Eigenfactor Bias Adjustment
A correction applied when estimating a factor covariance matrix from a limited history of returns, because the sample estimate systematically overstates the largest eigenvalues and understates the smallest ones.
Prerequisites: Covariance Matrix Estimation
When a factor risk model estimates a covariance matrix from a finite history of returns, sampling error doesn't spread evenly across all directions in the matrix. It systematically inflates the eigenvalues corresponding to the most volatile combinations of factors and deflates the eigenvalues of the least volatile combinations — even if the true underlying covariance matrix were perfectly stable, the sample estimate from limited data would still show this stretching. A portfolio optimizer that trusts the raw sample covariance matrix ends up underestimating risk in the directions the matrix says are calmest, which is exactly where an optimizer likes to load up leverage — producing portfolios that look low-risk in the model but carry more real risk than estimated.
Eigenfactor bias adjustment corrects this by simulating many resampled covariance matrices from the estimated factor model, measuring how much each eigenvalue is distorted on average across the simulations, and then rescaling the real matrix's eigenvalues to remove that known bias — shrinking the largest ones down and inflating the smallest ones back up before handing the matrix to a portfolio optimizer.
Sample covariance matrices systematically overstate risk in their most volatile directions and understate it in their calmest ones — eigenfactor bias adjustment corrects each eigenvalue by an amount estimated from resampling, preventing an optimizer from exploiting the model's artificially calm directions.
Further reading
- Menchero & Lee, 'Efficiently Combining Multiple Sources of Alpha', 2015 (eigenfactor bias correction discussion)