Ever thought about what the actual formula is for your loan payment schedule? Spreadsheet functions and financial calculators make it easy to avoid the details, but let's dig into it anyway. The formula
\[ A = P \left( \frac{r }{1-(1+r)^{-N}} \right) \]
looks sorta complicated.
Let $A$ be the monthly payment we're after, and $P$ is the initial principle. The monthly interest rate is $r$. That is, the loan grows by a factor of $(1+r)$ every month. We want to set up the payments so that after N months, the loan is paid and we're done.
So how do we do this? Every month the balance on the loan will be different, let's use $\alpha_m$ for the balance at month $m$. We know two things right away:
\[ \alpha_{0} = P \tag{initial balance} \]
\[ \alpha_{N} = 0 \tag{ending balance}\]
So we have
\[ 0 = \alpha_N = (1+r) \alpha_{N-1} - A \]
The last balance, $0$, is the penultimate balance increased by the interest and reduced by the monthly payment $A$. Let's rewrite in terms of $A$:
\[ A = (1+r) (\alpha_{N-1}) \]
\[ A = (1+r)( (1+r) \alpha_{N-2} - A ) \]
\[ A (1 + (1+r)) = (1+r)^2 \alpha_{N-2} \]
\[ A (1 + (1+r)) = (1+r)^2 ((1+r) \alpha_{N-3} -A)\]
\[ A (1 + (1+r) + (1+r)^2) = (1+r)^3 \alpha_{N-3} \]
We can take this all the way to N, where the form for $A$ will be:
\[ A \left( \sum_{n=0}^{N-1} (1+r)^n \right) = (1+r)^N \alpha_0 \]
The sum on the left is a geometric series that we know how to evaluate:
\[ A \left( \frac{1- (1+r)^N}{1-(1+r)} \right) = (1+r)^N P \]
Finally,
\[ A = \frac{rP(1+r)^N } {(1+r)^N -1}= P \left( \frac{r} {1-(1+r)^{-N}} \right) \]
