Why a Global Minimum Variance Portfolio Has Constant Covariance

Summary

The document explains why a global minimum variance portfolio has the same covariance with any fully invested asset or portfolio. Its algebra starts with the GMV portfolio weights, proportional to the inverse covariance matrix applied to a vector of ones. Multiplying those weights through the covariance matrix cancels the inverse, leaving a covariance equal to the reciprocal of the normalization constant, independent of the other portfolio's composition.

A qualitative argument gives the intuition: if two portfolios had different covariances with the GMV portfolio, a combination of them could be used to reduce variance further, contradicting the GMV definition. The result assumes the standard unconstrained minimum variance setup, a covariance matrix that can be inverted, and portfolios with weights summing to one. Constraints or other changes to the optimization problem can invalidate the stated weight formula and constant-covariance conclusion.

Key ideas

• In the standard unconstrained setup, GMV weights are proportional to the inverse covariance matrix times a vector of ones.
• This formula makes covariance with a fully invested portfolio equal to a constant independent of its weights.
• The constant is the reciprocal of the sum of all elements of the inverse covariance matrix.
• If covariances differed, diversification could produce a portfolio with lower variance, contradicting global minimality.
• The derivation relies on invertibility and the standard fully invested, unconstrained portfolio setup.

Tags

# Covariance of a GMV portfolio with any asset


# Covariance of a GMV portfolio with any asset












Why is that the covariance of a global minimum variance (GMV) portfolio in the efficient frontier with any asset is always the same?

## Answer by Mayou (score 2, accepted)

https://quant.stackexchange.com/a/8779

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 $

## Answer by Mayou (score 0)

https://quant.stackexchange.com/a/8781

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.