The problems referred to under the above designation are those which involve the double survivorship contingency of A living after the death of B before C. The following solutions of these cases are quite general, and consequently applicable to any mortality table whatever.

I have taken the problems in the order in which they appear in the work of Mr. Milne, whose notation I have also followed. By means of an obvious relation which exists between the value of the compound survivorship assurance and the corresponding annuity, the whole of these cases are reduced to the single fundamental annuity value combined with the values of simple survivorship assurances. Consequently, if we should ever possess a complete table of the values of the simple survivorship assurance and also of the values of the annuity we shall be able to calculate the numerical value in any case with the greatest facility.

The mode of using Barrett's method has now something of farrago about it: a little discussion may be useful, were it only to awaken attention to the confusion arising from changes made without adherence to the recognised laws of perspicuity.

The farrago to which I have alluded arises out of a struggle between the annuity, as commonly understood, and what I shall continue to call the annuity due. According to the technical meaning of the word annuity, the payment begins to grow at the moment of the grant: thus, a yearly annuity for 12 years, deferred 20 years, begins action at the end of 20 years, and grows the first payment in the 21st year. I shall vary the phrase used in my “Essay on Probabilities” (Cab. Cycl.) by allowing the words “annuity due for x years” to signify an annuity of x payments, the first immediately. Thus an annuity for 12 years, now granted, will in a year become an annuity due, also for 12 years.

In the correspondence that has recently taken place respecting the origin of Gompertz's formula expressive of the law of mortality, it has been stated, much to my surprise, that the investigation first given by that gentleman in the Philosophical Transactions of the Royal Society for the year 1825 was defective and inaccurate. I do not here propose to reopen the controversy on the subject of the originality of discovery, or to enter upon a renewed discussion of certain other allegations that have been so efficiently disposed of by Professor De Morgan and Mr. Sprague; but, after the statement to which I have more particularly alluded, it is only right that I should avail myself of the present opportunity to communicate the circumstance that Mr. Gompertz kindly presented me with a copy of his valuable Memoir in January, 1839, in which the various misprints had, in the most legible manner, been previously corrected by himself, and that the processes so corrected are perfectly accurate in every respect. In order that any one may be enabled to judge of this for himself, I shall here give a faithful transcription of that portion of the Memoir which contains the mathematical investigation of the formula, as it will occupy but little space.

As stated in the introduction to the Tables, they relate to 10,255 persons, as under—

And they embrace the experience of the first thirty-four years of the Society—reckoning, as was found most suitable, the first year's experience as the experience of the persons entering in the first year for one year from their respective entries, and so on. In this way it will be found that the experience does not include any of the persons entering in 1860, and none of the deaths in 1860 of persons who had been assured for more than a year from the time of payment of their annual premium in 1859. These deaths, and the entries of 1860, fall, according to the principle adopted, to be included in the experience of the thirty-fifth year of the Society.

The Act for taking the Census of England required that the 31,000 Enumerators employed should copy into as many books the householders' schedules and other particulars collected by them in their several districts. These books were to be placed, with the schedules, in the hands of the 2,197 Registrars, who were to subject them to a strict examination, and make all necessary corrections. This being accomplished, the books and other documents were to be transferred before the 30th April to the custody of the 631 Superintendent Registrars, who were required to test the accuracy of their contents by a further process of revision.

In the month of January, 1859, I had the honour to read before the Institute of Actuaries a paper upon the determination of the rates of premium for assuring against issue. Since that period the subject has lost none of its interest; on the contrary, its importance is yearly increasing, and it bids fair to form an important branch of assurance business. It is true that, out of the numerous inquiries made at Assurance Offices respecting the rates of premium for these risks, not 20 per cent, become the subject of policies absolutely granted, still it is desirable that there should be some sort of consistency in the premiums quoted. It would appear that considerable differences of opinion exist among actuaries upon the subject; and I feel, therefore, that I owe no apology to the Institute for departing from our usual routine subjects of rates of interest and mortality, and introducing once more the question of issue policies and the statistics of marriage.

The object of this paper is to endeavour to show that the true method of estimating the liabilities of an Insurance Company should be based upon the principle of re-insurance by its own rates of premium, interest being reckoned at 3 per cent.

In submitting the following observations, I feel it necessary, in the first place, to apologise for recurring to a subject which has been so frequently and ably treated by the President of the Institute of Actuaries. But as my method of dealing with this important branch of our business differs in some respects from the plan laid down by that learned writer, I trust I may be allowed to record the present attempt to demonstrate what in my opinion are the true principles which should guide us in determining the liabilities of Life Insurance Companies.

The rule which was given by Thomas Simpson for finding the value of an annuity on three lives is as follows:—let A, B, and C be the three lives, and let K be a single life, the annuity on which is equal to an annuity on the joint lives of B and C; then the value of an annuity on the joint lives of A and K is the required annuity on the joint lives of A, B, and C. The life A should be the youngest of the three.

