It is not my intention in this paper to travel over the same ground as that followed by Mr. Sprague in his paper in the 7th vol. of the Journal, p. 61, “On Certain Methods of Dividing the Surplus among the Assured in a Life Assurance Company; and on the Rates of Premium that should be Charged to Render them Equitable”, and that by Mr. Tucker, in the 9th vol., p. 245, “On the Rates of Premiums Required to Provide certain Periodical Returns to the Assured”, although the changes which have been introduced into the methods described by both writers, and the increased knowledge we possess of life assurance mortality statistics, would fully justify a revision and extension of their labours. Such a work, if undertaken, would fully repay the time spent upon it, and would prove a valuable addition to the information we already possess upon a very important subject.

Both the above-mentioned gentlemen enlarged upon what “H. A. S.”, in the 8th vol. of the Journal, p. 167, describes as “The incongruity existing between the rates of premium charged at certain ages and the benefits accruing therefrom”; and they have shown that when the net cost of the assurances and bonuses is calculated upon a certain basis, the figures differ, often very materially, from those of the premiums charged.

The records of the Colony of Victoria afford a field for scientific enquiry in many directions, and questions of interest to the physician, the economist, or the actuary, are to be found therein. Its separate existence is but of thirty years' duration, as it was on 1 July 1851 that the Queen's proclamation was issued erecting it into a separate colony, to be called after her Majesty's name.

Many social and economic questions concerning the colony have already been considered; and no longer ago than 1879, the present Government Statist, Mr. Henry Heylyn Hayter, read an exhaustive paper before the Statistical Society. The object of the present enquiry, however, has, so far as I am aware, not received previous attention; and as it is one of strictly professional and scientific interest, it is hoped that it may be appropriately considered by the members of this Institute.

That object is to deduce from the registered vital statistics of the colony a mortality table exhibiting the actual rate of mortality which has been experienced ; and to compare that table with other standard tables, in order to note the effect of the different conditions of life prevalent in Victoria. For it is to be expected that a colony, whose geographical extent is only slightly inferior to that of Great Britain, whose difference of temperature between summer and winter varies less than that of Lisbon or the favoured Riviera, and whose population is far greater than any other Australasian colony, would possess special characteristics calculated to mark their effect on the rate of mortality of the people.

The object of the present paper is simply to bring to the notice of the members of the Institute the results of an investigation into the mortality prevailing amongst the different classes of bonus policyholders in the above-named Company, and to point out what appears to be the practical bearing of these results. There are also one or two points in connection with the collecting of the experience which may be worth mentioning.

In the company in question (in common, I believe, with many societies) an option is given to the policyholders, on the occasion of the first division of profits in which they are to participate, of receiving the value of their bonus in one of three ways:—

I. Cash.

II. Reversionary Additions.

III. Reduction of future Premiums (either for life or for three years).

The selection has usually been understood to be made once for all, but, as a matter of fact, the option has been occasionally allowed at subsequent distributions.

There appeared to be some reason for anticipating a difference in the relative mortality of these three classes of policyholders in consequence of the exercise of this option. This seemed at least a point worth testing by an appeal to actual experience; and it was mainly with the view of instituting such a test that the investigation referred to was commenced.

We will now state, with the necessary technical detail, our reasons for the opinion we have already expressed, that it is not permissible to add to the assets, or otherwise bring into the valuation as a credit, the present value of the extra charge made upon half-yearly and quarterly premiums in excess of the aliquot part of an annual premium. This extra charge is made for three different purposes, (1) to cover the cost and trouble of collecting premiums at frequent intervals; (2) to make good the loss of interest on the portion of the premium which is not paid at the beginning of the year; and (3) when the half-yearly or quarterly payments are really premiums, and not instalments of yearly premiums, to compensate for the loss of premium which will occur in some cases through the life assured dying before the full year's premium for the current policy-year has been paid. The proper amount of the extra to be charged for (2) and (3) is easily determined. Since the value of an annuity payable in advance is 1 + a, ¾ + a, or ⅝ + a, according as it is payable yearly, half-yearly, or quarterly, the total yearly net premium for the age x is

according as it is payable yearly, half-yearly, or quarterly.

Mr. John Finlaison's method of graduation (Journal, vol. xxi, page 50) deals with an irregularity, which, for convenience, may be assumed to be unity, in the following manner:—It diminishes the irregularity to a small fraction on its first appearance at one end of the formula; conducts it along the formula by easy stages, as over a double inclined plane; and dismisses it gently at the other end. The method has the advantage of simplicity in working and check, but it is correct to first differences only.

Mr. Woolhouse's method is correct to third differences, and the irregularity glides over a curve; but the application of the formula requires sustained attention.

The object of this paper is to combine the facility of the one method with the smoothness and correctness of the other.

Let S be the sum of nine numbers, u 0 to u 8, increasing by third differences,

the central term,

whence .

I think the time has arrived when the subject indicated in the title of this paper may be fairly and fully considered. It is certainly one which must frequently have presented itself to the managerial mind; and there can be no reason why this question should not be discussed with as much philosophic calmness as any of the many theoretical problems, or points in practice, which continually present themselves for reflection, and perchance for decision.

