投资组合优化模型分析与算法实现
摘要
本论文研究了静态和动态的投资组合模型。
对于静态投资组合模型,分析了均值-方差模型、收益率服从正态分布的均值-VaR模型,重点研究了收益率服从非正态分布的均值-VaR模型。通常情况下,投资组合的收益率并非服从正态分布,因此研究收益率服从非正态分布的均值-VaR模型更具现实意义。在利用实际的证券收益率数据做仿真后,求解出各模型的最优投资组合并将相应的决策应用于实际投资中。实例显示,应用收益率服从非正态分布的均值-VaR模型较均值-方差模型、收益率服从正态分布的均值-VaR模型能获得更好的投资效果。
对于动态投资组合模型,分析了Merton的连续时间动态投资组合问题,重点研究了连续时间有阈值控制的期望贴现效用最优化的投资组合问题。有阈值控制模型的特点是风险资产的价格与市场状态相关,投资者按市场状态为其投资过程设置一个阈值,如果市场状态达到或超出阈值,该风险资产上的投资被停止。事实上,该模型是非常符合现实的,吸取一些大型金融机构失败的经验,充分的风险控制是必要的,对某些投资者来说按照市场的状态设定一个阈值来避免大的灾难性后果不失为一种好的方法。
文中利用matlab分别对基于解析解描述的有阈值控制模型和基于数值解描述的有阈值控制模型以及Merton投资组合模型进行仿真,其间应用了迭代算法、数值积分方法、偏微分方程求解函数等,实现了输入市场各相关参数及投资者对风险厌恶度,程序以投资者投资的效用函数最优为目标给出在风险资产和无风险资产上建议的投资组合比例,使投资者在面对复杂的金融市场进行投资时有一定的参考。文中还通过研究主要参数变化对投资组合的影响以及对按照该投资组合决策得到的终期财富情况的影响,给出建议的阈值控制区间参数的设置。
本文结合金融学、数学以及控制理论,从静态和动态的角度对复杂的金融市场形态进行模拟,求解出使资源优化配置的投资组合决策以及最终可能得到的投资效果。
We consider both static and dynamic portfolio models.
About static portfolio models, we consider Mean-Variance portfolio model, Mean-VaR portfolio model with normal distribution return rate and we focus on Mean-VaR portfolio model with abnormal distribution return rate. Usually the rates of return are not normal random variables. It is meaningful to consider Mean-VaR portfolio model with abnormal distribution return rate. After simulating with practical datum of return rate, we find out the optimal portfolio of all the models and apply them to real investment. The application shows that the investment result of mean-VaR portfolio model with abnormal distribution return rate is better than that of mean-variance portfolio model and that of mean-VaR portfolio model with normal distribution return rate. About dynamic portfolio models, we consider Merton’s dynamic portfolio model and continuous-time final-wealth optimal model with limit control where the price of risky asset is related to the state of the market. The latter one places a threshold according to the state of the market. Once the state of the market goes to or beyond the threshold, we stop investing.
Actually this model is very realistic. Learning from the failure of some financial corporations, it is essential to control the risk of investment. A threshold will help the investors to avoid more losses.
Matlab is applied to find out the portfolio policies of Merton’s portfolio model and models with limit control based on analytical description and numerical description. The arithmetic is composed with iterative arithmetic, numerical integral arithmetic, partial differential equation solution arithmetic & etc, with which, after entering the parameters of the model, suggested portfolio policy will be given out to the investors as a reference with the aim of investment utility optimization. After studying the influence of major parameters on the portfolio and final wealth, we get out the suggested threshold for the model.
Combined the knowledge of Finance, Mathematics and Control Theory, intricate market behavior is simulated with static and dynamic portfolio models. In this way, the optimal portfolio policy is worked out easily and the investors may find the likely final wealth ahead of time.
