Research On Uncertain Portfolio Selection Model And Decision With Background Risk And Mental Account

Portfolio selection theory is an important branch of modern finance, and also is the core content of financial area. In 1952, Markowitz published the article titled "Portfolio Selection", which was usually considered the inchoation of modern finance. He established the mean-variance model, which indicated that the portfolio selection study transferred from the qualitive research to quantitive research. Subsequently, the researchers make a further study on modern portfolio theory, and a variety of portfolio theory has came into being. In reality, since the security market is too complex, there are situations where people have none or no sufficient historical data. For example, newly listed securities have few historical data and in unexpected events there are no suitable historical data, either. Therefore, in these situations, the future security returns cannot be reflected by historical data and have to be given by experts’estimations according to their knowledge and judgement. This paper discusses a portfolio selection problem in such an uncertain environment. Below are the main content and innovation points:(1) In real life, investors do not only face portfolio risk but also 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. We discuss 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.(2) To reflect different attitudes towards risk that vary by goal in one portfolio investment, we apply mental account to the investment. Using uncertainty theory, we propose a new uncertain portfolio selection model with mental accounts and provide the determinate form of the model when security returns are normal uncertain variables. In addition, we discuss the shape of mean-variance efficient frontier of the subportfolios in the mental accounts. Further, we present the conditions under which the optimal aggregate portfolio is on the mean-variance efficient frontier when security returns are normal uncertain variables.(3) Futher we discuss the problem of uncertain portfolio selection with mental accounts and background risk, Using uncertainty theory, we propose an uncertain portfolio selection model with mental accounts and background risk and provide the determinate form of the model. Moreover, we discuss the shape and location of mean-VaR efficient frontier of the subportfolios with background risk and without background risk and compare the mean-VaR efficient frontier of the subportfolios with background risk with that without background risk. Further, we present the conditions under which the optimal aggregate portfolio is on the mean-VaR efficient frontier when security return rates are normal uncertain variables.(4) In order to make our model more close to reality, we consider whole hand transaction and transaction cost constraints into account and propose an uncertain portfolio selection model with whole hand transaction and transaction cost constraints. In order to help the investors to solve the model, the equivalent form of the model is given when returns of securities and background assets are all normal uncertain variables. We discuss impacts of whole hand transaction and transaction cost constraints on portfolio decision. It is concluded that when everything else is same, the expected optimal portfolio return with whole hand transaction and transaction cost constraints is smaller than that without whole hand transaction and transaction cost constraints.