关键词: 多阶段投资组合, 可信性测度, 均值-半方差, 基数约束, 离散近似迭代法

Abstract: Asset preservation and appreciation are crucial for investors. The mean-variance model proposed by Markowitz in 1952 settled the foundation of modern portfolio theory. In the mean-variance model, the returns on risky assets are assumed stochastic variables. Investors try to find an optimal portfolio by trading off maximum return and minimum risk in a static environment. This model provides guidance for investors to make scientific investment decisions, however, the financial market is complex, and many constraints need to be considered when investing in financial assets. Based on this, many scholars improved the mean-variance model from different dimensions.
Firstly, variance is used to measure the volatility of asset returns, which treats upward deviation and downward deviation from the mean of return as equally risky. For investors, only the downward deviation from the mean of returns is the risk, so using variance tends to overestimate the risk of a portfolio. Therefore, some scholars use downward deviation to measure portfolio risk, such as semi-variance, semi-absolute deviation, and so on.
Secondly, the financial market is as uncertain as ever. The return of the asset is affected by the economic and social environment, and other factors, which are not rational to be described by using stochastic variables. Especially for sub-new stock, which lacks efficient history data, the mathematic expectation of estimated return is not an unbiased measure of returns. And people always evaluate it according to expert opinions. Therefore, some scholars have extended portfolio optimal to the fuzzy environment and used fuzzy variables to describe the assets’ return, such as possibilistic variables, credibilistic variables, and uncertainty variables.
Thirdly, in real investment transactions, investment is a continuous process. In different periods, asset returns, and investors’ risk preferences are constantly changing, and long-term investors will adjust their investment portfolio positions in a timely manner as the environment changes. Therefore, portfolio problems have obvious multi-stage characteristics. In addition, investments are restricted by multiple constraints. Therefore, some scholars have extended single-period portfolio models to multi-period models and considered transaction costs, short-selling, cardinality constraints, and so on.
This paper uses credibilistic mean and credibilistic semi-variance to measure the assets’ return and risk respectively. Considering the transaction costs, budget constraints, threshold constraints, and cardinality constraints, we propose a credibilistic multi-period mean-semi-variance portfolio. Based on credibilistic theories, the model is converted to a dynamic optimization problem. Because of transaction costs and cardinality constraints, the model is a mixed integer dynamic optimization problem with path dependence. We use a discrete approximate iteration method to obtain the optimal portfolio strategy. To verify the effectiveness of the method, mathematical formulas are used to prove its convergence. Finally, we select thirty assets from the Shanghai Stock Exchange as the sample and give a numerical example to demonstrate the designed algorithm’s performance and the proposed model’s application. The sample interval is from April 2006 to March 2015. The history data is divided into five periods. This paper assumes that asset return is a triangular fuzzy variable, and evaluates the triangular distribution of asset return for each stage using the idea of quantiles, which was proposed by Vercher et al. in 2007. In the numerical example, we analyze the impact of cardinality constraints and risk aversion coefficient on terminal wealth. It can be concluded that the terminal wealth is positively correlated with the cardinality constraints, which means that when other constraints are controlled, the terminal wealth increases as the number of assets invested increases. It also can be obtained that the terminal wealth is negatively correlated with the risk aversion coefficient when other constraints remain unchanged, in other words, with the risk aversion coefficient becomes upper, the terminal wealth becomes lower.
Considering transaction costs, budget constraints, threshold constraints, and cardinality constraints, this paper proposes a creadibilistic multi-period mean-semi-variance portfolio. On the one hand, it enriches the theoretical research on multi-period portfolio models. On the other hand, due to considering the realistic constraints, this model is more practical and beneficial for investors to make scientific and reasonable strategies. However, with the development of digital finance, information types on the financial market are more and more abundant, such as history return, basic and technical information, video, photo, and so on. The factors that affect asset returns are no longer a simple linear relationship. Therefore, in the future study, it can be extended by using machine learning to predict asset return, exploring the impact of constraints change for terminal wealth.

Key words: multiperiod portfolio selection; credibilistic measure; mean-semivariance; cardinality constraints; discrete approximate iteration method