几类投资组合优化模型及其算法
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摘要
投资组合优化问题作为现代金融学的一个核心课题，主要研究如何在不确定情况下对金融资产进行合理配置与选择，从而实现收益率最大化与风险最小化间的均衡．1952年，美国经济学家HarryM.Markowitz在《TheJournalofFinance》杂志上发表了“PortfolioSelection”一文，首次使用证券收益方差度量风险，提出了均值-方差投资组合选择理论，被学术界公认为开创了现代投资组合理论的先河，奠定了定量化研究金融投资问题的基础．随着现代数学方法的发展及应用数学方法研究金融经济问题的金融数学的问世，使得现代金融投资理论开始摆脱纯粹经验化操作和单纯描述性研究的状态，进入了定量分析这一高级阶段，并为投资者进行投资决策提供了指导．当今世界经济飞速发展，金融危机和市场波动频繁出现，我国的资本市场虽然在改革开放之后得到长足发展，但还不太完善和成熟，使得投资者面临越来越多错综复杂的金融投资决策的理论和实践问题，对投资组合优化问题的研究也越来越具有重要的理论和现实意义．
本文从以下三个方面开展研究工作，一是带有基数约束的投资组合优化问题，二是带有交易费用的动态投资组合优化问题，三是投资组合随机优化模型的情景生产方法比较．主要工作如下：
1.人工蜂群算法是近几年提出的一种新的群智能算法，在求解多峰高维函数优化问题时体现出了更为优良的性质．考虑到人工蜂群算法的这一优点，利用人工蜂群算法研究了带有基数约束的投资组合优化模型．通过数值试验可以发现，人工蜂群算法在求解这一问题时，比别的智能优化算法体现出一些更优越的性态．
2.针对带有基数约束的投资组合优化问题，提出一种改进人工蜂群算法．在算法中，利用Deb选择策略使最优解满足约束条件，并引入新的搜索策略以提高算法的收敛速度；同时，使用Bolzmann选择概率来维护种群多样性，防止算法早熟．通过对测试问题的数值实验，表明使用该算法能获得更好的投资策略，有效分散投资组合风险，并说明该算法对于求解投资组合优化问题是快速有效的．
3.研究了存在固定交易费用和比例交易费用情况下的多阶段均值-方差投资组合优化问题．应用离散时间动态规划方法，给出了投资者的间接效用函数、无交易区域边界和有效前沿的解析解，从而确定了投资者的长期最优投资策略．通过数值试验描述了问题的求解过程，并说明了交易费用对有效前沿的影响．
4.研究了连续时间情形下，带有固定交易费用和比例交易费用的均值-方差投资组合优化问题．通过使用动态规划方法，推导出了原问题的Hamilton-Jacobi-Bellman方程，并得到了方程的显式解．从而，推导出原均值-方差问题的最优投资策略和有效前沿的表达式．数值试验给出了交易费用的变化对交易区域和有效前沿的影响，并说明了所给方法的可行性和有效性．
5.比较研究了四种情景生成方法在求解投资组合优化问题时的预测与决策效果．通过对比其样本内性质及样本外性质发现，情景生成方法与投资组合优化模型对于中国股票市场来说，在预测与决策方面是非常有效的工具．其中矩匹配方法较其他方法能更好的反映市场的下跌趋势，多变量GARCH方法能更好的反映市场的上涨趋势．
最后，列出了投资组合优化问题研究中有待进一步研究的几个问题．
Portfolio optimization, which addresses the ideal assignment of resources to thefinancial assets to balance the assets returns and the assets risks, is one of the coreresearch fields in modern financial management. In1952, Harry M. Markowitz, theAmerican economist, introduced the variance of the assets returns as the measure ofrisks in the paper “Portfolio Selection” which published in “The Journal of Finance”.This paper is considered as the beginning of the modern investment theory and thefoundation of numerical analysis of financial investment. Along with the improvementof the modern mathematical methods and the emergence of mathematical finance, thestudy of modern financial investment theory are no more only describing studies or pureempirical researches, but the numerical analysis guiding the behavior of investors.Today, the world’s economics grow rapidly, but the financial crisis and volatility arealso increased. And in our country, although the financial market has been developedgreatly, it is still not so perfect that the investors face more and more complicatedtheoretical and practical problems. So the researches on portfolio optimization becomemore and more important.
