Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns

• We first propose a hybrid portfolio optimization problem with mixture of random returns and uncertain returns.
• We give the analytical forms of variance of the portfolio return based on uncertain random variable.
• We present mean-variance models for hybrid portfolio optimization and translate them into convex quadratic programming.
• We consider the solution procedures and give the analytical solutions in the case with no more than two new securities.

The determination of security returns will be associated with the validity of the corresponding portfolio selection models. The complexity of real financial market inevitably leads to diversity of types of security returns. For example, they are considered as random variables when available data are enough, or they are considered as uncertain variables when lack of data. This paper is devoted to solving such a hybrid portfolio selection problem in the simultaneous presence of random and uncertain returns. The variances of portfolio returns are first given and proved based on uncertainty theory. Then the corresponding mean-variance models are introduced and the analytical solutions are obtained in the case with no more than two newly listed securities. In the general case, the proposed models can be effectively solved by Matlab and a numerical experiment is illustrated.