The least risky mix

Idea. The variance is a quadratic in $w$; its minimum is where the derivative vanishes.

Step 1: the variance. With ${\sigma }_A=0.2$, ${\sigma }_B=0.3$ and $\rho {\sigma }_A{\sigma }_B=0.03$: $$\begin{array}{rl}V(w)& =0.04w^2+0.09(1-w{)}^2\\ & +0.06w(1-w).\end{array}$$

Step 2: the minimum. In general, $$\begin{array}{rl}{w}^{\ast }& =\frac{{\sigma }_B^2-\rho {\sigma }_A{\sigma }_B}{{\sigma }_A^2+{\sigma }_B^2-2\rho {\sigma }_A{\sigma }_B}\\ & =\frac{0.09-0.03}{0.04+0.09-0.06}=\frac{0.06}{0.07}=\frac{6}{7}.\end{array}$$

So you put $\frac{6}{7}\approx 86\%$ in A and $\frac{1}{7}$ in B.

Check. The minimum variance is $\frac{{\sigma }_A^2{\sigma }_B^2(1-{\rho }^2)}{0.07}=\frac{0.0027}{0.07}\approx 0.0386$, a volatility of about 19.6%: lower than A alone, even though B is riskier. A little of B diversifies away part of A's risk because the correlation is less than $\frac{{\sigma }_A}{{\sigma }_B}=\frac{2}{3}$; above that value, the minimum-variance portfolio would be 100% in A or beyond.