The present dissertation makes a deep and systematic investigation on candidateconstraint portfolio optimization problem, dynamic portfolio optimization problem andscenario generation methods in portfolio optimization problem. The main researchesand creative results of this dissertation are shown as follows:
1. The Artificial Bee Colony algorithm, proposed for the high dimension andmulti-modal problem, is a recently introduced optimization algorithm. Considering theadvantages of Artificial Bee Colony algorithm, the cardinality constrainedmean-variance model is solved by using this algorithm. The experimental results showthat the proposed algorithm performs well for the portfolio optimization problem.
2. To tackle the cardinality constrained portfolio optimization problem, anmodified Artificial Bee Colony algorithm is designed. The Deb’s selection rule isintroduced to guarantee the feasibility of optimal solution. To improve the convergentspeed, a new search strategy is proposed. Furthermore, the Bolzmann selectionprobability is employed to maintain the population diversity. The experiment resultsindicate that the proposed algorithm is efficient and effective for the portfoliooptimization problem, which can obtain a better portfolio strategy and diversify theportfolio risk efficiently.
3. The general multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs are investigated. According to the dynamicprogramming approach, the optimal strategies, the boundaries of the no-transactionregion and the efficient frontier are given in the explicit form. Therefore, the long terminvestment strategies for the investors are given. Numerical result shows that themethod provided in this paper works well.
4. A mean-variance portfolio selection problem in continuous time with fixed andproportional transaction costs is investigated. Utilizing the dynamic programming, theHamilton-Jacobi-Bellman equation is derived, and the explicit closed form solution isobtained. Furthermore, the optimal strategies and efficient frontiers are also proposedfor the original mean-variance problem. Numerical experiments present the variation ofthe transaction region and efficient frontier with the transaction costs change, whichdemonstrate the proposed method performs effectively.
5. The effectiveness of forecasting and decision-making by using four scenariogeneration methods are compared. The results of in-sample and out-of-sample propertyof the portfolios, obtained by using these methods to generate the rates of return, showthat for the Chinese stock market, the scenario generation methods and optimizationmodel are useful for forecasting and decision-making. Moment matching method canbetter reflect the market’s downward trend and multivariate GARCH method can betterreflect the market’s upward trend.
At last, the questions of the portfolio optimization and its future developingtendency are summarized.
本文从以下三个方面开展研究工作，一是带有基数约束的投资组合优化问题，二是带有交易费用的动态投资组合优化问题，三是投资组合随机优化模型的情景生产方法比较．主要工作如下：
1.人工蜂群算法是近几年提出的一种新的群智能算法，在求解多峰高维函数优化问题时体现出了更为优良的性质．考虑到人工蜂群算法的这一优点，利用人工蜂群算法研究了带有基数约束的投资组合优化模型．通过数值试验可以发现，人工蜂群算法在求解这一问题时，比别的智能优化算法体现出一些更优越的性态．
2.针对带有基数约束的投资组合优化问题，提出一种改进人工蜂群算法．在算法中，利用Deb选择策略使最优解满足约束条件，并引入新的搜索策略以提高算法的收敛速度；同时，使用Bolzmann选择概率来维护种群多样性，防止算法早熟．通过对测试问题的数值实验，表明使用该算法能获得更好的投资策略，有效分散投资组合风险，并说明该算法对于求解投资组合优化问题是快速有效的．
3.研究了存在固定交易费用和比例交易费用情况下的多阶段均值-方差投资组合优化问题．应用离散时间动态规划方法，给出了投资者的间接效用函数、无交易区域边界和有效前沿的解析解，从而确定了投资者的长期最优投资策略．通过数值试验描述了问题的求解过程，并说明了交易费用对有效前沿的影响．
4.研究了连续时间情形下，带有固定交易费用和比例交易费用的均值-方差投资组合优化问题．通过使用动态规划方法，推导出了原问题的Hamilton-Jacobi-Bellman方程，并得到了方程的显式解．从而，推导出原均值-方差问题的最优投资策略和有效前沿的表达式．数值试验给出了交易费用的变化对交易区域和有效前沿的影响，并说明了所给方法的可行性和有效性．
5.比较研究了四种情景生成方法在求解投资组合优化问题时的预测与决策效果．通过对比其样本内性质及样本外性质发现，情景生成方法与投资组合优化模型对于中国股票市场来说，在预测与决策方面是非常有效的工具．其中矩匹配方法较其他方法能更好的反映市场的下跌趋势，多变量GARCH方法能更好的反映市场的上涨趋势．
最后，列出了投资组合优化问题研究中有待进一步研究的几个问题．
