The subject of War Mortality has not often been before the actuarial profession; there is a reference to the subject in Vol. iv of the Journal, and in Vol. vii there appear Mr. W. B. Hodge's papers on “The Mortality arising from Military Occupations”, which give statistics ending with the Crimean War. The next reference to the subject is in Vol. xxxiv of the Journal, where Messrs. Smee and Ackland's valuable paper is given; this was first published in pamphlet form some years before its appearance in the Journal. In the same volume Mr. McLaughlin contributed a paper on general naval and military statistics, dealing principally with the ten years 1886–95, and an abstract of the tables is given in the Transactions of the Second International Actuarial Congress. There is also the well-known paper by Mr. A. G. Mackenzie published in the Transactions of the Actuarial Society of Edinburgh in 1881; and, finally, last year Mr. A. T. Anderson read a paper before the Actuarial Society of New South Wales which dealt with the mortality during the war now under consideration.

In October 1900, on the morning of the first day of the autumnal meeting of the Actuarial Society of America, Mr. Emory McClintock, Actuary of the Mutual Life Insurance Company of New York, read a paper entitled “The Objects to be attained in Future Investigations of Mortality and Death Loss”, in which he stated that probably four-fifths of the publications of companies' general experience possessed only a “casual interest”, and that the same amount of effort might much better have been devoted to investigating special classes of business, adding that “it is “more important to the future interests of life insurance to learn “how fishermen compare with farmers, how physicians compare “with clergymen, and the like, than it is to gather all these “heterogeneous materials into one grand average in the form of “a new life table.”

Our Journal contains singularly little information on Vaccination—a brief remark by S. H. Ward, M.D., in 1860 (J.I.A., viii, 339); some paragraphs in a general paper by H. W. Porter, in the same year (J.I.A., ix, 151); and a paper by Dr. Sprague in 1877 (J.I.A., xx, 216), are the only references to the subject that I have been able to trace.

It is not proposed in the present paper to take up any original investigation, the main object being to bring up to date the papers submitted to the Institute in 1876 and 1877, and which appeared in J.I.A., xix, p. 381, and xx, p. 233. These papers were written not very long after the publication of the Twenty Offices' Experience, with the Monetary Tables based thereon, and to it they were almost entirely devoted. The peculiar rates of mortality prevailing amongst assured lives were investigated, by means of ten mortality tables calculated for central ages at entry, tables which were called by name “Analyzed.” That name has since been superseded by the name “Select”, and the tables themselves, together with the annuity tables based upon them, have become obsolete, their place having been taken by Dr. Sprague's much more complete, and much better, “Select” Tables. The Analyzed Mortality Tables, however, served their purpose, as the main object in taking the matter up was, to estimate the reserves of life offices when the mortality of lives assured is kept separate according to age at entry.

I have been invited by the Council of the Institute of Actuaries to read a paper dealing with the general bearings of the New Experience Tables. The Unadjusted Data of the Assurance Experience are set forth in the three volumes published by the Joint Committee of the Institute of Actuaries and the Faculty of Actuaries in Scotland, comprising (1) Whole-Life Assurances—Participating and Non-Participating—Male Lives; (2) Whole-Life Assurances—Participating and Non-Participating—Female Lives; (3) Endowment Assurances and Minor Classes of Assurances—Male and Female Lives. The Joint Committee have also kindly placed at my disposal, in the preparation of this paper, the graduated results, so far as available; and the Monetary Tables which have been computed, and which are in course of being printed for publication.

A striking indication of the little interest taken by the actuarial profession generally in the subject of Pension Funds is shown by the fact that in the many published volumes of the Journal of the Institute, hardly any information is to be found with reference to such Funds.

This lack of interest in Pension Funds was noticed some time ago by Mr. Ryan (page 470, vol. xxxi, of the Journal), who remarked that “ it was a very curious thing that the important “ question of the valuation of Widows Funds had on no occasion “ been treated by contributors to the proceedings of the Institute. “ In Dr. Sprague's 20-volume Index there was absolutely no “ reference to the question of Widows’ Funds. … He might “ add that the Index to the Journal contained no entry relating “ to superannuation schemes, which was again a matter of great “ public importance.’

