The Minimum-Variance Portfolio - The Efficient Frontier Step by Step

The leftmost frontier point minimizes exact variance.

highlighted = computed this step

Minimum-variance weight

With rho 0, the minimum-variance weight in asset A is 1/5. The remaining weight in asset B is 4/5.

w_A^*=\frac{\sigma_B^2}{\sigma_A^2+\sigma_B^2}=1/5

The leftmost portfolio

At that weight, exact variance is 1/125, return is 6.80%, and sigma rounds to 8.94%.

\sigma_{\min}^2=1/125,\quad r_{\min}=6.80\%

Exact variance, rounded sigma

The minimum-variance point is exact in variance and weights. Sigma is generally irrational, so it is rounded for display. This assumes the stated inputs and is descriptive, not investment advice.

\text{weights and variance exact; sigma rounded}