对于静态投资组合模型,分析了均值-方差模型、收益率服从正态分布的均值-VaR模型,重点研究了收益率服从非正态分布的均值-VaR模型。通常情况下,投资组合的收益率并非服从正态分布,因此研究收益率服从非正态分布的均值-VaR模型更具现实意义。在利用实际的证券收益率数据做仿真后,求解出各模型的最优投资组合并将相应的决策应用于实际投资中。实例显示,应用收益率服从非正态分布的均值-VaR模型较均值-方差模型、收益率服从正态分布的均值-VaR模型能获得更好的投资效果。
对于动态投资组合模型,分析了Merton的连续时间动态投资组合问题,重点研究了连续时间有阈值控制的期望贴现效用最优化的投资组合问题。有阈值控制模型的特点是风险资产的价格与市场状态相关,投资者按市场状态为其投资过程设置一个阈值,如果市场状态达到或超出阈值,该风险资产上的投资被停止。事实上,该模型是非常符合现实的,吸取一些大型金融机构失败的经验,充分的风险控制是必要的,对某些投资者来说按照市场的状态设定一个阈值来避免大的灾难性后果不失为一种好的方法。
文中利用matlab分别对基于解析解描述的有阈值控制模型和基于数值解描述的有阈值控制模型以及Merton投资组合模型进行仿真,其间应用了迭代算法、数值积分方法、偏微分方程求解函数等,实现了输入市场各相关参数及投资者对风险厌恶度,程序以投资者投资的效用函数最优为目标给出在风险资产和无风险资产上建议的投资组合比例,使投资者在面对复杂的金融市场进行投资时有一定的参考。文中还通过研究主要参数变化对投资组合的影响以及对按照该投资组合决策得到的终期财富情况的影响,给出建议的阈值控制区间参数的设置。
本文结合金融学、数学以及控制理论,从静态和动态的角度对复杂的金融市场形态进行模拟,求解出使资源优化配置的投资组合决策以及最终可能得到的投资效果。
We consider both static and dynamic portfolio models.
About static portfolio models, we consider Mean-Variance portfolio model, Mean-VaR portfolio model with normal distribution return rate and we focus on Mean-VaR portfolio model with abnormal distribution return rate. Usually the rates of return are not normal random variables. It is meaningful to consider Mean-VaR portfolio model with abnormal distribution return rate. After simulating with practical datum of return rate, we find out the optimal portfolio of all the models and apply them to real investment. The application shows that the investment result of mean-VaR portfolio model with abnormal distribution return rate is better than that of mean-variance portfolio model and that of mean-VaR portfolio model with normal distribution return rate. About dynamic portfolio models, we consider Merton’s dynamic portfolio model and continuous-time final-wealth optimal model with limit control where the price of risky asset is related to the state of the market. The latter one places a threshold according to the state of the market. Once the state of the market goes to or beyond the threshold, we stop investing.
Actually this model is very realistic. Learning from the failure of some financial corporations, it is essential to control the risk of investment. A threshold will help the investors to avoid more losses.
Matlab is applied to find out the portfolio policies of Merton’s portfolio model and models with limit control based on analytical description and numerical description. The arithmetic is composed with iterative arithmetic, numerical integral arithmetic, partial differential equation solution arithmetic & etc, with which, after entering the parameters of the model, suggested portfolio policy will be given out to the investors as a reference with the aim of investment utility optimization. After studying the influence of major parameters on the portfolio and final wealth, we get out the suggested threshold for the model.
Combined the knowledge of Finance, Mathematics and Control Theory, intricate market behavior is simulated with static and dynamic portfolio models. In this way, the optimal portfolio policy is worked out easily and the investors may find the likely final wealth ahead of time.
引文
[1]郭晓亭,蒲勇健,林略.风险概念及其数量刻画[M].数量经济技术经济研究,2004,2.
[2]王明涛.证券投资风险计量、预测与控制[M].上海:上海财经大学出版社.
[3]叶青.中国证券市场风险的度量与评价[M].北京:中国统计出版社,2001.
[4]朱淑珍.金融创新与金融风险——发展中的两难[M].上海:复旦大学出版社,2002.
[5] Markowitz H. Portfolio selection [J]. Journal of Finance, 1952, 7:77-91.
