Reduced, gives 24 Livres 18 Sous per Pound Sterling. The circular operation is therefore more advantageous than the direct exchange, as London pays, with £1 Sterling, 24 Livres 18 Sous, instead of 24 Livres 5 Sous. 567. Hitherto we have only examined one combination of the exchanges between several places in order to discover one result. In the following example different combinations of the same exchanges are examined, to find which is most favourable; and what is here performed with four places only, may be done with any greater number in the same manner; that is, by trial and comparison. London on Suppose the following to be the quotations of exchange: Amsterdam 35 s. Flemish per Pound Sterling. Madrid..... 38 d. Sterling per Dollar of Plate. Paris....... 24 Livres per Pound Sterling. 66 66 Amsterdam on London Paris on Madrid on .... 34 s. Flemish per Pound Sterling. Paris. 53 Grotes per Ecu of 3 Francs. Madrid..... 92 Grotes per Ducat of Plate. London..... 231 Livres per Pound Sterling. Madrid..... 16 Francs per Doubloon of Plate. Amsterdam. 54 Grotes Fl. per Ecu of 3 Francs. London.... 39 d. Sterling per Dollar of Plate. Amsterdam. 94 Grotes per Ducat of Plate. Paris........ 161⁄2 Francs per Doubloon of Plate. Now if London has a sum to receive in Madrid, what mode of operation will be most advantageous? By drawing directly on Madrid, London will receive 38 d. Sterling per Dollar. 1. Let London draw on Amsterdam, directing Amsterdam to draw on Paris, and Madrid to remit to Paris; then Reduced, gives the Dollar equal to 423 d. Sterling. 2. Let London draw on Paris, directing Paris to draw on Amsterdam, and Madrid to remit to Amsterdam; then 240 d. Sterling. Reduced, gives 391 d. per Dollar. 3. Let London draw on Madrid, and remit the bill to Paris to be negotiated, and let the returns be made in a bill on Amsterdam; then 4 Dollars 1 Doubloon 3 Francs 12 Grotes 35 s. Flemish = 1 Dollar of Plate, 16 Francs, 54 Grotes Flemish, 1 s. Flemish, 240 d. Sterling. Reduced, gives 414 d. Sterling per Dollar. 4. Let London draw on Madrid, and remit to be negotiated in Amsterdam, and order the returns to be made in a bill on Paris; then Reduced, gives 381 d. Sterling per Dollar, nearly. 5. Let London draw on Amsterdam, Amsterdam on Paris, and Paris on Madrid; then Reduced, gives 40,8 d. per Dollar. 6. Let Madrid remit to Paris, Paris to Amsterdam, and Amsterdam to London; then 240 d. Sterling. Reduced, gives 433 d. per Dollar, nearly. Several other combinations might be made, but the foregoing are sufficient for the present illustration. In recapitulating the different combinations here proposed, we find that Hence, it is evident, according to the rules laid down, (555,) that the 6th operation would be the most advantageous for the London drawer, as he would receive more Pence for the Dollar than by any other combination; but it must be observed that in this case, he would have to wait about six months for the remittance from Madrid through Paris and Amsterdam; whereas, by drawing directly on Madrid, he would receive the money immediately; consequently, interest should be deducted from the profit, besides commission, postage, &c. The Profit is, however, so great in this case, as far to exceed the additional expenses; thus: 38:43 100: 1155. Profit. 155 per cent. It is also evident, that if London was the debtor, and had to remit to Madrid, the lowest course of exchange, which is here the direct one, should be adopted. Compound Interest. 568. When the interest of a sum of money, at the time it is due, is, instead of being paid, added to the principal, and the sum considered a new principal on which interest is computed as before, and again when due, added to form a new principal, and so on, the accumulation of interest thence arising, is called Compound Interest. This accumulation is so much the greater as the increment of time for which it is calculated is shorter. Let P be the principal and the rate p. c. per annum ; then 100+r. XP, the amount for one year. 100: 100+r: :P : 100 Hence we say: Multiply the principal by the fraction 100+ the rate reduced to its lowest terms, or to a decimal, 100 or otherwise, as may be most convenient. The product or amount, which is the new principal, multi100+ r will evidently give the amount for the 100 plied by second year; and continuing thus, it is plain that PX 100+ r 100 = amount for 2 100 + ")" the General Rule: Multiply the principal by 100+ r 100 raised to a power signified by the number of years, half years, or other portions of time, for which the interest is calculated. 1. What is the compound interest of $340 for 5 years at 5 p. c. ? Lastly, 433,94 - 340 = $93,94, the interest required. Or, Multiply successively by the decimal amount of 1 unit. Thus: 340 X 1,05= $357 Amount for one year; then Or, multiply the logarithm of the amount of 1 unit by the number of years or times specified, and to the product add the log. of the principal, which will give the log. of the amount. Thus : 7. 1,05.........0,021189 × 5=0,105945 This log. gives the amount $433,934, which is very nearly the true result. The student will see, that to multiply the log. of 1,05 by 5, is (474) the same as to raise to the 5th power. Also (472), that to add the log. of 340, is the same as to multiply (1) by that number. 2. What is the compound interest of $340 for 5 years at 5 p. c., the interest payable half-yearly. Ans. $95,23. The student may verify his results as in Ex. 1. 3. What is the amount of £587 16 s. Sterling at compound interest for 6 yrs. 9 mos. at 6 p. c.? Ans. £871 6 s. 61 d. Use of Geometrical Progression in the Calculation of Compound Interest.* 569. From the preceding article, where P is assumed as the principal, and the rate p. c., it is evident that P: PX ·100+r 100 2 r PX (100+): P× (1006+ ")", &c., is a geometrical progression. Hence we infer, that taking the principal for the first term, every series in the calculation of compound interest is a geometrical progression. *See Table, art. 465. |