The system pursued by each Office in the distribution of surplus may be divided into three parts, namely—

1. The class of members amongst which surplus is distributed;

2. The actual distribution of it amongst that class; and

3. The interval which elapses between successive distributions.

It is impossible to obtain a comprehensive view of the subject without considering separately these three distinct parts; and, since it forms no part of the present purpose to estimate the relative advantages of different Offices, no attempt will be made to follow the numerous combinations or systems of distribution produced by differences in practice under each of the above heads.

When a mass of results are to be calculated and added up, it generally happens that a small fraction is of little consequence in each item, except as the little of which many make a mickle. The money calculator knows nothing of the principle that it is very hard, next to impossible, practically quite impossible, to get a mickle out of positives and negatives, when both are sure to occur in equal numbers in the long run, and with equal quantities. Generally speaking, the items of money calculations are determinable, and accuracy to the penny below the true result is a matter of contract. But in life contingencies, in which a certain amount of inaccuracy is known to lurk in the fundamental tables, no real truth is gained by tacking shillings, pence, and farthings, on to hundreds of pounds. If the nearest pound be always taken, granting that the calculated result shall be thereby a little altered, no man alive can say whether that alteration shall be approach to, or recession from, what we should call absolute truth, if our tables were perfect: it is as likely to be one as the other.

I Hope, from time to time, to be able to give some further tables bearing on the subject, and to state how far I think subsequent experience and new information confirm or modify the interim conclusions which I ventured to submit in the present state of our knowledge; and I shall be very glad that my recent observations may receive attention, and correction where required, from critics abler than their author.

It is to a Mr. George Barrett, of whom nothing besides is publicly known, that we are indebted for the principle of the Commutation Tables, and for the method of computing, by means of them, the values of benefits depending on the contingencies of human life. The method was first introduced to public notice, after it had been refused a place in the Transactions of the Royal Society, by Mr. Baily, in an Appendix to the second edition of his Doctrine of Life Annuities, published in 1813. Mr. Griffith Davies, in a work on life contingencies, published in 1825, by certain additions to the tables, and alterations in their structure, according to Professor De Morgan, “increased the utility and extended the power of the method to an extent of which the inventor had not the least idea.” Mr. Barrett's method was also briefly noticed in the Appendix to Mr. Babbage's Treatise on Life Assurance. The method, as improved by Mr. Davies, has since been treated, and a very large collection of tables adapted to it, for both one and two lives, has been given, by Mr. Jones, in his work on annuities, in the Library of Useful Knowledge. But by far the most valuable papers on the subject are two in the Companion to the Almanack, for 1840 and 1842, by Professor De Morgan, which contain the materials of many thousand formulae, applicable to almost every case that can occur. There is also some notice of the method in the article “Reversions,” in the Penny Cyclopædia, which article likewise is the production, we believe, of Professor De Morgan.

Of the many valuable contributions by which the transactions of this Institute have been enriched by the labours of Mr. Samuel Brown, not the least interesting is the paper on the investments of Assurance Companies, read before the Institute on the 25th January, 1858. I feel much diffidence in again reverting to this question; but as Mr. Brown, at the outset of his paper, admits that the subject is too extensive to be exhausted by a single discussion, it has occurred to me that an attempt to arrive at some general principles on which the funds of Life Assurance Societies should be invested, and to consider how such principles can best be reduced to practice, might, perhaps, be the means of exciting some useful discussion.

In the third of my papers entitled “Contributions to the History of Insurance and of the Theory of Life Contingencies” after treating upon the earliest examples which, after a somewhat tedious search, I could find of tontine associations practically carried out—examples confined, as it will be recollected, to some comparatively small projects of the Dutch municipalities of Kampen, Middelburg, and Groningen, employed for raising capital sums, by the grant of tontine life annuities, in 1670 and 1671—the circumstance was mentioned, that the credit of the invention was properly accorded to the Italians, or more particularly to the Italian Tonti, whose name has been so effectually incorporated into several languages through the word “tontine.”

The premium paid on a policy of assurance is divisible into two parts—the first, or net premium, representing the annual contribution absolutely necessary, on a calculation of averages, to provide a stipulated sum on death; and the second being an arranged contribution towards the expenses of the Office.

Policies of life assurance are of many kinds—subject to the payment of premiums in various ways, and to risks of different descriptions.

A policy has several values, according to the purposes to which it is applied, the necessities of the holder, and the intention of the purchaser.

The following pages are intended as a guide to those who, not having made the nature and uses of a policy a subject of study, may yet have occasion to form an estimate of its value for some special purpose.

In the Assurance Magazine (Oct., 1860, and July, 1861) there have appeared two papers written by me to show in what respect the law of human mortality described by Mr. Gompertz in 1825 differs from the law described by me in 1832. In the same papers I have also shown in what respect the formula of Mr. Gompertz, , for the number living in terms of the mortality according to age, differs from my formula for the same purpose, which is , on putting instead of , as assumed by Mr. Gompertz.