Gentlemen,—Permit me, in the first place, to offer you my most sincere thanks for the honour you have conferred upon me in electing me President of your Society for this Session. It is an honour which I highly value, although I am fully sensible of my inability to discharge, as I should wish to do, the duties which it involves.

When considering as to a subject on which to address you, I was led to think that an examination of those portions of the Life Assurance Companies Act which relate to valuations might probably prove interesting to you, and I accordingly selected that as the basis of my paper for to-night.

It is to be noted that the aim of the Act is limited mainly to securing publication by the Companies of their accounts and valuation statements in a clear and intelligible manner and on a uniform system, and that there is an important difference between its principles and those embodied in the legislation of the United States of America, under which a test-valuation is prescribed, and practically a Government certificate of solvency is provided.

Recent investigations in the theory of life contingencies have thrown new light on the treatment of problems involving the probability of survival of several lives. For the exact solution of such problems by older methods, there are required complete tables of annuities (or other integral functions) on the number of lives to be dealt with. The time and labour necessary for the computation of tables of annuities on three or more joint lives, for every combination of ages, are so great as to have hitherto effectually prevented their construction; while tables of annuities on all combinations of quinquennial ages, like the joint-life tables given by Price and Milne, or the tables of annuities on three joint lives published by Filipowski in 1850, cannot generally be applied without a troublesome and uncertain process of interpolation. When, therefore, cases involving more than two lives have occurred, it has been usual in dealing with them to employ methods of approximation, leading to more or less error in the final result. Of late years, however, it has been found possible to express, with considerable exactness, the rate of mortality during the greater period of adult life, as a function of the age, by means of an hypothesis having at the same time important a priori considerations in its favour. Corresponding expressions have been deduced, representing the values of annuities on any number of joint lives, and relations have been established which enable us to determine the values of such annuities from tables of moderate extent, with considerable certainty and accuracy.

I Have endeavoured to draw together one or two of the popular arguments for ascertaining the sum which ought to be given now, in consideration of another sum, say of £1 or £100, to be paid at death; or, in the language which is peculiar to the Actuarial profession, to determine the Single Premium for the Assurance of £1.

The systems of Life Assurance calculation which were in practice when Price, Baily, and Milne brought forward their most valuable volumes, have in a great measure been superseded by a different but a more powerful arrangement, thanks to the labours of Barrett, Davies, and Jones; and here permit me to follow these illustrious names with that of David Chisholm. But while by this system a saving of time and mental exertion is afforded to the Actuaries of the present generation, I very much fear that a knowledge of first principles is less likely to be attained by the student under that cabalistic system, than if trained under the simpler system of Milne. The columnar or commutative system of calculation facilitates the expertness of the calculator at the expense, if I may use the expression, of his understanding; while the other system is much more apt to bring under his notice the exact nature of the problem, and the reasons involved in his formula for determining its solution.

Gentlemen,—It is usual for your President, at the opening of a session, to address to the younger members of the Society some words of advice and encouragement, indicating the direction their studies should take to qualify them for the important positions which in time they may be called upon to fill. This has been so ably done by some of my distinguished predecessors in this chair, and the number of new entrants and students now bears so much less a proportion to the number of older members of the Society than formerly, that I intend to depart in great measure from this time-honoured practice, and to devote the opportunity with which I am favoured, mainly to an endeavour to add to the information you possess on a subject that I am sure is of interest to you, viz., The comparative amount of reserves obtained by valuations on different approved Tables of Mortality, and by different rates of interest.

It is my intention to place before you some results of valuations made by the Actuaries of The Scottish Widows' Fund to determine these differences on the actual business of that Society to 1873, which I think I may say contains risks of all possible durations in combinations that cannot differ much from those in other prosperous Offices, and I believe that I shall be able to show, that if the combinations did differ considerably, the results on a business of long standing would not be material. With this latter object I shall, so far as is practicable, compare the results of these valuations with the results obtained by Messrs. Manly, King, and Valentine, on valuations of the risks in hypothetical Offices so ingeniously and laboriously compiled by the two first-named gentlemen.

