A correct measure of the Rate of Mortality in the First Year of Assurance is of considerable value. It enables the actuary to calculate the limit to which initial expense may grow without incurring the risk of loss through lapse at the end of the first year ; while its more obvious purpose is to estimate the immediate effect of the Selection by offices through their medical examinations and restrictive rules. In the first year we see the full force of the skill of medical officers pitted against the desire on the part of the lives assured to make their best appearance. We need not suggest that they try to deceive the doctor, but it is often necessary for him to apply tactful cross-questioning in order to elicit important information.

A mortality table shows the number living at each age x, and the number dying in the following year; and the ratio of the latter to the former, is the probability of dying in a year, and is called the rate of mortality at age x. This function varies from age to age by finite steps; but it is not to be supposed that the actual intensity of mortality remains constant throughout each year and changes suddenly at the end of each completed year of life. On the contrary, it is a function which varies by an indefinitely small amount with each indefinitely small interval of time; and the measure of its value at any instant, like that of any other varying rate, is the change that would be produced in a unit of time if the rate were to remain constant throughout that unit, and equal to its value at the instant in question. For example, when we speak of a railway train travelling at the rate of 60 miles an hour, we do not mean that it has travelled 60 miles in the past hour, or that it will travel 60 miles in the next; but that, if its rate were to remain the same for an hour, it would travel 60 miles in that time, the unit of time in this case being one hour. Similarly, the intensity of mortality at any instant among a large number of persons, is not measured by the number who actually die in the following year (the unit of time in the case of a mortality table being one year), but by the number who would die in that year if all the conditions remained unchanged throughout the year; that is to say, it is measured by the number who would die if they did not deteriorate in any way with their increasing age or otherwise, and if the number of persons under observation remained unaltered, the places of those who died being constantly filled by fresh lives subject to the same intensity of mortality.

From the published list of the papers read to the late Actuarial Society, it would appear that, notwithstanding its interest and importance, little or no attention was paid to the subject of reversions and other allied securities. This is probably accounted for by the fact that numerous able papers have been read from time to time to the Institute of Actuaries, and the field may have been considered as pretty well exhausted. The Institute papers, however, for the most part confine themselves to particular branches of the subject, many of them almost entirely to questions of formula ; and to a person beginning the subject, the variety and detail of such papers must be perplexing, not to say alarming. Especially will this be the case if the student is seeking information as to the actual business handling of reversionary transactions, for the theoretical is more frequently in evidence than the practical. It seemed, therefore, that it might not be useless to have a paper of the nature of an introduction to the practical working of reversions—a paper which might form a base for attacking the subject in its more complex developments—and the following notes have been written on these lines. I do not know whether anything of the nature of an apology is owing in respect of a paper which naturally embodies much that will appear commonplace to those familiar with the subject; but I take it that the aim of the Faculty is not only to stimulate original research, but also to encourage any work which is likely to be useful to junior members.

The distinguishing feature of the New Experience which differentiates it from all others, is the division of the observations into various classes of assurance; and when our President did me the honour of asking me to contribute a paper, I thought I could not do better than invite your attention to the premiums which, according to that experience, ought to be charged for these classes, as it is of the greatest importance to insurance companies generally that the risks which they undertake should be at adequate rates. Some of us have suspected, others have known, that the fate of mortality varied according to the class of assurance, and that one table of mortality applied to all was not satisfactory; but it is now possible for the first time to arrive at definite conclusions on the subject, and to determine in many cases what the difference between them really is.

In the Paper which I had the honour of submitting to you at the close of last session, “On Premiums deduced from the Mortality Experience of British Life Offices”, I expressed the hope that at a future time I might have an opportunity of inviting your consideration to the other branch of the subject, namely, policy values. This hope I am now able to realise.

I have based my calculations upon the “Grouped Select and Ultimate” rates of mortality obtained from the unadjusted observations in the manner explained in the paper referred to. The period during which selection was assumed to exist was, it will be remembered, ten years; and the observations at five consecutive ages at entry were grouped together, and the results assumed to relate to the central age of the group. A correction to first differences was made, as the average age of the group differed in most cases from the central age at entry, owing to the number of entrants varying at each age. The net premiums are given in that paper, and the annuity values from which they were deduced have been used in the calculation of the policy values. Any additional annuity values required were calculated in the same way as before, a correction to first differences being made for the age, when necessary.Notwithstanding that the unadjusted observations have been used, the results will be found to be very regular.

The Council of the Faculty of Actuaries lately proposed to reprint my paper “On Life Assurance Book-keeping”, read before the Actuarial Society of Edinburgh in 1886. In revising it for this purpose, I have introduced a set of specimens of the various books referred to, with relative entries, which will, I believe, make the paper easier to follow. Their introduction has made it necessary, in order to avoid repetition, to make considerable alterations in the text. I have also, without interfering with the general system described, made such improvements in the books and entries described, as have been suggested to me by the experience of the last sixteen years. Having regard to those alterations, it would not be satisfactory to describe this paper as a second edition of my paper of 1886, and I have therefore given it a new title. The specimen books and entries have been prepared by Mr. John Notman, F.F.A., to whom I am indebted for valuable assistance in the work of revision.

Every one who has had to deal with the valuation of life interests, reversions, and entails, or with investigations involving Marriage Rates, must have been impressed with the carefully compiled and very complete tables which have been furnished by Dr. Sprague, dealing with the combined probabilities of marriage and mortality. These relate entirely to the marriage and mortality probabilities of bachelors and widowers of the Peerage ; and, so far as I am aware, no similar tables have been published relating to females. I find that in the discussion on Dr. Sprague's paper of 1879 (J.I.A., xxi, 448), Mr. C. J. Bunyon stated that about fifteen or sixteen years previously he had formed a table precisely similar to Dr. Sprague's, showing the probabilities of death and marriage among 10,000 unmarried people, up to the age of 60; only his table was not a table of bachelors, but of spinsters.

Although the question of the valuation of Provident and Pension Funds has recently been so exhaustively treated by Mr. Manly in his two papers appearing in the Journal of the Institute of Actuaries (vol. xxxvi, p. 207, and vol. xxxvii, p. 193), I have been rash enough on the present occasion to undertake to contribute a few practical notes with reference to the subject, with the view of further elucidating, if possible, the effect of the assumption of varying rates of mortality before and after the pension age, of secessions, of different scales of salaries, and of different rates of interest.

Before bringing to your notice the short memorandum which I have prepared on the subject of mortality in the tropics, I feel that I owe the Faculty an apology for submitting to them so fragmentary a paper as the present.

I had hoped, by recapitulating the history of mortality statistics in India and some other tropical countries from the earliest date at which these became available, to have traced down to the present day the progress that has been made and the lives which have been saved in consequence of improved manner of living and enlarged knowledge and better treatment of disease; but other duties intervened which prevented my fulfilling my purpose, and my intended paper has contracted into these few notes exhibiting the experience of my own company.