96 MATHEMATICAL THEORY OF INVESTMENT Suppose, for example, a man wishes to pay a debt of $5000, bearing interest at 6 % effective, in five equal annual install- ments, the first installment to be paid one year hence. The annual payment is required. The payments constitute an an- nuity whose annual rent R' is required. By formula (!'), R = $5000 . ——-06——- 1-(1.06)-6 = $5000 x.237396 = $1186.98. EXAMPLES 1. Find the formula for the annual rent of an annuity that 1 will purchase when the annuity is payable p times a year and interest is convertible annually. 2. A man makes a cash payment of $2000 on a farm purchased for $10,000, and wishes to pay the balance, with interest at 5%, in five equal annual installments. What will the annual installment be ? 3. Find the formula for the annuity, payable p times a year, for n years that can be purchased for 1, when interest is at nominal rate j, convertible in times a year. 38. Fundamental relation between -'- and — ^ ^l THEOREM. The annual rent of the annuity that 1 will purchase, diminished by the annual rent of the annuity that will amount to 1, is equal to j^, where j^ is defined by the equation 3w =P [(1 + ^ -1]- (see <3')' § 20>-) To prove the theorem it must be proved that ^-^^W (1) By §§ 36 and 37, ^ ^ */ «5 0 ' ^_^p[CWy^v\^_^__ a^ 1 - (1 + i)~n 1 - (1 + 0-" i