4 months, 4007. on 5 months, and 1,1207. on 6 months; find the equated time of payment. 23. At what advance on cost must a merchant mark his goods, so that, after allowing 6 per cent. of his sales for bad debts, 7 per cent. of the cost for expenses, and an average credit of 6 months (money being worth 6 per cent.), he may make a clear gain of 15 per cent. on the first cost of the goods? SECTION VI.-COMPOUND INTEREST AND ANNUITIES. ART. 32.-Rate of Interest. Interest is said to be compound when the principal is allowed to grow by the continual addition of the interest at the end of a specified interval of time. Hence to the specification of the rate an additional specification is added -payable yearly, or half-yearly, as the case may be. If the rate of interest is given in terms of the year, as, for example, 5£ interest per 100£ principal per year, while the interval at which the interest is to be added to the principal is the half-year, we ought first to convert the rate into terms of the half-year by dividing by 2. In the above case, or 2.5£ interest per 100£ principal per half-year, £ interest per £ principal per half-year. Suppose that we have reduced the rate of interest to the form r£ interest per £ principal per interval, where the interval referred to is the interval between the times at which the interest becomes due; then (1+r)£ at beginning of 2nd interval = £ at beginning of 1st; and 1 £ at beginning of 1st = £ at beginning of 2nd. 1 + r ART. 33.-Future and Present Value. Suppose that a sum of money is lent at compound interest, for the first interval at p£ interest per £ principal per interval, for the second at q£ interest per £ principal per interval, for the third at then we have r£ interest per £ principal per interval; (1+p)£ at beginning of 2nd interval = £ at beginning of 1st, (1+q)£ at beginning of 3rd interval = £ at beginning of 2nd, (1+r)£ at beginning of 4th interval = £ at beginning of 3rd. Hence by rule Art. 10, (1 + p)(1 + q)(1 + r)£ at beginning of 4th interval = £ at beginning of 1st, or (1+p)(1 + q)(1+r)£ at end of 3 intervals = £ at beginning of 3 intervals. If the rates of interest are the same, that is, if p=q=r, the rate of growth or of future value becomes (1+r)3£ at end of 3 intervals = £ at beginning. And generally for n intervals (1 + r)"£ at end of n intervals = £ at beginning. The reciprocal rate-the rate of present value-is 1 (1+r)n £ at beginning of n intervals = £ at end. When the period during which the interest accumulates comprises an integral number of intervals and a fraction of an interval, the calculation for the fraction of an interval is the same as for simple interest. or ART. 34.-True Discount. The rate of increment is— {(1+r)" -1}£ increment = £ at beginning, The latter put into the form £ discount = £ debt due at end of n intervals (1 + is the rate for calculating true discount when compound interest is reckoned. if we take only the first two terms of this expansion, we get for the rate of discount (1 − 1)£ discount = £ debt at end of n intervals, 1 which is the same as the rate derived from reckoning interest simply. (Art. 30.) ART. 35.-Annuity. By an Annuity is meant a uniform payment made at equal intervals of time, generally a year or a half-year. It is specified in terms of £ paid per year, with an additional specification of payable yearly, or half-yearly, as the case may be. We shall suppose that it is payable yearly, but the same reasoning applies to any other interval of payment. ART. 36.-Amount of an Annuity. To find the rate for the amount at the end of n years of an annuity payable yearly, and which has been paid at the end of each year in the period. By the previous Article, (1+r)"-1£ at end of n years = £ paid at end of 1st year, The above terms form a geometric progression, of which (1+r) is the common ratio; hence their sum is (1+r)" - 1 r Hence (1 + r)" − 1 £ at end of n years = £ paid per year, for n years. The reciprocal rate is r £ paid per year for n years = £ at end of n years; (1 + r)" − 1 from it we find what annuity paid for n years is equivalent to a given sum paid at the end of the n years. ART. 37.-Reversion of an Annuity. Suppose that an annuity will begin to be reaped s years hence, and will continue to be reaped for n years. What is its present value? By the previous Article, (1 + r)" − 1 £ at end of period = £ per year. 1£ at present = (1 + r)"+8£ at end of period; (1+r)~+,£ at present = £ per year. 1 When the present time is the beginning of the period of n years, s=0, and 1 1 (1 +r)” £ at beginning of n years = £ per year. r ART. 38.-Perpetual Annuity. By a perpetual annuity is meant an annuity which is payable for an indefinitely great number of years, or half-years, as the case may be. We have seen that the value at the beginning of the period of payment is 1 1 (1+r)”£ at beginning = £ per year for n years. r When n, the number of years, is very great 1/(1+r)" approximates to 0; hence, for a period containing a very large number of years of payment, 1£ at beginning = £ per year for ever. Suppose that the annuity has been paid for a finite number of years, the value of the remainder when it begins to be paid is still 1£ at present = £ per year for ever. The present value of any permanent property is connected with its annual value by the above rate. The value 1/r gives the number of years' purchase. It is considered as so many years, which, multiplying the annual income, gives the present worth of the property. Similarly, according to the preceding article, is the number of years' purchase for a uniform annual payment extending over n years. In the following table G denotes any unit of value. The table of entries, though short, is sufficient to give, with the aid of one multiplication, the entry for any year up to 59. For example, take 36 years. Multiply together the values for 30 and for 6 by the method of contracted multiplication. (See Example 2 following.) RATE OF IMPROVEMENT OF MONEY AT COMPOUND INTEREST. |