Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/89

1000 nominees will be found by taking twenty-two times the value of an annuity for 1 year, eighteen times the value of an annuity for 2 years, fifteen times the value of an annuity for 3 years, and so on, the last term being the value of 1 annuity for 91 years, and adding them together ; and the value of an annuity on one of the nominees will then be found by dividing by 1000. Before leaving the subject of De Wit, we may mention that we find in the correspondence a distinct suggestion of the law of mortality that bears the name of Demoivre. In De Wit s letter, dated 27th October 1671 (Ass. Mag., vol. iii. p. 107), he speaks of a "provisional hypothesis" suggested by Hudde, that out of 80 young lives (who, from the context, may be taken as of the age 6) about 1 dies annually. In strictness, therefore, the law in question

might be more correctly termed Hudde s than Demoivre s.

De Wit s report being thus of the nature of an unpub lished state paper, although it contributed to its author s reputation, did not contribute to advance the exact know ledge 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 Dr Edmund Halley, F.R.S. In the Philosophical Transactions, Nos. 196 and 198 (January and March 1693), he gave the first approxi mately correct mortality table (deduced from the records of the numbers of deaths and baptisms in the city of Breslaii), and showed how it might be employed to calculate the value of an annuity on the life of a nominee of any age. His method of procedure exactly agrees with the formula (1) above given; and while he confesses that it requires a series of laborious calculations, he says that he had sought in vain for a more concise method. His papers, which are full of interest, are reprinted in the eighteenth volume of the Assurance Magazine.

Previous 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 annui ties rose put 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. JEmilius Macer (230 A.D.) 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 one of the most eminent commentators on the Justinian Code 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 ) 43 44 44 45 16 15 upwards ) 5

Here also we have no reason to suppose that the element of interest was taken into consideration ; and the assump tion, 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 1 7th century. Fuller information upon the early history of life annuities will be found in the article " Annuities on Lives, History of," in Mr Walford s Insurance Cyclopaedia.

Demoivre, in his Treatise on Annuities, 1725, showed that it was unnecessary to go through the whole of tho calculation indicated by the formula (1) or (2) for each age, and that the value of an annuity at any age might be deduced by a simple process from that at the next older age. This may be demonstrated as follows : If it were certain that a person of any age, say 39, would live for a year, then the value of an annuity on his life would be such a sum as would increase at interest in a year to the value of an annuity on a life one year older, say 40, increased by a present payment of 1 ; that is, putting a for the value of an annuity and 3 a for that on a life one year older, the value would be v (1 + l a). But it is un certain that the life will exist to the end of a year, and the value of the annuity must therefore be reduced in the proportion of this uncertainty, or be multiplied by the probability that the given life will survive a year. Putting then p to denote this probability, we have a = vp (1 + l a). This formula may also be demonstrated algebraically. We have seen that

[ math ]

where z is the difference between the age of the given life and that of the oldest in the mortality table. (Assuming the present age to be 39, then in the English Table No. 3, Males, 2 will be!07-39 &eq 68.) In the same way, we have

H H Hence a &eq v

—the same result as already proved.

If we suppose the present age to be x, we may put the formula in the shape

• &eq * -r- l (i + *+i); *

but it will be found preferable to omit the subscript x whenever this can be done without risk of confusion.

This formula has been commonly attributed to Simpson, who in 1742 published his Doctrine of Annuities and Re versions ; but, although he certainly showed that it is applicable to annuities on the joint duration of two or more lives, the first discovery of it is undoiibtedly due to Demoivre. (See Farren s Historical Essay on the Use and early Progress of the Doctrine of Life Contingencies in England, p. 46.) The formula appears to have been in dependently discovered by Euler, and was given by him in a paper in the Memoirs of the Royal Academy of Sciences at Berlin, for the year 1760.

Mr Peter Gray has shown in his Tables and Formula?,

1849, how Gauss s logarithmic table may be advantage ously employed in calculating the values of annuities

by the above formula. That table gives us the value of