Instructor:Chris Bemis

Random Matrix in FinanceUnderstanding 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

2 3

Data300 stocks 546 weeks

Analysisσ, λ, Q

ReconstructionOptimize mean variance

1 Data

• Bouchard’s idea

• Marchenko-Pustur Law

AnalysisEigenvalue Decomposition of Fully Allocated MVO

Data Selection300 stocks Х 546 weeks

Criterion:

•Return history over 10 years of weekly data

•Biggest market capitalization

DataFiltered Variance-Covariance Matrix

Data Selection300 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

1

2

3

4

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=118.3564

Analysis Total variance explained by noise

3 Reconstruction

•Filtered Variance-Covariance Matrix

•An Example of Mean-Variance Optimization

ReconstructionTheory

ReconstructionTheory

AnalysisFiltered Variance-Covariance Matrix

ReconstructionCalculated Filtered Optimal Weight

ReconstructionCalculated Filtered Optimal Weight

Weight from filtered Sample• Less volatility• Lower concentration• No extreme shorting

Weight from Sample• Bigger volatility• Higher concentration• Extreme shorting

ReconstructionComparison the weight

ReconstructionSample Weight and Filtered Weight Comparison

ReconstructionSample Weight and Filtered Weight Comparison

Expected Return from Sample Covariance Matrix is

Expected Return from Sample Covariance Matrix is

ReconstructionCumulative Value of Filtered Portfolio and Sample Portfolio Per Month

ReconstructionCumulative Value of Filtered Portfolio and S&P 500 Per Month

Questions