The method proposed by the late Sir J. W. Lubbock ov approximating to the value ov a life annuity, appears hitherto to hav been very littl notist by mathematicians and not at all by practical men. It was demonstrated by him in the Cambridge Philosophical Transactions for the year 1829, in a paper entitld “On the Comparison of various Tables of Annuities”; and a short account ov it is also givn in the articl Probability, publisht by the Useful Knowledge Society, and very offen bound up with David Jones's work on Annuities: see § 56, p. 36. The former paper was reprinted in the Assurance Magazine (vol. v, p. 277). The original is disfigurd by numerous misprints, which ar all faithfully reproduced in the reprint; and when we add that very littl numerical illustration ov the application ov the method is givn, we hav probably said enuf to account for its attracting so littl attention in any quarter. Considerd as a means ov approximating to the value ov an annuity, the formula has been superseded by the simpler one givn by Mr. Woolhouse; but I believ it may still be practically useful in som cases.

The determination of the probable effect, upon the increase of population, of the extinction of small pox, is a problem which has been discussed at some length by three of the great continental mathematicians of the last century, namely, D. Bernouilli, D'Alembert, and Laplace. I propose, first, to solve the problem by what may be termed the theory of the composition of decremental forces, and then to reproduce (by way of comparison) from Mr. Todhunter's History of the Theory of Probabilities, the solutions given by the three eminent mathematicians in question.

Since the passing of the Life Assurance Companies Act, 1870, the accounts and valuation statements lodged by the various companies with the Board of Trade, in accordance with the requirements of the Act, have been printed and published annually. The valuation returns afford, in the majority of cases, sufficient information to enable an actuary to form a tolerably correct opinion of the financial strength or weakness of the companies to which they refer. In some cases, however, owing to the special character of the facts brought out in the schedules, it is important that the actuary should have before him, when testing the sufficiency of the reserve, some facts, relating to the growth of the business, and the purposes to which the expenditure has been applied, which are not elicited by the Act. For instance, he should know the annual amount of new business which has been transacted, and the annual amount of assurances surrendered and dropped, as well as the claims by death, in order to enable him to form an opinion as to the average duration of the existing policies. He should also have a division of the expenses, showing those applicable to new business and those applicable to the management of business already acquired. An approximation to these particulars may generally be made by the actuary after a study of the reports and other voluntary publications of the companies; so that, on the whole, it may be said that the Act affords to the public the means of judging, through the medium of the actuarial profession, of the relative positions of the various offices as regards stability and profitableness.

The question with which I have headed this paper is one that may seem startling enough. It has been so widely taken for granted that a large new business must be an unmixed good to all concerned in a life company, that to debate whether it be so or no, will seem in the eyes of many to be a very idle thing. And yet, though I have been considering the subject for a long time, I cannot answer the question unhesitatingly in the affirmative. In saying this, it ought to be pointed out, that the matter is regarded from a purely practical point of view. The problem is not ought a large new business to benefit policyholders? but, as a matter of fact, does it? Even if the question were answered in the negative, it would not follow that no new business, or an insufficient quantity of it, would better suit the interests of the policyholders. Waste must be supplied, and a certain degree of magnitude maintained in life companies.

I have somewhere read that the experience of life assurance companies furnishes the only certain means of ascertaining the law of mortality, since by this means only is it possible to keep in view through a lengthened period and trace up to death, each of a selected number of persons. As far as the possibility is concerned, we may admit that the law of mortality of the members may be deduced in this manner by proper methods. It is important, however, to bear in mind that, in the case of assurances, we have to deal with selected lives; and the influence of this selection, though it may be disregarded in the later years of assurance, is of considerable importance in the earlier years, so that, unless special precautions be taken, the mortality of the general population cannot be correctly obtained. The mortality may be more nearly determined if an account be kept of the rejected proposals, as is done in England, and the deaths noted which happen among those cases. This, I certainly think, should be done universally, as it must be of practical use to the companies to know how far the rules by which they are guided in accepting or rejecting proposals are justified. No such account is kept, as far as I know, either in Germany or Austria. We may otherwise eliminate the influence of selection, by omitting from the calculation the earlier years of assurance, during which this influence is felt.