1. Covariance Estimation For Markowitz Portfolio Optimization Ka Ki Ng Nathan Mullen Priyanka Agarwal Dzung Du Rezwanuzzaman Chowdhury 14/7/2010

2. Outline: 1.Michaud’s Resampling combined with Ledoit’s estimators 2.Weight Descriptor 3.Return Analysis 4.Standard error 5.Conclusion and Future Research 24/7/2010

3. Resampling combined with Ledoit’s 1.Estimate (μ, Σ) from the observed data using estimators in Ledoit’s paper 2.Propose the distribution for the returns, e.g., Returns ~ N(μ, Σ) – meaning prices follow log-normal distributions 3.Resample n (large) of Monte Carlo scenarios 4.Solve the optimization problem for each Monte- Carlo scenario 5.Resampling allocation computed as the average of all obtained allocations. 3/10/20103

4. 4 Performance Std (unconstrained ) Resampling Combined (unconstrained) Identity18.1218.18 Constant Correlation13.1019.9 Market Model11.0821.64 PCA9.1321.34 Shrinkage to identity10.8318.20 Shrinkage to market8.949.46

5. 3/10/20105 Performance Std (constrained ) Std Resampling Combined (constrained) Identity18.1118.18 Constant Correlation15.1919.9 Market Model12.5920.11 PCA10.3620.43 Shrinkage to market9.539.02 Shrinkage to identity10.3719.3

6. 3/10/20106 Weight Descriptors – Lowest Weight Ledoit’s valuesOur values Identity0.090.08 Constant Correlation-0.17-0.14 Industry Factor-1.08-1.20 Market Model-0.41-0.36 PCA-0.84-0.81 Shrinkage to identity-1.23-1.07 Shrinkage to market-1.01-0.89

7. 3/10/20107 Weight Descriptors – Highest Weight Ledoit’s valuesOur values Identity0.090.08 Constant Correlation2.862.51 Industry Factor2.812.94 Market Model2.11.87 PCA2.952.66 Shrinkage to identity1.191.03 Shrinkage to market3.813.0

8. Rate of Returns on Investments Average annual arithmetic return – No reinvesting, i.e. start each year with the same amount of money where Average annual geometric return – Time weighted – With reinvesting 84/7/2010

9. Return Analysis New features for return analysis in software 1.Average annual return (both geometric and arithmetic) 2.Plot of annual return 3.Histogram of monthly return with Gaussian fit 94/7/2010

10. Compare to S&P 500 Compared the returns of all 16 estimators (for both unconstrained and 20% constrained cases) and S&P 500 – The comparison was incomplete when it was presented two weeks ago Downloaded the S&P 500 data from CRSP Added an option for S&P 500 comparison in the software – Value-Weighted Return – Equal-Weighted Return 104/7/2010

11. Annual Return Plots 114/7/2010

12. Histogram of Returns Typical returns have heavier tails than a Gaussian distribution Gaussian fit 124/7/2010

13. Average Annual Geometric Return 134/7/2010

14. Analysis Return of identity matrix ≈ S&P 500 Equal-weighted return Problems 1.20% constrained results are much lower than 20% 2.Most importantly, 20% constrained results are lower than the unconstrained cases Possible reasons for the error 1.q = 0.0153 is wrong 2.Not on efficient frontier 3.Most likely: Models based on previous data applying to future data. Bad predictor of future expected returns μ 14 "For expected returns, we just take the average realized return over the last 10 years. This may or may not be a good predictor of future expected returns, but our goal is not to predict expected returns: it is only to show what kind of reduction in out-of-sample variance our method yields under a fairly reasonable linear constraint." 4/7/2010

15. Sample Output Text File 154/7/2010

16. Standard error Method 1: Assuming returns to be iid gaussian Method 2: Get N bootstrap samples for monthly returns assuming uniform distribution Method 3: Get N bootstrap samples for monthly log-returns assuming log-normal distribution Bootstrap results are stable for N=1000 164/7/2010

17. Standard error- Conclusion 1.Bootstrap technique confirmed our earlier belief that the standard errors presented in the paper are monthly. Though we agree it makes more sense to present annualized standard errors when standard deviations are annualized. 2. Our monthly SE results for uniform distribution bootstrap closely match with Ledoit’s (even for Identity and Psuedo-inverse estimators). 174/7/2010

18. Results: Unconstrained Standard Error (monthly) LedoitBootstrap Uniform Bootstrap Lognormal IID Gaussian Identity 0.440.420.230.22 Constant Correlation 0.190.180.16 Pseudo inverse 0.23 0.15 Market Model 0.160.170.14 Industry factors 0.17 0.110.12 PCA 0.160.190.12 Shrinkage to identity 0.170.190.12 Shrinkage to market 0.150.180.11 184/7/2010

19. Results: Constrained Standard Error (monthly) LedoitBootstrap Uniform Bootstrap Lognormal IID Gaussian Identity 0.420.380.240.23 Constant Correlation 0.29 0.19 Pseudo inverse 0.320.310.170.16 Market Model 0.27 0.150.16 Industry factors 0.23 0.13 PCA 0.220.250.13 Shrinkage to identity 0.210.240.120.13 Shrinkage to market 0.200.230.12 194/7/2010

20. Results: Unconstrained Standard Error (annualized) LedoitBootstrap Uniform Bootstrap Lognormal IID Gaussian Identity 0.441.450.800.76 Constant Correlation 0.190.620.55 Pseudo inverse 0.230.800.52 Market Model 0.160.590.48 Industry factors 0.170.590.380.42 PCA 0.160.660.42 Shrinkage to identity 0.170.690.42 Shrinkage to market 0.150.620.38 204/7/2010

21. Results: Constrained Standard Error (annualized) LedoitBootstrap Uniform Bootstrap Lognormal IID Gaussian Identity 0.421.320.830.80 Constant Correlation 0.291.000.66 Pseudo inverse 0.321.070.590.55 Market Model 0.270.940.520.55 Industry factors 0.230.800.45 PCA 0.220.870.45 Shrinkage to identity 0.210.830.420.45 Shrinkage to market 0.200.800.42 214/7/2010

22. Project goals and achievements Ledoit and Wolf [2002] 1.Implemented 8 estimators including ‘Shrinkage to market’ 2.Results for standard deviation closely match those in the paper. 3.Fixed calculations for constrained portfolio 4.Solved standard error calculation puzzle 5.Added functionality for Return Analysis 6.Weight Analysis 224/7/2010

23. Project goals and achievements Benninga & Disatnik [2002] 1.Implemented 8 estimators including ‘two block estimator’ 2.Ran simulations for different time periods 3.Results for standard deviation closely match those in the paper. 234/7/2010

24. Future work: 1.Matlab code documentation and cleaning 2.Project report 3.Any further analysis Prof. Pollak suggests 244/7/2010