_Sinking fund_ bonds are paid as a whole at the end of their term, interest being paid annually, or at some other fixed regular period, upon their face value. The name arises because of the custom of establishing a sinking-fund into which a certain proportion of the debt is to be paid annually, and this loaned out so that at the end of the period it will amount to the face of the bonds. Since there is always time lost between the collection and loaning of the sinking fund money the interest derived therefrom will not usually be the same as that of the bonds. For this reason and from the further fact that sinking funds are frequently drawn upon for other purposes than that for which they were created this type of bonds is less economical than either of the other two types. The sinking fund which must be raised annually to discharge a debt of _P_ dollars in _n_ payments, if it can be loaned at _i_ per cent, is given by the formula:[197] _i_ Sinking fund = ------------------- . _P_ (1 + _i_)^{_n_} - 1 To illustrate the use of the formula let the debt be $10,000, the average rate that can be expected from the sinking fund 4 per cent, and the time five years. Substituting in the formula, .04 _S_ = ------------- . $10,000 (1 + .04)⁵ - 1 To solve, the denominator is first evaluated: Log (1 + .04)⁵ = 5 log 1.04 = 5 × 0.017033 = 0.085165 Taking the antilog, (1 + .04)⁵ = 1.21665 and (1 + .04)⁵ - 1 = 0.21665 Then 0.4 × $10,000 _S_ = ------------- = $1846.27. 0.21665 Annuity tables, which may be seen at nearly any bank or brokers’ office, or in Bulletin 136, U. S. Department of Agriculture, give the annuity which will amount to 1 in five years at 4 per cent as 0.1846271; this multiplied by $10,000 gives $1846.27. To the nearest cent the following tabular statement shows the growth of the sinking funds: ----+------------+---------------+---------------+------------------ |Sinking-fund| |Annual Payments|Total Sinking-fund |at Beginning|Interest during| into | at End Year| of Year | Year | Sinking-Fund | of Year ----+------------+---------------+---------------+------------------ 1 | 0. | 0. | $1,846.27 | $ 1,846.27 2 | $1846.27 | $ 73.85 | 1,846.27 | 3,766.39 3 | 3766.39 | 150.66 | 1,846.27 | 5,763.32 4 | 5763.32 | 230.53 | 1,846.27 | 7,840.12 5 | 7840.12 | 313.61 | 1,846.27 | 10,000.00 ----+------------+---------------+---------------+------------------ If this loan, the bonds, bore 5 per cent interest the cost to the borrower would have been the principal plus the interest on principal less the interest on the sinking fund: $10,000 + $2500 - $768.65 = $11,731.35;