Why is that the covariance of a global minimum variance (GMV) portfolio in the efficient frontier with any asset is always the same?
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Here is the full math proof. Let g be the GMV portfolio and p be another asset.
We have:
$$ \begin{align*} Cov(x_g, x_p) &= E[{w_g}^T (x- \overline{x}) {(x- \overline{x})}^Tw_p]\\ &= {w_g}^TE[(x- \overline{x}) {(x- \overline{x})}^T]w_p\\ &= {w_g}^T\Sigma w_p \\ &= (\displaystyle\frac{{i}^T {\Sigma}^{-1}}{C})\Sigma w_p\\ &= \displaystyle\frac{{i}^Tw_p}{C}\\ &= \displaystyle\frac{1}{C} \end{align*} $$
where $C = 1^T {\Sigma}^{-1} 1 $
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$\begingroup$ Am I correct in my reading of this proof that $w_{p}$ can be an arbitrary portfolio, rather than just a specific asset? $\endgroup$ – John Aug 16 '13 at 19:23
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$\begingroup$ Yes, this is true for any asset or portfolio. Note that $\displaystyle\frac{1}{C}$ is also the covariance of the gloval MVP with itself. $\endgroup$ – Mayou Aug 16 '13 at 19:45
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$\begingroup$ Can explain how u get from line 2 to 3 and 3 to 4? $\endgroup$ – lakesh Aug 17 '13 at 0:22
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$\begingroup$ From 2 to 3, we apply the basic definition of the variance-covariance matrix. From 3 to 4, we have that the vector of weights for the global MVP is the solution to the minimization problem: $min_w w^{'} \Sigma w$ s.t. $w^{'}1 = 1$. The solution is then expressed as: $w_g = \displaystyle\frac{{\Sigma}^{-1}1}{1^T \Sigma 1}$. Replace $w_g$ in line 3 by the latter expression. $\endgroup$ – Mayou Aug 19 '13 at 13:03
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Here is a more qualitative proof: Imagine that the global MVP had two distinct covariances with two other portfolios. This means that additional diversification using these 3 assets would result in a portfolio with a variance lower than that of the global MVP. This would be contradictory to the fact that the global MVP has the lowest possible return variance for a given covariance matrix $\Sigma$. Therefore, the covariance of the global MVP with any other asset or portfolio is constant.