[6] Konno H..Piecewise linear risk functions and portfolio optimization[J].Journal of the Operations Research Society of Japan,1990,33(2):139-156.
[7] Konno H.,Yamazaki H..Mean absolute deviation portfolio optimization model and its applications to Tokyo stock market[J].Management Science,1991,37(5):519-531.
[8] Mao J.C.T..Models of capital budgeting,E-V vs E-S[J].Journal of Financial and Quantitative Analysis,1970,5:657-675.
[9] Swalm R.O..Utility theory-insights into risk taking[J].Harvard Business Review,1966,44:123-136.
[10] Mansini R.,Speranza M.G..Heuristic algorithms for the portfolio selection problem with minimum transaction lots[J].European Journal of Operational Research,1999,114:219-233.
[11] Speranza M.G..Linear programming models for portfolio optimization[J].Finance,1993,14:107-123.
[12] Young M.R..A minimax portfolio selection rule with linear programming solution[J].Management Science,1998,44:673-683.
[13] Cai X.Q.,Teo K.L.,Yan X.Q.,Zhou X.Y..Portfolio optimization under a minimax rule[J].Management Science,2000,46:957-972.
[29] Buffington J, Elliott R J. American options with regime switching[J]. Int. J. theoret. Appl. Finance, 2002, 5: 497-514.
[30] ?akmak U, ?zekici S, Portfolio optimization in stochastic markets[J]. Math Methods Oper Res 2006, 63: 151-168.
[31] Richard S. Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model[J]. J. Financial Econ., 1975,1(2): 187-203.
[32] Karatzas I, Wang H. Utility maximization with discretionary stopping[J]. SIAM J. Control Optim., 2000,39: 306-329.
[33] Bouchard B, Pham H. Wealth-path dependent utility maximization in incomplete markets[J]. Finance Stochast, 2004, 8: 579-603.
[34] Kraft H, Steffensen M. Portfolio problems stopping at first hitting time with application to default risk[J]. Math. Meth. Oper. Res., 2006, 63: 123-150.
[35]唐磊.现代资产组合理论及模型综述:[EB/OL]. [2005-05-26]. http://www.chinavalue.net/Article/Archive/2005/5/26/5361.html.
[36]李伯德.最优投资组合的数学模型与案例分析[J].兰州商学院学报:2006,22(2):97-99.
[37]戴玉林.马科维兹模型的分析与评价[J].金融研究:1991,9:57-63.
[38] J.H. Evans,S.H.Archer.分散风险与离串趋势的减少:一项经验分析[J].金融杂志,1968,12.
[39]张慧毅,徐荣贞,蒋玉洁. VaR模型及其在金融风险管理中的应用[J].价值工程, 2006, 8:51-53.
[40] Gordon J. Alexander, Alexandre M. Baptista. Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis[J]. Journal of Economic Dynamics & Control, 2002, 26: 1159-1193.
[41]郭丹,徐伟,雷佑铭.机会约束下的均值-VaR组合投资问题[J].系统工程学报, 2005 20(3):256-260.
[42] Laurent E. G., Maksim O., Francois O.. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach[J]. Operation Research, 2003, 15(4): 543-556.
[43] Jarque-Bera检验[EB/OL]. http://baike.baidu.com/view/1229229.htm.
[44]叶中行,林建忠.数理金融[M].北京:科学出版社,1998:30.
[45]卫淑芝.随机市场环境下动态最优投资组合模型研究[D].上海:上海交通大学数学系,2009.
[46]裘宗燕.从问题到程序[M].北京:机械工业出版社,2005:258.
[47]定积分的近似计算:学习家园[EB/OL]. [2006-12-09]. http://www.learn51.com.cn/knowledge/hq_kb_view.asp?zt=MBA&d_id=1811&sort1=166&sort2=551.
[48] Matlab Help Introduction to PDE Problems
[2]王明涛.证券投资风险计量、预测与控制[M].上海:上海财经大学出版社.
[3]叶青.中国证券市场风险的度量与评价[M].北京:中国统计出版社,2001.
