Research On Improved Estimation Of Covariance Matrix For Portfolio Risk Measurement

Since Markowitz(1952)proposed the mean variance model,portfolio se-lection has always been a topic of concern.Among many models dealing with portfolio selection,the moment-based mean variance model and the global min-imum variance model are the two most commonly used models because of their simplicity,intuition and effectiveness.In the mean variance model,the optimal portfolio weight depends on the mean and covariance matrix of asset return-s,while in the global minimum variance model,the optimal portfolio weight only depends on the covariance matrix of asset returns.Therefore,covariance matrix plays an important role as a measure of risk in both mean-variance model and global minimum variance model.Usually,sample covariance ma-trix is used as the estimation of covariance matrix.However,a large number of academic studies show that there are some shortcomings in using sample covariance matrix as risk measurement in real-world data.Therefore,based on the mean variance model and the global minimum variance model,this paper improves the performance of the sample covariance matrix as a risk measure in the portfolio from three aspects:(1)In the framework of mean variance model,it is assumed that excess asset returns are independent of each other and follow the normal distribution.The optimal portfolio weights derived from the mean variance model depend on the covariance matrix of the asset returns.In practice,the sample covari-ance matrix is used instead of the real covariance matrix to obtain the available portfolio weights.However,these exists estimation error in the sample covari-ance matrix and the shrinkage method can effectively reduce the estimation error in the sample covariance matrix.Therefore,in the framework of mean variance model,based on Cholesky decomposition of sample covariance matrix,we propose to use Stein-type shrinkage estimation strategy to determine the optimal portfolio weight.At the same time,by using the maximum expected utility of investors,the effective shrinkage parameter domain is theoretically given.(2)In the mean variance model,the optimal portfolio weight depends on the mean and covariance matrix of asset returns.Normally,the available port-folio weights are obtained by using sample mean and sample covariance matrix to replace the mean and covariance matrices of asset returns respectively.How-ever,there are estimation errors in sample mean and sample covariance matrix,which makes the mean variance model more sensitive to estimation errors.In addition,it is found that the estimation error in sample mean are much larg-er than those in sample covariance.Therefore,we use the global minimum variance model which only depends on the covariance matrix to replace the mean variance model.Also,a large number of literatures show that the asset returns are non-normal distribution.Therefore,we remove the harsh assump-tion that the asset returns follow the normal distribution and only assume that the covariance matrix of asset returns exists.However,estimation error still exists in the sample covariance matrix.At the same time,the volatility of the financial market will cause the time-varying of the financial asset risk and the correlation between assets.The sample covariance matrix can not capture this dynamic change very well.It is found that the estimation error can be effectively reduced by using the linear combination of the inverse covariance matrix and the target matrix,which can get higher risk adjusted return and lower level risk in portfolio selection,and time-varying covariance matrix can properly and effectively capture the time-varying of the financial asset risk and the correlation between assets.In view of the advantages of the linear combination of the inverse covariance matrix and the target matrix and the time-varying covariance matrix,we propose to shrink the time-varying inverse conditional covariance matrix in order to enhance the performance of the port-folio selection under the global minimum variance model.Because the inverse of matrix plays an important role in portfolio selection,we give a new method to improve the accuracy of inverse matrix calculation.(3)In the global minimum variance model,financial asset risk and the correlation between assets are measured by covariance matrix.The covari-ance matrix can be decomposed into diagonal variance matrix and Pearson correlation coefficient matrix.In the global minimum variance model,variance measures the positive and negative returns,and takes the summation of the positive and negative returns as the total risk.However,investors only care about negative returns.In addition,Pearson correlation can only measure the linear correlation,and it can not well describe the characteristics of fat tail and nonlinear correlation of financing assets data.Therefore,it is inappro-priate to use the traditional covariance matrix as a measure of the financial asset risk and the correlation between assets.Literature and simulation results show that semivariance can effectively describe the negative return risk,and distance correlation can effectively fit the problem of thick tail and nonlinear correlation of financial data.Therefore,considering short-selling constraints,we propose a hybrid covariance matrix based on semivariance and distance dependence to measure risk and improve the performance of portfolio.In this paper,three risk measurement methods are compared and evalu-ated by using simulated data and real-world data.The results show that our methods are reasonable and effective.