Portfolio Selection Using Z -Number Theory: Two Solution Methodologies

The Z -number theory is a new concept in fuzzy logic. This theory describes the uncertainty of information where any Z -number is displayed by a pair of fuzzy numbers. The first component of the Z -number creates a restriction on the uncertain variable through a fuzzy number, while the second component of the Z -number explains the reliability of the first component through another fuzzy number. Because there is no any accurate image of the future of financial markets, the Z -number theory can be widely used in the financial markets widely. In this paper, the theory of Z -number is investigated in portfolio selection problem using the utility function. The Z -number is almost always converted to classical fuzzy number in the existing literature. Although this approach reduces the computational complexity, converting Z -number to classical fuzzy number causes the loss of significant information. Hence, we propose two practical models with Z -number approach and optimize them both with and without converting Z -number to classical fuzzy number. Optimization without converting Z -number causes the information about utility of assets and their reliability to be available in each stage. However, optimization with converting Z -number causes portfolio selection model to be transformed into a linear programming model. Furthermore, a practical method for obtaining the utility of assets in shape of the Z -numbers is introduced. In addition, liquidity constraint and the maximal fraction of the capital allocated constraint are considered in portfolio selection models. Eventually, two numerical examples from NYSE and NASDAQ Stock Market are provided, and the results are compared with each other.