Reverse seed interest calculation?

Question

Assuming an apartment was bought at 1.89 million $.

After 5 years it was sold at 2.5 million $.

The percent difference is : 2.5/1.89 = 1.32 => 32% increase.

But how can I find the interest (x) value where :

After 1 years : 1.89 * 1.X
After 2 years : (1.89 * 1.X) * 1.X
After 3 years : ((1.89 * 1.X) * 1.X) *1.x
...
...   5 years : 2.5 million $

Is there any formula to calculate reverse interest seed?

Answer 1

With a $p\,\%$ annual increase, an initial amount $K$ turns into $K'=K\cdot q^n$ after $n$ years (cf. your repeated factors 1.x), where $q=1+\frac p{100}$. Here we know $K=1.89$, $K'=2.5$ (both in units of mega-dollars) and $n=5$. Then $$ q=\sqrt[n]{\frac{K'}K}=\sqrt[5]\frac{2.5}{1.89}=1.057537\ldots$$ and hence $p=100(q-1)\approx 5.75$. Thus we are speaking of roughly $5.75\,\%\text{p.a.}$ of interest.

Remark: The "naive" way of simply dividing the $32\,\%$ by the $5$ years yields $6.4\,\%$, which is wrong, but at least it is somewhat a ballpark estimate - perhaps good enough for a mental estimate, and gets significantly wronger if we consider more years or higher annual interest rates.