Gentlemen,—I have to thank you for the honour you have conferred upon me in electing me to be your Honorary President—an honour which I appreciate all the more when there are so many gentlemen connected with the profession better qualified to adorn the post. Had I consulted your interests, I certainly should have declined the office, but the honour was too great to be refused, even although the acceptance sacrificed your advancement.

Having resolved to accept the position, the difficulty which presented itself to my mind was, on what subject to address you. At first I felt inclined to take as my topic the Education of an Actuary, but on reflection I found that it would be impossible to add anything to what had been said by my predecessors; and however satisfactory this would have, been to me, I felt it might not be so to you; so I at length concluded that, as a scientific paper would be out of place, I could not do better than give you a few practical notes relating to the actuarial profession.

The subject of War-Risk does not seem to have been discussed in actuarial circles since Mr. W. B. Hodge's papers on “The Mortality arising from Military Operations” appeared in the seventh volume of The Assurance Magazine. This admirable investigation, published in 1858, ends with the Crimean War; and considering the subsequent revolutionary changes in military science and tactics, a glance at the results of more recent campaigns may not be without interest.

The divergences of mortality characterising these campaigns preclude, as might be expected, the discovery of any general law of war mortality, and consequently the scientific calculation of extra premium to meet the corresponding risk. It seems as if, in the case of war, the Actuary must take a view of the whole surrounding circumstances, somewhat similar to that of a surveyor in fixing a premium for a fire insurance risk. Details of climate, of the season of the year, of the resources of the enemy, of the probable duration of the war, and of many other circumstances, alter the complexion of a military risk most materially, and must be carefully considered. Perhaps the most important question is the probable mortality from disease; a secondary consideration, the probable mortality from battle.

Gentlemen,—I thank you very heartily for the honour which you have conferred upon me by electing me your President for this session. It is an honour of which any one might be proud: and one for which, bestowed on me, who have so recently come amongst you, I have reason to be especially grateful.

In following the now time-honoured custom of delivering an Introductory Address to you on assuming this Chair, I had to consider how I could best fulfil the duty imposed npon me. I have not, as you are aware, as some of your former Presidents have done, devoted the whole, or even the greater part, of my life to actuarial study, and I was unwilling therefore to take, as the subject of these remarks, any matter of purely actuarial doubt and difficulty. But I have all my life been engaged in general business, and frequently been occupied with subjects which required some knowledge of finance. In these I have been ever met by the unknown quantities,—the value of money, and the future rate of interest. In everything relating to the success of Insurance Offices a knowledge of these values is likewise of paramount necessity, for, however accurately your facts as to mortality are stated, however skilfully your formulæ are framed, if a mistake is made as to the value of money or rate of interest, error will be the result.

1. Capital is wealth appropriated to reproductive employment, and, in the words of the late J. S. Mill, “the gross profit on capital may be distinguished into three parts, which are respectively the remuneration for risk, for trouble, and for the capital itself, and may be called insurance, wages of superintendence, and interest.”

It is with the last of these, Interest, that we have at present to do.

2. In civilized communities wealth is measured in money, and therefore in our investigations it is of money that we shall speak; but evidently it is immaterial, in laying down general propositions, whether the standard units are of one actual value or of another, so long as we remain consistent with ourselves, and use the same standard throughout. We shall therefore, as the most convenient course to suit all currencies, treat of units of value, without specifying to what currency these units belong.

Gentlemen,—I beg leave to thank you for the honour you have done me in again requesting me to act as your Honorary President. This was an honour which I accepted with much pleasure; for I am always glad to meet the younger men of my profession, to discuss with them points of principle or practice, and, as far as practicable, to assist them in their endeavours to obtain a better knowledge of the theory of our subject, or (it may be) to extend its boundaries by original investigations.

After much consideration as to the subject on which I should address you to-night, I decided not to take up some intricate or new point of theory, but rather to put together a few practical remarks suggested by my business experience. I do not take this course because I consider the theory of our subject is thoroughly workt out. On the contrary, I am of opinion that there are many important questions which still remain to be elucidated,—subjects which will well repay careful study, and which will reflect credit upon those who investigate them exhaustively.

