Occasional allusions to copyhold tenure are met with in works upon actuarial science, and in exercises for students; and the admirably lucid paper by Mr. Peter Gray upon the Theory of Successive Lives, which appeared in the second volume of the Journal, is likely to be remembered as having a bearing upon the subject. A brief account of the intricacies which beset this tenure, and of the attempts which have been made to cope with the arithmetical difficulties to which these intricacies give rise upon proceeding to convert the tenure into freehold, may not be without interest for some of the readers of the Journal.

It may at first sight appear strange that while life insurance more especially, and some other branches of insurance at least theoretically, are based upon exact statistical details, fire insurance has never attained to anything like such scientific exactitude. But it will be remembered that, in its first inception, fire insurance was to be a State or municipal institution, wherein it is customary for the insured to pay a certain nominal rate at the time of entering into the contract; and at the end of the financial year the cost of insurance, as measured by the losses and expenses, is ascertained in the aggregate, and an assessment made over the whole property of the town or district for the balance. Under these conditions, the ascertainment of an exact rate in advance is not necessary. Then, again, many of the early fire offices were based upon the “contribution plan.” Here also a nominal payment is made in advance, or annually, and a call or assessment is made either annually or on the happening of any serious calamity. Here too the necessity for exact rates did not arise.

The opinions exprest at several recent meetings ov the Institute ov Actuaries apear to indicate that ther is a general agreement among practical actuaries, that the Institute Tabl HM(5) represents very faithfully the mortality among insured lives after the first benefit ov selection has worn of,—say, after the laps ov five years from the date ov insurance;—and that therefore life office valuations shoud, in the case ov policies ov five or more years' standing, be to a great extent based upon that tabl. When we consider to what extent it shoud be uzed, we see at once that we shoud calculate by means ov it the values ov the sums payabl by the offis and ov the sums receivabl, that is, ov the gross premiums; we shoud also, ov cours, uze it in converting the divisibl surplus into reversionary bonuses. But it is not a suitabl tabl for the calculation ov the net premiums; and net-premium valuations therefore cannot with propriety be made by means ov it alone. The mortality during the first five years being considerably less than that shoen by the tabl in question, the true initial anuities (or anuities at the date ov asurance) wil be greater than those givn by it for the same age, and the net premiums consequently less. It folios then that, if we wer to value the HM(5) net premiums, we shoud hav too large a credit, and the liability bro't out woud be too smal.

In the reports ov life insurance companies ther is offen found a statement to the efect that the value ov the liabilities has been calculated by a tabl which exhibits a hiher rate ov mortality than has been experienst. The inference sugested, and aparently intended to be drawn, must be that such a cours is more prudent than a valuation made by a tabl that woud represent with tolerabl accuracy the mortality experienst. In other words, it is implied that, the hiher the rate ov mortality exhibited by the tabl uzed in the valuation, the greater wil be the values of the policies obtaind by its use. This may at first siht seem to be a very reasonabl conclusion, but those who hav made numerical comparisons between the values ov policies as givn by different tabls, wil not need to be told that it is wholly eroneos. This is by no means a new observation. Remarks to this efect wil be found in a paper by Mr. Jellicoe in the Assurance Magazine, vol. 3, p. 186: see also a paper in the same volume, p. 143, by the late Mr. Hopf. More recently the point is wel bro't out in Mr. Manly's Messenger Prize Essay, vol. 14, p. 304.

The importance of this subject may be indicated by the fact that more than half a million a year is paid by the British Life Offices to their assured, as surrender values on the discontinuance of their policies. Competition also, and the increasing interest which is being taken in the subject by the insuring public, have a tendency to induce companies to make these surrender values a prominent feature in their prospectuses; perhaps also it may be an inducement to some companies to offer large values in order to bid for larger patronage.

Not much seems to have been written on the subject of the Mortality of the West Indies as compared with that of the mother country. The following contribution, being the experience of the Scottish Amicable Life Assurance Society in these colonies during the last thirty years, may therefore, perhaps, be useful as well as interesting. The Directors of the Society have cordially adopted the suggestion that the figures should now be made public.

Perhaps the most important lesson many of you can learn—one that is frequently lost sight of by persons who are solely or mainly engaged in theoretical studies—is that the researches which you are pursuing, by the aid of a variety of complicated symbols, are, after all, not merely elaborate mathematical puzzles to be solved principally for the purpose of awakening or sharpening the intellect, but have, for you at least, an eminently practical aim as well, and are insisted on as part of the education of an Actuary, with a view to their serving as the framework on which, when afterwards called upon, he must be prepared to base actual business transactions involving large sums of money, or, it may be, to lay down principles for the regulation in all its various details of a life assurance office.

In valuing interests that involv the probabilitys ov marryge and ov leaving issue, we require to hav the means ov ansering such questions as the folloing among others. What is the probability that a bachelor ov a givn age wil (1) marry, or (2) die unmarryd, in an asignd year from the present time? or (3) be alive and stil unmarryd after the laps ov a givn number ov years ? In ordinary life contingency calculations, the corresponding questions as to the probabilitys ov life and deth ar anserd by means ov the figurs containd in the familiar mortality tabl (formerly cald a tabl ov mortality, but by Dr. Farr a life tabl); and the great practical convenience ov that tabl sugests that, when we hav to deal with chances ov marryge as wel as ov deth, we shoud at the outset construct on analogos principls a combined marryge and mortality tabl.

