I just saw a youtube video where they mentioned that a closed-form solution exists, bypassing the need for an amortization table. This was framed in the context of having multiple loans to service and just wanting that ratio. They said they wouldn't be providing a solution, so I am not even sure where to start.
They basically wanted a formula
$f(r, term) = \frac{principal}{interest}$ where:
- r is the apr/12
- term is the specific month we are in (1, 2, 3, ..., n) since the principal & interest amounts changes each month
To say that ${principal}/{interest} = f(r, term)$ is a bit oversimplified. If $term$ is counted from the time when the loan originated, you will need at least one more piece of information, such as the original length of the loan.
I think it is simpler to write $$ \frac{\Delta principal}{interest} = g(r, k) $$ where $k$ is the number of remaining payments, including the one for which you want to find the ratio ${\Delta principal}/{interest}$, and $r$ is the rate of interest per payment period. (And I really would rather write ${\Delta principal}/{interest}$ instead of ${principal}/{interest}$, since "$principal$" seems like it ought to be the amount of principal still owed, in which case ${principal}/{interest}$ is just $1/r$.)
The principal still owed on the loan just prior to the final $k$ payments is the present value of a stream of $k$ equal payments of the amount $Pmt$, which is $$ PV = \frac{1 - (1 + r)^{-k}}{r} Pmt. $$ Therefore $$ \frac{Pmt}{PV} = \frac{r}{1 - (1 + r)^{-k}}. $$ But $$ \frac{Pmt}{PV} = \frac{interest + \Delta principal}{PV} = r + \frac{\Delta principal}{PV} $$ and therefore $$ \frac{\Delta principal}{PV} = \frac{Pmt}{PV} - r = \frac{r}{1 - (1 + r)^{-k}} - r = \frac{r}{(1 + r)^k - 1}. $$
Finally, $$ \frac{\Delta principal}{interest} = \frac{\left(\dfrac{\Delta principal}{PV}\right)} {\left(\dfrac{interest}{PV}\right)} = \frac{\left(\dfrac{r}{(1 + r)^k - 1}\right)}{r} = \frac{1}{(1 + r)^k - 1}. $$
I notice, however, that in the video they were actually talking about $$ \frac{\Delta principal}{Pmt}. $$ This can be derived from $$ \frac{\Delta principal}{Pmt} = \frac{\left(\dfrac{\Delta principal}{PV}\right)} {\left(\dfrac{Pmt}{PV}\right)} = \frac{\left(\dfrac{r}{(1 + r)^k - 1}\right)} {\left(\dfrac{r}{1 - (1 + r)^{-k}}\right)} = \frac{1}{(1 + r)^k}. $$