Approximation of monthly payment using Taylor expansion

Question

I am trying to understand what does APR(annual percentage rate) and how it is calculated. Thanks to Wikipedia, I got the formula of monthly payment for a fixed rate multi-year mortgage in the following format.

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

Where $P_{0}$ represents the total amount borrowed from the lender and $r$ represents the monthly interest rate, $N$ is the totaly amount of terms/periods.

I know in this scenario, $r$ is a relatively small number like 1~5% where N could be a fairly big number for long term mortgages like 30*12 = 360 month..

I am trying to use Taylor expansion to simplify the formula:

$$(1+r)^{-N} = 1 + r(-N) + r^2\frac{(-N)(-N-1)}{2!} ...$$

If I only keep the first term, it will be

$$c \approx \frac{P_{0}}{N}$$

If I keep the first two terms, it will be:

$$ c \approx \frac{2P_0}{N(2-r(N+1))} $$

I used both those two approximations to calculate the mortgage and they are both totally off the chart, can anyone help explain under what scenario I can do this type of approximation and is there a practical approximation I can use to simply the monthly payment formula.

Answer 1

You're totally right about this Taylor expansion:

$$ 1-(1+r)^{-N} = Nr+ \frac{1}{2} N(N+1) r^2 + ... $$

but you cannot put these terms in c since it's in denominator. So you need to find Taylor expansion of

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

As a hint, the first term is $\frac{P_0}{N}$ and the second one is $P_{0}\frac{(N+1)}{2N}r$.