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Instructor: Chris Bemis

Random Matrix in Finance Understanding and improving Optimal Portfolios. Instructor: Chris Bemis. Mantao Wang, Ruixin Yang, Yingjie Ma, Yuxiang Zhou, Wei Shao, Zhengwei Liu. Purpose and Phenomenon of Project. The impact of near-zero eigenvalues in mean-variance optimization.

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Instructor: Chris Bemis

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

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

  3. Data 300 stocks 546 weeks Analysis σ, λ, Q Reconstruction Optimize mean variance CONTENTS 1 2 3

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

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

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

  13. Analysis Ideas

  14. Analysis Minimization

  15. Analysis Minimization

  16. Fitting result Q = 1.1494

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

  27. Reconstruction Sample Weight and Filtered Weight Comparison

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

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