One such subject is the very practical one of the proper allowance to be made for the surrender of a policy upon an under-average life. According to general practice, every such policy that has been in force for not less than (say) three years, is considered as entitled to a surrender value.

When this Society elected its first Honorary President, and in so doing paid a fitting compliment to one who had taken the leading part in its formation, his reply to the request that he would undertake the office was, “I have never in my life received any mark of approbation and esteem which has given me greater pleasure.”

I do not know that any words of mine could more fittingly express the feelings with which I received your invitation to take this Chair; but, if I were to add anything to the expression, it would be to say how sensible I am of the honour of holding a position which has been filled by so many eminent members of our profession.

If during the considerable period of my connection with the Society I have been able to show in any useful way my warm interest in its welfare, I am more than repaid by finding that I have gained in some measure your confidence and esteem.

In considering how best to discharge the duty which a time-honoured and excellent custom has associated with the office of President, I was led to think that your time might be profitably occupied in taking a glance at our past history as a Society, and at our present position, and trying to gather from the survey some useful lessons for the future.

Gentlemen,—I thank you for again electing me your President, —it is an honour which I highly esteem. You may rest assured that I will, to the utmost of my power, endeavour to perform the duties of the chair, I trust, to your satisfaction.

It is customary on the opening of a session for your President to say a few words by way of “Address” on some subject, of his own selection, which has more or less relation to the general business of the Society. As I had last winter the pleasure of reading a paper which occupied a good deal of my time, I have chosen a subject for which the materials are almost all at hand, but which when strung together may, in their continuity, possess an interest which will not be unwelcome to the members of an Actuarial Society. I propose to consider one or two incidents in the History and Statistics of Life Assurance.

The History of Life Assurance is exceedingly interesting. Bestowing all our attention upon the engagements of the passing hour, we are very apt to forget the arduous up-hill struggles of the many schemes of the past two centuries. If an account should ever be written, as I believe it will, of the career of those futile efforts, it should teach us to be very careful, and watchful of our position. There is in our time, it sometimes appears to me, rather an excess of that spirit of competition which I fear tends to weaken, and not to strengthen. Life Assurance is not entirely one of the exact sciences. The ordinary trader may easily, by applying one or two principles of accounting, exhibit the solvency or insolvency of his business; but it is almost impossible, except in the mass, to fathom the Liabilities of the business of Life Assurance. An ordinary debt is owing now, or a bill will certainly mature on a given date, but our debts accrue in the future, and are not individually dated. In the maturing of a multitude of them we have the utmost confidence.

The Theory of Logarithms was first made known to the world by John Nepair, Baron of Marchiston (now Merchiston), in a volume published at Edinburgh in 1614, under the title, “Mirifici Logarithhmorum Canonis descriptio Ejusque usûs in utrâque Trigonometriâ, et etiam in omni Logisticâ Mathematicâ, Amplissimi, Facillimi, et expeditissimi, explicatio.” This work was translated into English by Edward Wright, and published at London in 1618, with the table reduced by one place.

In presenting to the Members of the Actuarial Society of Edinburgh the results of my investigation into so practical a matter as that described in the title of this paper, I feel that whatever may be their opinion with regard to the manner in which I have proceeded, and the methods which I have employed in my researches, I am at least absolved from the necessity of apologising for the nature of my subject. A glance at the syllabus of the Society for the current Session is sufficient to establish the fact that scientific inquiries upon theoretical points, and the discussion of matters of practical importance, are alike welcomed as acceptable contributions to its Transactions. But, if I may be permitted to anticipate in a slight measure the course of this investigation, as set forth in the following pages, I would take the opportunity of expressing the hope that my paper may be regarded as possessing some slight interest from the theoretical as well as the practical point of view. It will be my aim to afford such statistics of the Annuity business transacted in this country as the valuable returns to the Board of Trade—an endless source of statistical information upon every point connected with the operations of our Life Assurance Companies—will supply, more particularly with respect to the valuation of such contracts; to construct from data thus obtained a common base, which will enable the relative reserves according to certain mortality tables to be measured—or, as it is technically termed, a “Model Office;” and, finally, to employ this instrument in endeavouring to show the financial results yielded by the application of each of the tables in question.