1. Covariance Estimation For Markowitz Portfolio Optimization Team = {Ka Ki Ng, Nathan Mullen, Priyanka Agarwal,Dzung Du, Rez Chowdhury} Presentation by Rez

2. Outline 1. Covariance estimator code implementation this week 2. Overview of each estimator implemented –Results (std. dev. values) 3. More Results (some extra plots) 4. Conclusion & Future Work…

3. This week’s achievements Total of 14 estimators from 2 papers Implemented 3 more estimators from Ledoit & Wolf paper –Only 2 estimators left from this paper… Principals Components and Industry Factors –PCA code is almost working… some bugs :(

4. This week’s achievements Disatnik and Benninga has 7 estimators –We already had one working. –This week we implemented the remaining 6… Extra stuff that almost worked: -Almost got all the estimators to work with Ledoit and Wolf’s constraint of 20% expected return on portfolio’s… some bugs that should be very easily fixable… -Almost got all the estimators to work with short sales constraint… also should be relatively easy to debug…

5. Ledoit’s Standard Error Values Tried many things… Seems that Ledoit has developed his own method for estimating standard error for stock returns that are not necessarily assumed to be gaussian… The method and the code is buried in his big list of papers We have some leads and may get this to work…

6. Some methodology work Developed an algorithm to NOT look into the future for stock picking like Ledoit –It should also hold cash (“risk-free” rate) positions for stocks that drop out in investment horizon just like Ledoit –Yet to be implemented, but hopefully in the future…

7. Identity Estimators Covariance matrix is scalar multiple of identity matrix –Ledoit uses the mean of the diagonal values from the sample matrix for this… –Ledoit GMVP std dev = 17.75 –Our GMVP std dev = 18.43

8. Constant Correlation -Every pair of stocks has the same correlation coefficient. -N + 1 parameters (N variance, 1 covariance) -Ledoit GMVP std dev = 14.27 -Our GMVP std dev = 13.22

9. Shrinkage to Identity The scalar multiple of identity matrix is the shrinkage target – Ledoit GMVP std dev = 10.21 –Our GMVP std dev = 9.87 Shrinkage factor is stable around 0.1. One can argue this implies robustness of the model in some sense…

10. Note on these results… Our implementation gives slightly better (lower) risk values than Ledoit –Why? We look into the future (Benniga method) –If stocks don’t drop out, variance (volatility) is reduced –Also, Ledoit’s using cash positions should be a factor »The cash positions can be relatively pretty big…

11. Benninga estimators Diagonal estimator –Simply the diagonal of the sample matrix. Everything else is zero. Stalking Horse. Terrible Estimator, but att least it’s invertible… Benninga GMVP std dev = 13.12 Our GMVP std dev = 15.01

12. Shrinkage to constant correlation matrix The Constant Corr matrix is the shrinkage target –Benninga GMVP std dev = 8.52 –Our GMVP std dev = 9.11 Shrinkage factor is sort of stable around 0.7… Maybe one can argue this implies robustness of the model in some sense…

13. Random average of sample and single index Uniformly random variable alpha goes from (0.5,1) –Why (0.5,1) instead of (0,1)? –Benninga uses Ledoit and Wolf observation that there is more estimation error in the sample matrix than there is specification error in the single-index matrix. So AT MOST sample gets half the weight Whereas market index matrix gets AT LEAST half the weight Benninga GMVP std dev = 8.51 Our GMVP std dev = 9.08

14. Portfolio of sample, single index, and constant corr Equally weighted –Benninga GMVP std dev = 8.47 –Our GMVP std dev = 9.00

15. Portfolio of sample, single index, and constant corr, diagonal Equally weighted –Benninga GMVP std dev = 8.46 –Our GMVP std dev = 8.98

16. Portfolio of sample, single index, diagonal Equally weighted –Benninga GMVP std dev = 8.39 –Our GMVP std dev = 8.97

17. Observation on these results The ranking of estimators based on risk is the same, with the very simple diagonal portfolio estimator being a very close second best! Our standard deviation values are slightly higher though… –Why? –Our CSRP data window is 1970 to 1995 (Ledoit and Wolf) –Main reason: Benninga and Disatnik window is from 1964 to 2003. –Also our periods go from August to July »Benninga’s goes from January to December

18. Future Work We should definitely also run the models for the Benninga window of observation. All the values for the Benninga estimators are expected to match then… »The End. - Thanks!