Portfolio optimization, which addresses the ideal assignment of resources to thefinancial assets to balance the assets returns and the assets risks, is one of the coreresearch fields in modern financial management. In1952, Harry M. Markowitz, theAmerican economist, introduced the variance of the assets returns as the measure ofrisks in the paper “Portfolio Selection” which published in “The Journal of Finance”.This paper is considered as the beginning of the modern investment theory and thefoundation of numerical analysis of financial investment. Along with the improvementof the modern mathematical methods and the emergence of mathematical finance, thestudy of modern financial investment theory are no more only describing studies or pureempirical researches, but the numerical analysis guiding the behavior of investors.Today, the world’s economics grow rapidly, but the financial crisis and volatility arealso increased. And in our country, although the financial market has been developedgreatly, it is still not so perfect that the investors face more and more complicatedtheoretical and practical problems. So the researches on portfolio optimization becomemore and more important.
The present dissertation makes a deep and systematic investigation on candidateconstraint portfolio optimization problem, dynamic portfolio optimization problem andscenario generation methods in portfolio optimization problem. The main researchesand creative results of this dissertation are shown as follows:
1. The Artificial Bee Colony algorithm, proposed for the high dimension andmulti-modal problem, is a recently introduced optimization algorithm. Considering theadvantages of Artificial Bee Colony algorithm, the cardinality constrainedmean-variance model is solved by using this algorithm. The experimental results showthat the proposed algorithm performs well for the portfolio optimization problem.
2. To tackle the cardinality constrained portfolio optimization problem, anmodified Artificial Bee Colony algorithm is designed. The Deb’s selection rule isintroduced to guarantee the feasibility of optimal solution. To improve the convergentspeed, a new search strategy is proposed. Furthermore, the Bolzmann selectionprobability is employed to maintain the population diversity. The experiment resultsindicate that the proposed algorithm is efficient and effective for the portfoliooptimization problem, which can obtain a better portfolio strategy and diversify theportfolio risk efficiently.
3. The general multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs are investigated. According to the dynamicprogramming approach, the optimal strategies, the boundaries of the no-transactionregion and the efficient frontier are given in the explicit form. Therefore, the long terminvestment strategies for the investors are given. Numerical result shows that themethod provided in this paper works well.
4. A mean-variance portfolio selection problem in continuous time with fixed andproportional transaction costs is investigated. Utilizing the dynamic programming, theHamilton-Jacobi-Bellman equation is derived, and the explicit closed form solution isobtained. Furthermore, the optimal strategies and efficient frontiers are also proposedfor the original mean-variance problem. Numerical experiments present the variation ofthe transaction region and efficient frontier with the transaction costs change, whichdemonstrate the proposed method performs effectively.
5. The effectiveness of forecasting and decision-making by using four scenariogeneration methods are compared. The results of in-sample and out-of-sample propertyof the portfolios, obtained by using these methods to generate the rates of return, showthat for the Chinese stock market, the scenario generation methods and optimizationmodel are useful for forecasting and decision-making. Moment matching method canbetter reflect the market’s downward trend and multivariate GARCH method can betterreflect the market’s upward trend.
At last, the questions of the portfolio optimization and its future developingtendency are summarized.
引文
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