The point may indeed arise—whether I am the proper person to introduce the topic. I take the individual responsibility of deciding in the affirmative. I have, on the one hand, been as frequently assailed by the insurance press, as any one, and, on the other, received as much kindness and friendly recognition as any man can desire, and more than I claim to deserve. It may be that in either case the extreme has been reached, or passed. I have the advantage of having been a writer upon the press, insurance and general, from the days of my youth, and I say at once that my sympathies are largely on that side. But I think that the familiarity which draws me to the side of its virtues, also renders me, at least in some degree, cognizant of its short-comings. I have the further advantage of having been on various occasions consulted by managers on the one hand, and by editors on the other, upon the points which I now proceed to discuss.

A considerable number of investigations on mortality in times past has already been published, and without scarcely any exception they all go to show the same thing, that the mortality now-a-days is much smaller than it used to be. Some of these results are, however, only found by a sort of induction, in considering the conditions of life and the state of medical science in past centuries, and most of the statistical facts do not go farther back than to 1750; for more remote periods, the statistical information which has been gathered is scanty, untrustworthy, and dispersed.

I therefore undertook the task of investigating a distinct class of society during as long a period as possible, hoping thus to be able to confirm or correct our notions on the variations of mortality. The materials were taken from an elaborate biographical work by the Rev. S. V. Wiberg, published 1870—71, with a supplement from 1879.

I confess that the general impression of the results was that of a disappointment. In fact, the materials appeared so incorrect, that distinct numerical conclusions were not admissible. Still, I have thought it of interest to publish the chief results, hoping that, by-and-by, more correct materials may be at hand, and that the readers will consider the results here laid down as merely preliminary.

It is not to be expected, at this period of time, that anything new remains to be discovered in the Doctrine of Annuities, especially by one not conversant with analysis; yet a useful application may occasionally be found of the results of former investigations, the more so should such have remained in comparative neglect.

The announcement of the relation which subsists between an annuity payable yearly, and when it is payable half-yearly, quarterly, monthly, &c., the rate of interest continuing the same, is due to the illustrious De Moivre, and was made by him in the second edition of his treatise of Annuities on Lives, published in 1743. It was reproduced in the subsequent editions of the same work, and is to be found, with some errors of the press corrected, in that appended to the third edition of his Doctrine of Chances, 1756.

Although I have not strictly followed De Moivre, having computed my factors before I found they were described in his work, I here transcribe his rules with the demonstration of them, remarking that, the notation he employs being at variance with that to which we are now accustomed, and likely, therefore, to cause embarrassment to the student, I have substituted the modern notation throughout.

The problem of finding the rate of interest corresponding to a given annuity-value, has engaged considerable attention at various times; and many formulas have been proposed for obtaining approximate solutions by the aid of existing tables of annuities. Most of these will be found in Mr. Sutton's paper on the Rate of Interest yielded by Foreign Loans (J.I.A., xix, 77). Of the 5 formulas there given, (2) is probably the most generally useful, (3), (4), and especially (5) (Prof. De Morgan's formula), involving a good deal of calculation. The object of the present short paper is to suggest 2 new formulas, which give very accurate results, and at the same time require but little work in their practical application.

Let a be the value of an annuity for n years certain, at a rate of interest i, (which it is required to find), and let a1, a2, a3, be the tabulated annuity-values for the same term at the rates i1, i1 + h, i1 + 2h, where a lies between a1 and a2, and consequently i between i1 and i1 + h.

The rapid growth of Accident Assurance, the passing of the Employers' Liability Act, and the periodically recurring discussions respecting schemes for a National Accident Fund, have created a demand for statistics on the subject of accidents, which cannot be said to have been satisfied.

The Tables given in this paper are believed to be unique as regards this country. The object of their compilation was to show the Rates of Fatal Accidents in various occupations. They are based upon a comparison of the male population in each occupation in England and Wales, as enumerated at the Census of 1871, with the average of the number of the violent deaths occurring to male persons following that occupation in England and Wales during the three years, 1870, 1871 and 1872. For the extraction of the particulars relating to these deaths from the records of the General Register Office, special permission was obtained by Dr. Farr; and it is with his cordial approval that the facts are now laid before the members of the Institute.

The Census of 1871 was taken on the 3rd April of that year. The population enumerated was therefore slightly in excess of the mean population to which the average of the deaths for the years 1870, 1871 and 1872 would have reference. The difference would, however, be slight; and for the present purpose it was thought that the population as enumerated might be considered suitable.

The ordinary method of approximating to three-life annuities is as follows:—Let the lives be aged respectively x, y, z, (x being the youngest). Find an age w such that at the given rate of interest aw=ayz, then axyz=axw nearly. Baily diminishes the value thus found by .05, but I cannot say if this correction is usually made. It will generally be found to be too small if the lives are very young, but much too large at the older ages.

It has been shown that the above method (neglecting now the correction referred to) is strictly accurate in any mortality table where the mortality conforms to Mr. Gompertz's hypothesis, and where consequently the force of mortality can be represented throughout by the expression

(B and q being constants). The reason for this is, that the force of mortality for two joint lives, or, x, x + t, may be represented by a similar expression

(where qt′ = qt + 1).

It follows from this that if we have

we have also

and, the force of mortality in the joint status being equal to that for the single life, throughout the table, the two annuities are identical, not only as to their total value, but also as to the value of each successive payment.