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, $\rho_{A, B}=?$

  • $\begingroup$ 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? $\endgroup$ – Kermittfrog Nov 18 '20 at 7:13
  • $\begingroup$ 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? $\endgroup$ – Kermittfrog Nov 18 '20 at 7:28

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 $$ \frac{|1^{\top}\Sigma^{-1}\mu|}{\sqrt{(1^{\top}\Sigma^{-1}1)(\mu^{\top}\Sigma^{-1}\mu)}}. $$

  • $\begingroup$ I also got the same answer last month $\endgroup$ – develarist Nov 18 '20 at 0:53
  • 2
    $\begingroup$ @develarist the good thing about math is that it doesn't go stale. $\endgroup$ – steveo'america Nov 18 '20 at 1:18
  • $\begingroup$ I think that this answer does not imply the correlation between any portfolio and the MVP. $\endgroup$ – Kermittfrog Nov 18 '20 at 7:26
  • $\begingroup$ 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. $\endgroup$ – Kermittfrog Nov 18 '20 at 16:42

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