[4]朱淑珍.金融创新与金融风险——发展中的两难[M].上海:复旦大学出版社,2002.
[5] Markowitz H. Portfolio selection [J]. Journal of Finance, 1952, 7:77-91.
[6] Konno H..Piecewise linear risk functions and portfolio optimization[J].Journal of the Operations Research Society of Japan,1990,33(2):139-156.
[7] Konno H.,Yamazaki H..Mean absolute deviation portfolio optimization model and its applications to Tokyo stock market[J].Management Science,1991,37(5):519-531.
[8] Mao J.C.T..Models of capital budgeting,E-V vs E-S[J].Journal of Financial and Quantitative Analysis,1970,5:657-675.
[9] Swalm R.O..Utility theory-insights into risk taking[J].Harvard Business Review,1966,44:123-136.
[10] Mansini R.,Speranza M.G..Heuristic algorithms for the portfolio selection problem with minimum transaction lots[J].European Journal of Operational Research,1999,114:219-233.
[11] Speranza M.G..Linear programming models for portfolio optimization[J].Finance,1993,14:107-123.
[12] Young M.R..A minimax portfolio selection rule with linear programming solution[J].Management Science,1998,44:673-683.
[13] Cai X.Q.,Teo K.L.,Yan X.Q.,Zhou X.Y..Portfolio optimization under a minimax rule[J].Management Science,2000,46:957-972.
[29] Buffington J, Elliott R J. American options with regime switching[J]. Int. J. theoret. Appl. Finance, 2002, 5: 497-514.
[30] ?akmak U, ?zekici S, Portfolio optimization in stochastic markets[J]. Math Methods Oper Res 2006, 63: 151-168.
[31] Richard S. Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model[J]. J. Financial Econ., 1975,1(2): 187-203.
[32] Karatzas I, Wang H. Utility maximization with discretionary stopping[J]. SIAM J. Control Optim., 2000,39: 306-329.
[33] Bouchard B, Pham H. Wealth-path dependent utility maximization in incomplete markets[J]. Finance Stochast, 2004, 8: 579-603.
[34] Kraft H, Steffensen M. Portfolio problems stopping at first hitting time with application to default risk[J]. Math. Meth. Oper. Res., 2006, 63: 123-150.
[35]唐磊.现代资产组合理论及模型综述:[EB/OL]. [2005-05-26]. http://www.chinavalue.net/Article/Archive/2005/5/26/5361.html.
[36]李伯德.最优投资组合的数学模型与案例分析[J].兰州商学院学报:2006,22(2):97-99.
[37]戴玉林.马科维兹模型的分析与评价[J].金融研究:1991,9:57-63.
[38] J.H. Evans,S.H.Archer.分散风险与离串趋势的减少:一项经验分析[J].金融杂志,1968,12.
[39]张慧毅,徐荣贞,蒋玉洁. VaR模型及其在金融风险管理中的应用[J].价值工程, 2006, 8:51-53.
[40] Gordon J. Alexander, Alexandre M. Baptista. Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis[J]. Journal of Economic Dynamics & Control, 2002, 26: 1159-1193.
[41]郭丹,徐伟,雷佑铭.机会约束下的均值-VaR组合投资问题[J].系统工程学报, 2005 20(3):256-260.
[42] Laurent E. G., Maksim O., Francois O.. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach[J]. Operation Research, 2003, 15(4): 543-556.
[43] Jarque-Bera检验[EB/OL]. http://baike.baidu.com/view/1229229.htm.
[44]叶中行,林建忠.数理金融[M].北京:科学出版社,1998:30.
[45]卫淑芝.随机市场环境下动态最优投资组合模型研究[D].上海:上海交通大学数学系,2009.
[46]裘宗燕.从问题到程序[M].北京:机械工业出版社,2005:258.
[47]定积分的近似计算:学习家园[EB/OL]. [2006-12-09]. http://www.learn51.com.cn/knowledge/hq_kb_view.asp?zt=MBA&d_id=1811&sort1=166&sort2=551.
[48] Matlab Help Introduction to PDE Problems