It is a common thing to find in the reports of life assurance companies a statement to the effect that the actual deaths and claim payments have been under the expectation, or have represented a certain percentage of the expectation. The records of life assurance literature would appear to show that little attention has been given to the methods on which these comparisons are based, the only contribution to the Journal bearing directly on the point being a paper by Mr. G. M. Low (J.I.A., xviii, 195). But as the matter possesses a considerable and growing importance, the following note has been prepared with the object of suggesting a method by which such estimates may be computed according to a plan which seems reasonable and defensible on scientific grounds.

I do not pretend that much which is original will be found in this paper. Previous writers, notably the late Sir J. W. Lubbock, and Messrs. Woolhouse, Sprague, and G. F. Hardy, have, so far as actuarial science is concerned, nearly exhausted the subject of Formulas of Approximate Summation; and from time to time examples have been given of the practical use of the formulas, which have indicated the great facilities they afford to computers. But something in the way of numerical illustration remains to be done in order to bring the formulas into more general use, and to supply that want is my present object.

The state of the law in respect to Joint-Stock Companies, and associations for business purposes generally, remained in the same unsatisfactory condition which I have described in the previous section down to this time. Those great Insurance Associations, with millions of capital subscribed, in many cases, by the merchant princes of the land, for purposes admittedly the most beneficent, were simple partnerships, almost without legal recognition, except for purposes tending to their detriment or destruction. They could be attacked or pulled down by legal process readily enough; but they could only protect themselves against fraud, or recover their just debts by the most cumbersome of processes. What they had a right to expect, as institutions designed to aid largely in the accumulation of national wealth, was protection in carrying out their laudable designs; but of this they had not a vestige. Not only was every holder of stock in a proprietary company primarily and personally liable to his last shilling for the engagements of the partnership, but every policy-holder in a mutual society, being a member of, and therefore a partner in such society, was equally liable for all its engagements.

It would be a very useful and interesting task to examine the numerous mortality experiences which have been publisht of late years; to compare them carefully with each other, so as to see how far they agree and in what respects they differ; and to note what contributions (if any) they have made to the general theory of life insurance. On the present occasion I will state the results of my examination of the recently publisht experience of the abovementioned company. The experience extends over 20 years, and contains 124,526 policy-years, the numbers at risk ranging from 301·5 policies in the year 1866 to 13,927 in the year 1885.