I am trying to understand a step in a textbook I am working with. The example involves two investments with the same expected return $\mu$.
We have, however, different variances for the investments ($\sigma_1$ and $\sigma_2$). Let $R(\pi$) be a random variable that outputs the return in terms of $\pi$. That is $0 \leq \pi \leq 1$ and if the first investment receives $\pi$ percent of the investment, the second receives $1-\pi$. Thus, $Var(R(\pi))=\pi^2\sigma_1^2 + (1-\pi)^2\sigma_2^2$.
Now this is the step I don't understand:
Minimizing this with respect to $\pi$ gives the optimal portfolio:
$$\pi_\text{opt}=\frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2}.$$
How was the equation derived? I tried taking the derivative with respect to $\pi$ of the variance but this did not yield the above equation.
EDIT: I think I understand a bit better. I guess I am wondering if there is a general standard derivation applied here or if this was more of an intuitive approach taken by the author. To me it seems that we find the "weight" of the variance of the second investment, the higher the weight the more we allocate to investment $1$ to offset the variance (risk) of investment $2$.
$$ g(p) = p^2\sigma_1^2 + (1 - p)^2\sigma_2^2 $$ $$ g'(p) = 2p\sigma_1^2-2(1-p)\sigma_2^2 = 0 $$ $$ p(\sigma_1^2+\sigma_2^2)=\sigma_2^2\to p = \frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2} $$ $$ g''(p) = 2(\sigma_1^2+\sigma_2^2) >0 $$