基于最小扰动相关去噪法股票投资组合风险优化

Optimization of stock portfolio risks based on correlation filtering using minimum perturbations of eigenvectors

金融相关矩阵的计算是构建金融投资组合的基础.为解决金融相关矩阵的"维数灾祸"问题,进而促进金融投资组合风险的优化,受基于随机矩阵理论(RMT)和特征向量的Krzanowski稳定性的KR去噪法的启发,对收益相关矩阵特征值增大时的特征向量最小扰动进行了数学推导,并将以该扰动衡量的特征向量的Krzanowski稳定性引入到RMT去噪法中,进而建立对金融收益相关矩阵去噪的KRMIN方法.KRMIN法对KR法的算法进行了两方面的优化.一方面,KRMIN法对KR法的特征值设定方法进行了扩展;另一方面,KRMIN法采用模拟退火算法计算特征值.理论研究表明,由于在收益相关矩阵特征向量的稳定性和特征值算法准确性上的优势,KRMIN方法将获得比KR法更好的组合风险优化效果.通过bootstrap方法,开展了将LCPB法、PG+法、KR法和KRMIN法用于不同数量股票的投资组合优化的实证研究.结果表明:LCPB法、PG+法、KR法和KRMIN法都能通过股票收益相关矩阵去噪而带来投资组合风险的优化;基于收益相关矩阵特征向量的Krzanowski稳定性的KR法和KRMIN法的组合风险比其他两种方法更低;由KRMIN法得到的收益相关矩阵的特征向量稳定性和组合风险优化效果好于KR法.

Calculating financial correlation matrices is the basis of constructing investment portfolios. To avoid ＂curse of dimensionality＂ of financial correlation matrices and thus facilitate optimization of financial portfolio risk, inspired by the KR filtering method based on random matrix theory （RMT） and Krzanowski stability of correlation matrix eigenvectors, this study got the minimum perturbation of a certain eigenvector in a correlation matrix when the corresponding eigenvalue changed and introduced the eigenvector stability of correlation matrix measured by the minimum perturbation into RMT filtering. Thus the KRMIN method used for financial correlation denoising was established. The KRMIN method is based on improvement on the KR method. For the KRMIN filter, the setting method of new eigenvalues replacing noisy ones is gotten by extending that of the KR method, and the new eigenvalues are computed by simulated annealing. Theoretical studies have shown that because eigenvectors of earnings correlation matrices are more stable and calculation of eigenvalues is more accurate, the KRMIN filter can produce better portfolios than the KR filter. By means of bootstrapping technique, the empirical study used LCPB, PG＋, KR and KRMIN methods for portfolio optimization of different number of stocks. It proves that all the methods can result in optimization of stock portfolio risks by filtering return correlation matrices. Portfolio risks of KR and KRMIN methods considering Krzanowski stability of eigenvectors of earnings correlation matrices are lower than those of the other two methods. And in contrast to the KR method, the KRMIN method can cause greater stability of eigenvectors of earnings correlation matrices and lowerportfolio risks.