Log 1.05 = -0.021189 -5 Log 1.05 = -0.105945 = 9.894055-10 Log⁻¹(9.894055-10) = 0.783529 1 - 0.783529 = 0.216471 Log Annual Payment = Log _i_ - Log 0.216471 + Log _P_ = Log 0.05 - Log 0.216471 + Log 10,000 = (8.698970 - 10) - (9.335398 - 10) + 4.000,000 = 3.363572 Annual Payment = Log⁻¹ 3.363571 = $2309.748. The following table shows the repayment of the loan by annual payments of $2309.75: ----+------------+------------+----------------+------------- | Principal | | | | Owing at | |Principal Repaid| |Beginning of|Interest for| at End |Total Payment Year| Year | Year | of Year | for Year ----+------------+------------+----------------+------------- 1 | $10,000.00 | $500.00 | $1,809.75 | $2,309.75 2 | 8,190.25 | 409.51 | 1,900.24 | 2,309.75 3 | 6,290.01 | 314.50 | 1,995.25 | 2,309.75 4 | 4,294.76 | 214.74 | 2,095.01 | 2,309.75 5 | 2,199.85 | 109.99 | 2,199.75 | 2,309.74 | +------------+----------------+------------- |Totals | $1548.74 | $10,000.00 | $11,548.74 ----+------------+------------+----------------+------------- Since it is more convenient to have the bonds in even hundreds of dollars and the interest in dollars some adjustment from the theoretical amounts are usually made but such that the annual payments will be near the theoretical. Sometimes, too, the bonds are made smaller for the first few years then gradually increase so that the natural growth in population and wealth may bear its proportional burden. One adjustment for the example just given is shown: ----+------------+------------+----------------+------------- | Principal | | | Year| Owing at | |Principal Repaid| |Beginning of|Interest for| at End |Total Payment | Year | Year | of Year | for Year ----+------------+------------+----------------+------------- 1 | $10,000 | 500 | $1,800 | $2,300 2 | 8,200 | 410 | 1,900 | 2,310 3 | 6,300 | 315 | 2,000 | 2,315 4 | 4,300 | 215 | 2,100 | 2,315 5 | 2,200 | 110 | 2,200 | 2,310 | +------------+----------------+------------- | Totals | $1,500 | $10,000 | $11,550 ----+------------+------------+----------------+------------- =Total Cost by the Three Kinds of Bonds.=--The total cost of a loan, as shown by the following table taken from Bulletin 136, U. S. Department of Agriculture, is generally greatest under the sinking fund plan and least under the serial. The serial, too, is the simplest to compute.