# Correlation between mean-variance efficient portfolios

If the covariance solution between the returns series of the minimum-variance portfolio ($$A$$) and any other portfolio along the efficient frontier ($$B$$) is

$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}}$$ What is the derivation of the closed-form analytical solution for the correlation between those portfolios, $$\rho_{A, B}=?$$

• Hi @develarist, why don't you simply plug in all your portfolios (MVP, Tangency, target-return) into the formulas for correlation and see what drops out? Can you show where your intuition breaks or what component you are missing? – Kermittfrog Nov 18 '20 at 7:13
• Hint: What is the optimal weight vector given a target return $M$? Plug that into the (co)variance formula $w(M)^T\Sigma w_{MVP}$. Simplify as much as possible knowing the definitions $a,b,c$. That should get you somewhere, no? – Kermittfrog Nov 18 '20 at 7:28

## 1 Answer

Just divide covariance by the square roots of the two variances. In this case you would want $$\frac{1/a}{\sqrt{\frac{1}{a}\frac{c}{b^2}}},$$ which takes value $$\frac{|1^{\top}\Sigma^{-1}\mu|}{\sqrt{(1^{\top}\Sigma^{-1}1)(\mu^{\top}\Sigma^{-1}\mu)}}.$$

• I also got the same answer last month – develarist Nov 18 '20 at 0:53
• @develarist the good thing about math is that it doesn't go stale. – steveo'america Nov 18 '20 at 1:18
• I think that this answer does not imply the correlation between any portfolio and the MVP. – Kermittfrog Nov 18 '20 at 7:26
• Sorry, bad English: I think the question asks for the correlation between any portfolio on the frontier and the MVP, not (only) the Tangency portfolio. – Kermittfrog Nov 18 '20 at 16:42