This article will show the steps to obtain the formula for the mortgage/loan with a fixed rate.
Here is a fancy image to scare people off.
Let’s dive in. The standard way of seeing a loan is that the principal is growing due to the interest, and the client makes recurrent deposits that cover the interest and slowly reduce the principal. This is the standard way of formulating the interaction. You can find an explanation of this formula here: Derivation of the loan/mortgage monthly payment formula.
Another way of seeing it is to consider that the loan grows with interest. At the same time, the money you deposit prevents/counter-balance the interest, so we can model the funds you deposit with an imaginary growth at the same speed as the loan. At this point, you can imagine two piles: one is the growing loan the other is the pile with your contribution that will grow with the same interest as the loan. In this model, the loan will be paid when the two piles have the same amount.
So we have the left side, which is the growing loan, P is the principal, r is the interest and n is the number of periods:
And the right side, where we make the recurrent deposit. Let N be the amount you deposit each period, r is always the interest rate and n is always the number of periods. In the image below, there is the calculation of the amount you have after each period due to the recurring deposit and the compound interest.
We take the last line and make the needed simplifications:
The only passage that requires explanation is the rewriting of the summation, which is a case of the geometric (or exponential) series that can be proven by induction. The image below shows the series and the special case where i start at 1.
We can now see the complete equation:
