1. Instructor: Chris Bemis Random Matrix in Finance Understanding and improving Optimal Portfolios Mantao Wang, Ruixin Yang, Yingjie Ma, Yuxiang Zhou, Wei Shao, Zhengwei Liu

2. Purpose and Phenomenon of Project Finding optimal weights Covariance matrix Marchenko-Pustur to fit data PCA reconstruction The impact of near-zero eigenvalues in mean-variance optimization

3. 1 23 Data 300 stocks 546 weeks Analysis σ, λ, Q Reconstruction Optimize mean variance

4. 1 Data Bouchard’s idea Marchenko-Pustur Law

5. Analysis Eigenvalue Decomposition of Fully Allocated MVO

6. Data Selection 300 stocks Х 546 weeks Criterion: Return history over 10 years of weekly data Biggest market capitalization

7. Data Filtered Variance-Covariance Matrix

8. Data Selection 300 stocks Х 546 weeks Why some of eigenvalues close to 0? Some original return data are extremely small Random effect Collinearity among 300 stocks The impact of near-zero eigenvalues in MVO

9. 2 Analysis of Results Empirical distribution of eigenvalues Marchenko-Pustur Law Analysis

10. Correlation Matrix Best Fit M-P Distribution Filter Noisy Data Goals: To eliminate the random noise in the covariance matrix Analysis Procedures

11. Procedure 1 2 3 4 Correlation Matrix Distribution of Eigenvalues Best Fit M-P Distribution Filter Noisy Data Analysis Procedures

12. Analysis Ideas Random & Not Random Marchenko-Pastur Law

13. Analysis Ideas

14. Analysis Minimization

15. Analysis Minimization

16. Fitting result

17. Analysis

18. Analysis of largest λ The largest eigenvalue λ=118.3564

19. Analysis Total variance explained by noise

20. 3 Reconstruction Filtered Variance-Covariance Matrix An Example of Mean-Variance Optimization

21. Reconstruction Theory

22. Reconstruction Theory

23. Analysis Filtered Variance-Covariance Matrix

24. Reconstruction Calculated Filtered Optimal Weight

25. Reconstruction Calculated Filtered Optimal Weight

26. Weight from filtered Sample Less volatility Lower concentration No extreme shorting Weight from Sample Bigger volatility Higher concentration Extreme shorting Reconstruction Comparison the weight

27. Reconstruction Sample Weight and Filtered Weight Comparison

28. Reconstruction Sample Weight and Filtered Weight Comparison Expected Return from Sample Covariance Matrix is

29. Reconstruction Cumulative Value of Filtered Portfolio and Sample Portfolio Per Month

30. Reconstruction Cumulative Value of Filtered Portfolio and S&P 500 Per Month

31. Questions