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Compound Wishart Matrices and Noisy Covariance Matrices: Risk Underestimation

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  • Beno^it Collins
  • David McDonald
  • Nadia Saad

Abstract

In this paper, we obtain a property of the expectation of the inverse of compound Wishart matrices which results from their orthogonal invariance. Using this property as well as results from random matrix theory (RMT), we derive the asymptotic effect of the noise induced by estimating the covariance matrix on computing the risk of the optimal portfolio. This in turn enables us to get an asymptotically unbiased estimator of the risk of the optimal portfolio not only for the case of independent observations but also in the case of correlated observations. This improvement provides a new approach to estimate the risk of a portfolio based on covariance matrices estimated from exponentially weighted moving averages of stock returns.

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  • Beno^it Collins & David McDonald & Nadia Saad, 2013. "Compound Wishart Matrices and Noisy Covariance Matrices: Risk Underestimation," Papers 1306.5510, arXiv.org.
    • Handle: RePEc:arx:papers:1306.5510
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    References listed on IDEAS

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    1. Pafka, Szilárd & Kondor, Imre, 2003. "Noisy covariance matrices and portfolio optimization II," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 319(C), pages 487-494.
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    3. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    4. Giulio Biroli & Jean-Philippe Bouchaud & Marc Potters, 2007. "The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy," Papers 0710.0802, arXiv.org.
    5. Tse, Y. K., 1991. "Stock returns volatility in the Tokyo stock exchange," Japan and the World Economy, Elsevier, vol. 3(3), pages 285-298, November.
    6. Akgiray, Vedat, 1989. "Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts," The Journal of Business, University of Chicago Press, vol. 62(1), pages 55-80, January.
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