| Portfolio theory has been an important part of modern financial theory and one of difficult and hot issues in scientific research nowadays. Its central problem is how to allocate and utilize capital assets under risk. The authors apply fuzzy mathematics and optimization theory to study portfolio selection problems systemically and deeply, we try to establish a new framework for investment analysis.This dissertation consists of eight chapters.In Chapter 1, the academic background and importance are briefly addressed, and the same time the developments of some classical portfolio selection model and fuzzy portfolio selection models are also introduced.In Chapter 2, Fuzzy Mean-Variance portfolio selection model are studied. In sec-tion 2.1, Fuzzy set was applied to portfolio selection model, the fuzzy mean variance portfolio selection model was proposed(FMV),and the fuzzy programming problem can be transformed into a linear optimal problem (PMV)with an additional quadratic con-straint by fuzzy maths theory. For such problems there are no special standard algo-rithms. A cutting plane algorithm was proposed to solve (PMV), and then the nonlin-ear programming problem can be solved by sequence linear programming problem. A numerical example was given to illustrate the behavior of the proposed model and al-gorithm. In section 2.2,a portfolio selection problem with possibilistic return rates was considered. An efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an ordinary parameter linear programming problem by introducing possibilistic mean value and possibilistic variance, a numerical example is given to il-lustrate the behavior of the proposed model. In section 2.3, a portfolio selection problem with fuzzy expected return rates was considered. A efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an multi-objective parameter lin-ear programming problem by the knowledge of fuzzy math,Fuzzy two-stage algorithm was applied to solve it. a numerical example was given to illustrate the behavior of the proposed model. In section 2.4, a interval number fuzzy portfolio selection model is proposed in a method, which Markowitz portfolio selection model is converted into fuzzy linear programming model with fuzzy constraint and profit rates and risk rates of security is described by interval fuzzy number, then a interval programming problem is converted into a parameter linear programming problem by the knowledge of interval math. Finally, a numerical example is given to illustrate the validity of the method.In chapter 3, a portfolio selection problem with fuzzy return rates is dealt. A pos-sibilistic mean safety-first portfolio selection model was proposed. Specially, by fuzzy order, we present a mathematical programming model with possibilistic constraint. The possibilistic programming problem can be solved by transforming it into a linear pro-gramming problem.In chapter 4, fuzzy mean-variance, safety-first portfolio selection model is pro-posed when we consider the factor of variance, at the same time, the model is converted into an quadratic programming one with possibilistic mean of fuzzy order and fuzzy number. For there no special standard algorithms to such problems, we introduce a cutting plane algorithm to solve them.In chapter 5, Fuzzy Mean absolute deviation portfolio selection is studied. In sec-tion 5.1, we study multi-objective portfolio selection(MADL), In model,the risk was taken as the sum of the absolute deviation of the risk assets instead of covariance. liquid-ity was depicted by turnover.Problem(MADL)is a multi-objective linear optimal prob-lem, Fuzzy two-stage algorithm was applied to solve it.The optimum solution of an example about this portfolio model was given with this new algorithm. In section 5.2 the thesis dealt with a mean absolute deviation portfolio selection problem with fuzzy return rates under fuzzy liquidity constraint, a new possibilistic programming approach based on possibilistic mean and fuzzy liquidity has been proposed, the problem can be reduced to a linear programming by possibility theory. A numerical example of portfolio selection problem is given to illustrate our proposed approach.In chapter 6, Fuzzy Mean?portfolio selection is studied. In section 6.1, mean-?portfolio selection model was presented, and Fuzzy two-stage algorithm was applied to solve it. In section 6.2, mean-?portfolio selection model under the fuzzy constraint was discussed, the fuzzy programming problem was turned into parameter linear pro-gramming by means the knowledge of membership function, thus satisfied portfolio selection model can be obtained on the basis of the investors subjective wish to select the parameter. In section 6.3,interval fuzzy number mean-?portfolio selection model was discussed, interval number satisfied level was taken, by means of the knowledge of fuzzy numbers we obtained the solution of the model.In chapter 7, the portfolio selection based on entropy is given, problem is a multi-objective linear optimal problem, Fuzzy two-stage algorithm is applied to solve it.The optimum solution of an example about this portfolio model is given with this new al-gorithm. Later, interval fuzzy entropy portfolio selection model was considered, and parameter programming of the model was obtained after interval number inequality was transformed into clear number inequality using interval number satisfied level.Thus the model can be solved according to risk preference of investor.In chapter 8, Fractional Utility function Portfolio Selection model is presented. About the largest return and the smallest risk of the portfolio selection problem, a utility function that balances the returns and risks is advanced, and then the portfolio selec-tion model is built based on the nonlinear fractional programming, in order to solve the model, an genetic algorithm is proposed,and the numerical example of it are given. |
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