Correlation Matrices Analysis Of Stock Returns: Based On The Random Matrix Theory

The research on the correlation of stock returns has its theoretical value and practical value as well. The former can be shown in that the stock market, a complex dynamic system, can be better known; and the latter can be shown in its importance in asset allocation and investment risk estimates. However, it is difficult to get a correlation matrix which is reliable and can stand the test of practice. Market conditions change over time, and this will cause changeable correlation between two stocks. Meanwhile the finite length of time series available to estimate cross correlations will introduces "measurement noise". Due to the reasons above, there is much uncertainty and a great deal of noise information in the empirical correlation matrices.Therefore, the study of the properties of the correlation matrices is very meaningful.The paper analyzes the statistic properties of correlation matrices for stock returns first and finds that there is a great deal of noise information in empirical correlation matrices.In order to distinguish noise information from true information,200 stocks are selected from HS300 as sample and the daily return of the 200 stocks in three years (2007-2009) are used to calculate the correlation matrix of their return. And then the statistical difference between the empirical correlation matrix of return and the random correlation matrix is tested. By calculating the eigenvalues of correlation matrix, it is found that most eigenvalues fall into the predictive scope of random matrix theory, and about 3% of them are above the upper limit of the predicted value, and there is a strong linear relationship between the largest eigenvalue and the correlation coefficient. Then the nature of the eigenvectors of the Correlation matrix is discussed and it is found that the distribution of all the other eigenvector elements is relatively close to the normal distribution except several eigenvectors which are corresponding with large eigenvalue and there is a strong relationship between the vector element which is corresponding with the largest eigenvalue and the correlation coefficient.Finally, the application of random matrix theory in portfolio is discussed. And the empirical correlation matrix is filtered with the random matrix method: the differences of the mean-variance model modified by the random matrix theory in actual investment in different cases (in the short selling case and in the contrary case) are compared. And the conclusion is that:in the case of allowing short selling, the filter matrix can significantly improve the investment results of the model, and in case of no short selling, there is no significant improvement.The conclusions of the dissertation are as follows:1,In the A share market, there is much noise information in the correlation matrix of stock returns and the largest eigenvalue reflects the main information of the market.2,The correlation coefficient of the A share market as a whole is relatively large, and the consistency of the volatility of the stock market as a whole is much obvious:the prices rise and fall together, and the effectiveness of the market is relatively poor and the ability to spread risk is relatively weak.3,The method of filtering the correlation matrix using the random matrix theory is invalid in the A-share market investment.