sufficiency of the premiums which the company is to receive in consideration of its risks. It is therefore of great consequence that the company should ascertain from time to time how the rate of mortality actually experienced compares with that indicated by the table on which the calculations are founded, in order that there may be proper grounds for being satisfied that the basis is a correct one. For it is not sufficient that the mortality table founded on should be known to represent faithfully the death-rate prevailing among the general population, or even among assured lives generally.

In 1793 the first Act “for the encouragement of Friendly Societies” was passed, and its preamble gave as the reason for passing it, that the encouragement of such societies was likely to be attended with very beneficial results, “by promoting the happiness of individuals, and at the same time diminishing the public burdens”. The encouragement given by the State consisted of exemptions from fees and stamps, facilities in recovering debts and settling disputes, and exemption of members from removal under the Poor Law until they became actually chargeable to the parish. The next Act of any consequence was that of 1819. Another Act had become needed. Since the passing of the former Act the population had increast from 9,000,000 to 11,500,000, but the amount annually expended in poor-law relief had risen to more than £7,500,000—within half a million of the amount annually expended now with nearly a doubled population. In other words, while since 1750 the population had increast by one half, the amount of poor-law relief had increast tenfold, or twenty times as fast as population.

There is nothing immutable in this world, and certainly life companies are no exception to the general rule; for in them, from year to year, we find fluctuations of every kind—in amount of new assurances—in expenses of management—in value of assets—in realized interest—in the death claims—in fact, in every branch and item of their business. Of all these variations there is none so important as that concerning the mortality experienced. The fluctuations in the other elements of a life office's affairs are trifling as compared with this, and can without difficulty be prepared for. But in the perturbations of the death claims from year to year, we are dealing with a matter that cannot be calculated on in advance —an element which, if violent, may endanger the safety of the company itself; and, even when this is not the case, will often prove a serious inconvenience to it. I have called violent mortality fluctuations inconvenient to a life office. To men acquainted with the practical as well as the theoretical side of assurance, the word will not seem exaggerated. They, at least, know how important it is that the bonus to policyholders and the dividend to shareholders, do not retrograde. The moment the bonus tide begins to ebb, there is an outcry; and variations in the mortality may thus create disgust in those already connected with an office, and at the same time add another difficulty to the already sufficiently difficult task of its agents and canvassers.