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

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

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

1 Data Bouchard’s idea Marchenko-Pustur Law

Analysis Eigenvalue Decomposition of Fully Allocated MVO

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

Data Filtered Variance-Covariance Matrix

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

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

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

Procedure Correlation Matrix Distribution of Eigenvalues Best Fit M-P Distribution Filter Noisy Data Analysis Procedures

Analysis Ideas Random & Not Random Marchenko-Pastur Law

Analysis Ideas

Analysis Minimization

Analysis Minimization

Fitting result

Analysis

Analysis of largest λ The largest eigenvalue λ=

Analysis Total variance explained by noise

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

Reconstruction Theory

Reconstruction Theory

Analysis Filtered Variance-Covariance Matrix

Reconstruction Calculated Filtered Optimal Weight

Reconstruction Calculated Filtered Optimal Weight

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

Reconstruction Sample Weight and Filtered Weight Comparison

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

Reconstruction Cumulative Value of Filtered Portfolio and Sample Portfolio Per Month

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

Questions