or the interest on the loan plus the sinking-fund payments: $2500 + $9231.25 = $11,731.35 _Serial Bonds_ are such that a fixed amount of the principal is retired at definite periods of time. Usually the amount retired is an aliquot part of the whole. The payments to be made at any particular time is the fixed portion of the principal plus the interest on the unpaid portion up to that date. The periods of retirement are usually annual or semi-annual. Assuming the principal to be _P_ and that one nth part of it is paid each year, the formulas are: ( 1 ( 1 - _k_)) Annual payment for the _k_th year = _P_(--- + _i_(1 + -------)). (_n_ ( _n_ )) ( 1 - _k_) Interest for the _k_th year = _Pi_(1 + -------). ( _n_ ) Total amount of interest to the end of ( 1 - _k_) the _k_th year = _Pik_(1 + -------). ( 2_n_ ) Total amount of interest and principal ( 1 ( 1 - _k_)) paid up to the end of the _k_th year = _Pk_(--- + _i_(1 + -------)). (_n_ ( 2_n_ )) The following table shows how a debt of $10,000 bearing 5 per cent interest would be discharged by equal annual payments in five years: ----+------------+------------+----------------+------------ |Principal at| |Principal Repaid| |Beginning of|Interest for| at end of |Total Annual Year| Year | Year | Year | Payment ----+------------+------------+----------------+------------ 1 | $10,000 | $ 500 | $ 2,000 | $ 2,500 2 | 8,000 | 400 | 2,000 | 2,400 3 | 6,000 | 300 | 2,000 | 2,300 4 | 4,000 | 200 | 2,000 | 2,200 5 | 2,000 | 100 | 2,000 | 2,100 | +------------+----------------+------------ |Totals | $1,500 | $10,000 | $11,500 ----+------------+------------+----------------+------------ _Annuity Bonds_ are those wherein a uniform periodic payment is made to discharge the debt in a given time. The formula for the necessary payment to discharge a debt of _P_, with interest rate _i_ in _n_ years is, _i_ Annual payment = -------------------- . _P_. 1 - (1 - _i_)^{_-n_} Results may be taken from books of tables already referred to or by means of logarithms the formula may be solved. For example let it be required to discharge a debt of $10,000 in five equal payments, the rate of interest being 5 per cent. Solution: (1 + _i_)^{_-n_} = 1.05⁻⁵.