We consider the problem of the statistical uncertainty of the correlation matrix in the optimization of a financial portfolio,and focus on the application of cluster and random matrix theory(RMT) in Markowitz's portfolio model,where the estimation of the correlation matrix is unavoidably associated with a statistical uncertainty due to the finite length of the asset return time series.Hierarchical cluster analysis is found that it is quite robust with respect to measurement noise due to the finiteness of sample size.Single linkage method and average method adopted here are both hierarchical cluster analysis.Single linkage cluster is a kind of hierarchical cluster analysis.In the single linkage procedure,the distance between cluster p and cluster q is defined by the nearest distance between the elements in cluster p and the element of cluster q.Average linkage is anther hierarchical cluster analysis adopted here,whose distance between cluster p and cluster q is defined as the average distance of all elements of p and q. By using the hierarchical cluster procedure,the number of distinct elements of the correlation coefficient matrix turns from n(n-1)/2 to n-1.Then the correlation coefficient matrix is a positive ultrametric matrix.We construct the portfolio by solving the Markowitz optimization problem by using the ultrametric matrix rather than the original correlation matrix.The random matrix theory(RMT) takes the stock market as a complex system,in which the elements act on each other in unknown ways.It needs three steps to use RMT to filter the correlation coefficient matrix:(1) measure the chaotic degree of the system with the original correlation coefficient matrix;(2) calculate the eigenvalues of the chaotic degree;(3) remove the meaningless part of the eigenvalues.To test the effect of the filters,we use the data of 2004-2007 of the 50 stocks composing the SSE 50 index to calculate the correlation coefficient matrix and apply the hierarchical cluster analysis procedure and RMT to filter it.Based on the filtered matrix and original matrix,we use the Markowitz portfolio model.Then we compare the reliability,risk,effective size and investment behavior of the portfolios and the results are as follow:(1) The realized risk and the predict risk of the RMT portfolios are both the smallest and these of the average linkage portfolios are the largest,and these of the Markowitz's portfolios and the single linkage portfolios are between them;(2) The reliability of the single linkage portfolios are remarkably the highest and the average linkage portfolios are the lowest,thus,those of the Markowiz's portfolios and the RMT portfolios are close;(3) The effective size of the RMT portfolio is apparently larger then the other three portfolios,and in the three portfolios,the effective size of the average linkage portfolios are the smallest,which means the diversification of the average linkage portfolios are worst of all;(4) The average linkage portfolios has the highest risk and lowest return.We cannot charge the investment behaviors of the RMT portfolios and the Markowiz's portfolios are because their risk and return are both in different extent,but we see the RMT portfolio's risk and return are both the smallest and the Markowitz's portfolio's are both the largest.We draw a conclusion that the statistic uncertainty of the correlation coefficient matrix can be avoided after the matrix is filtered by single linkage cluster analysis and RMT,and the portfolios educed from the filtered matrix is better than that from the original matrix.In the improved portfolios,the RMT portfolio is better than the cluster portfolios.Single linkage cluster analysis and RMT can improve the mean-variance model while the average linkage cannot. |