Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/83

whatever monetary unit he pleases, whether pound, dollar, franc, thaler, &c. It is much to be desired that this course should be followed in any tables that may be published in future.

The annuity, it will be observed, is the totality of the payments to be made (and received), and is so understood by all writers on the subject; but some have also used the word to denote an individual payment (or rent), speaking, for instance, of the first or second year s annuity, a practice which is calculated to introduce confusion, and should therefore be carefully avoided.

The theory of annuities certain is a simple application of algebra to the fundamental idea of compound interest, According to this idea, any sum of money invested, or put out at interest, is increased at the end of a year "by the addition to it of interest at a certain rate; and at the end of a second year, the interest of the first year, as well as the original sum, is increased in the same proportion, and so on to the end of the last year, the interest being, in technical language, converted into principal yearly. Thus, if the rate of interest is 5 per cent,, 1 improved at interest will amount at the end of a year to 1 05, or, as we shall in future say, in conformity with a previous remark, 1 will at the end of a year amount to 1 05. At the end of a second year this will be increased in the same ratio, and then amount to (1 05) 2 . In the same way, at the end of a third year, it will amount to (1 05) 3 , and so on.}}

Let i denote the interest on 1 for a year ; then at the end of a year the amount of 1 will be 1 + i. Reasoning as above, at the end of two years the amount will be (1 +i), at the end of three years (1 + i) 3 , and so on. In general, at the end of n years the amount will be (1 +i)"; or this is the amount of 1 at compound interest in n years. The present value of a sum, say 1, payable at the end of n years, is such a sum as, being improved at compound interest for n years, will exactly amount to 1. We have seen that 1 will in n years amount to (1 +t)", and by pro portion we easily see that the sum which in n years will amount to 1, must be r- , or (1 + i) n . It is usual to (l-M)" put v for r-, so that v is the value of 1 to be received at the end of a year, and v* the value of 1 to be received at the end of n years.

{{ti|1em|Since 1 placed out at interest produces i each year, we see that a perpetuity of i is equal in value to 1 ; hence, by proportion, a perpetuity of 1 is equal in value to - . At i 5 per cent, ieq -05, and - = 20; or a perpetuity is worth 1> 20 years purchase : at 4 per cent., it is worth 25 years purchase, ( eq 25 j : at 3 per cent., it is worth 33J years purchase, (.^ eq 33^.

Instances of perpetuities are the dividends upon the public stocks in England, France, and some other coun tries. Thus, although it is usual to speak of 100 con sols, this 100 is a mere conception or ideal sum; and the reality is the 3 a year which the Government pays by half-yearly instalments. The practice of the French in this, as in many other matters, is more logical. In speak ing of their public funds, they do not mention the ideal capital sum, but speak of the annuity or annual payment that is received by the public creditor. Other instances of perpetuities are the incomes derived from the deben ture stocks now issued so largely by various railway com panies, also the feu-duties commonly payable on house property in Scotland. The number of years purchase which the perpetual annuities granted by a government or a railway company realise in the open market, forms a very simple test of the credit of the various governments or rail ways. Thus at the present time (May 1874) the British per petual annuity of 3, derived from the 3 per cent, consols, is worth 93, or 31 years purchase; and a purchaser thus obtains 3-226 per cent, interest on his investment. Other examples are given in the subjoined tables, the figures in which are deduced from the Stock Exchange quotations of the irredeemable stocks issued by the various governments:

No. of Years	No. of Years' Purchase.
British,	31⋅00
Dutch,	23⋅69
Swedish,	21⋅20
Russian,	20⋅40
French,	19⋅67
Brazilian,	20⋅00
Portuguese,	15⋅42
Argentine,	13⋅50
Austrian,	13⋅40
Italian,	13⋅00
Turkish,	9⋅50
Spanish,	6⋅67
Venezuelan,	3⋅50

The following are a few other examples of perpetuities:—

Interest per cent, yielded to a Purchaser. 3-23 4 22 4-72 4-90 5-08 5-00 6-48 7-41 7-46 7-69 10-53 15-00 23-57 Name, Metropolitan Board of "Works Stock, 27 50 London and N."W. Railway Deben- C1.L 1 ture Stock, North British Railway Debenture r*i_ 1 Stock, Edinburgh Water Annuities, 22 53 Interest per cent, yielded to a Purchaser 3-64 3-83 4-25 4-43

We may mention in passing that the more usual practice of foreign governments when borrowing is not to grant the lender a perpetual annuity, but to issue to him bonds of say 100 each, bearing an agreed rate of interest, these bonds being usually issued at a discount, and redeemed at par by annual drawings during a specified term of years. We have seen that the present value of any sum payable at the end of n years is found by multiplying it by (l+i)"; hence the value of a perpetuity of 1 deferred n years is r Now an annuity for n years is clearly the difference between the value of a perpetuity to com mence at once and a perpetuity deferred n years ; its value is therefore - - r eq r ; or putting a for the value of the annuity, we have

i-i. f l (!+*)"" l-(l + t)-" ... , ,, I/ If If

By means of this equation, having any two of the three quantities, a, i, n, we can determine the third either exactly or approximately. Thus for n we have

log -(I- to) log/l + t)

There is no means of determining the value of i exactly, but it may be found to any degree of accuracy required by methods of approximation which our limits will not allow us to describe.^[1]

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1. A similar difficulty meets us in many other cases, as, for instance, when we wish to determine the rate of interest, having given the amount of an annuity for a given term, or the value of an annuity 01 perpetuity deferred for a certain number of years. The reader who wishes to pursue this subject is referred to Francis Baily s Doctrine of Interest and Annuities, 1808, in the appendix to which the formulas of previous authors are examined, and new ones, which are at once simpler and more correct, are demonstrated. Particular cases of the problem are considered in Turnbull s Tables of Compound Interest and Annuities, p. 132, and in various papers in the Assurance Maga zine, among which may be specified De Morgan s paper " On the Deter mination of the Rate of Interest of an Annuity," vol. viii. p. 61, and a letter by J. M Lauchlan in the eighteenth volume. The analogous problem of determining the rate of interest in the bonds of foreign governments above mentioned, debentures, and similar securities, has been fully treated of by Gray, Ass. Mag., xiv. 91, 182, 897 ; and by Makeham, xviii. 132.