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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报

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具无条件协方差矩阵的最优资产组合有效前沿的几何刻画

王筱凌1,2,   王玉文3,4,   刘冠琦4   

  1. 1. 黑龙江财经学院金融学院，哈尔滨  150025

    2. 泰国格乐大学国际学院，曼谷  10220

    3. 哈尔滨石油学院数理教研部，哈尔滨  150027

    4. 哈尔滨师范大学数学科学学院，哈尔滨 150025

  • 收稿日期:2022-12-21 接受日期:2023-04-27 出版日期:2025-10-15 发布日期:2025-10-15
  • 通讯作者: 王玉文 E-mail: wangyuwen1950@aliyun.com
  • 基金资助:
    国家自然科学基金 (12101163)；黑龙江省教改(重点)委托项目 (SJGZ20190031)；黑龙江财经学院校级课题(青年) (XJQN202559).

Geometric Characterization of Efficient Frontier of Optimal Portfolio with Unconditional Covariance Matrix

WANG Xiaoling1,2,   WANG Yuwen3,4,   LIU Guanqi4   

  1. 1. College of Finance, Heilongjiang University of Finance and Economics, Harbin 150025

    2. International College, Krirk University, Bangkok 10220

    3. Department of Mathematics and Physics Teaching and Research, Harbin Institute of Petroleum, Harbin, 150027

    4. School of Mathematical Sciences, Harbin Normal University, Harbin 150025
  • Received:2022-12-21 Accepted:2023-04-27 Online:2025-10-15 Published:2025-10-15
  • Contact: Y. Wang. E-mail address: wangyuwen1950@aliyun.com
  • Supported by:
    The National Natural Science Foundation of China (12101163); the Key Teaching Reform Entrusted Project of Heilongjiang Province (SJGZ20190031); the School-level Project of Heilongjiang University of Finance and Economics (Youth) (XJQN202559).

摘要：

通过协方差矩阵的Moore-Penrose广义逆，针对基于均值–方差准则的任意有限个风险资产组合问题，在给定预期收益的条件下，建立了最小方差投资组合的最优策略和方差的表达式。由此，针对两种不同情况，通过广义逆进行分析，分别给出资产组合的有效前沿的几何刻画。所获结果不仅包含了文献中有关协方差矩阵在正定条件下的结果，也包含了当$n=2$且两个风险资产完全负相关时，其资产组合的有效前沿几何刻画的经典结果。

关键词: 资产组合, 均值–方差, 有效前沿, 协方差矩阵, Moore-Penrose逆

Abstract:

In this paper, by using the Moore-Penrose generalized inverse of the covariance matrix, for the mean-variance selection problem of any finite kinds of risk assets portfolio, the expression of the optimal strategies and the variance for the minimum variance portfolio under the given portfolio expected return are given. By proceeding the generalized inverse analysis, we further deduced the geometric characterization of the efficient frontier of the portfolio under two different circumstances, respectively. The conclusions not only include the results of the covariance matrix under the positive definite condition in the literature, but also include the classical results of the geometric characterization of the effective frontier of the portfolio when $n=2$ and the two risky assets are completely negatively correlated.

Key words: portfolio, mean-variance, effective frontier, covariance matrix, Moore-Penrose generalized inverse

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