ANNUITIES 95 EXAMPLES 1. A man gives a mortgage for $10,000, to be repaid in 6 years with interest payable annually at 6 %. If the interest is paid promptly, what sum must be set aside each year to repay the principal when it falls due, provided the money set aside can be invested at 4 % ? 2. What sum must be set aside annually to provide for the rebuilding, after 25 years, of a bridge costing $25,000, provided the money set aside can be invested at 4 % ? 37. The annuity that 1 will purchase. The annuity whose present value is 1 is usually spoken of as " the annuity that 1 will purchase." It plays a large part in the solution of many important problems. PROBLEM. To determine the annual rent of an annuity that 1 will purchase. Let R be the required annual rent. Since the present value of an annuity whose annual rent is It is Ra^, the value of £ will be determined from the equation Sa^=l. Consequently, J^=7^=l-'-vn ^ is the required formula. To find the annuity that a given sum A will purchase, it is only necessary to multiply both sides of equation (1) by A. The resulting formula is S=A-^^- (v) It is not difficult to determine the formula when interest is convertible oftener than once a year, or when the annuity is payable several times a year. Formula (1), or its equivalent, (!'), is one form of the amorti- zation equation of Euler. It is used in determining the annual installment that will provide for the payment of an interest- bearing debt when principal and interest are to be paid in a series of equal annual installments.