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On the Tables of Single and Annual Assurance Premiums published by the late Mr. William Orchard, and on a Theoretical Table of Mortality proposed by him
Published online by Cambridge University Press: 18 August 2016
Extract
The values which it has been heretofore customary to tabulate for actuarial purposes, in connection with specified rates of mortality and interest, are those of annuities on single and joint lives at all ages; and from these, by operations more or less complex, and with the aid of subsidiary tables, the actuary has had to form such other values as might be required in the solution of any particular problem in hand. Of the values thus requiring to be formed, those of the single and annual premiums for assurance occur perhaps more frequently than any others; and hence the desirableness of simplifying as much as possible the operations by which these are deduced from the corresponding annuities. To effect this simplification was the object of Mr. Orchard's work, in which, as will be admitted by all competent to form an opinion on the subject, the author has been completely successful. The arithmetical operation heretofore necessary has been entirely superseded, and the required values are found simply by inspection.
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- Research Article
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- Copyright © Institute and Faculty of Actuaries 1855
References
page 183 note * I had written here,“ few even of the most experienced arithmeticians;” but a very little consideration has induced me to alter the expression to its present form. I believe that one result of experience in arithmetical operations is to beget caution, since it reveals the existence of liabilities to error not previously suspected.
page 184 note * Singularly enough, at the meeting of the Institute of Actuaries next succeeding that at which Mr. Orchard presented a description and specimens of his tables, there was exhibited in print a copy of “ Conversion Tables, by William Wood, F.I.A., Secretary to the Scottish Amicable Life Assurance Company.” The object of Mr. Wood's tables was precisely the same as that of Mr. Orchard's. The idea on which they are founded also is the same; but it is earned out by Mr. Wood in a different way, and is applied by him to fewer rates of interest, while his results are presented in certainly a much less convenient form. See p. VIII.
page 184 note † Page 47, 3 per cent. The single premium corresponding to 15 16 should be 52·932.
page 185 note * Lacroix, Traité Elémentaire du Calcul des Probabilités, pp. 199, 200.
page 185 note † Phil. Trans. 1826. (See also the recently published Life of Dr. Young, by Dr. Peacock, chap, xiii.)
page 185 note ‡ I copy the following memorandum made by myself, under date May 3rd, 1845:—
“ The following formula I copied from an original letter of Mr. Babbage to Mr. Baily (1823), accompanying a presentation copy of Baron Maseres' Scriptores Optici. It represents, nearly, the Swedish Table of Mortality. It is of course empirical:
page 186 note * Phil. Trans. 1825.
page 186 note † Life Tables, founded upon the discovery of a Numerical Law, &c., 1832.
page 186 note ‡ Fifth Report of the Registrar-General, Appendix, pp. 342, &c.
page 188 note * It will be found that the quadratic formed by equating to each other the two values of lx has for its roots 80 and 81.
page 197 note * I am indebted to Mr. Charles Watkins, of the Pelican Life Office, for the formation of the column of annuities at 6 per cent.
page 197 note † These are the irregularities arising from its want of graduation, and the extraordinary increase in the value of life during the ten years that follow age 87. Mr. Milne's remarks on this last-named point (p. 554) appear to me to be altogether at variance with the sound judgment which is his general characteristic.