Variance Reduction - Two-Asset Portfolios Step by Step

Lower correlation reduces exact portfolio variance.

highlighted = computed this step

Less than perfect correlation

When rho is less than one, the cross term is smaller than in the same-move case. For the same fifty-fifty portfolio, variance falls from 9/400 to 1/80 at zero rho, and to 1/400 at rho -1.

\rho<1\Rightarrow \sigma_p^2\text{ can fall}

Exact comparison

The exact comparison is variance, not rounded sigma. The rounded sigma values are 15.00%, 11.18%, and 5.00% for the three rho cases.

\sigma_p=\sqrt{\sigma_p^2}

Diversification input

Diversification depends on the stated sigma and rho inputs. In the real world those inputs are estimated from data with uncertainty. This is descriptive, not investment advice.

\text{diversification is model-input dependent}