In the course of a paper read before the Institute in January 1885 (J.I.A., xxv, 141), I suggested a certain method of making approximate valuations on different bases of the liabilities of a life assurance company under its whole-life policies. Shortly afterwards I had occasion to make a valuation of the liabilities of a particular company, and I took the opportunity of testing the suggested method of approximate valuation by actual comparison of the results arrived at by its application with those derived from a rigidly accurate class valuation, and I have thought that it would be interesting to the members of the Institute to have an explanation of the process employed in making use of the approximate method, and a statement of the results of the comparison.

In the Diary for 1850 the following; mathematical query was proposed by the Rev. T. P. Kirkman, of Croft, near Warrington:—

Two answers were printed in the Diary for 1851; one by the proposer and another by Messrs. Samuel Bills, of Hawton; Thomas Jones, of Chester; Thomas Wainman, of Burley; and W. H. Levy, of Shalborne.

The principal circumstances affecting the value of individual live, have reference to sex, age, personal peculiarities, habits, occupation, residence, family history, previous ailments, and present health. These are brought before the medical referee, in the printed forms of examination and inquiry adopted by Life Assurance Societies, in a series of questions calculated to elicit the information required. In the present treatise they will be considered under several heads.

Almost a quarter of a century ago—in 1958 to be precise—one of my predecessors devoted the main part of his Presidential Address to a discussion of state and occupational pensions. He spoke at a time when price inflation was 2¾%, growth in the national income had been averaging 2% p.a. in real terms and the number out of work was half a million. The redemption yield on long-dated Government stock was about 5¼% and the dividend yield on the Actuaries' Investment Index of Ordinary Shares was 5¾%—we did not yet have the reverse yield gap that recognizes the endemic nature of inflation.

The President in his inaugural address last session, expressed the opinion that “the time is ripe for another discussion on surrender-values.” In accordance with that suggestion, I have ventured to bring to your notice some considerations bearing on the subject of surrender-values and the principles which, it would appear, should underlie their calculation.

The paper is an attempt to bring out the exact relationship between what has been called the ‘modified osculatory process’ and the older formulae of osculatory interpolation. The conditions upon which the modified process is stated to be based are shown to be insufficient in themselves to produce a determinate formula even after the degree has been made as low as possible, and a general formula is derived which satisfies these conditions and of which the formulae hitherto employed are particular cases. The general formula contains, as might be expected, certain arbitrary constants which reflect the relaxation of certain of the conditions insisted on by the older formulae.

Illustrative graduations are performed by means of the general formulIllustrative graduations are performed by means of the general formula

and the results are compared with previous graduations of the same data by Jenkins's fifth difference formula (which may be obtained by putting b=0 in the general formula), viz.:—

This is the particular case of the general formula which has hitherto been used in practice.

On a former occasion I had the honour to lay before the Institute a historical sketch of this subject, from the earliest period of authentic records down to that of the legal establishment of the practice of taking interest, in the reign of Elizabeth.

The facts then adduced showed that very high nominal rates of interest were obtained in the ancient world, and during the middle ages; but such rates are, by no means, to be taken as measures of the profits made upon loans, which must always have been seriously diminished, and often entirely swallowed up, by the exactions and spoliations to which the lender was exposed, and the difficulties he experienced in enforcing his claims, from defective laws, and the prejudices entertained against him by the community.

Until within a very recent period the doctrines of probability have been applied to scarcely any great public purpose except the Assurance of human life. In that particular department of knowledge, a number of labourers have been for many years busily occupied in gathering the facts, endeavouring to classify events, and trace some general laws. In later years, researches have been extended into the effects of different classes of diseases upon health and longevity. Still more recently, the calculations of fidelity risks, of railway and other accidents, of bankruptcy, and of other circumstances affecting the person of an individual, or his commercial dealings, have been made with a view of proving their capability of being estimated for Insurance.

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.

The object of the following notes is to state somewhat more fully and formally a method (first published in the Insurance Record for 10 March 1905) of approximating to the reserves required for whole-life assurances, with due regard to the influence of selection; and to illustrate the practical application of the method.