Extension of the random matrix theory to the L-moments for robust portfolio selection

Abstract

In this paper, we propose an approach for selecting stocks from a large investment universe by studying information on the eigenvalues of the correlation matrix. For this purpose, we use a robust measure of moments called L-moments, and their extensions to a multivariate framework. The random matrix theory allows us to extract factors which contain real information from the estimator of the correlation matrix obtained using the L-moments (henceforth the Lcorrelation matrix). An empirical study of the American market shows the coherence of such an approach and highlights the consistency of the Lcorrelation matrix in comparison with the sample correlation matrix. For both estimators of the correlation matrix, it seems that the largest eigenvalue corresponds to the market, and that the other eigenvalues which contain information partition the set of all stocks into distinct sectorial groups. An analysis of the group of stocks shows that the selected stocks obtained from the Lcorrelation matrix outperform those obtained from the sample correlation matrix in terms of the Sharpe ratio, although the sample correlation matrix provides a well-diversified portfolio in terms of volatility in an out-of-sample investment approach.

Keywords:

• Asset allocation
• Correlation structures
• Econophysics
• Empirical finance
• Multi-factor models
• Portfolio allocation
• Portfolio analysis
• Portfolio constraints

JEL Classification:

• C1
• C10
• C13
• G1
• G10
• G11

Acknowledgements

I am grateful to Thierry Chauveau, Emmanuel Jurczenko and Thierry Michel for help and encouragement in preparing this work. I also thank Rama Cont, Frank Fabozzi and Gilles Zumbach and the anonymous referees for their feedback, the participants of the EFMA09 Symposium on Risk Management, and the participants of the International Finance Research Forum09.

Notes

¹Haugen (1999) shows that, under certain conditions, the global minimum variance portfolio provides a better Sharpe ratio than the S&P500 index using the S&P500 universe.

²For instance, the one-factor model of Sharpe (1963) allows for building a structured estimator of the covariance matrix.

³The time-domain estimators take into account all observed returns to build an estimator (the sample covariance matrix, for instance), in contrast the state-domain estimators consider only historical returns close to the actual return.

⁴They are only linearly influenced by large deviations.

⁵In some ways the problem of interpreting the correlations between individual stock-price changes is reminiscent of the difficulties experienced by physicists in the fifties in interpreting the spectra of complex nuclei. Large amounts of spectroscopic data on the energy levels were becoming available but were too complex to be explained by model calculations because the exact nature of the interactions was unknown. The random matrix theory has been developed in this context (Wigner 1956, Dyson 1962, Dyson and Mehta 1963, Mehta 1991) to deal with the statistics of energy levels of complex quantum systems. With the minimal assumption of a random Hamiltonian, given by a real symmetric matrix with independent random elements, a series of remarkable predictions were made and successfully tested on the spectra of complex nuclei. Deviations from the universal predictions of the random matrix theory identify system-specific, non-random properties of the system under consideration, providing clues about the nature of the underlying interactions.

⁶Sometimes, the inverting covariance matrix does not even exist.

⁷The null hypothesis states that the correlation matrix in the market is the identity matrix, which does not correspond to the empirical evidence in the market.

⁸The corresponding Lcorrelation matrix is not symmetric following our formula. However, the derived Lcorrelation matrix is a regular matrix.

⁹Plerou et al. (2002) show that eigenvectors corresponding to eigenvalues smaller than the theoretical lowest bound contain, as significant participants, pairs of stocks having the largest correlation coefficient value in the data sample.

¹⁰This result differs from the observations of Plerou et al. (2002). They find large values of the inverse participation ratio on both sides of the theoretical distribution, suggesting a random band matrix structure.

¹¹This is a mix of consumer staple and consumer discretionary.

¹²They use a larger universe of stocks in intra-daily and daily frequencies.

¹³The deviating eigenvectors correspond to the eigenvectors obtained from eigenvalues which deviate from the theoretical bound.

¹⁴The initial sample window goes from 05/29/1981 to 05/17/1991. The overlap matrix is computed by incrementing the initial sample window of one week, one year, two years, until 17 years.

¹⁵Note that the number of eigenvalues of the Lcorrelation matrix when the whole sample is considered is equal to seven in the S&P500 universe.

¹⁶The value of the theoretical upper bound depends on the number of historical returns and the number of stocks in the investment universe.

¹⁷The lowest eigenvalues are not taken into account because they only influence a small number of stocks and produce no empirical evidence.

¹⁸Since

where (resp. ) corresponds to the Lcorrelation coefficient between (resp.) with respect to (resp. ), we recover the Lvariance–covariance matrix from the following expression:

¹⁹The ratio between the number of assets in the database and the size of the estimation window remains constant for the two databases.

²⁰There is no cash considered in our expression of the Sharpe ratio.