Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/641

 27. Ex. 2. What is the present value of an annuity of L. 50 for 21 years, receivable in equal half-yearly payments, when money yields an interest of 2½ per cent. every half year?

By Table II. it appears, that an annuity of L. 1 for 42 years, when the interest of money is 2½ ''per cent. per annum'', will be worth L. 25·8206 (25); 25 times this sum, or L. 645, 10s. 3½d. is therefore, the required value, and exceeds the value when the interest and the annuity are only payable once a-year by L. 4, 9s. 1d. (18).

28. The excess of an annuity-certain above the interest of the purchase-money, is the sum which, being put out at the time of each payment becoming due, and improved at compound interest until the expiration of the term, will just amount to the purchase-money originally paid.

But, while everything else remains the same, the longer the term of the annuity is, the less must its excess above the interest of the purchase-money be, because a less annuity will suffice for raising the same sum within the term. Therefore, the proportion of that excess to the annual interest of the purchase-money, continually diminishes as the term is extended; and when the annuity is a perpetuity, there is no such excess (8).

29. The reason why the value of an annuity is increased by that and the interest being both payable more than once in the year, is, that the grantor loses, and the purchaser gains, the interest produced by that part of each payment, which is in excess above the interest then due upon the purchase-money, from the time of such payment being made, until the expiration of the year.

Hence it is obvious, that the less this excess is, that is, the longer the term of the annuity is (28), the less must the increase of value be.

And when the annuity is a perpetuity, its value will be the same, whether it and the interest of money be both payable several times in the year, or once only.

30. When the annuity is not payable at the same intervals at which the interest is convertible into principal, its value will depend upon the frequencies both of payment and conversion; but its investigation without algebra, would be too tong, and of too little use, to be worth prosecuting here.

II. OF ANNUITIES ON LIVES.

31. When the payment of an annuity depends upon the existence of some life or lives, it is called a Life-annuity.

32. The values of such annuities are calculated by means of tables of mortality, which show, out of a considerable number of individuals born, how many upon an average have lived to complete each year of their age; and, consequently, what proportion of those who attained to any one age, have survived any greater age.

The fifth Table at the end of this article is one of that kind, which has been taken from Mr Milne’s Treatise on Annuities, and was constructed from accurate observations made at Carlisle by Dr Heysham, during a period of 9 years, ending with 1787.

33. By this table it appears, that during the period in which these observations were made; out of 10,000 children born, 3203 died under 5 years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to 5 years of age, 6460 survived their 10th year.

But the mortality under 10 years of age, has been greatly reduced since then, by the practice of vaccination.

This table also shows, that of 6460 individuals who attained to 10 years of age, 6047 survived 21. And that of 5075 who attained to 40, only 3643 survived their 60th year.

34. There is good reason to believe (as has been shown in another place), that the general law of mortality, that is, the average proportion of persons attaining to any one age, who survive any greater age, remains much the same now among the entire mass of the people throughout England, as it was found te be at Carlisle during the period of these observations; except among children under 10 years of age, as was noticed above (33).

If this be so, it will follow, that of 6460 children now 10 years of age, just 6047 will attain to 21; or rather, that if any great number be taken in several instances, this $$\left ( \frac{6047}{6460} \right )$$ will be the average proportion of them that will survive the period.

And if 6460 children were to be taken indiscriminately from the general mass of the population at 10 years of age, and an office or company were to engage to pay L. 1, eleven years hence, for each of them that might then be living; this engagement would be equivalent to that which should bind them to pay L. 6047 certainly, at the expiration of the term. Therefore, the office, in order that it might neither gain nor lose by the engagement, should, upon entering into it, be paid for the whole, the present value of L. 6047, to be received at the expiration of 11 years; and for each life, the $1⁄6460$th part of it; that is, the $6047⁄6460$th part of the present value of L. 1 to be received then.

But when the rate of interest is 5 per cent. the present value of L. 1, to be received at the expiration of 11 years, is L. 0·584679; therefore, at that rate of interest, there should be paid for each life $$\frac{6047 \times 0{\cdot}584679}{6460}$$ = L. 0·5473.

And the present value of L. 100, to be received upon a life now 10 years of age attaining to 21, will be L. 54·73, or L. 54, 14s. 7d.

In the same manner it will be found, that reckoning interest at 4 per cent. the value would be L. 60, 16s. 1d.

35. This is the method of calculating the present values of endowments for children of given ages; and the values of annuities on lives may be computed in the same manner.

For, from the above reasoning it is manifest, that if the present value of L. 1, to be received certainly at the expiration of a given term, be multiplied by the number in the table of mortality against the