in his paper " On the Calculation of Single Life Contingencies," which appeared in the Companion to the Almanac for the year 1840, and which is reprinted in the Assurance Magazine, xii. 328. His explanation of the term is to the
following effect: Taking any two ages, say 30 and 40,we have, according to the English Table No. 3, Males see appended table (3),—D M = 125464, N., = 2385610;
D 40 - 83406, N 40 = 1374058.
Transpose the numbers opposite each age to the other age; then whatever may be the present age (less than 30)—
A person might now give up £83,406, due at the age of 30, to receive £125,464, if he live to be 40.
A person might now give up an annuity of £1,374,058, to be granted at the age of 30, to receive in return another of £2,385,610 to be granted at the age of 40, if he should live so long.
"These proportions are independent of the present age of the party, and show that the most simple indication of the tables is the proportion in which a benefit due at one age ought to be changed, so as to retain the same value and be due at another age. They might, therefore, withgreat propriety, be called Commutation Tables."
It is clear that this property will not be altered if all thequantities in the D column, and consequently those in the N column, are increased or diminished in a constant ratio.
A " D and N table" may be used, not only to find the value of annuities, immediate, deferred, and temporary, but alsoto find the annual premium that should be paid for a given number of years as an equivalent for a deferred annuity. If the annuity is deferred n years, and the annual premium of equal value is to be paid for m years, it will be ^T i_ vT* The table may also be used to find the single and annual premiums for insurances, immediate, deferred, or temporary. The single premiums are—
1. For an ordinary insurance, - -^y;
2. For an insurance deferred n years, ^-Ty;
3. For a temporary insurance for n years,
The annual premiums payable during life for the same benefits are found by substituting N. e _ 1 for D x in the denominator; and the annual premiums payable for m years, by putting N,_, - N xini _ l in the denominator instead of D.,..
Before quitting this subject, we should mention that in practice other columns are added to the table besides the D and N columns. A column, S, is given for the purpose of calculating the values of increasing annuities; a column, M, for calculating the values of assurances; and a column, 11, for calculating the values of increasing assurances. An explanation of the M column belongs to the subject INSURANCE; for an account of the S and R columns, we refer the reader to the works and papers on life insurance contingencies, in which the D and N (or commutation) method is described; particularly to those of David Jones, Cray, and De Morgan.
The earliest known specimen of a commutation table is contained in William Dale's Introduction to the Study of the Doctrine of Annuities, published in 1772. A full account of this work is given by Mr F. Hendriks in the second number of the Assurance Magazine, pp. 15–17. Dale's table, as there quoted, differs from the one above described in that it commences only at the age of 50, and that he has tabulated l x v - K instead of Ijf. Hesays, " These calculations being made for the use of the societies in particular who commence annuitants at the age of 50, it was not thought necessary to begin the tables at a younger age." He gives, however, another table based on different mortality observations, commencing at the age of 40; and in this case he tabulates l j .v* to . His table also differs from the common form in that it is adapted to find the values of annuities payable by half-yearly instalments.
The next work in which a commutation table is found is William Morgan s Treatise on Assurances, 1779. In this work the values of - z - ^ c l are tabulated, and not those of l x v*; but, as above mentioned, the properties of the table are not altered by the change. Morgan gives the table as furnishing a convenient means of checking the correctness of the values of annuities found by the ordinary process. It may be assumed that he was aware that the table might be used for the direct calculation of annuities; but he appears to have been ignorant of its other uses.
The first author who fully developed the powers of the table w r as John Nicholas Tetens, a native of Schleswig, who in 1785, while professor of philosophy and mathematics at Kiel, published in the German language an Introduction to the Calculation of Life Annuities and Assurances. This work appears to have been quite unknown in England until Mr F. Hendriks gave, in the first number of the Assurance Magazine, pp. 1-20 (Sept. 1850), an account of it, with a translation of the passages describing the con struction and use of the commutation table, and a sketch of the aiithor s life and writings, to which we refer the reader who desires fuller information.
The use of the commutation table w r as independently developed in England apparently between the years 1788 and 1811 by George Barrett, of Pet worth, Sussex, who was the son of a yeoman farmer, and was himself a village schoolmaster, and afterwards farm steward or bailiff. .In the form of table employed by him, the quantity tabulated is not Ijf, but l x (\ + i)" x , where 10 is the last age in the mortality table used. It has been usual to consider Barrett as the originator in this country of the method of calculating the values of annuities by means of a commu tation table, and this method is accordingly sometimes called Barrett s method. (It is also called the commuta tion method and the columnar method. ) Barrett s method of calculating annuities was explained by him to Francis Baily in the year 1811, and was first made known to the world in a paper written by the latter and read before the Royal Society in 1812.
By what has been universally considered an unfortunate error of judgment, this paper was not recommended by the council of the Royal Society to be printed, but it was given by Baily as an appendix to the second issue (in 1813) of his work on life annuities and assurances. Bar rett had calculated extensive tables, and with Baily s aid attempted to get them published by subscription, but with out success; and the only printed tables calculated accord ing to his manner, besides the specimen tables given by Baily, are the tables contained in Babbage s Comparative View of the various Institutions for the Assurance of Lives, 1826. It may be mentioned here that Tetens also gave only a specimen table, apparently not imagining that per sons using his work would find it extremely useful to have a series of commutation tables, calculated and printed ready for use.
of Life Contingencies, a work which contains, among other tables, two arranged on the plan we have above explained, the idea of them having been confessedly derived from
Baily s explanation of Barrett s tables. The method wa?,