The following problem, which has actually occurred in practice, presents several points of interest. It is hoped, therefore, that it may be found worthy of the attention of the Members of the Institute :–

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%

The benefitto be paid for consists of two parts—first, the annuity, and secondly, the return of premium. The values of these are to be found separately.

Notwithstanding the very large amount of leasehold property, which in the course of every year is bought and sold in this country, and notwithstanding the extensive transactions—of almost hourly occurrence—in the public market, in Government and other temporary Annuities, the subject of the rate of interest which any given purchase will yield the buyer is very imperfectly understood, even by those most deeply interested in the inquiry, unless they happen to be at the same time well versed in actuarial computations.

It is not unfrequently imagined by a buyer, that if he purchase a leasehold property or a temporary Annuity at a price corresponding with the value of an Annuity at a given rate of interest, (say 5 per cent.,) that he has made a purchase which will pay him 5 per cent., or which, in other words, will enable him to spend 5 per cent, per annum on his outlay, and at the same time replace his capital undiminished at the expiration of the term.

In the year 1823 a plan was projected, by the late Dr. George Pinckard, for establishing an Assurance Office which should undertake, as its leading feature, to grant Policies on Lives deviating so much from the common standard of health as to render them wholly inadmissible at any Office then existing. In pursuance of this plan, the Clerical, Medical, and General Life Assurance Office was established early in 1824. In the absence of any trustworthy data which might serve as a guide, the projector considered that the object in view could be accomplished, with safety, only by the aid of more extended and diversified medical knowledge than was obtainable by Offices dependent on the unassisted judgment of one or two Medical Examiners. In forming the Board of Directors, therefore, he expressly provided, that of the 17 gentlemen of whom it was composed, 8 at least should be members of the Medical Profession.

In any Company or Society for the assurance of lives—let it be constituted in whatever manner it may—it is evident that certain rates of mortality and interest are always prevailing; and that such rates are more or less constant throughout any given period, as the causes on which they depend are more or less stationary.

Now, the annual amount required by any such Company to meet its engagements, will entirely depend upon the particular rates which happen to obtain in it; and hence it becomes a question of the utmost importance to such bodies, to determine as nearly as possible how they are situated in this respect.

To trace the succession of events which have given rise to the duplex system of assurance in this country, known as the Participating, and Non-participating, would not be altogether uninteresting or unprofitable. There is little doubt, that the immoderate premiums required at the outset by the few who were disposed to embark in the then novel speculation of life assurance, suggested the idea of mutual association, as a means of achieving the same objects on more advantageous terms; and thus the participating system owed, to a certain extent, its origin to the non-participating. But a few years' experience soon sufficed to demonstrate to the keen eye of competition, that the payments required under the new arrangements were unnecessarily high; and that, after a large reduction in them, they might be acted upon, not only with safety, but with a considerable profit to boot.

The mode in which the Property and Income Tax of this country is levied, has created so much dissatisfaction, and is so universally acknowledged to be inequitable, that I am induced to call the attention of the Members of this Institute to the subject, with a view to arrive at some elucidation of the principles upon which direct taxation should be based: being satisfied that there is no tribunal before which the consideration of such a matter could be more properly brought, or where its peculiar features could be more thoroughly and efficiently investigated.

I propose to bring under your consideration this evening, the investigation of a question which has been a good deal discussed from time to time; and which, I observe, is attracting some notice at the present day. It is one of some interest and importance to persons engaged in our pursuits, and I therefore trust that the endeavour to throw a little more light upon it will not be altogether unacceptable.

The question I allude to has reference to the mode of valuing the risks of a Life Assurance company. The actuaries, with some exceptions, of what I may, perhaps, without impropriety call the old school, contending that the same elements should be used in valuing as those upon which the Table of Premiums is constructed. So that, for instance, an office charging premiums deduced from the Northampton Table, should, according to their view, make its valuations in accordance with the data from which those premiums are deduced; whilst some of the more modern practitioners advocate the using, in these valuations, tables based upon what may be considered—as far as our present knowledge goes—a true rate of mortality, and a true rate of interest.

In a Paper read a short time ago before the Institute, I endeavoured to show the inexpediency of using, for the purposes of valuation, Tables constructed on the principle which characterises the Northampton Table. I did not intend to pursue this particular branch of the subject further ; but, finding that more difference of opinion exists in reference to it than I was aware of, and considering that the practice of making valuations by the Northampton Table, and by Tables framed upon similar principles, is still carried on to a great extent, I am led to offer a few more observations upon the matter previously to entering upon the consideration of some other questions, the correct solution of which materially depends upon a right understanding of this one.

I cannot better introduce the subject of this paper than by a quotation from Mr. Morgan's Preface to his Tables of the experience of the Equitable Society. He writes as follows, viz.:—

“In a body of lives of the same age, all selected as healthy from the general mass of mankind, it is obvious that the rate of mortality must be considerably less for the first ten or twenty years after selection, than amongst those from whom they are thus chosen. As, however, these selected lives advance in age, their general health and the rate of mortality amongst them will naturally approximate to the common standard.

