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The Maximum Diversification Index

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Abstract

We propose a new method to assess the risk diversification potential of a given investment set, using only the information content of the covariance matrix of returns. Namely, we extend Rudin and Morgan’s (2006) work to numerically solve for the ‘Maximum Diversification Index’ by means of a genetic algorithm. Using stock returns data from the S&P-500 index, we show that the MDI can be efficiently implemented to delimit a large set of investable assets by eliminating those subjects that do not improve the diversification characteristics of the underlying portfolio pool. Indeed, a subset of the S&P-500 stocks obtained using the MDI procedure preserves the mean-variance properties of the initial dataset as shown by the ex-post efficient frontiers.

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Notes

  1. Merton (1987) also states that this is true even for the largest institutions for which the number of individual stocks held in a single portfolio represent only a small fraction of the total number of securities available (p. 506).

  2. We rely on a numerical solution as the objective function is not differentiable and there is no closed-form solution to the optimization problem.

  3. It can be verified that for equation (3) to take the value of 1, the first element of the component weight vector must be very close to unity, that is W1≈1, in which case the remaining elements automatically tend to have values close to zero, w2⩽i⩽N ≈ 0.

  4. Specifically, we have prepared an R (R Development Core Team, 2011) package for both calculating and optimizing the PDI. The R code as well as the datasets are available from the authors upon request.

  5. It can be noticed that the empirical PDIs are close to, but slightly below, n although one would expect it to be equal to the number of ‘truly independent’ assets in the portfolio pool, which is by construction equal to n in our case. Note, however, that we generate computationally ‘random’ data arrays and, therefore, the term ‘uncorrelated’ must be understood in statistical sense and not in terms of absolute orthogonality.

  6. As an alternative way to enhance the number of matches, it is conceivable to select a single configuration, say T=300, N=250 and n=25, increase the number of iterations, and run the simulations using an improved GA instead of the more general one adopted here.

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Authors and Affiliations

  1. University of Paris-Est, IRG (EA 2354), Place de la Porte des Champs, Créteil, 94010, France

    Erkin Diyarbakırlıoğlu

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  1. Erkin Diyarbakırlıoğlu
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  2. Mehmet H Satman
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Correspondence to Erkin Diyarbakırlıoğlu.

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Diyarbakırlıoğlu, E., Satman, M. The Maximum Diversification Index. J Asset Manag 14, 400–409 (2013). https://doi.org/10.1057/jam.2013.28

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  • DOI: https://doi.org/10.1057/jam.2013.28

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