Duplication Invariance in the Most-Diversified Portfolio
Summary
The document explains duplication invariance in the Most-Diversified Portfolio (MDP): adding a second asset with returns identical to an existing asset should not make the portfolio treat that exposure as a new diversification opportunity. The clarification is that duplication means copying an asset’s return behavior, which adds a matching row and column to the covariance structure, rather than introducing an unrelated asset that changes the risk relationships.
Under this setup, the MDP can split its allocation between the original and duplicate while keeping their combined weight at the level assigned to the original asset before duplication. The discussion relates this behavior to redundant equations in the optimization’s first-order conditions and contrasts it with methods that may increase total allocation to the repeated exposure. It is a conceptual explanation of a property cited from prior work, not a new proof or a broad empirical comparison of portfolio methods.
Key ideas
- A duplicated asset has the same returns as an asset already in the portfolio universe.
- The duplicate creates a repeated row and column in the covariance matrix.
- MDP duplication invariance means the combined allocation to both copies remains equivalent to the original asset’s allocation.
- The optimization’s first-order conditions contain redundancy when an identical asset is added.
- The discussion explains a cited property but does not present a new proof or empirical study.
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# Proof for the Duplication Invariance property for the Most-Diversified portfolio # Proof for the Duplication Invariance property for the Most-Diversified portfolio 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! ## Answer by Stefan Voigt (score 1, accepted) https://quant.stackexchange.com/a/32301 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. ## Answer by Kasit (score 1) https://quant.stackexchange.com/a/32317 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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