On reading over my paper after it was published, I did not feel quite satisfied as to the accuracy of my approximate formula, given on page 235, for the adjustment of in respect of the excess of contributions included in the Table as constructed, so I have had the values of and , calculated. The constant summation of these columns, namely, and , will give the columnar values to be used for making accurate adjustments.

The great value of quadrature formulæ for the calculation of survivorship and other benefits involving three or more lives has been already illustrated in the pages of the Journal. From certain remarks made by Mr. G. F. Hardy and Mr. George King in the discussion which followed the reading of Mr. King's paper (J.I.A., xxvi, pp. 297, 301), I inferred that it would be of interest to examine the errors involved in these formulæ, and to compare the results yielded by them and by other methods of approximation, such as the substitution of an equivalent life. Mr. Sheppard has shown (Proceedings of the London Mathematical Society, xxxii, p. 258) how most of the best-known quadrature formulæ can be readily derived from the Maclaurin Summation Theorem, with expressions for the errors in terms of differential coefficients. In a recent paper (Proceedings of the London Mathematical Society, xxxiv, p. 335), I have obtained similar formulas by direct integration from Everett's Interpolation Theorem, with expressions for the errors in terms of central differences.

The subject of Temporary Assurances has been suggested as one that would be of interest to Members of the Institute of Actuaries, and the present paper is an attempt to examine the mortality among lives who have been assured by this class of policy. All the particulars that have been used are given in the “Combined Experience of Assured Lives—Endowment Assurances and Minor Classes of Assurance” (C. & E. Layton, 1900), but for convenience of reference Table I has been extracted and gives the experience in a slightly altered form for the durations 0–4 years, and Table II shows the groups into which the data were thrown so as to render them less uneven. The reason for omitting the experience beyond duration “4” was that it is generally assumed for practical purposes that selection has worn off by that time.

The point in this case was whether the Society was liable to be assessed in respect of interest, dividends, and rents, arising from investments in foreign countries.

The Society had large funds invested in various forms in foreign countries, in some of which countries the Society carried on the business of Life Assurance, and in others of which it carried on no business of any kind. It was admitted that only a part of the income arising from these investments was remitted to Great Britain, the rest of such income being either reinvested or remitted direct to other foreign countries for investment, or applied in payment of establishment and other expenses in the country where the interest was earned, or remitted direct to other foreign countries for the general purposes of the Society.

The Joint Committee on Mortality Investigation asked Dr. Sprague some time ago to graduate the Annuity Experience by the Graphic Method with the view of determining the period during which selection lasts; but as he was too fully occupied to allow of his personally undertaking the proposed graduation, the work was undertaken by the writer on principles arranged in consultation with Dr. Sprague.

With a view to preparing the way, in some humble degree, to a generally satisfactory method of distribution hereafter, I beg leave to submit a few brief observations upon the merits and demerits of certain plans already in operation. Unhappily, I cannot at present, in the existing state of our enquiries, present any constructive notions capable of being fused into a symmetric system.

Life insurance in Denmark has certain peculiarities, which may be interesting perhaps also to foreign actuaries.

In 1871 the Danish Government thought to satisfy all the life insurance necessities of the country by establishing by law, and with the full guarantee of the State, one single institution, “Statsanstalten for Livsforsikring.” By this all legislation with respect to private life insurance was supposed to be rendered superfluous.

It may well seem an act of no small temerity to attempt a fresh paper on a subject so often and so ably treated in essays read before this Institute as that of the net premium method of valuation. I am fully conscious of this, and the more so that I cannot profess to be able to add anything appreciable to what has already been said, either expressly or by implication, upon the matter. It will be admitted, however, that the practical importance of the subject, its vital bearing upon the conduct of the business of life assurance, can scarcely be exaggerated. Mr. J. Hill Williams, commenting upon Mr. A. H. Bailey's paper on “The Pure Premium Method of Valuation”, in March 1878 (J.I.A., xxi,, 126), said “The subject of this paper is one upon “which we can never have too much writing or too much “discussion, for it is the very life of our science, and upon the “views which we take of this most important question depends “the safety of many of the greatest financial institutions in the “country”; and with the spirit of such remarks we shall probably all agree. That being so, I think I am right in saying further that we have had no express discussion of this topic at our Institute for several years past. It has, doubtless, on many occasions been referred to—as a thing so intimately intertwined with our daily practice could hardly help being—in connection with other matters primarily before us; but it has not been the actual and avowed theme for consideration.