基于模糊信息处理的组合投资决策方法研究
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摘要
组合投资决策的研究对象是一个复杂系统,而且也是一个需要人参与判断评估的系统。研究这样一个系统,不可避免地需要面对不确定信息的处理问题。本文在分析了组合投资理论研究的发展动态之后,以模糊信息处理为切入点,围绕组合投资决策过程的两个核心问题—预期收益率的获取和风险的估计,从模糊性的角度出发,研究模糊收益率信息的获取,表示,探讨组合投资决策模糊性建模的理论和方法。具体地,本项研究取得的主要成果有:
(1)从实际应用的角度出发,研究结合领域专家意见获取风险资产预期收益率的三种方法。第一,利用专家的经验知识和掌握的信息,通过咨询,由领域专家对风险资产预期收益率给出区间判断,再经过集值统计综合集成得到表示预期收益率的方法。第二,在采集了领域专家对预期收益率的模糊数判断信息之后,对判断信息进行集成处理,提出利用模糊数的平均数指标表示预期收益率的方法。这两种方法的特点是可操作性强,并能消除不同专家在做出判断时可能产生的片面性和偏差,充分发挥专家群体的经验和智慧。第三,以风险资产的历史数据作为投资者的不完全信息,将风险资产的历史数据与专家的经验判断结合起来,由领域专家对每一个历史数据做出可能性程度的判断,并以其作为样本点,给出了根据最佳贴近样本点这一准则获取预期收益率的可能性分布的方法。在实际的投资决策领域中,专家和投资者的意见是非常重要的,利用可能性程度、区间判断、模糊数来反映专家和投资者掌握的信息和经验,显得更贴近于实际,更加合理。
(2)采用时间序列预测方法,研究模糊信息的时间序列预测问题,通过模糊数空间中的距离,建立了模糊环境中对应的最小二乘回归模型,证明了回归模型解的存在性和唯一性,给出了确定模型模糊参数的计算公式,提出了从整体上测定模糊观测值在回归方程周围分散程度的估计标准误差,推导了计算估计标准误差的算式,并利用贴近度指标评价了单个序列值的拟合优劣程度。本文提出的模型,既是普通最小二乘回归方法的推广,又能最大限度地利用已知信息进行合理的分析。依据此模型,可以解决收益率的模糊时间序列预测问题。
(3)从收益率受多因素影响的角度,将这个多因素共同作用的系统看成一个模糊系统,研究模糊收益率信息的多元线性回归预测问题,建立其回归预测模型,研究模型解的性质,给出了从整体上测定模型拟合程度的估计标准误差和从个体上评价模
The research object for portfolio selection is a complex system, in which people's thinking and judgment are needed to make decision. It is inevitable to relate to the processing of uncertain information because of the complexity of the problems themselves, the vagueness in people's thinking and judgment, and the influence of various uncertain factors existing in boundary environment around the systems. After the review of the development of portfolio selection theories, considering the case in which fuzziness must be treated, this paper concentrates on two core problems of the decision process from the angle of fuzziness, one is the acquirement of anticipation profit rate and another is the estimate of risk, and discusses the techniques for modeling the portfolio selection with fuzzy information. Concretely, this paper concludes the following main results:
(1) Three kinds of methods for acquiring anticipation profit rate based on experts' judgment are presented. The first one is the method of interval judgment, in which the judgment intervals for anticipation profit rate are gathered and processed by using expert's experience and the information they mastered. The second one is the average synthetic method. The experts' judgment is given as fuzzy number, and the average index is presented for synthesizing the experts' judgment. The method presented here is strongly maneuverable and can dissolve the deviation in judgment from different experts. The third one takes the historical data of assets as incomplete information and models the anticipation profit rate as possibility distribution by combining the historical data with experts' judgment. Since the experts' knowledge in actual investment is very important. It is more suitable for using possibility grade, interval judgment and fuzzy number to reflect the experts' judgment.
