A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices

Olivier Ledoit and Michael Wolf


Many economic problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator - the sample covariance matrix - is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte-Carlo confirm that the asymptotic results tend to hold well in finite sample.

This is based on the paper I sent on the academic job market under the title "Portfolio Selection: Improved Covariance Matrix Estimation". It constituted the first chapter of my 1995 Finance PhD thesis at MIT: Essays on Risk and Return in the Stock Market. I was invited to present it at UCLA, the University of Chicago, Wharton and Yale, all of which offered me tenure-track positions as Assistant Professor of Finance. I also presented it at the Q Group, who awarded me the Roger F. Murray prize.

The code in Matlab for the estimator proposed in the paper can be downloaded for free from the website of my co-author Michael Wolf in the Department of Economics of the University of Zurich.

Journal of Multivariate Analysis, Volume 88, Issue 2, February 2004, pages 365-411

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