Covariance Estimation For Markowitz Portfolio Optimization

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

2. Outline • Work Done Last Week • Some Results • Conclusion and Future Work

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:

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

5. PCA Estimator

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

7. Horse Race

8. Horse Race (PCA) 3/10/2010 8

9. More Horse Race Mean return of non-constrained portfolio: 0.7% – 1.2%

10. More Horse Race

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.

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

13. Resampling (Bootstrapping)

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

15. Robust Allocation 3/10/2010 15

16. THANK YOU!