(2) The fuzzy time series forecasting technique is developed to model the anticipation profit rate. Similar to traditional least squares, the fuzzy analogue by the distance defined on fuzzy number space is proposed. It is shown that the model has unique solution and the solution can be given by an analytic expression. In order to measure the dispersion between the fuzzy observed data and the estimated regression equation, an index, called standard deviation of estimate, is given and the formula for computation is derived. And also,
another index is presented for evaluating the goodness of fit between the observed value and estimated value. Using the model we can deal with time series problems with fuzzy observation data. (3) From the viewpoint that anticipation profit rate is influence by multi-factors, a multidimensional linear regression model is developed to fit the fuzzy observed values. The properties of solution are studied, and the analytic expression is given. Also, two indexes, called as standard deviation of estimates and goodness of fit respectively, are presented for estimating the fitting results. The model is strictly verified by theory and can be used to forecasting the anticipation profit rate. (4) Based on the research for acquiring fuzzy anticipation profit rate, the model for portfolio selection is put forward by taken the degree deviated from the central point as the measure of risk. Further, the properties for the solution are explored, and a sufficient and necessary condition about the solution is obtained. Finally, the relationship between the expected return and risk is researched, and some conclusions are gotten. In actual application of the model, the optimal portfolio can be calculated on each given level, and thus the decision can be made. (5) To synthesize the information on every level, a model for optimal portfolio selection is proposed. First, regarded fuzzy profit rate as the probability distribution of a variable, the mean of fuzzy number is defined by possibility distribution and necessity distribution, which reflect the upper and lower bound of probability distribution of fuzzy number. Moreover, the linear property of mean is proved and the decision model is given. After the existence and uniqueness of the optimal portfolio are proved, the corresponding situation without no-negative restraint is discussed and the solution is derived. Finally, the properties of the solutions are researched and the unique solution is reached through a simple calculated way. The characteristic of this model is that the information on every level is synthetically considered and the given information is brought into full use. (6) In order to give attention to the profit rate and the risk at the same time, a one-objective decision method is given by compromised with the minimum of investment risk on the maximum of profit. First of all, the ratio of profit to risk is taken as decision object, and the model is established. Next, the existence of the solution is proved, and
solving the decision model is converted into solving the eigenvector of a matrix. Finally, the algorithm for seeking the solution is given. Different from Markowitz's mean-variance model, the techniques presented in this paper try to model experts' knowledge from the angle of fuzziness, while Markowitz's model deals with the data according to the statistic viewpoints. Our models, based on the fuzzy information processes, can be used to make decision for portfolio selection in fuzzy environment. The models are rigorously justified and have the actual applied value and meaning.
(1)从实际应用的角度出发,研究结合领域专家意见获取风险资产预期收益率的三种方法。第一,利用专家的经验知识和掌握的信息,通过咨询,由领域专家对风险资产预期收益率给出区间判断,再经过集值统计综合集成得到表示预期收益率的方法。第二,在采集了领域专家对预期收益率的模糊数判断信息之后,对判断信息进行集成处理,提出利用模糊数的平均数指标表示预期收益率的方法。这两种方法的特点是可操作性强,并能消除不同专家在做出判断时可能产生的片面性和偏差,充分发挥专家群体的经验和智慧。第三,以风险资产的历史数据作为投资者的不完全信息,将风险资产的历史数据与专家的经验判断结合起来,由领域专家对每一个历史数据做出可能性程度的判断,并以其作为样本点,给出了根据最佳贴近样本点这一准则获取预期收益率的可能性分布的方法。在实际的投资决策领域中,专家和投资者的意见是非常重要的,利用可能性程度、区间判断、模糊数来反映专家和投资者掌握的信息和经验,显得更贴近于实际,更加合理。
(2)采用时间序列预测方法,研究模糊信息的时间序列预测问题,通过模糊数空间中的距离,建立了模糊环境中对应的最小二乘回归模型,证明了回归模型解的存在性和唯一性,给出了确定模型模糊参数的计算公式,提出了从整体上测定模糊观测值在回归方程周围分散程度的估计标准误差,推导了计算估计标准误差的算式,并利用贴近度指标评价了单个序列值的拟合优劣程度。本文提出的模型,既是普通最小二乘回归方法的推广,又能最大限度地利用已知信息进行合理的分析。依据此模型,可以解决收益率的模糊时间序列预测问题。
(3)从收益率受多因素影响的角度,将这个多因素共同作用的系统看成一个模糊系统,研究模糊收益率信息的多元线性回归预测问题,建立其回归预测模型,研究模型解的性质,给出了从整体上测定模型拟合程度的估计标准误差和从个体上评价模
The research object for portfolio selection is a complex system, in which people's thinking and judgment are needed to make decision. It is inevitable to relate to the processing of uncertain information because of the complexity of the problems themselves, the vagueness in people's thinking and judgment, and the influence of various uncertain factors existing in boundary environment around the systems. After the review of the development of portfolio selection theories, considering the case in which fuzziness must be treated, this paper concentrates on two core problems of the decision process from the angle of fuzziness, one is the acquirement of anticipation profit rate and another is the estimate of risk, and discusses the techniques for modeling the portfolio selection with fuzzy information. Concretely, this paper concludes the following main results:
(1) Three kinds of methods for acquiring anticipation profit rate based on experts' judgment are presented. The first one is the method of interval judgment, in which the judgment intervals for anticipation profit rate are gathered and processed by using expert's experience and the information they mastered. The second one is the average synthetic method. The experts' judgment is given as fuzzy number, and the average index is presented for synthesizing the experts' judgment. The method presented here is strongly maneuverable and can dissolve the deviation in judgment from different experts. The third one takes the historical data of assets as incomplete information and models the anticipation profit rate as possibility distribution by combining the historical data with experts' judgment. Since the experts' knowledge in actual investment is very important. It is more suitable for using possibility grade, interval judgment and fuzzy number to reflect the experts' judgment.
