Research On Portfolio Decision Based On Processing Of Fuzzy Information

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, andsolving 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.