Interest — Chambers's Encyclopaedia

Interest is the consideration paid for the use of money. The interest of £100 for one year is called the rate per cent.; the money lent, the principal; and the sum of any principal and its interest, the amount. The current or market rate of interest varies from a variety of causes, the chief of which are the relation existing between the accumulation of money and the demands of borrowers, the prevailing rate of profits on trade, and the security and duration of the loan. In Great Britain the price of the public funds indicates the interest obtainable for a permanent loan with no risk of loss, while the 'bank rate'—i.e. the minimum rate at which the Bank of England will discount bills—represents the interest for temporary loans with less undoubted security. In the former case, as with fixed annuities, the nominal rate of interest never varies; but the real return to the investor depends on the price he has to pay for the capital. Thus, if the price of 2½ per cent. consols be 91¾, the actual return will be 3 per cent.

Interest is computed on either of two principles. SIMPLE INTEREST is charged on the principal alone for any length of time. The computation of simple interest is easy, resolving itself into a mere question of proportion: thus, having given the interest on £100 for 1 year, to find the interest on any other sum for any period. Various ingenious devices are made use of to save labour in these calculations, especially by bankers, and are given in most handbooks. COMPOUND INTEREST is the charge made where—the interest not being paid when due—it is added to the principal, forming the amount upon which the subsequent year's interest is computed. The rules for most readily making computations by compound interest can only be effectively expressed algebraically, and, using $i$ to represent the interest of £1 for one year, and $n$ the number of years, we annex a few of the elementary formulas for £1, from which the result for any sum is obtained by simple multiplication.

(1) Amount of £1 for a given time at compound interest.—At the end of the first year the principal (£1) with its interest ( $i$ ) will become $1 + i$ . At the end of the second year the amount will be $(1 + i) + i(1 + i)$ , or more simply $(1 + i)^2$ , and, generally, the amount of £1 in $n$ years is $(1 + i)^n$ . Example: To find the amount of £6 in 20 years at 5 per cent. interest. Here $i$ is .05 and $n$ is 20, whence the required amount is $6 \times 1.05^{20} =$ (by logarithms) $6 \times 2.65 = £15.18$ .

(2) Present value of £1 due $n$ years hence.—Since £1 becomes $1 + i$ in one year, by proportion $\frac{1}{1 + i}$ , otherwise written $(1 + i)^{-1}$ or $v$ , will become £1 in the same time, and hence the present value of £1 due $n$ years hence is $(1 + i)^{-n}$ (or $v^n$ ).

At 5 per cent. simple interest a sum of money doubles itself in 20 years, while at compound interest with the same rate it takes less than 15 years. In 100 years £1 at 5 per cent. simple interest becomes £6; at 5 per cent. compound interest it becomes £131, 10s., or thereby.

(3) ANNUITIES CERTAIN.—Amount of an Annuity of £1 in $n$ years.—At the end of the $n$ years the last year's annuity will be due, and therefore worth £1; the second-last will be worth one year's interest in addition, or $1 + i$ ; the third (reckoning backwards), $(1 + i)^2$ ; and so on to the first year's annuity, which will amount to $(1 + i)^{n-1}$ . The amount required is therefore the sum of the geometrical series $1 + (1 + i) + (1 + i)^2 + \dots + (1 + i)^{n-1}$ ; or, $\frac{(1 + i)^n - 1}{i}$ .

(4) Present Value of an Annuity.—This is easily found from (3), as the result there found must evidently be the present value, improved at compound interest—i.e. multiplied by $(1 + i)^n$ . Hence the present value is $\frac{1 - (1 + i)^{-n}}{i}$ ; or, $\frac{1 - v^n}{i}$ .

Tables for the four classes of values above described, based on various rates of interest, are given in most works on annuities and other handbooks; and various useful results, besides those immediately intended, can readily be deduced from them.—The calculation of Life Annuities is complicated by the element of the probability of life, and is treated under ANNUITY.

INTEREST, IN LAW.—The charging of interest was formerly looked upon with great disfavour, and was either forbidden or restricted by the Usury Laws (q.v.), which were not finally repealed till 1839. In English law there is no obligation imposed on the debtor to pay any interest whatever, though the sum has been long due and often demanded. The creditor can always sue for his debt, which is his proper remedy, but he derives no benefit from giving time to his debtor. Therefore, if interest is to be paid, this must be, as a general rule, by virtue of express agreement. A tacit agreement, however, would be presumed and given effect to where it could be proved to be a custom between the parties, or the usage of a particular trade to allow interest. Thus, by the usage of merchants, it has always been usual, when an action has been brought to recover the amount of a bill of exchange or promissory-note, for the jury to add interest from the time it was due. In the case of money due upon an award by an arbitrator interest is due from the day when the award was made. Where money is due on a bond also interest is added from the day it ought to have been paid; and if a surety has to pay money for his principal he can recover it back with interest. In all other cases, if there was no express agreement about interest, none could be claimed. By 3 and 4 Will. IV. chap. 42, sec. 28, a jury may now add interest at the ordinary rate on all debts or sums certain, which are made payable under some written instrument at a certain time; and even if not due under a written instrument, then if a written demand has been made, expressly giving notice that interest will be charged from and after the date of the demand if not paid then, interest will also be due. But even in these last cases it is discretionary in the jury to give the interest, and therefore it is not claimable as a matter of course. As regards compound interest, it is a fortiori not claimable in any case, except where it has been expressly stipulated for, or where there is in some particular trade a definite custom to pay interest, and such custom must always be proved. The courts generally name 4 per cent. when interest is decreed for, but sometimes 5 per cent.; and where funds have been misapplied the Court of Chancery charges compound interest at 5 per cent. Pawnbrokers are allowed to charge interest not exceeding a fixed sum. See PAWNBROKING.

In Scotland the law has always been much more liberal in allowing interest to be claimed on outstanding debts, for there the converse principle was acted on, that on nearly all debts whatever interest was claimable either by statute or by common law. Thus, interest is due on bills of exchange, on the amount contained in a horning or charge to pay, on sums paid by cautioners, on the price of lands sold, on money advanced at request, on the price of goods sold if the usual time of credit has expired, and generally on all debts when payment is due and has been demanded. In certain cases principal and interest to a fixed date are accumulated into a capital sum on which interest runs; and the House of Lords, on appeal, may give decree for compound interest. The courts charge penal interest at the rate of 20 per cent. against factors and trustees who illegally retain trust funds in their own hands.

In the United States the legal rates of interest vary in the different states and territories, from 5 per cent. in Louisiana to 12 in Wyoming, but in the majority 6 per cent. is the legal rate. In most of the states there are penalties for usury, ranging from forfeiture of the excess of interest to forfeiture of principal and interest. But a higher rate, and in many cases any rate, is allowed by contract.