Monthly mortgage payment

 The formula of the monthly mortgage payments for a fixed-rate loan can be found online as $$\begin{equation} M = P\frac{r(1+r)^n}{(1+r)^n-1}\,,\tag{1}\end{equation}$$ where

•  $M$ is the mortgage payment.  
• $P$ is the principal, i.e., the initial amount borrowed. 
• $r$ is the monthly interest rate. For annual interest rate $2.5\%$, $r=2.5\% / 12$.
• $n$ is the number of payments. For 30-years fixed rate mortgage, $n=30 * 12 = 360$.

Here we provide a derivation of the above formula. Let $b_i$ be the owned balance in the $i$-th month. Initially, $b_0=P$ and we choose the constant monthly payment $M$ such that $b_n=0$. We then have the recursion relation between two consecutive month as $$\begin{equation}b_{i+1}=b_i(1+r) - M\,.\tag{2}\end{equation}$$

A trick to solve (2) is to write Eq. (2) into the following form $$\begin{equation}b_{i+1}+C = \left(b_i+C\right)(1+r)\,,\tag{3}\end{equation}$$ so that $b_i+C$ is a geometric sequence: $$\begin{equation}b_i + C = (b_0 + C) (1+r)^i\,.\end{equation}$$ Note that $b_0=P$ and $b_n=0$, we thus have the equation $$\begin{equation} C = (P+C) (1+r)^n\,,\end{equation}$$ from which we can solve $$\begin{equation}C= \frac{P(1+r)^n}{(1+r)^n-1}\,.\end{equation}$$

Finally, by comparing Eq. (2) and (3), we know that $M=Cr$, which then has the same form as Eq. (1).

After proving the formula (1), let's do a simple exercise. Suppose the 30-years fixed rate mortgage rate is $2.5\%$ and the loan amount is $P_1$. For exactly the same monthly mortgage payments, what is the new loan amount $P_2$ when the 30-years fixed rate mortgage rate is increased to $6.0\%$?

Using Eq. (1), for the same $M$, we have the relation $$\begin{equation} \frac{P_2}{P_1} = \frac{r_1 (1 + r_1)^n}{r_2 (1 + r_2)^n}\frac{(1 + r_2)^n-1}{ (1 + r_1)^n-1}\,.\end{equation}$$ The surprise is the final numerical result: after submitting $r_1=2.5\%/12$ and $r_2=6.0\%/12$, we can calculate $$\begin{equation} \frac{P_2}{P_1} = 0.659\,.\end{equation}$$ The new loan amount is only two thirds of the old loan amount after the interest rate increases!