De Witt’s report being thus of the nature of an unpublished state paper, although it contributed to its author’s reputation, did not contribute to advance the exact knowledge of the subject; and the author to whom the credit must be given of first showing how to calculate the value of an annuity on correct principles is Edmund Halley. He gave the first approximately correct mortality table (deduced from the records of the numbers of deaths and baptisms in the city of Breslau), and showed how it might be employed to calculate the value of an annuity on the life of a nominee of any age (see Phil. Trans. 1693; Ass. Mag. vol. xviii.).
Previously to Halley’s time, and apparently for many years subsequently, all dealings with life annuities were based upon mere conjectural estimates. The earliest known reference to any estimate of the value of life annuities rose out of the requirements of the Falcidian law, which (40 B.C.) was adopted in the Roman empire, and which declared that a testator should not give more than three-fourths of his property in legacies, so that at least one-fourth must go to his legal representatives. It is easy to see how it would occasionally become necessary, while this law was in force, to value life annuities charged upon a testator’s estate. Aemilius Macer (A.D. 230) states that the method which had been in common use at that time was as follows:—From the earliest age until 30 take 30 years’ purchase, and for each age after 30 deduct 1 year. It is obvious that no consideration of compound interest can have entered into this estimate; and it is easy to see that it is equivalent to assuming that all persons who attain the age of 30 will certainly live to the age of 60, and then certainly die. Compared with this estimate, that which was propounded by the praetorian prefect Ulpian was a great improvement. His table is as follows:—
| Age. | Years’ Purchase. | Age. | Years’ Purchase. |
| Birth to 20 | 30 | 45 to 46 | 14 |
| 20 ” 25 | 28 | 46 ” 47 | 13 |
| 25 ” 30 | 25 | 47 ” 48 | 12 |
| 30 ” 35 | 22 | 48 ” 49 | 11 |
| 35 ” 40 | 20 | 49 ” 50 | 10 |
| 40 ” 41 | 19 | 50 ” 55 | 9 |
| 41 ” 42 | 18 | 55 ” 60 | 7 |
| 42 ” 43 | 17 | 60 and upwards | 5 |
| 43 ” 44 | 16 | ||
| 44 ” 45 | 15 |
Here also we have no reason to suppose that the element of interest was taken into consideration; and the assumption, that between the ages of 40 and 50 each addition of a year to the nominee’s age diminishes the value of the annuity by one year’s purchase, is equivalent to assuming that there is no probability of the nominee dying between the ages of 40 and 50. Considered, however, simply as a table of the average duration of life, the values are fairly accurate. At all events, no more correct estimate appears to have been arrived at until the close of the 17th century.
The mathematics of annuities has been very fully treated in Demoivre’s Treatise on Annuities (1725); Simpson’s Doctrine of Annuities and Reversions (1742); P. Gray, Tables and Formulae; Baily’s Doctrine of Life Annuities; there are also innumerable compilations of Valuation Tables and Interest Tables, by means of which the value of an annuity at any age and any rate of interest may be found. See also the article [Interest], and especially that on [Insurance].
Commutation tables, aptly so named in 1840 by Augustus De Morgan (see his paper “On the Calculation of Single Life Contingencies,” Assurance Magazine, xii. 328), show 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. 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 F. Hendriks in the second number of the Assurance Magazine, pp. 15-17. William Morgan’s Treatise on Assurances, 1779, also contains a commutation table. 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 was 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 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 construction and use of the commutation table, and a sketch of the author’s life and writings, to which we refer the reader who desires fuller information. It may be mentioned here that Tetens also gave only a specimen table, apparently not imagining that persons using his work would find it extremely useful to have a series of commutation tables, calculated and printed ready for use.
The use of the commutation table was independently developed in England-apparently between the years 1788 and 1811— by George Barrett, of Petworth, Sussex, who was the son of a yeoman farmer, and was himself a village schoolmaster, and afterwards farm steward or bailiff. It has been usual to consider Barrett as the originator in England of the method of calculating the values of annuities by means of a commutation table, and this method is accordingly sometimes called Barrett’s method. (It is also called the commutation 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. Barrett had calculated extensive tables, and with Baily’s aid attempted to get them published by subscription, but without success; and the only printed tables calculated according 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.
In the year 1825 Griffith Davies published his Tables of Life Contingencies, a work which contains, among others, two tables, which are confessedly derived from Baily’s explanation of Barrett’s tables.