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On the Construction and Use of Commutation Tables for Calculating the Values of Benefits depending on Life Contingencies
Published online by Cambridge University Press: 18 August 2016
Extract
It is to a Mr. George Barrett, of whom nothing besides is publicly known, that we are indebted for the principle of the Commutation Tables, and for the method of computing, by means of them, the values of benefits depending on the contingencies of human life. The method was first introduced to public notice, after it had been refused a place in the Transactions of the Royal Society, by Mr. Baily, in an Appendix to the second edition of his Doctrine of Life Annuities, published in 1813. Mr. Griffith Davies, in a work on life contingencies, published in 1825, by certain additions to the tables, and alterations in their structure, according to Professor De Morgan, “increased the utility and extended the power of the method to an extent of which the inventor had not the least idea.” Mr. Barrett's method was also briefly noticed in the Appendix to Mr. Babbage's Treatise on Life Assurance. The method, as improved by Mr. Davies, has since been treated, and a very large collection of tables adapted to it, for both one and two lives, has been given, by Mr. Jones, in his work on annuities, in the Library of Useful Knowledge. But by far the most valuable papers on the subject are two in the Companion to the Almanack, for 1840 and 1842, by Professor De Morgan, which contain the materials of many thousand formulae, applicable to almost every case that can occur. There is also some notice of the method in the article “Reversions,” in the Penny Cyclopædia, which article likewise is the production, we believe, of Professor De Morgan.
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- Research Article
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- Copyright © Institute and Faculty of Actuaries 1863
References
page 84 note † For some account of Mr. Barrett, see vol. iv., p. 185, of this Journal.—ED. A. M.
page 92 note * By an expression for the general term of a series, is meant, an expression in which a variable quantity is introduced, and which, by giving any particular value to the variable, gives the term of the series corresponding to that value. Thus, in the above general expression, Dxlx v x which denotes the age, is the variable; and if we give to it a particular value, we have immediately the term of the series corresponding to that value. For instance, if x = 20, the expression becomes
D20=l20 v 20,
and this is the value in column D corresponding to age 20.
page 93 note * It will be afterwards seen that there is no necessity for employing those particular powers of v, which have been made use of in the construction of the table. The only condition as to these powers, which is indispensable to the possession by the table of the required properties, is, that their indices shall form an increasing arithmetical series, of which the common difference is unity. The powers that have been chosen possess the advantage of imparting to the expressions for the values of the numbers in columns D and C, a symmetry that would not otherwise have belonged to them.
page 94 note * Algebraically considered, it is not necessary to make this limitation.
page 99 note * By this symbol is meant, and the phrase may in reading be substituted for it, “a life now aged x years.”
page 99 note † We have assumed above that the number of purchasers of endowments will be the number represented by the mortality table to be alive at the age at which the purchase is made; but the only assumption as to their number which it is necessary to make is, that this number will be sufficient to secure an average mortality proportional to that represented in the table. If we had assumed any other number, we should have had to find, by a proportion, the number of survivors at the advanced age. But, by the assumption in the text, this operation is saved. In both cases the result would be the same. The value of a fraction depends, not on the absolute magnitude of its terms, but on their relative magnitude or ratio. The value of 3–4ths is the same as that of 6–8ths or 9–12ths, because the ratio of 3 to 4 is equal to that of 6 to 8 or 9 to 12.
The same remarks will apply to the other benefits.