Uncertain portfolio selection with mental accounts and background risk

Li Xue,Hao Di

doi:10.3934/jimo.2018124

Abstract

In real life, investors face background risk which may affect their portfolio selection decision. In addition, since the security market is too complex, there are situations where 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 with background risk in such an uncertain environment. In the paper, in order to reflect different attitudes towards risk that vary by goal in one portfolio investment, we apply mental accounts to the investment. 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 efficient frontier of the subportfolios with background risk and without background risk. Further, we present the conditions under which the optimal aggregate portfolio is on the efficient frontier when return rates of security and background asset are all normal uncertain variables. Finally, a real portfolio selection example is given as an illustration.

Highlights

Since the proposal of the mean-variance theory by Markowitz [20], quantitative analysis on portfolio selection has been being a hot research topic, and a lot of models such as mean-variance models [4], mean-semivariance models [21] and mean-VaR models [2] have been developed
Since the security market is too complex, there are situations where the future security returns cannot be reflected by historical data and have to be given by experts’ estimations according to their knowledge and judgement
In order to reflect different attitudes towards risk that vary by goal in one portfolio investment, we have applied mental account to the investment

Summary

Introduction

Since the proposal of the mean-variance theory by Markowitz [20], quantitative analysis on portfolio selection has been being a hot research topic, and a lot of models such as mean-variance models [4], mean-semivariance models [21] and mean-VaR models [2] have been developed. The mean-VaRU efficient frontier of any subportfolios with background risk is an upward concave broken line when the security return rates ξi and the return rate of background asset rb are normal uncertain variables. Since security return rates are all normal uncertain variables, according to (14) and the equation (5), the expected return and standard deviation of the optimal subportfolio X1 are E1 = xses + xtet and δ1 = xsρs +

Results

Conclusion