1. Covariance Estimation For Markowitz Portfolio Optimization Ka Ki Ng Nathan Mullen Priyanka Agarwal Dzung Du Rezwanuzzaman Chowdhury 3/10/20101

2. Outline Work Done Last Week Some Results Conclusion and Future Work 3/10/20102

3. Portfolio Selection Problem Given N stocks with mean return μ and covariance matrix Σ Markowitz’s portfolio selection framework: where q is the expected return level (constrain). Closed form solution: 3/10/20103

4. Work done Implement PCA estimator Run constrained portfolio selection for all our covariance matrix estimators. Resampling (Bootstrapping) Approach 3/10/20104

5. PCA Estimator 3/10/20105

6. Backtesting Time window: same as Ledoit and Wolf’s paper – Use NYSE and AMEX stocks from August 1962 to July 1995 – For each year t from 1972 to 1994 In-sample period: August of year t-10 to July of year t for estimation Out-of-sample period : August of year t to July of year t+1 3/10/20106

7. 7 Horse Race Ledoit’s Std (unconstrained ) Our Std (unconstrained) Identity17.7518.42 Constant Correlation14.2713.22 Pseudoinverse12.3712.10 Market Model12.0011.15 PCA10.319.51 Shrinkage to identity10.219.87 Shrinkage to market9.558.95

8. 3/10/20108 Horse Race (PCA) Number of Principal ComponentsStd (in %) 310.0488 59.5122 109.1124 259.0689 358.9262 508.8409

9. 3/10/20109 More Horse Race Ledoit’s Std (constrained ) Our Std (constrained) Pseudoinverse12.3712.10 Market Model12.0011.15 Shrinkage to market9.558.95 Ledoit’s Std (20%-constrained ) Our Std (20%-constrained) Identity17.94107.98 Constant Correlation16.30127.52 Market Model13.77118.46 PCA11.3099.96 Shrinkage to identity11.1195.27 Shrinkage to market10.43 88.06 Mean return of non-constrained portfolio: 0.7% – 1.2%

10. 3/10/201010 More Horse Race Ledoit’s Std (constrained ) Our Std (constrained) Pseudoinverse12.3712.10 Market Model12.0011.15 Shrinkage to market9.558.95 Ledoit’s Std (20%-constrained ) Our Std (2%-constrained) Constant Correlation16.3017.20 Market Model13.7714.34 PCA11.3011.49 Shrinkage to identity11.1111.36 Shrinkage to market10.4310.53

11. Another Approach ??? Main disadvantage of the classical MV- portfolio optimization: extremely sensitive to the [unknown] input estimates of mean and covariance matrix. Small change in mean or covariance estimates lead to significant change in weights. 3/10/201011

12. Michaud’s Resampling (Bootstrapping) 1.Estimate (μ, Σ) from the observed data 2.Propose the distribution for the observed data, e.g., L ~ N(μ, Σ) 3.Resample n (large) of Monte Carlo scenarios 4.Solve the optimization problem for each MC scenario 5.Resampling allocation computed as the average of all obtained allocations 3/10/201012

13. Resampling (Bootstrapping) 3/10/201013 ShrinkageShrinkage + Resampling100 Shrinkage + Resampling200 88.0636.2836.19

14. Future Work Implement remaining estimators Check the constrained portfolio problem, compare to similar results from literature Clean up MATLAB codes 3/10/201014

15. Robust Allocation 3/10/201015

16. THANK YOU!