Endowment assurance can boast of no great antiquity, though its constituent elements, the temporary assurance and the pure endowment, both date from early times. Traces of the latter are found as long ago as the 16th century; thus, in a pamphlet by one Thomas Wilson, published in 1572, and entitled “A Discourse upon Usurie, by waie of Dialogue and oracions, for “the better varietie and more delight of all those that shall read “this treatise”, it is stated that, “Whosoever lendeth such a summe “of money (£100), and hath a childe of one yeere, shall have for “his childe, if the same childe doe live till he be full fifteene years “of age, 500li (£) of money: but if the childe die before that “time, the father to lose his principal for ever.” It is evident that the rate of mortality in those days must have been alarmingly high, either among children or among the institutions granting these benefits. A form of assurance, similar in some respects to the above, called “Apprenticeship Assurance”, was introduced towards the beginning of the 18th century. In a prospectus dated 1710 it is stated that, “Anyone who now is, or that is to be an “apprentice, may by paying 1s. entrance and 2s. 6d. per quarter, “gain, probably from £40 to £50 up to £500 or £600.” After reading this astonishing statement, one cannot but admit that the tontine and other estimates of American Companies, about which we hear so much in the present day, pale their uneffectual fire before the similar extravagant statements made by our predecessors in the reign of Queen Anne.

One of the most striking features in the history of Life Assurance business during the past few years is the marked increase—both absolute and proportional—which has been shown in the number and amount of policies effected under the Endowment Assurance plan. There is every reason to expect that this increase will continue, and that ere long Endowment Assurances will divide honours with whole life policies as regards importance in the periodical valuations. It is consequently very desirable so to improve the methods of valuing such policies as to minimize the amount of labour involved, and it will become more and more necessary to abandon methods which answered very well so long as Endowment Assurances were looked upon as “Special Policies”, but which are extremely cumbersome when applied to a large mass of contracts. The shorter methods that have hitherto been suggested, though admirably adapted for use in test or check valuations, involve an error which many actuaries will consider too appreciable to be neglected, more especially as the approximate methods nearly always bring out a reserve which is smaller than the true reserve, and it accordingly becomes desirable to seek some process that will be free from this objection.

The following is a method of obtaining an approximate value of the annuity , based on the assumption that the mortality follows Makeham's formula. The numerical results will be found sufficiently close for practical purposes.