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Computing Maximum Diversification Weights Without a Risk-Free Asset

Article Quant Q&A · Author: Dirty Dan
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Summary

The document explains how to compute a maximum diversification portfolio using only risky assets. Given a nonsingular covariance matrix, the procedure takes its inverse, multiplies it by the vector of asset volatilities, and normalizes the resulting weights so they sum to one. This provides a direct way to form fully invested weights without adding a risk-free asset to the optimization.

The answer presents the formula as equivalent to the approach in cited maximum-diversification research, but gives no empirical performance comparison or worked numerical example. The method assumes the covariance matrix is invertible and that there are no portfolio constraints. The discussion does not address what to do if calculated weights are negative, how estimation error affects the result, or whether the resulting portfolio meets practical limits such as long-only investing. Rescaling to a sum of one is part of the stated procedure, though the question’s broader comparison with strategies that exclude a risk-free asset is not explored in depth.

Key ideas

  • The method starts with the inverse of the risky assets’ covariance matrix.
  • Multiplying that inverse by the vector of asset volatilities produces unnormalized portfolio weights.
  • Normalizing the resulting weights to sum to one gives a fully invested portfolio.
  • The stated calculation assumes a nonsingular covariance matrix and no constraints.

Tags

Portfolio constructionStatisticsRisk management
Full text
# Maximum Diversification Strategy without risk free asset


# Maximum Diversification Strategy without risk free asset












I am currently dealing with the Maximum Diversification Strategy and I am trying to understand the synthetic universe approach from Choueifaty et al.(https://www.tobam.fr/wp-content/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pdf)

Specifically I would like to know if you guys think that it is still mathematically correct to omit the risk free asset (since other investment strategies which I am comparing it to, don't have the possibility to invest in a risk free asset) and rescale the portfolio weights at the end of the optimaziation simply to 1, without changing his fundamental approach?

Best regards

## Answer by Alex C (score 3, accepted)

https://quant.stackexchange.com/a/45341

Yes, it is easy to find the MDP of N risky assets if you have the covariance matrix V (assumed non-singular) if there are no constraints:

Step 1. Compute the inverse of the covariance matrix: $CINV = V^{-1}$

Step 2. Find the standard deviations $\sigma$ by taking the square roots of the diagonal elemnts of $V$

Step 3. Find $X = CINV \times\sigma$ i.e. multiply the matrix CINV by the column vector $\sigma$. So $X$ consists of weighted row sums of CINV, with the $\sigma_i$s as weights

Step 4. Normalize X so the elements sum to 1, i.e take $W=\frac{X}{\sum_i x_i}$. These are the MDP portfolio weights.

It is quite simple, the clearest explanation I found is in Theron and Van Vuuren: The maximum diversification investment strategy: A portfolio performance comparison (link), it is also equivalent to what is described in the Chouefaty article.

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.