In considering the origin and development of the friendly society system, it is necessary to recognize the twofold purpose that these institutions have fulfilled. In the view of the actuary they are merely associations for the mutual assurance of certain benefits, generally in the case of the sickness or death of any of the members. This, however, represents only one side, and that by far the most recent, of the work of these institutions. A not less prominent and much older feature is to be found in the opportunities for social intercourse, companionship, and mutual help, which such associations naturally afford—wants common to man in every state of society, and as old as civilization itself. Hence, from the earliest times, small communities have existed, drawn together by the common interests, tastes, or pursuits of the members, such communities being originally purely social in character, subsequently charitable and mutually protective, and only in recent times financial.

In proceeding to make a valuation of a class of policies effecting whole-life assurances, the first step is to collect and arrange the amounts of sums assured, inclusive of bonuses, and the corresponding annual premiums under the present ages of the lives assured: that is, under the ages brought up to the date of the proposed valuation. The whole of the data for the class of policies under consideration is thus condensed, and finally exhibited, in a list showing the several amounts for every separate year of present age. The subsequent valuation of these numbers by the proper factors, so as to obtain the respective present values and thence the totals of the present values, involves necessarily a certain amount of calculation, the accuracy of which it becomes very desirable to check; more especially if the figures are considerable and there are many of such computations to make.

Some months ago I had occasion to enquire of half-a-dozen offices what were their rates of single premium for a certain survivorship assurance. The highest rate quoted exceeded the lowest by 40 per-cent of the latter. This fact appears to me to afford a good illustration of the different estimates formed by actuaries of the value of the risk run under survivorship assurances. The discrepancy probably arises mainly from the use of different tables of mortality in calculating the risk premiums. It will be seen later on, that the differences in survivorship premiums as found from different mortality tables are in some instances very considerable. I am inclined to think, too, that some actuaries, through a feeling of uncertainty as to whether they have correctly estimated the risk premiums, load them very heavily in order to be on the safe side.

The Institute of Actuaries has always been ready to welcome contributions from members able to give information regarding the mortality found to prevail among special classes of lives, and among lives resident in other parts of the world than Great Britain. The authors of the present paper having had occasion to look closely into the subject of West Indian mortality, chiefly in connection with the financial affairs of the Barbados Mutual Life Assurance Society, have therefore willingly responded to an invitation to lay before the members of the Institute some account of the statistics which they have been able to gather, and of the conclusions at which they have arrived. They desire, in the first place, to express their thanks to the directors of the Barbados Mutual Society for permitting them to publish the results of the society's mortality experience, and to Mr. Spencer C. Thomson, the manager of the Standard Life Office, for some valuable statistics relating to the experience of that office regarding West Indian mortality.

There is one portion of the history of life assurance which I suppose could never be accurately written: I refer to the origin of the various forms of assurances, of the various plans of distributing or appropriating the surplus, and the authors and dates of discovery of many of the formulas; in fact, the history of the science of life assurance in contradistinction to the history of the business and its development.

Nineteen years ago, in November 1868, Mr. Bailey communicated to the Institute (J.I.A., xiv, 437) some figures which showed in a very striking manner, the wide differences of opinion which formerly prevailed among actuaries, as to the terms upon which advances should he made upon the security of reversionary life interests. In the discussion which followed the reading of my paper “On the Valuation of Reversionary Life Interests”, he is reported to have said: “As showing the expediency of further “discussing the subject, he instanced a case which was submitted “several years ago to five well known actuaries, where the “redemption money for the contingent annuity was fixed hy A “at £282,000, by B at £237,000, by C at £184,000, by D at “£154,000, and by E at £47,000.” The differences here shown are certainly very remarkable, and not very creditable to the profession ; but, judging from certain figures which have lately come under my notice, I am inclined to think that the opinions of actuaries do not now differ so widely as formerly.

In a paper which I read before the Society in 1879, I gave the results of an investigation I had made with the object of determining the probability, that a man marrying at any age over 40, will, or will not, have children; and I have now extended the same enquiry to men of all ages. The statistics upon which my former conclusions rested, related to 339 marriages entered into by peers and their near relations above the age of 40, and were extracted from Lodge's Peerage for the year 1871. For all statistical enquiries this work is, in my opinion, greatly preferable to any of the other works that give records of the British Peerage. The principal ground for this opinion is, that Lodge usually gives the dates of birth of the daughters of each family, as well as those of the sons; whereas other Peerages generally omit those dates, and place the names of the daughters (without any dates) after the names of the sons, so that it is impossible to tell whether they are older or younger than their brothers. In the case of a few families this practice is also adopted in Lodge, presumably by special desire of the head of the family; and it is a matter of regret that this, together with other circumstances to be presently mentioned, diminishes the value of the book for statistical purposes.

In the course of a paper read before the Institute in January 1885 (J.I.A., xxv, 141), I suggested a certain method of making approximate valuations on different bases of the liabilities of a life assurance company under its whole-life policies. Shortly afterwards I had occasion to make a valuation of the liabilities of a particular company, and I took the opportunity of testing the suggested method of approximate valuation by actual comparison of the results arrived at by its application with those derived from a rigidly accurate class valuation, and I have thought that it would be interesting to the members of the Institute to have an explanation of the process employed in making use of the approximate method, and a statement of the results of the comparison.

Mr. G. F. Hardy has contributed a valuable paper on this subject (J.I.A., xxiv, p. 95) and (p. 103) has done good service by introducing a theorem by Gauss for the accurate integration of algebraic functions limited to any stated degree, and by showing how the same may be applied to the approximate integration of other functions. Again, Mr. George King (J.I.A., p. 276) has made good use of one of the best of Mr, Hardy's formulas in his Numerical Calculation of the Values of Complex Benefits, practically elucidating the important advantages gained by its adoption. I have read these papers with much interest and have thereby been induced to investigate the matter still further, from my own point of view, in its especial relation to assurance calculations, and with results that may be considered worthy of communication.

A number of the schedules collected by Mr. Ansell of the National Life Assurance Society in 1871, and from which were deduced the interesting results contained in his Statistics of Families in the Upper and Professional Classes, were, with the permission of the Directors of that Society, lent to Mr. Sprague, who wished to extract from them certain information bearing upon an investigation he was making. These schedules had been filled up by members of the clerical, medical, and legal professions, and by a large number of other gentlemen and noblemen in England and Wales; and they contain, in addition to full particulars as to the children of the marriage, the dates of the births and the marriage of the parents, and their state at marriage, whether bachelor or widower, spinster or widow. Mr. Ansell has not investigated the law which the ages of the wives follow, when a number of men of a given age marry; and at Mr. Sprague's suggestion, and upon principles indicated by him, I undertook this novel enquiry.

The Registrar-General's mortality statistics prove beyond doubt that a marked decline has occurred in the national death rate in recent years. There appears, however, to prevail a singular reluctance to accept this decline of mortality as evidence of the improved health and sanitary condition of the English people, or to recognize that it necessarily implies an increase of the mean duration of life in England.