A method of adjusting mortality tables was originally devised by me expressly for the purpose of accomplishing the graduation of the Institute Tables of Mortality, which operation had been entrusted to me by the Experience Committee. The resulting adjustments proved to be very satisfactory, and a full explanation of the process adopted is given in the Institute of Actuaries' Life Tables and in the Journal of the Institute, vol. xv, p. 389.

The calculations which I have made respecting the rates of mortality in New South Wales are founded upon the returns of the Registrar-General for the years 1860-75, and upon two censuses taken in the years 1861 and 1871. There is every reason to believe that the returns of births and deaths are as accurate as can be expected in any case, and that the results of the censuses are as trustworthy as is usually the case. The returns for the year 1876 were not published when I commenced the calculations; and I have not gone farther back than 1860, because during the year before, Queensland with a considerable population was separated from New South Wales, producing of course an awkward break in the registrations.

Often as the method of estimating the liabilities of life assurance companies has been discussed by the Institute, there seem to me to be some reasons why it will not be entirely useless to devote another evening to the subject. The Act of 1870 has now been sufficiently long in force that, with one or two exceptions, all the existing companies have furnished the particulars required by the 5th and 6th schedules, which give, in considerable detail, the methods and results of their valuations. A tolerably minute examination of these documents, as they have from time to time appeared, has satisfied me that much difference of practice still exists even among those who would consider themselves in agreement on the main principles; and the 105th number of the Journal contains a paper by Mr. McCandlish, which, being an address ex cathedrâ to the Edinburgh students from the President of the Actuarial Society, may have an authority to which, as it seems to me, some of its doctrines are hardly entitled, and which therefore should not pass altogether unchallenged.

Although much has been written in recent years on the comparative merits of different methods of valuation, I am not aware that any attempt has yet been made to ascertain the ultimate bearing on the Pure Premium Method, of the facts that have been recently brought forward as to the pressure of the expenses and the effects of selection in an assurance company. Speaking generally, it will, I think, be found that proposals to diminish the pure premium reserve on account of the necessarily heavy expenses of new business, have been made without reference to the mortality profit that may be expected to arise from the recent selection of the lives, and vice versa. This being the case, it occurred to me that it might be of considerable practical interest to make an investigation in which effect would be given to the influence both of expenses and of selection, for it would enable us to estimate how far the one neutralized the other. In order to render the inquiry more clear, however, it will be well to begin by considering each separately.

A survivorship annuity of £100, payable half-yearly till death, on (26) after (32), is to be paid for by an annual premium, returnable if (26) die before (32). Required the annual premium, which shall give to the grantor a profit of 15 per-cent ; the rates of mortality and interest assumed as those which will actually obtain, being Carlisle 4 per-cent.

As the formulas obtained by Prof. Pell differ from those generally in use among actuaries, it seems desirable for the sake of future students of the subject to point out that he has inadvertently fallen into an error which will necessitate considerable alteration in the reasoning on pages 142 to 144 of his paper. He uses k to denote what he calls “the absolute rate of mortality at the exact age x”, and k′ for the same at age x + 1; evidently meaning the same quantity that British actuaries now by common consent call the force of mortality. This term is embodied in the Scheme of Notation appended to the “Institute of Actuaries Life Tables”, regarding which the Council of the Institute say in the preface to the volume, that they strongly recommend the notation for general adoption, in the expectation that its use will contribute materially to advance the study of life contingencies, by promoting exactness of thought and expression. It would always be convenient, and occasionally save much trouble and confusion to readers, if all the writers in this Journal would invariably adhere to the recognized notation, so far as they can make it answer their purpose for the time being.

After I had written the first draft of an article on the above subject, my attention was drawn to two articles by Mr. King “On Mortality amongst Assured Lives.” I had felt much diffidence in saying anything having the appearance of adverse criticism of the valuable works issued by the Institute; but now that I observe that Mr. King's articles, which are in some measure founded on opinions similar to my own, have been received with marked approval, I feel more confidence in stating plainly what appear to me defects in the HM and HM(5) Tables. But whatever I may say respecting them, I trust that my assurance will be accepted that I entertain feelings of the highest respect for the eminent knowledge and ability of the distinguished persons concerned in their preparation, with the most hearty and grateful appreciation of the benefits which they have conferred on the profession. I need hardly say that nearly all that I know of the subject of mortality amongst assured lives, is due to them.

It is one of the fundamental theorems of the Doctrine of Probability that, “If a compound event consists in the concurrence of two or more independent simple events, that is, events the happening or failing of any of which has no influence on the happening or failing of any of the others, the probability of the compound event is equal to the product of the probabilities of the simple events.”

Mr. Artemas Martin has solicited my opinion in regard to the following problem, the true solutions of which have, he informs me, been much discussed in America. I give, first of all, Mr. Martin's solution, and append thereto my own remarks in regard to the accuracy, or otherwise, of the solution.