Why is that the covariance of a global minimum variance (GMV) portfolio in the efficient frontier with any asset is always the same?
Answer
Here is the full math proof. Let g be the GMV portfolio and p be another asset.
We have:
Cov(xg,xp)=E[wgT(x−¯x)(x−¯x)Twp]=wgTE[(x−¯x)(x−¯x)T]wp=wgTΣwp=(iTΣ−1C)Σwp=iTwpC=1C
where C=1TΣ−11
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