Diversification

The Riskless Mix

Perfect negative correlation can create a zero-variance portfolio.

highlighted = computed this step

Minimum-variance weight

With rho -1, the minimum-variance weight in asset A is 1/3. The remaining weight in asset B is 2/3.

w∗=σBσA+σB=1/3w^*=\frac{\sigma_B}{\sigma_A+\sigma_B}=1/3

Riskless mix

At that weight, the exact variance is 0 and sigma is 0.00%. The two risky assets combine to zero risk in this idealized case. This zero-risk mix returns 7.33%, below the 8.00% of the fifty-fifty portfolio: eliminating risk here comes with a lower return, not a free lunch.

σp2=0\sigma_p^2=0
Riskless mixThe minimum-variance weight is recomputed from stated inputs.Riskless mix at rho=-1ItemExactDisplayWeight in A1/31/3Weight in B2/32/3Portfolio return11/1507.33%Variance00Sigma00.00%

Idealized endpoint

Perfect negative correlation is an idealization. Real-world sigma and rho are estimated from data with uncertainty, and rho is rarely exactly negative one. This is descriptive, not investment advice.

rho equals negative one is idealized\text{rho equals negative one is idealized}