#6 贷款计算公式的推导
数学 2023-10-17等额本息
设：
- 总金额 A
- 月利率 R
- 贷款期限 N
- 每月还款 X
那么：
第 1 期剩余金额：$Q_1 = A(1+R)-X$
第 2 期剩余金额：$Q_2 = Q_1(1+R)-X$
。。。
第 n 期剩余金额：$Q_n = Q_{n-1}(1+R)-X$
推导一下：
$$
\begin{aligned}
Q_2 &= Q_1(1+R)-X \
&= (A(1+R)-X)(1+R)-X \
&= A(1+R)^2-X(1+R)-X \
&= A(1+R)^2-X(1+(1+R))
\end{aligned}
$$
$$
\begin{aligned}
Q_3 &= Q_2(1+R)-X \
&= (A(1+R)^2-X(1+(1+R)))(1+R)-X \
&= A(1+R)^3-X(1+(1+R))(1+R)-X \
&= A(1+R)^3-X(1+R)^2-X(1+R)-X \
&= A(1+R)^3-X(1+(1+R)+(1+R)^2)
\end{aligned}
$$
$$
\begin{aligned}
Q_k &= A(1+R)^k-X(1+(1+R)+ \cdots +(1+R)^{k-1})
\end{aligned}
$$
根据等比数列公式 $ S_n=\frac {a_1(1-q^n)} {1-q} $（q 不等于 1）：
$$
\begin{aligned}
1+(1+R)+ \cdots +(1+R)^{k-1} &= \frac {1 \times (1-(1+R)^k)} {1-(1+R)} \
&= \frac {1-(1+R)^k} {1-(1+R)}
\end{aligned}
$$
代入上面的推导方程中：
$$
\begin{aligned}
Q_k &= A(1+R)^k-X \cdot (1+(1+R)+ \cdots +(1+R)^{k-1}) \
&= A(1+R)^k - X \cdot \frac {1-(1+R)^k} {1-(1+R)}
\end{aligned}
$$
k = N 的时候：
$$
\begin{aligned}
Q_n &= A(1+R)^N - X \cdot \frac {1-(1+R)^N} {1-(1+R)} = 0 \
A(1+R)^N &= X \cdot \frac {1-(1+R)^N} {1-(1+R)} \
X &= \frac {A(1+R)^N} {\frac {1-(1+R)^N} {1-(1+R)}} \
&= \frac {A(1+R)^N} {\frac {(1+R)^N-1} {R}} \
&= \frac {AR(1+R)^N} {(1+R)^N-1}
\end{aligned}
$$
也就是：
$$
\begin{aligned}
每期还款金额 = 贷款金额 \times 月利率 \times \frac {(1+月利率)^{还款期数}} {(1+月利率)^{还款期数}-1}
\end{aligned}
$$
等额本金
设：
- 总金额 A
- 月利率 R
- 贷款期限 N
每月还款计算：
$$
\begin{aligned}
X_1 &= {\frac A N} + A \cdot R \
X_2 &= {\frac A N} + (A - ({\frac A N})) \cdot R \
X_3 &= {\frac A N} + (A - ({\frac A N}) \cdot 2) \cdot R \
X_N &= {\frac A N} + (A - ({\frac A N}) \cdot (N - 1)) \cdot R
\end{aligned}
$$
也可以视作：
$$
\begin{aligned}
X_1 &= {\frac A N} + {\frac N N} \cdot A \cdot R \
X_2 &= {\frac A N} + {\frac {N - 1} N} \cdot A \cdot R \
X_3 &= {\frac A N} + {\frac {N - 2} N} \cdot A \cdot R \
X_{N-1} &= {\frac A N} + {\frac 2 N} \cdot A \cdot R \
X_N &= {\frac A N} + {\frac 1 N} \cdot A \cdot R
\end{aligned}
$$
总还款金额：
$$
\begin{aligned}
T &= A + A \cdot R \cdot {\frac {(1 + 2 + \cdots + N)} N} & 总还款金额\
&= A + A \cdot R \cdot ({\frac {N + 1} {2}}) \
I &= A \cdot R \cdot ({\frac {N + 1} {2}}) & 总利息 \
\end{aligned}
$$
