Some months ago I had occasion to enquire of half-a-dozen offices what were their rates of single premium for a certain survivorship assurance. The highest rate quoted exceeded the lowest by 40 per-cent of the latter. This fact appears to me to afford a good illustration of the different estimates formed by actuaries of the value of the risk run under survivorship assurances. The discrepancy probably arises mainly from the use of different tables of mortality in calculating the risk premiums. It will be seen later on, that the differences in survivorship premiums as found from different mortality tables are in some instances very considerable. I am inclined to think, too, that some actuaries, through a feeling of uncertainty as to whether they have correctly estimated the risk premiums, load them very heavily in order to be on the safe side.

Nineteen years ago, in November 1868, Mr. Bailey communicated to the Institute (J.I.A., xiv, 437) some figures which showed in a very striking manner, the wide differences of opinion which formerly prevailed among actuaries, as to the terms upon which advances should he made upon the security of reversionary life interests. In the discussion which followed the reading of my paper “On the Valuation of Reversionary Life Interests”, he is reported to have said: “As showing the expediency of further “discussing the subject, he instanced a case which was submitted “several years ago to five well known actuaries, where the “redemption money for the contingent annuity was fixed hy A “at £282,000, by B at £237,000, by C at £184,000, by D at “£154,000, and by E at £47,000.” The differences here shown are certainly very remarkable, and not very creditable to the profession ; but, judging from certain figures which have lately come under my notice, I am inclined to think that the opinions of actuaries do not now differ so widely as formerly.

Mr. G. F. Hardy has contributed a valuable paper on this subject (J.I.A., xxiv, p. 95) and (p. 103) has done good service by introducing a theorem by Gauss for the accurate integration of algebraic functions limited to any stated degree, and by showing how the same may be applied to the approximate integration of other functions. Again, Mr. George King (J.I.A., p. 276) has made good use of one of the best of Mr, Hardy's formulas in his Numerical Calculation of the Values of Complex Benefits, practically elucidating the important advantages gained by its adoption. I have read these papers with much interest and have thereby been induced to investigate the matter still further, from my own point of view, in its especial relation to assurance calculations, and with results that may be considered worthy of communication.