the average growth rate/ Calculate the annualized return for the entire period

Question

The returns from an investment are $2\%$ in Year $1$, $5\%$ in Year $2$, and $21.8\%$ in the first half of Year $3$. Calculate the annualized return for the entire period.

I know the average growth rate formula, but don't know how to apply it to the third year.

Answer 1

Here;s a start:

If you invested $P$ dollars, it would grow to $P(1.02)(1.05)(1.218)=1.304478P$ in $2.5$ years.

So we need to know what constant annual rate, $r$, would produce the same growth.

We need to solve $(1+r)^{2.5}=1.304478$.

Try to take it from there.