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In Properties of the most diversified portfolio by Choueifaty, he shows that the Most-Diversified portfolio (MDP) has three quantitative properties.

The Paper: http://www.tobam.fr/wp-content/uploads/2014/12/04.2013_JIS_TOBAM-Properties-of-the-Most-Diversified-Portfolio.pdf

Specifically for duplication invariance, he proves that it exists by stating that "the introduction of a redundant asset leads to a redundant equation in the first-order equations associated to the MDP program" [footnote 17].

I dont really understand how having an extra asset (which changes the covariance matrix) introduce a redundant equation.

I hope someone can clarify my doubts. Thank you!

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The introduction of a redundant assets means, that one of the existing assets is duplicated. So, in other words, you do not introduce an extra asset which changes the covariance matrix, but instead you simply assume there is a new asset available which has the same return as one of the assets which are already available (in the paper asses A is duplicated).

What is the effect of duplicating an asset? Well, the MDP adjusts for that and recognizes that the duplicated asset is not adding anything and therefore adjusts the portfolio weights: The both assets (A and the duplicate A) obtain in sum the same fraction of wealth at would be the case if only A is existent and no duplicated asset exists to invest in. As the authors show in the Table on page 10, other methods may not recognize that the assets are duplicate and instead put more wealth in both of them.

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  • $\begingroup$ yeah I understand the intuition behind it, but is there a more formal representation in terms of the matrix notation? The introduction of a duplicated asset effective changes the covariance matrix from (using the example of sigma(A)=0.2, sigma(B)=0.1, corr(A,B)=0.5 C=[0.02,0.01; 0.01,0.01] to C1=[0.04,0.01,1; 0.01,0.01,0.01; 1,0.01,0.04] $\endgroup$ – Kasit Feb 8 '17 at 7:47
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Ah nvm, I'm embarrassed that this actually took me so long to understand.. but adding a repeated asset basically adds a row identical to the original asset in the covariance matrix, thus introducing a redundant equation in the programming problem.

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