(2) The fuzzy time series forecasting technique is developed to model the anticipation profit rate. Similar to traditional least squares, the fuzzy analogue by the distance defined on fuzzy number space is proposed. It is shown that the model has unique solution and the solution can be given by an analytic expression. In order to measure the dispersion between the fuzzy observed data and the estimated regression equation, an index, called standard deviation of estimate, is given and the formula for computation is derived. And also,
another index is presented for evaluating the goodness of fit between the observed value and estimated value. Using the model we can deal with time series problems with fuzzy observation data. (3) From the viewpoint that anticipation profit rate is influence by multi-factors, a multidimensional linear regression model is developed to fit the fuzzy observed values. The properties of solution are studied, and the analytic expression is given. Also, two indexes, called as standard deviation of estimates and goodness of fit respectively, are presented for estimating the fitting results. The model is strictly verified by theory and can be used to forecasting the anticipation profit rate. (4) Based on the research for acquiring fuzzy anticipation profit rate, the model for portfolio selection is put forward by taken the degree deviated from the central point as the measure of risk. Further, the properties for the solution are explored, and a sufficient and necessary condition about the solution is obtained. Finally, the relationship between the expected return and risk is researched, and some conclusions are gotten. In actual application of the model, the optimal portfolio can be calculated on each given level, and thus the decision can be made. (5) To synthesize the information on every level, a model for optimal portfolio selection is proposed. First, regarded fuzzy profit rate as the probability distribution of a variable, the mean of fuzzy number is defined by possibility distribution and necessity distribution, which reflect the upper and lower bound of probability distribution of fuzzy number. Moreover, the linear property of mean is proved and the decision model is given. After the existence and uniqueness of the optimal portfolio are proved, the corresponding situation without no-negative restraint is discussed and the solution is derived. Finally, the properties of the solutions are researched and the unique solution is reached through a simple calculated way. The characteristic of this model is that the information on every level is synthetically considered and the given information is brought into full use. (6) In order to give attention to the profit rate and the risk at the same time, a one-objective decision method is given by compromised with the minimum of investment risk on the maximum of profit. First of all, the ratio of profit to risk is taken as decision object, and the model is established. Next, the existence of the solution is proved, and
solving the decision model is converted into solving the eigenvector of a matrix. Finally, the algorithm for seeking the solution is given. Different from Markowitz's mean-variance model, the techniques presented in this paper try to model experts' knowledge from the angle of fuzziness, while Markowitz's model deals with the data according to the statistic viewpoints. Our models, based on the fuzzy information processes, can be used to make decision for portfolio selection in fuzzy environment. The models are rigorously justified and have the actual applied value and meaning.
引文
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[2] M. Rothschild, J. Stiglitz. Increasing risks II: its economic consequences. Journal of Economic Theory, 1971, 3: 66-84
[3] W. Sharpe. Simple model for portfolio analysis. Managements Science, 1963, 9(3): 277-293
[4] W. Sharpe. Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance, 1964, 19: 425-442
[5] J. Lintner. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 1965, 47: 13-37
[6] J. Mossin. Equilibrium in a capital asset market. Econometrics, 1966, 34: 768-783
[7] Merton, R., An intertemporal capital asset pricing model, Econometrics, 1973, 41(5), 867-888
[8] G. Larry, T. Wang. Intertemporal asset pricing under knightian uncertainty. Econometrics, 1994, 62(30): 283-322
[9] H. Konno, H. Shirakawa and H. Yamazaki. A mean-absolute deviation-skewness portfolio optimization model. Annals of Operations Research, 1993, 45: 205-220
[10] C. Pornchai, D. Krishnan, and H. Shahid et al. Portfolio selection and skewness: Evidence from international stock markets, Journal of Banking & Finance, 1997, 21: 143-167
[11] G. Gennotde, A. Jung. Investment strategies under transaction cost: the finite horizon case, Management Science, 1994, 40: 385-404
[12] Z. F. Li, S. Y. Wang and X. T. Deng. A linear programming algorithm for optimal portfolio selection with transaction costs. International Journal of Systems Science, 2000, February 179-223
[13] 胡国正, 李楚霖. 考虑交易费用的证券组合投资的研究. 预测, 1998, 5: 66-67
[14] 陈华友, 许义生. 含交易费用的证券组合投资的多目标规划模型. 运筹与管理, 1999, 8(3): 57-60
[15] J. K. Sengupta. Mixed strategy and information theory in optimal portfolio choice. International Journal Systems Science, 1989, 20(2): 215-227
[16] X. T. Deng. Portfolio management with optimal regret ratio. In: Proceedings of international conference on management science. Hong Kong, 1998: 289-295
[17] R. EL-Yaniv. Competitive solutions for online financial problems. ACM Computing Surveys, 1998, 30: 28-68
[18] H. M. Rudolf, J. Wolter and Z. Heinz. A linear model for tracking error minimization. Journal of Banking and Finance, 1999, 23: 85-103
[19] S. Al-binali. A risk-reward framework for the competitive analysis of financial games, Algorithumica, 1999, 25: 99-115
[20] R. Bell, T. M. Cover. Game-theoretic optimal portfolios. Management Science, 1998, 34: 724-733
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