Whenever constructing a quant portfolio or managing portfolio risk, the risk model is at the heart of the process. A risk model, usually estimated with a sample covariance matrix, has 3 typical issues.
- Not positive-definite, which means it’s not invertible.
- Exposed to extreme values in the sample, which means it’s highly unstable through time and will be exploited by the optimizer.
- Ex-post tracking error is always larger than the ex-ante tracking error, given a stochastic component in the holdings, which means investors will suffer from unexpected variances, either large or small.
Issue 1 is a pure math problem, but issue 2 and 3 are more subtle and more related to each other. A common technique called shrinkage has been devised to solve these issues. The idea behind is to add more structure to the sample covariance matrix by taking a weighted average between itself and a more stable alternative (e.g. a single-factor model or a constant correlation covariance matrix). Two main considerations are involved in the usage of shrinkage: 1. what’s the shrinkage target, i.e. the alternative? 2. what’s the shrinkage intensity, i.e. the weight assigned to each matrix?
Links above provide details about these considerations. I did several tests to show how the differences between ex-ante and ex-post tracking errors vary when using different shrinkage targets and intensities.The test is done with 450 stocks that were both in the S&P500 by the end of Oct 2016 and had been listed for at least 10 years. An equal-weighted portfolio is formed using 2-year weekly data and is rebalanced every month.
The test is done with 450 stocks that were both in the S&P500 by the end of Oct 2016 and had been listed for at least 10 years. An equal-weighted portfolio is formed using 2-year weekly data and is rebalanced every month. The shrinkage intensity changes from 10% to 90% by 10% throughout the test. The spreads between the ex-ante and ex-post variances are recorded each week .
As shown above, the Ledoit-Wolf approach (single-factor, optimal intensity as derived by L&W) creates the least estimation error among all other approaches tested. Interestingly, the sample covariance matrix approach shows higher ex-ante risks than ex-post, which violates the theory mentioned above. This is possibly because in this test the ex-ante variances always stay constant for four weeks while the ex-post variances change every week, which amplifies the actual spread if we believe that they should move together over time.