It will be found convenient to reduce broken sums of money and periods of time to decimals of a pound and of a year, respectively. The amount of the above sum at the end of the given time will be PRESENT VALUE AND DISCOUNT AT SIMPLE INTEREST. The present value of any sum s due t years hence is the principal which in the time t will amount to s. The discount upon any sum due t years hence is the difference between that sum and its present value. PROBLEM III. To find the present value of s pounds due t years hence, simple interest being calculated at the rate r. By formula (2) we find the amount of a sum p at the end of t years to be S = pptr Consequently p will represent the present value of the sum s due t years hence, and we shall have -S Р =3 +tr (3) for the expression required. PROBLEM IV.—To find the discount on s pounds due t years hence, at the rate r, simple interest. *r is the interest of L1 for one year. To find the value of r when interest is calculated at the rate of LA, or L4.75 per cent. per annum, we have the following proportion. Since the discount on s is the difference between s and its present value, we shall have Required the discount on £100, due 3 months hence, interest being calculated at the rate of 5 per cent. per annum. PROBLEM V. To find the amount of an annuity a continued for t years, simple interest being allowed at the rate r upon the successive payments. At the end of the first year the annuity a will be due, at the end of the second year a second payment a will become due, together with a r, the interest for one year upon the first payment, at the end of the third year a third payment a becomes due, together with 2 a r, the interest for one year upon the two former payments, and so on, the sum of all these will be the amount required, Hence, adding these all together for the whole amount, s = ta+ar (1 + 2 + 3 + (-1)) + Or, taking the expression for the sum of the arithmetical series, 1 + 2 + 3 PROBLEM VI.—To find the present value of an annuity a payable for t years, simple interest being allowed at the rate r. It is manifest that the present value of the annuity must be a sum such, that, if put out to interest for t years at the rate r, its amount at the end of that period will be the same with the amount of the annuity. Hence, if we call this present value p, we shall have by Probs. I and V. PROBLEM VII.—To find the amount of a sum p laid out for t years, compound interest being allowed at the rate r. At the end of the first year the amount will be, by Problem II. ppr, or p (1 + r) Since compound interest is allowed, this sum p (1+r) now becomes the principal, and hence, at the end of the second year, the amount will be ? (1+r), together with the interest on p (1+r) for one year; that is, it will be p(1+r) + p r (1+r), or p (1+r)2 The sum p (1+r)2 must now be considered as the principal, and hence the whole amount at the end of the third year will be p(1+r)2+pr (1+r), or p (1+r)2 And, in like manner, at the end of the tth year we shall have s = p(1+r) ' (7) Any three of the four quantities, s, p, r, t, being given, the fourth may always be found from the above equations. Example 1. Find the amount of £15. 10s. for 9 years, compound interest being allowed *It is unnecessary to give any examples on this rule, as the purchase of annuities at simple interest can never be of practical utility. Thus, if we wished to ascertain by this formula the present value of an annuity of L50, to continue for 40 years, calculating interest at 5 per cent., we should find it to be L1316 13s. 4d. But the interest of L1316 13s. 4d. at the same rate is upwards of £65 per annum continued for ever. at the rate of £3 per cent. per annum. The interest payable at the end of Find the amount of £182. 12s. 6d. for 18 years, 6 months, and 10 days, at the rate of 3 per cent. per annum, compound interest; the interest being payable at the end of each year. In this case, it will be convenient, first, to find the amount at compound interest of the above sum for 18 years, and then calculate the interest on the result for the remaining period. Again, to find the interest on this sum for the short period, we have = 2.5304856 = log. 339.224. The whole amount required will therefore be s+str = £339.224+£6.26172 Example 3. Required the compound interest upon £410 for 24 years, at 4 per cent. per annum, the interest being payable half yearly. In this case, the time t must be calculated in half years; and since we have supposed r to be the interest of £1 for one year, we must substitute which will be the interest of £1 for half a year; the formula (7) will thus become 2 t 2' S = p (1 + 1) 2 The interest must be the difference between this amount and the original prin cipal, £400 was put out at compound interest, and at the end of 9 years amounted to £569 6s. 8d; required the rate of interest per cent. |
