Sunday, March 4, 2018

mean variance - 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



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=(iTΣ1C)Σwp=iTwpC=1C



where C=1TΣ11


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