Uncertain portfolio selection with background risk

In real life, investors face background risk which may affect their portfolio selection decision. In addition, there are situations where background asset return and the security returns have to be given by experts' evaluations because of occurrence of unexpected incidents in economic and social environment or lack of historical data. This paper discusses an uncertain portfolio selection problem in which background risk is considered and the returns of the securities and the background assets are given by experts' evaluations instead of historical data. Using uncertainty theory, we propose a new uncertain portfolio selection model with background risk. To enable the users to solve the problem with currently available programing solvers, the crisp form of the model is provided. In addition, we discuss the optimal solution of the model when the returns of the securities and the background asset return obey normal uncertainty distributions, and compare the optimal portfolio with background risk with that without background risk. It is concluded that when everything else is same, the expected optimal portfolio return with background risk is smaller than that without background risk. Finally, a numerical example is given as an illustration.