In a paper which I read before the Institute of Actuaries in April 1887 (J.I.A., xxvi, 357) I gave some tables which seemed to me likely to be of use, in the case of certain life offices, for the purpose of forming estimates of the amounts of bonuses arising on ordinary whole-term with-profit policies. The offices contemplated were mutual societies, making their periodical valuations every five years by the HM Table, at 3 per-cent interest. Life offices differ so much in regard to rates of premium charged and rate and incidence of expenses, and also in regard to the rate of interest earned by their funds, that it appeared to me it would be desirable to construct tables in such a form that the bonuses, which would fit the circumstances of any particular office, might, without difficulty, be calculated from them, rather than to calculate bonuses for a hypothetical office. An illustration was given, it is true, but this was subsidiary to the main purpose of the paper, and was furnished with the object of showing how the tables might be applied.

It will be within your recollection that in the course of the discussion which followed the reading of Mr. Sunderland's paper in December last (see p. 257 ante), I stated that while endeavouring to obtain information with regard to American insurance business, in connection with, the subject of that paper, I found to my astonishment that a large amount of business was apparently being carried on in that country upon lines more or less resembling a fallacious principle which had long ago been exploded in this country.

I believe there was a general concensus of opinion that the discussion which followed the reading of my paper at the sessional meeting of the Institute on 17 December 1888 (J.I.A., xxvii, 362) as an eminently practical one, and that another evening might usefully be devoted to a further discussion of many of the questions which were then raised.

With the object, therefore, of starting the discussion again, I propose to examine, in a general way, some of the views which were then expressed; and to better carry out that object, I shall consider the subject under four heads. (1) Whether, in the formula , Px, should be the pure premium according to the mortality table and rate of interest used in the valuation, or some other value. (2) Whether d should be taken at the rate of interest used in the valuation, or at some other rate. (3) To what extent special refinements can be introduced, and how far they may be neglected in practice. (4) The superiority of ray tables for practical use over other methods.

The Norwegian General Widows' Fund (Den norske almindelige enkekasse) is a public institution, guaranteed by the Norwegian Government, and consisting of two sections. The older of these, which ceased to receive new members in 1848, contains (1) the members of the Norwegian Widows' Fund, established on 20 March 1814, and (2) the Norwegian members of the old Danish-Norwegian Widows' Fund, who were transferred to it in 1821 in accordance with the Treaty of Kiel.

The problem of estimating the lord's interest in a copyhold property which is subject to a fine on death and alienation, in other words, the enfranchisement value of such a property, has been a constant source of perplexity to Surveyors.

Its chief difficulty arises from the fact that the contingency of alienation has to be taken into account, in addition to the contingency of death, which is alone involved in ordinary actuarial problems.

I believe that the first publisht paper on the subject of premiums for insurance against issue was that of Mr. Archibald Day, which was read before the Institute on 31 January 1859 (J.I.A., viii, 127). In this paper he gave from the Census Statistics the numbers of bachelors and widowers living in 1851, and the numbers of marriages of bachelors and widowers in the same year; and from these he deduced the probabilities of a bachelor and of a widower of any age marrying in a year, Then by means of the last-mentioned probabilities, combined with the probabilities of life given by the English Table No. 1, he calculated the value, at 3 per-cent interest, of £1 payable at the end of the year in which a husband shall contract a second marriage. The formula he used for this purpose may be written npH(1−npw)ϕH+n−1 × vn, where H and W represent the respective ages of husband and wife, and ϕ is the probability of a widower marrying in a year. It may be remarked in passing, that this formula is only approximately correct, inasmuch as there is a contingency that is not provided for in it, namely, that the husband may be alive and his wife dead, but that he may already have married a second time.

The two following papers contain contributions to the statistics of insanity in Scotland, which have been compiled with care and accuracy. They may therefore prove of service to actuaries who may be called upon to deal with problems where the ordinary vital contingencies are complicated by the element of insanity.

Although, as I have said, observations upon assured lives (in my opinion) afford, upon the whole, by far the best examples of the typical characteristics of the normal law of mortality, yet it is evident that all such observations are nevertheless affected to a considerable extent, first, by the selection exercised by the office in regard to the admission of lives, and, secondly, by the selection exercised by the policyholders in the option which they exclusively enjoy of terminating the contract of assurance whenever their convenience or caprice may induce them to avail themselves of the privilege. While the form of selection last mentioned is probably much the more important of the two in its effect upon the aggregate mortality, yet, for obvious reasons, its effect is of minor importance considered as a “disturbance” of the normal law which expresses the relation connecting rates of mortality at successive ages with each other. The number of persons withdrawing from time to time at particular ages must necessarily be, more or less, proportional to the number existing at those ages among the entire body, but there can be no connection whatever between the number of persons admitted from time to time and the corresponding existing number. Hence, although the effect of withdrawal upon the aggregate rate of mortality is considerable, its operation upon the mortality table is throughout pretty uniform, and consequently it does not distort the table to anything like the same extent as the effect of selection in regard to admissions. For instance, at the earlier ages, say under 30, the lives recently admitted necessarily constitute nearly the whole of the lives upon which the observations at these ages are based; but this is evidently not the case at the middle and higher ages, the proportion of lives newly admitted being at this part of the table comparatively small.

Referring again to Art. 4 of Gompertz's treatise, let us denote by a0dx the actual probability of dying in the infinitely small time dx, at the initial age of the mortality table, and by axdx the abstract chance of death (in the time dx) at the end of x years; that is to say, the chance considered “independently of” (as Gompertz expresses it) or as “abstracted from” the deterioration resulting from increased age, then, according to Gompertz's law, the actual probability of dying, in the time dx, at the latter age will be axqxdx, in which expression q only is an arbitrary constant denoting the rate of deterioration. If, in axqxdx, we put x=0, it becomes a0dx, which coincides with the expression first assumed. Hence, Gompertz's law, which supposes that “the “vital force or recuperative power loses equal proportions in equal “times”, is expressed, in its general form, by the equation μxdx=axqxdx, or, which is the same thing, by μx=axqx.

As the marriage rate among widowers depends not only on their age, but also upon the length of time for which they have been widowers, it was necessary to form the combined marriage and mortality table for widowers on principles analogous to those adopted by Mr. Sprague in the construction of his Select Life Tables, where the rate of mortality among assured lives for each age at entry was, after the lapse of five or a less number of years, joined on to the HM(5) rate of mortality. The first step, therefore, was to form a combined marriage and mortality table for chronic widowers—that is to say, those among whom the marriage rate has reached its ultimate magnitude; in other words, to find (wl)x, the number of widowers alive and still unmarried at any age, and number marrying, (wm)x, and dying, (wd)x, in the following year.

I have long had an idea that the ordinary formula for the value of an annuity,

might be transformed into some such form as the following:

where the coefficients would represent the condensed effect of the mortality, and the powers of i would make the calculation easy and rapidly convergent. I have lately succeeded in making the proposed transformation.