That the mean duration of life amongst the indigenous inhabitants of one part of the globe differs more or less from that which obtains amongst those of other parts, there is every reason to believe. It is probable that a difference exists even when comparing the most civilized nations of Europe one with another; and still more may we look for discrepancies when we carry our investigations amongst the natives of more remote and less civilized lands.

The climate, extent of cultivation, modes of life, state of public morals, each and all of them have a tendency to affect the longevity of the human race, and will not fail to make their influence felt in proportion as their good or evil energies are called into existence.

A good deal of misconception prevails, on the part of many interested in the science of life contingencies, in regard to the expressions usually assigned as the measures of the probabilities of the different orders of survivorship among two or more lives. It seems to be supposed that these expressions are little better than rude approximations, adopted from convenience or necessity, depending on no definite principles, and consequently giving results but little entitled to confidence. It appears to me that these notions are attributable, in no small degree, to the manner in which the expressions in question have been spoken of by the writers, Messrs. Morgan and Baily, by whom they were first deduced.

To discover in the doctrine of life contingencies, convenient, and at the same time nearly true approximations, to be employed in determining the values of Survivorships, instead of the long and complex formulæ at present required for their solution, is an inquiry of no small importance to the practical actuary, more especially when it is remembered, that, even if the exact formulæ be employed, the results will in the greater number of cases be themselves mere approximations, in consequence of the defective nature of some of the elementary values employed in the solution

In a Postscript to the introductory matter of the Assurance and Annuity Tables, compiled and published by Messrs. Smith, Orchard, and myself in April, 1851 (and which Postscript was written by me), it is stated that I had “devised a new table, which, while answering all, and more than all, the purposes of Tables V. and VI. [which show the single and annual premiums for a survivorship assurance of £1 for every combination of two ages], by means of a single value for each pair of ages, and thus occupying little more space than one of them, would also be very much less laborious in its construction than the former of those tables. The new tables would, in fact, consist of series of columns supplementary to Mr. Jones's Commutation Tables for Two Lives, and possessing all the properties, mutatis mutandis, with respect to the formation of temporary and deferred, as well as present, whole life benefits and payments, that give to tables of the form of those mentioned their great value.”

My object in the following paper is to give, in a few leading problems, a general and somewhat elementary view of the Theory of Probabilities, exhibiting the immediate dependence of the doctrine of Life Contingencies on the laws of probability and combination, and showing how apparently complicated problems, involving two and three lives, can be deduced directly from their very simplest and most elementary forms of expression.

From the nature of the subject-matter proposed to be treated, it will be at once perceived, that this paper is intended rather for the instruction of the student in this branch of mathematical analysis, than for the use of the proficient; and I claim for it no higher distinction.

During the last session of the Institute of Actuaries, I had the honour to read a paper on the Value of the Principle of Selection as exercised by the Offices in granting Assurances upon Lives, and to submit some calculations which I had made with reference to that subject, in which calculations the lives admitted at particular ages were kept apart from others during their subsequent existence. I have now to offer to the Institute an extended set of Tables of a similar character; and, after stating the course which I have followed in constructing them, I shall make a few observations on the value of selection as exercised by the policy-holder against the Company.

The present paper was undertaken on happening to peruse very lately Messrs. Gray, Smith, and Orchard's laborious and useful work, entitled Assurance and Annuity Tables, according to the Carlisle rate of Mortality, at 3 per Cent. In the introductory section of that work, when referring to the larger Tables V. and VI., containing the single and annual premiums for survivorship assurances, it is there very justly remarked, “Considering the frequency with which occasion arises for the functions here tabulated, and the tedious nature of the operations requisite when their values have to be formed from other tables, it may, at first sight, seem.

In laying before the members of the Institute of Actuaries this hasty sketch of the Progress of the Assurance of Life and Property on the Continent, I beg to offer a few words of explanation on what induced me to attempt it.

Understanding that several actuaries and managers of Foreign Assurance Companies were likely to be in England at this time, without any general acquaintance with, or means of introduction to, those who are engaged in similar pursuits and studies in this country, the council of the Institute were led to propose a general meeting and conference, and to offer to them the hospitality and attentions which all classes and professions are vying with each other in paying to our distinguished foreign visitors.

The discovery of some general Law of Mortality, by means of which the values of the contingencies, depending upon human life, might be expressed in some simple formulæ, without the necessity of performing the numerous calculations which are required even for the chances of a single life, and still more in the various cases, in which two or more lives are involved, has occupied the attention and labours of many scientific men. But the result has not hitherto been satisfactory, in spite of the skill with which their investigations have been conducted, and the unwearied patience which they have brought to the task. The failure has probably arisen from the want of a sufficient number of observations, the authenticity of which could be relied upon; or for want of that accuracy in the data, without which it is in vain to attempt to find the traces of a regular law, disturbed as it must be by the peculiar circumstances under which the observations were made, or by the ignorance, and, in some cases it may be, by the wilful errors of those who have been employed to collect the facts.