Bactracking to find compound interest

Question

I'm trying to find what percentage 5000 dollars compounding monthly over 120 months will be if the final sum will be 7000 dollars.

So: 7000=5000(1+r/12)^120

When working backwards to find r I always get the percentage = 0.2% but I did trial and error and know the actual answer is about 3.37%

How do I work backwards to find the correct r?

Answer 1

We have $$1.4=\left(1+\frac{r}{12}\right)^{120}.$$ Now we can use a calculator to find $(1.4)^{1/120}$. I get about $1.0028079$. Subtract $1$, multiply by $12$. I get about $0.0336944$.

Remarks: $1.$ I did this on an ordinary calculator. But it could also have been done by Google. Just type in (1.4)^(1/120). It gives $1.00280787001$.

You can even let Google do the whole calculation. Type in 12*((1.4)^(1/120)-1).

$2.$ Note that we have calculated the nominal annual rate $r$. The effective annual rate is somewhat larger. It is $\left(1+\frac{r}{12}\right)^{12}-1$. In our case, to find the effective annual rate, we do not need to compute $r$. We can simply find $(1.4)^{1/10}-1$. This turns out to be about $0.0342197$. The difference between nominal and effective becomes significantly larger when interest rates are high.

Answer 2

A1 = A0 + rA0
A2 = A0 + rA0 + r(A0 + rA0)
   = A0(1 + r)^2

At = A0(1 + r)^t

(At/A0)^(1/t) - 1 = r

(7/5)^(1/10) - 1 = r = 0.0342

I calculated r using python3 (7/5)**(1/10) - 1.