This paper studies a portfolio optimization problem in which some candidate securities possess sufficient transaction data and the others are newly listed and lack enough data. Their corresponding returns are assumed to be random variables and uncertain variables, respectively. Accordingly, the total return on a portfolio becomes an uncertain random variable. In this paper, we first define value-at-risk of uncertain random variable and discuss its mathematical properties as well as numerical solution procedure. Then we employ it to measure the risk associated with uncertain random returns and formulate the corresponding portfolio optimization models with uncertain random returns. An active-set method is used to solve the proposed models and a numerical example is given to illustrate its application.
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