Improved estimation of covariance matrix for portfolio optimization

1. Improved estimation of covariance matrix for portfolio optimization • Priyanka Agarwal • Rez Chowdhury • Dzung Du • Nathan Mullen • Ka Ki Ng

2. Progress so far: • Simulations for 12 estimators presented last week • Implementation of Benninga’s two block estimator • Speed Optimization • Cleaning and documenting the code for the software • 5. Portfolio with constraints • 6. PCA estimator

3. Motivation behind shrinkage and portfolio of estimators • Covariance matrix not invertible. • Optimal portfolio with large short-sale positions. • Large N (number of assets) as compared to T (time-series) - This makes the matrix ill-conditioned with large off-diagonal elements amplifying the estimation error.

4. Simulations • Stocks in NYSE only (as in Disatnik/ Benninga) • First portfolio formed on Jan 1974; last portfolio formed for 2003 • Total 360 monthly returns for each of the 6 simulations • In-sample period: 120 months and 60 months • Out-sample period: 12, 24 and 36 months • Compare the performance of shrinkage estimators vs. portfolio of estimators

5. Table 1.a

6. Table 1.b

7. Table 1.c

8. Table 2.a

9. Table 2.b

10. Table 2.c

11. Conclusion • Simulations show consistency of our codes. • Performance improvement using shrinkage estimator and portfolio of estimators is within the same range. • Portfolio of estimators is simpler to use and implement. • Shrinkage estimator gives rise to a new type of error.

12. Two Block Estimator • Motivation: To overcome the drawbacks of the short sale constraint • Discontinuity imposed on the relation between asset statistics and optimal asset weights • Solution obtained is numerical and not analytical • Result: Produces a positive GMVP in an unconstrained optimization

13. Two Block Estimator - Methodology • Estimated covariance matrix with two blocks. • Each block has the sample variance on the diagonal. • Pair of stocks within the same block have the same covariance (1 and 2 ). • Covariance between stocks from different blocks equals a third constant . • Disatnik and Benninga characterizes conditions on the covariances when unconstrained GMVP is positive. (refer to the paper)

14. Two Block Estimator – Arbitrary example • Each block with the same number of stocks divided based on permno. • 1 = (0.99) min(si2) • 2 = (0.99) min(si2) •  = (0.99) min(1, 2 )

15. Two Block Estimator – Results Portfolio with two block estimator performs the best amongst all with no constraints.

16. Two Block Estimator – Improvements • One block with stocks with positive beta and another with negative beta stocks. (positive 1 and 1 while negative ) • More than two blocks * We may implement this if time permits (this was not tested by Disatnik, Benninga 2006)

17. Speed Optimization • Worst case scenario: ~ 75% faster • Old code: ~3700 seconds • Optimized code: ~900 seconds • 2. 60 months in-sample takes longer to run than 120 months in-sample. • 3. ‘Out of memory’ issues with 60 months in-sample simulations

18. Backup

19. Estimate of shrinkage constant (shrinkage to market)

20. Future actions: • Fix covariance matrix estimation with constraints • Implement PCA and more than one factor industry models • Speed Optimization • Look into issue with shrinkage to constant correlation estimator • Fix memory issues with 60 months in-sample simulation • If time permits, implement a more financial oriented two-block estimator (this was not implemented by Disatnik, Benninga 2006)