It is now some years since a paper was presented to the Institute dealing with the comparative bonus-earning powers of whole life and endowment assurances. In view of the importance of the subject I have ventured to submit this paper, which I hope may lead to a useful and interesting discussion. The matter has been so ably dealt with by the late Mr. Sunderland, Mr. Lidstone and Mr. Chatham that I have the greatest hesitation in presenting my own views, and I must therefore beg the Members of the Institute to receive this paper in the same kindly spirit which they have always manifested towards new contributors.

My object in the paper has been to show the relative bonuses which should be allotted to whole life and endowment assurance policies having special regard to

(1) The expenses of the office;

(2) Variations in the rate of interest earned; and

(3) Mortality.

In Table A are given the office premiums which have been employed in this investigation. They are based on the H[M] 4 per-cent Table, with a loading for a £1 per-cent compound reversionary bonus, together with 10 per-cent and a constant of 3s. per-cent for expenses and commission—

It has often occurred to me that but scant justice has been done to the application of the infinitesimal calculus to the theories of compound interest and life contingencies. This is, perhaps, in some measure due to the popular relegation of the differential and integral calculus to the realms of the so-called “higher mathematics.” There are, of course, two aspects of the case to be borne in mind. On the one hand, it is necessary to present the subjects in such a form as will be best suited to the student who is commencing to study them. For this purpose experience shows that a start should be made with particular cases, leaving the generalization until such time as the student shall have obtained a grasp of first principles sufficient to enable him to view the subjects in their general aspect. On the other hand, however, there is no doubt that to the reflective mind there comes a time when the desire is felt to invert the process and deduce the formulæ in their logical sequence from a fundamental general hypothesis.

So much has been written on the algebra of formulæ for graduating an irregular series by means of successive summations, and on the theoretical errors introduced by this process, that there is possibly some danger of the underlying principles, on which all such formulæ are founded, being overlooked or insufficiently understood. In the present note an attempt is made to deal with this matter from first principles, with the object of rendering clear how such formulæ Work and why it is that they produce, with greater or less success according to the nature of the particular formula, a smooth succession of values from rough and irregular data.

Mr. George King's recent investigation of the magnitude of the error introduced into certain functions of a mortality table following Makeham's law when these are re-graduated by summation formulas (J.I.A., xli, 54), has again attracted attention to the subject of the graduation of mortality tables, and has recalled the lively controversy which took place some years ago between Dr. Sprague and the late Mr. Woolhouse over the merits and demerits of the methods championed by those distinguished authorities. Not the least interesting contribution to the discussion which followed the reading of Mr. King's paper was furnished by Dr. Sprague himself, who wrote in defence of certain criticisms expressed by him with regard to methods of summation generally and Mr. Woolhouse's formula in particular. Dr. Sprague remarked that he had “objected to Mr. Woolhouse's “formula of graduation,on two grounds, which it has in common “with all other summation formulas; that is to say, (1) because “it does not remove the irregularities of the original series of “facts, but only reduces them, and (2) because it distorts the “law of the facts to a greater or less extent.” He went on to say that the results of his re-graduation of the HM (Text-Book) Table by Woolhouse's formula had led him to the conclusion that the second objection might be disregarded in practice, but that there still remained objection (1), which he considered by far the more important.