Portfolio Optimization Methods Based On New Risk Measure And Sharpe Ratio

Fuzzy random variable is a useful tool to describe twofold uncertainty. Inthis paper, fuzzy random variables were studied in both theory and applicationaspects. In fuzzy random theory, we study the convergence modes of fuzzy randomvariables in equilibrium theory. First we introduce several convergence concepts forsequences of fuzzy random variables, such as convergence in equilibrium measure andconvergence in equilibrium distribution. Then, we discuss the convergence criteriafor the convergence modes. On the basis of the convergence criteria, we establish theconvergence relations among the convergence modes. Finally, we define the integralof fuzzy random variable with respect to equilibrium measure, and establish thedominated convergence theorem and bounded convergence theorem for sequences ofintegrable fuzzy random variables.In the application of equilirium theory, we propose a portfolio optimizationmethod in fuzzy random environment. In this method we introduce a new riskmeasure–quadratic deviation, and build a model to minimize the risk per rewardto evaluate the performance of portfolio selection, which based on the new mea-sure and sharpe ratio. Hence we find the the balance between risk and reward. Inour model, the expectation of securities portfolio is used to measure rewards, thequadratic deviation is used to measure the risk of portfolio, and the ratio–risk-rewardis used to evaluate the performance of portfolio selection, that is the risk for perreward. For the proposed fractional programming model, we discuss its convexitybefore the deterministic expression is given, so that the local optimal solution isthe global optimal solution. Since the calculation about the quadratic deviation offuzzy random variable is difcult in general case, we discuss some equivalent for-mulas when the return ratio follows some special distributions. After we give thedeterministic expression, we turn the fractional programming model into a convexprogramming problem so that it can be solved easily by optimization software. Atlast we give numerical examples to explain the reasonable and efectiveness of theproposed method, and illustrate the performance of investment proportion by vari-ous values of risk-free rates and compare the performance with random case.