The study of statistics is so peculiarly connected with the vocation and intellectual pursuits of the actuary, that I feel convinced there is no member of the Institute who will not be interested in a full report of the very important meeting recently held at Brussels. I apprehend that the real business of the actuary is the reasoning on all events to which the mathematics of probability can be applied, and reducing the conclusions to a form in which they can be practically used for the public benefit. His study is the doctrine of averages; and though his functions have been hitherto confined in a great measure to subjects relating to the assurance of human life, there are evidently many other topics to which, in time, his attention ought to be directed. The discovery of the mathematical laws of events will eventually be recognized as the sole means of bringing uncertainty to certainty, and out of irregularity deducing order.

In looking over some papers of the late Francis Baily, at the request of his sister, for the purpose of advising on their disposal, I found in one packet a number of letters addressed to him by the late George Barrett, the inventor (in this country) of the method for calculating life contingencies which now goes by his name. I also found the manuscripts of Baily’s three works—on leases, on interest, and on life assurances: these are now in their proper place, the library of the Institute of Actuaries.

From Baily’s methodical habit of arranging his papers, I feel confident that the letters alluded to are all which he received from Barrett: unless, indeed, some passed which he preferred not to preserve. The interest with which actuaries look upon the method published in these letters makes me think it desirable to give some account of them in a work which all actuaries consult.

In most branches of mathematics, the actual use of fundamental processes in the form which the first definitions suggest, is often supplanted, either by processes of greater skill, or by the use of pure reasoning. In the subject of which this paper treats, there has not been much attempt to connect formulæ by reasoning. The actual exhibition of successive annual results has been the only method extensively employed; and it throws the required total result into a series of terms: this series is either algebraically summed, or calculated term by term for insertion in a table.

The present paper is intended to show that this summation of algebraical series may be dispensed with, at least in questions of annuities certain : and also that common points of principle, which the ordinary methods leave altogether out of sight, will reduce many questions of life annuities to an absolute coincidence of form with the corresponding questions of ordinary annuities.

I. While so much improvement has recently taken place in the arrangement and construction of various tables for facilitating calculations founded on existing data, very little has been done in the way of investigating and correcting the data themselves; and it is feared that the question of the rate of mortality among select lives is still involved in the greatest doubt and obscurity.

II. It is not proposed in the present paper to go farther than to show that the rate of mortality, during the first year of selection, of select assured lives is so materially different from what it has hitherto been represented, as to lead to the inference that the data from which the erroneous deduction has been made cannot be true data for the ascertainment of the value of selection. To investigate the rate of mortality of select lives at separate ages, I conceive to be of the utmost importance for the elucidation of truth, and the proper direction of sanatory inquiries; but I do not consider that sufficient data at present exist for the determination of this, and these can only be obtained by a united inquiry. I shall be very happy if the present observations have any effect in hastening such an investigation, which sooner or later must be entered upon.

The question is frequently asked, in what respect competition in life assurance business differs from that in any other trade or profession. It is contended, that in trade the effect of competition is to produce a better article at a lower price—to awaken new tastes and desires in the public, and so to create a market, which, notwithstanding the diminution in the price, may afford larger profits to the merchant, by the greater quantity of the commodity which he can sell or the greater frequency with which his capital may be turned. But there is nothing analogous to this in life assurance, for in truth it is nothing more than a redistribution of the capital of the public—a deduction of the portion which one man may save out of his profits in the course of a long life, for the benefit of the family of another who does not live to save it.

In the paper I shall have the honour of reading to you this evening, upon some points connected with the education of an actuary, I shall endeavour to describe some of the more important duties and responsibilities of that office—to offer some observations to the student, as to his preparation for the profession of an actuary—and to suggest to the Council of the Institute some reasons for enlarging the sphere of the examinations that they may think proper to prescribe.

I shall not detain you with any account of the ordinary duties of an actuary, which are of course well understood in this room, but shall at once proceed to the consideration of the foundation of all actuarial knowledge—mathematics.

In discussing the subject of assurance valuations and the most approved methods of making them, I have often had occasion to draw attention to the great importance of ascertaining with the utmost possible accuracy the rates of mortality and interest which have been found to prevail in any case under consideration, so as to determine what rates of premium are really required to provide, independently of other exigencies, for the sums assured, and how far the particular rates charged are sufficient or more than sufficient for that purpose. I have insisted the more strenuously on the necessity of this proceeding, from the conviction that in almost every Association some peculiarity in the prevailing mortality will be found to exist, distinguishing it from that of its neighbour, and arising from the different character of such influences as the condition of life of the persons whose lives are assured, the mode of selection, the greater or less predominance of the male sex and of assurances made by persons on their own lives over such as are effected on the lives of others, the constitution of the Company, and so on.

The recommendations made to and approved by the Select Committee of 1852 on Friendly Societies, with reference to the abolition or reduction of the duties payable in respect of policies for life assurance, have been acted upon more promptly than could well have been hoped for. The duties, it is true, have not been partly abolished in the way suggested, but the reduction now effected is so considerable as to make them fall very lightly upon those who will hereafter have to pay them.

The Report of the Select Committee of the House of Commons on the subject of “Decimal Coinage,” of 1st August 1853, sets entirely at rest any doubt or question as to the great advantages and facilities which would be afforded by the adoption of a system of decimal numeration and decimal coinage.

That the change will accordingly be made, I feel confident; and as the basis of the new arrangement, as well as the method of carrying it out, are of vast importance to the public at large, and in business generally, I have considered it a fit subject of deliberation for this Institute. It may be said that we should have taken an earlier and more prominent part in originating and promoting a change of system, of the importance of which we had individually, I may safely assume, been long convinced; but I am inclined to think that we have wisely reserved our opinions, and that they will be more valuable in the present stage of the discussion (now that the Report of the Select Committee, and the evidence taken before them, has been published), than they would have been earlier in the day.

In order perfectly to obtain its object, a table of mortality must exhibit the actual mortality at the different ages; and to be of any practical use, we should be able from the results of the past to predict events for the future.

The actual mortality can only be directly determined by knowing the population, as well as the number of deaths, which occur at each age. There are, however, very few countries in Europe in which these two elements are sufficiently ascertained. Even in Belgium, the division of the population by ages was but imperfectly known till the census of 1846, the results of which were only published about the end of 1848. It was therefore necessary, in order to form the tables of mortality, to pass over the important element of the existing population, and frame them from the registers of deaths. In this manner, the tables of mortality which I have given successively since 1827 have been constructed on the hypothesis of a stationary population.

Although this document has already appeared in several publications devoted to the discussion of life assurance matters, we think our readers will consider, with us, that the importance of it justifies its republication in the pages of this Journal.

To the Institute of Actuaries it cannot but be gratifying to observe that the recommendations made by the Committee tally as nearly as possible with the resolutions, so far as they go, adopted at the Special General Meetings held on the 12th and 19th April, in the present year. Thus the first resolution was to the effect that the Act of 1844 has created an invidious distinction between the Offices established prior to 1844 and those established since, and that the Act in question ought to be forthwith repealed, and provision made for putting all existing Offices on an equal footing.