Capital and Interest Computation

Capital Interest
\(a\)       0
\(a\times (1-r)\)       \(a\times r\)
\(a\times (1-r)^2\)       \(a\times (1-r)\times r\)
\(a\times (1-r)^3\)       \(a\times (1-r)^2\times r\)
…       …
  • Total Interest
\[\begin{align*} a \times r [ 1 + (1-r) + (1-r)^2 + (1-r)^3 + \cdots + (1-r)^{35} ] &= I \\ a \times r \cdot \frac{(1-r)^{36}-1}{1-r-1} &= I \\ a \cdot [1-(1-r)^{36}] &= I \\ a \cdot (1-r)^{36} &= a - I \\ (1-r)^{36} &= \frac{a-I}{a} \\ 1-r &= \sqrt[36]{\frac{a-I}{a}} \\ r &= 1 - \sqrt[36]{\frac{a-I}{a}} \end{align*}\]

or

\[\begin{align*} (1-r)^{36} &= \frac{a-I}{a} \\ 36 \ln{(1-r)} &= \ln{\frac{a-I}{a}} \\ \ln{(1-r)} &= \frac{\ln{(\frac{a-I}{a})}}{36} \\ 1-r &= \exp{[\frac{\ln{\frac{a-I}{a}}}{36}]} \\ r &= 1 - \exp{[\frac{\ln{\frac{a-I}{a}}}{36}]} \end{align*}\]

Fixed monthly payment for a fixed rate mortgage

  • c: montly payments.
  • r: montly interest rate (since the quoted yearly percentage rate is not a compounded rate, the montly percentage rate is simple the yearly percentage rate divided by 12.)
  • N: number of monthly payment, called the loan “term”.
  • p: amount borrowed, known as loan’s principal.

In the standardized calculations used in US:

\[c = \frac{r\cdot p}{1-(1+r)^{-N}}=\frac{p\cdot r\cdot (1+r)^N}{(1+r)^N-1}\]

Derivation

Amount owned this month = amount owned from the previous month + interest on this amount - fixed amount paid every month

1st term:

\[(1+r)\cdot p - c\]

2nd term:

\[[(1+r)\cdot p - c](1+r)-c = (1+r)^2p-[1+(1+r)]\cdot c\]

3rd term:

\[[(1+r)^2p-[1+(1+r)]\cdot c](1+r)-c=(1+r)^3p-[1+(1+r)+(1+r)^2]c\]

…

Nth term:

\[(1+r)^N\cdot p - [1+(1+r)+(1+r)^2 + \cdots + (1+r)^{N-1}]\cdot c = (1+r)^Np-\frac{(1+r)^N-1}{1+r-1}\cdot c\]

As at the Nth term, the payment is done, we have

\[\begin{align*} 0 &= (1+r)^Np-\frac{(1+r)^N-1}{1+r-1}\cdot c \\ \frac{(1+r)^N-1}{r}\cdot c &= (1+r)^N\cdot p \\ c &= \frac{p\cdot r\cdot (1+r)^N}{(1+r)^N-1} \end{align*}\]

or

\[\frac{r\cdot(1+r)^N}{(1+r)^N-1}-\frac{c}{p} = 0\]