0
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ 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
1
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.