Is it possible to solve for $t$ in $\frac{at(1+t)^d}{(1+t)^d-1} = m $?

Question

This equation gives us a mensuality of a loan in function of the initialAmount ($a$), the duration in month ($d$) and the mensual rate ($t$). $$ \frac{a\cdot t\cdot(1+t)^d}{(1+t)^d-1} = m $$

I want to isolate $t$ on this equation. In fact I want to write a function who take $a$, $d$, and $m$, and return $t$.

I don't know if this is possible, I tried some things but it become too complicated for me. I searched on internet and found a website (symbolab) to solve this, but it just show me that "Steps are currently not supported for this problem".

Is there a way to get $t$, knowing all the others parameters ?

Answer 1

You need to get the solution numerically using, for instance, Newton's method. The interest rate $t$ is a root of $f(t) = (m-a t)(1+t)^d+m$ and Newton's method would consist in starting from an initial guess $t_0$ and building the sequence $$ t_{k+1} = t_k+\frac{f(t_k)}{f'(t_k)}. $$

Example: When $a=1000$, $d=12$ and $m = 83.7854$ the interest rate is close to 1%. If you take an initial guess of 10%, you get 14 correct decimal digits with only 4 iterations.