Covariance Estimation For Markowitz Portfolio Optimization

1. Covariance Estimation For Markowitz Portfolio Optimization Ka Ki Ng Nathan Mullen PriyankaAgarwal Dzung Du RezwanuzzamanChowdhury

2. Outline: • Michaud’s Resampling combined with Ledoit’s estimators • Weight Descriptor • Return Analysis • Standard error • Conclusion and Future Research

3. Resampling combined with Ledoit’s Estimate (μ, Σ) from the observed data using estimators in Ledoit’s paper Propose the distribution for the returns, e.g., Returns ~ N(μ, Σ) – meaning prices follow log-normal distributions Resample n (large) of Monte Carlo scenarios Solve the optimization problem for each Monte- Carlo scenario Resampling allocation computed as the average of all obtained allocations. 3/10/2010 3

4. Performance 3/10/2010 4

5. Performance 3/10/2010 5

6. Weight Descriptors – Lowest Weight 3/10/2010 6

7. Weight Descriptors – Highest Weight 3/10/2010 7

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

9. Return Analysis • New features for return analysis in software • Average annual return (both geometric and arithmetic) • Plot of annual return • Histogram of monthly return with Gaussian fit

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

11. Annual Return Plots

12. Histogram of Returns • Typical returns have heavier tails than a Gaussian distribution • Gaussian fit

13. Average Annual Geometric Return

14. Analysis • Return of identity matrix ≈ S&P 500 Equal-weighted return • Problems • 20% constrained results are much lower than 20% • Most importantly, 20% constrained results are lower than the unconstrained cases • Possible reasons for the error • q = 0.0153 is wrong • Not on efficient frontier • Most likely: Models based on previous data applying to future data. Bad predictor of future expected returns μ "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."

15. Sample Output Text File

16. Standard error Method 1: Assuming returns to be iidgaussian 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

17. Standard error- Conclusion • 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).

18. Results:

19. Results:

20. Results:

21. Results:

22. Project goals and achievements • Ledoit and Wolf [2002] • Implemented 8 estimators including ‘Shrinkage to market’ • Results for standard deviation closely match those in the paper. • Fixed calculations for constrained portfolio • Solved standard error calculation puzzle • Added functionality for Return Analysis • Weight Analysis

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

24. Future work: • Matlab code documentation and cleaning • Project report • Any further analysis Prof. Pollak suggests