The necessity for a reliable investigation into the mortality experienced by life tenants under reversions has long been felt.

So far as I am aware, the only published experience is that given by Mr. Neil Campbell in a paper read before the Faculty of Actuaries in 1902 (T.F.A., vol. i, p. 79), although doubtless individual offices have at various times examined the mortality in respect of their own purchases. The investigation by Mr. Campbell was of considerable service in indicating the trend of the mortality, but was unfortunately of limited value inasmuch as it was based upon only 226 lives producing 2,097 years of exposure to risk.

The problems of vital statistics are of the utmost importance as affecting in many ways the welfare of the people. In connection with the study of such problems the value of reliable life tables applicable to the general population is well recognized. The preparation of national life tables presents various difficulties, however, if the results are to command respect for their reliability. It is therefore in the hope that British actuaries will be interested to see how these difficulties have been met in another country that the following notes have been prepared.

The history of life insurance presents some curious changes in the character of the factors which have from time to time taken precedence as matters of anxiety to those responsible for the successful conduct of the business. In the early days it was anticipated that it was the rate of mortality which would give the greatest possibility of variation, and which was most likely to undermine the foundation of the structure. In time, however, this bogey was laid by the heels. Then, in the final years of last century the fall in the rate of interest was a matter of great concern in view of the persistent fall which had occurred over a long period, the danger being that the rate earned on accumulated funds would fall below that necessary to maintain solvency.

The question whether it is necessary to use select mortality rates in the calculation of office reserves was pronounced by Mr. R. P. Hardy during the discussion on the papers submitted by Messrs. King, Diver and Ackland to be a highly debatable one.

The discussion disclosed two lines of thought on the subject, one admitting that a select valuation was theoretically correct, the other questioning the advisability of making a select valuation at all, the arguments in favour of the use of an aggregate table being crystallized in Mr. Warner's statement that the logical complement of a select table valuation would be the employment of a rate of interest representing to the nearest penny what had been actually realized during the quinquennium. Considered as a warning against undue striving after exactitude this is unanswerable, but when the calculation of the office reserves is viewed as a preliminary to the declaration of bonus, the conclusion might be drawn that an aggregate table valuation so entirely meets the case as to permit the full distribution of the resulting surplus.

A year ago I had the honour of submitting to the Institute a paper on “A Short Method of Constructing Select Mortality Tables”, which has been published in the Journal, vol. lii, pp. 286–341. For convenience it may be referred to as “the former paper.” A definition of the short method was given as follows:

ALL the methods of group valuation submitted to the Institute in the last decade require the calculation of the second sum of the quantities valued, whether in the form of multiplications by —4½, —3½, … 3½, 4½, i.e., the second sum about the centre, or the more convenient second sum about the end. The object of this paper is to show that the α, β method can be transformed so as to avoid this difficulty.

Many investigations have been made in the past to ascertain the surplus, or the bonuses, which would arise with a given valuation basis, from different assumptions as to the actual experience and premiums of an office ; but there is little on record regarding similar efforts which may have been made to see how the surplus year after year would be varied by the introduction of different valuation bases, the other two factors remaining constant.

The remainder form of Newton's formula of interpolation with divided differences has been known for at least 37 years; yet actuaries have taken hardly any notice of the complete formula which ought to be one of their principal tools. It will, therefore, not be superfluous to begin by reproducing the very elementary proof from first principles.