There is no subject of greater importance in the conduct of the business of Life Assurance than that of the valuation of policies. It is a matter which comes before the actuary from day to day and from year to year, whether in the form of fixing the amount to be returned to an individual policyholder on the cancelment of his contract, or in that of estimating what reserve should be made in the aggregate for a mass of policies existing at the close of a valuation period. Both of these questions have been discussed before the Institute on many previous occasions, and it is not necessary to enter into them now. The task I have set before myself is the humbler one of dealing with results already arrived at. I do not propose to enquire what are the reasons for preferring one mortality table or rate of interest to another, or on what principle of calculation the surrender-value of a policy should be based. I accept the results brought out by the use of any mortality table or rate of interest agreed on, and I propose to consider how these results may be represented by means of diagrams and models, and whether by this means we can arrive at any clearer understanding of the relation to one another of the reserves brought out by different standards of valuation.

In December 1879 I read before the Institute a paper (J.I.A. xxii, 117) on the probability that there will hereafter be issue of a marriage hitherto childless. The statistics on which I based my conclusions, were those given by Dr. Matthews Duncan in his work, Fecundity, Fertility, and Sterility; and by Mr. Charles Ansell, Jr., in his Statistics of Families of the Upper and Professional Classes. But I was not altogether satisfied with the results I obtained; for, as I pointed out, the two sets of statistics differ from each other in some material respects, and I was unable to give any satisfactory explanation of the difference, or to say which set was the more trustworthy. For the purpose I had in view both sets had one great defect, namely, that they gave no information as to the number of childless marriages which corresponded to those that became fruitful at different intervals as stated in the tables. I have now, by means of a careful examination of the records of the British Peerage families, obtained a fresh set of statistics; and I purpose on the present occasion applying them to the solution of the same problem as I discussed in 1879.

In introducing a consideration of this subject to the Institute, I do not purpose putting forward any new theories; many papers have been written upon the different methods of valuation and the modes of distributing profits, which have been most valuable contributions for determining the best theoretical basis to be adopted. I need only refer to the papers of Messrs. Jellicoe, Sprague, Bailey, Meikle, Manly, Valentine, King, and others, all of which are to be found in the Journal of the Institute, in order to remind you how fully these subjects have been considered at different times. The present paper, however, is intended to investigate the practical effect of applying the respective theories, not upon any theoretical assumption, but as they have to be applied in actual practice, and to see how far they are influenced by the different rates of premium actually charged by the different companies.

Amicable Society.—We now arrive at the period of the formation of the Fourth Mutual Contribution Life Office established in London. This was designated the Amicable Society for a Perpetual Assurance; and it was destined, if not to be perpetual, to live long enough to become famous. It differed in some material respects from its predecessors, the chief distinction being that the contribution to its funds was fixed at £6. 4s. per annum. The first 2,000 members to pay, in addition, 5s. to the registrar and 5s. to the joint stock of the office; and all subsequent members were to pay 10s. to the joint stock. The founder and the registrar of the society was John Hartley; and he was to receive, permanently, 1s. per quarter out of each contribution.