# 本息平均攤還

## 公式

${\displaystyle P_{0}}$

${\displaystyle P_{1}=P_{0}+P_{0}*r-c}$ ( 本金 + 本期利息 - 本期償還 )
${\displaystyle P_{1}=P_{0}(1+r)-c}$ (等式 1)

${\displaystyle P_{2}=P_{1}(1+r)-c}$

${\displaystyle P_{2}=(P_{0}(1+r)-c)(1+r)-c}$
${\displaystyle P_{2}=P_{0}(1+r)^{2}-c(1+r)-c}$ (等式 2)

${\displaystyle P_{3}=P_{2}(1+r)-c}$

${\displaystyle P_{3}=(P_{0}(1+r)^{2}-c(1+r)-c)(1+r)-c}$
${\displaystyle P_{3}=P_{0}(1+r)^{3}-c(1+r)^{2}-c(1+r)-c}$

${\displaystyle P_{N}=P_{N-1}(1+r)-c}$
${\displaystyle P_{N}=P_{0}(1+r)^{N}-c(1+r)^{N-1}-c(1+r)^{N-2}....-c}$
${\displaystyle P_{N}=P_{0}(1+r)^{N}-c((1+r)^{N-1}+(1+r)^{N-2}....+1)}$
${\displaystyle P_{N}=P_{0}(1+r)^{N}-c(S)}$ (等式 3)

${\displaystyle S(1+r)=(1+r)^{N}+(1+r)^{N-1}....+(1+r)}$ (等式 5)

${\displaystyle S(1+r)-S=(1+r)^{N}-1}$
${\displaystyle S((1+r)-1)=(1+r)^{N}-1}$
${\displaystyle S(r)=(1+r)^{N}-1}$
${\displaystyle S=((1+r)^{N}-1)/r}$ (等式 6)

${\displaystyle P_{N}=P_{0}(1+r)^{N}-c(((1+r)^{N}-1)/r)}$
${\displaystyle P_{N}}$（最後一期之餘額）會是 0（明顯地）因為已經還清。
${\displaystyle 0=P_{0}(1+r)^{N}-c(((1+r)^{N}-1)/r)}$

${\displaystyle c=(r(1+r)^{N}/((1+r)^{N}-1))P_{0}}$

${\displaystyle c=(r/(1-(1+r)^{-N}))P_{0}}$