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Covariance Estimation For Markowitz Portfolio Optimization

Covariance Estimation For Markowitz Portfolio Optimization. Team = {Ka Ki Ng, Nathan Mullen, Priyanka Agarwal,Dzung Du, Rez Chowdhury} Presentation by Rez. Outline. 1. Covariance estimator code implementation this week 2. Overview of each estimator implemented Results (std. dev. values)

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Covariance Estimation For Markowitz Portfolio Optimization

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  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!

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