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Minimizing Value-at-Risk methodology under fuzzy-stochastic approach

This paper describes new models for portfolio selection under uncertainty with risk and vagueness aspects which are solved by fuzzy-stochastic methodology. And the corresponding optimal portfolio is derived using minimizing Value-at-risk(VaR). The VaR concept has been extended to the portfolio value-at-risk measure used for managing risk and returns under a multiple-asset portfolio. An approach to modeling risks by VaR under imprecise and fuzzy conditions is discussed. It is supposed that the input data and problem conditions is difficult to determine as real numbers or as some precise distribution functions. Thus, vagueness is modeling through the fuzzy numbers. A numerical example of a portfolio selection problem is given to illustrate our proposed approaches.