Honey, I Shrunk the Sample Covariance Matrix

• Authors: Olivier Ledoit and Michael Wolf.
• BSE Working Paper: 92 | September 15
• Keywords: Covariance matrix , markowitz optimization , shrinkage , tracking error
• JEL codes: C13, C51, C61, G11, G15
• Covariance matrix
• markowitz optimization
• shrinkage
• tracking error

The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization. It contains estimation error of the kind most likely to perturb a mean-variance optimizer. In its place, we suggest using the matrix obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients towards more central values, thereby systematically reducing estimation error where it matters most. Statistically, the challenge is to know the optimal shrinkage intensity, and we give the formula for that. Without changing any other step in the portfolio optimization process, we show on actual stock market data that shrinkage reduces tracking error relative to a benchmark index, and substantially increases the realized information ratio of the active portfolio manager.