摘 要

伴随市场经济的持续完善与健全, 应用数学规划基本理论来解决投资组合情况的需求逐渐增多. 分析数学规划于投资组合领域内的运用, 目的在于为投资人员争取最多的收益, 帮助投资人员在能够承担风险范围内获得最多的收益, 或是在保障固定收益的情况下使得投资组合所承担的风险最小, 帮助投资人员对于具体的投资行为实现资产的最佳配置, 促使资产获得最多的收益，具备极高的实践价值.

拉格朗日乘子法被普遍应用在数学规划领域, 可用以解决带约束的相关问题. 此种方式的核心观念为以原本问题的拉格朗日函数为切入点, 加之适宜的罚函数, 进而计算出没有约束限制的优化子问题.

此次研究借助拉格朗日乘子法来研究投资组合优化中的均值-方差模型. 首先考虑无交易成本的均值-方差风险度量模型, 结合具体案例, 利用拉格朗日乘子法进行求解, 分别给出在风险最低且满足一定收益率的条件下的投资方案, 以及在收益最大且满足一定风险下的投资方案.

其次考虑有交易成本的均值-方差风险度量模型, 结合具体案例, 利用拉格朗日乘子法进行求解, 给出在风险最低且满足一定收益率的条件下的投资方案, 并对投资风险进行预估.

最后, 本文就均值-方差模型进行总结并提出展望, 在选择投资组合时, 应当运用恰当的模型和方法, 尽可能在最大的收益率以及最小的风险下, 找出最合适的投资组合比例.

关键词: 拉格朗日乘子法; 投资组合优化; 均值-方差风险度量模型.

The application of Lagrange multiplier method in portfolio optimization

ABSTRACT

With the continuous improvement and perfection of the market economy, the demand of applying the basic theory of mathematical planning to solve the portfolio situation is increasing gradually. Analysis of the use of mathematical programming in the field of portfolio, aims to staff for most of the investment profit, help investment people to get the most revenue risk scope, or in the case of ensuring fixed income makes the minimum risk portfolio, help investment people for specific investment behavior to achieve the optimal allocation of assets, assets get the most benefits, has high practical value.

Lagrange multiplier method is widely used in the field of mathematical programming to solve problems with constraints. The core idea of this method is to take the Lagrangian function of the original problem as the entry point, plus the appropriate penalty function, and then calculate the optimization sub-problem without constraints.

In this study, Lagrange multiplier method is used to study the mean-variance model in portfolio optimization. Firstly, the mean-variance risk measurement model without transaction cost is considered. Combining with specific cases, the Lagrange multiplier method is used to solve the problem. Then, the investment scheme with the lowest risk and certain return rate is given, and the investment scheme with the highest return and certain risk is given.

Secondly, the mean-variance risk measurement model with transaction costs is considered, and the Lagrange multiplier method is used to solve the problem in combination with specific cases. The investment plan with the lowest risk and certain rate of return is given, and the investment risk is estimated.

Finally, this paper summarizes the mean-variance model and puts forward the prospect. When selecting the investment portfolio, appropriate models and methods should be used to find out the most appropriate investment portfolio proportion with the maximum return rate and the minimum risk.

Key words: Lagrange multiplier method; Portfolio optimization; Mean-Variance risk measurement model.

目录

第一章 前言 5

1.1 研究背景及意义 5

1.2 国内外研究现状 5

1.3 本文将做的工作 7

第二章 预备知识 7

2.1 拉格朗日乘子法 7

2.1.1 等式约束优化问题的乘子法 8

2.1.2一般约束优化问题的乘子法 9

2.2 投资组合理论 12

2.3 投资组合模型 14

第三章 无交易成本的均值-方差风险度量模型 15

3.1无交易成本的均值-方差风险度量模型 15

3.2 案例分析与数值求解 15

第四章 有交易成本的均值-方差风险度量模型 17

4.1 有交易成本的均值-方差风险度量模型 17

4.2 案例分析与数值求解 18

第五章 结论与展望 19

5.1 结论 19

5.2 展望 19

参考文献 21

致 谢 22