Compound Interest - Derive compound frequeny ($n$) and interest rate ($r$), rate from $A$, $P$ and $T$.

Question

Working from this formula:

$$A\left( t\right) =P\left( 1+\dfrac {r}{n}\right) ^{nt}$$

I'm trying to arrive at a system of equations to determine possible $r$ (float) and $n$ (integer) combinations, when one knows $A$, $P$ and $t$.

I started out by going down the logarithm path, but then realised I ended up with $n$ in the logarithm base, which has left me somewhat stumped. Any ideas?

Answer 1

You only have one equation $$A=P\left( 1+\dfrac {r}{n}\right) ^{nt}$$ and two unknown variables $n$ and $r$. So, you cannot get each of them.

If gt6989b did, defining $a=\frac 1t\log\left(\frac AP \right)$, the equation becomes $$a=n \log\left(1+\frac rn \right)$$ and then, if you know $n$, you get $t$ easily $$r=n \left(e^{a/n}-1\right)$$ which is gt6989b's answer.

If you know $r$, the solution is much more complex since given by $$n=-\frac{a \,r}{a+r\,W\left(-\frac{a }{r}e^{-\frac{a}{r}}\right)}$$where appears Lambert function.

Answer 2

You have $$ ln(A/P) = nt \ln(1 + r/n) $$ so $$ r = n \exp \left(\frac{\ln(A/P)}{nt}\right) - n = n \exp \left(\ln\left((A/P)^{1/(nt)}\right)\right) - n = n \left[(A/P)^{1/(nt)} - 1\right] $$