Abstract: In the mean-variance portfolio model, the covariance matrix is likely to be singular since the multi-collinearity and correlation can arise from the increase of financial assets, especially when considering a large-scale portfolio. In view of this situation, we reconsider the mean-variance portfolio problem under singular covariance matrix. A new approach based on generalized inverse matrix is proposed as a remedy for the deficiency of conventional methods in which covariance matrix is constrained to be invertible. The analytic solutions of frontier portfolio and portfolio frontier are derived with different interest rates for borrowing and lending, which extending successfully the classic Markowitz portfolio model.

Key words: financial engineering, portfolio, moore-penrose generalized inverse, different interest rates for borrowing and lending

摘要： 随着金融资产种类的增加,特别是考虑大规模投资组合问题时,很可能出现资产间的多重共线性或相关性,从而出现协方差阵奇异的情况。然而,目前关于投资组合的均值—方差分析大都是在协方差阵正定的条件下得到的,因此,不适用于奇异协方差阵的情形。针对这一问题,利用广义逆矩阵研究了协方差阵奇异时的均值—方差投资组合模型,在不同借贷利率条件下得到了前沿组合和组合前沿的解析解,突破了传统方法中要求协方差阵可逆的限制,推广了经典Markowitz模型。

关键词: 金融工程, 证券组合, Moore-Penrose广义逆, 不同借贷利率