Abstract: Investment can help people resist inflation and achieve the maintenance and appreciation of wealth. How to effectively allocate financial assets is a central issue. The mean-variance portfolio model proposed by Markowitz in 1952 settled the foundation for the development of portfolio theory. The model estimates the return and risk by probability theory, achieving the optimal allocation of assets by trading off returns and risks, with the objective function of maximizing returns under established risk constraints or minimizing risks under return constraints. With the extensive research of scholars, the research content and application of static portfolio theory are gradually enriched and improved.
The static portfolio model assumes that investors hold the initial investment portfolio until the end of the investment period without any adjustments. However, factors such as investors’ preferences, and policies change over time, make it difficult to describe the dynamics of the factors by static portfolio models. In actual financial transactions, investment is a continuous process, and investors will adjust their portfolio positions based on the investment environment. Therefore, the multi-period portfolio theory has gradually attracted the attention of scholars.
Scholars have conducted extensive research on multi-stage portfolio optimization, but most of them provide optimal pre-commitment strategies. The pre-commitment strategy is that when investors formulate a strategy during the initial period, they must ensure that the pre-formulated strategy is executed throughout the subsequent investment period. Each period of the strategy depends on the initial state of the investment. However, in multi-period investments, investors need to adjust their strategies based on existing information. It is obvious that the pre-commitment strategy is not the optimal time-consistent strategy. The time-consistent strategy is that investors can adjust their investment strategies with changes in market information. That is to say, the time-consistent strategy satisfies the separability and has no aftereffect in the sense of dynamic programming. There have been transaction costs and cardinal constraints in multi-period portfolios. Therefore, realistic constraints should be considered when studying portfolio optimization. This makes it difficult to solve the optimal time-consistent strategy.
This paper uses mean and variance to measure the return and risk of portfolios, respectively. Considering transaction costs, borrowing and lending constraints, and threshold constraints, we propose a multi-period mean-variance portfolio model. Since the variance is not separable in the sense of the dynamic programming principle, this model is an optimization control problem with time inconsistency. Because of transaction costs, the multi-period portfolio selection is a dynamic optimization problem with path dependence. To seek a time-consistent strategy, we convert the model into a dynamic programming problem approximately by using a nested conditional expectation mapping and design a novel discrete iteration method to obtain the optimal portfolio time-consistent strategy. In order to keep the effectiveness of the novel discrete iteration method, we first prove its linearity and convergence. Then, an example is given to illustrate the behavior of the proposed model and the designed algorithm. This article randomly selects 20 stocks from the Shanghai Stock Exchange for investment, with sample data from April 1, 2006 to March 30, 2017. The moving average method is used to estimate the returns for the next five periods. In the experiment, this article analyzes the impact of weight parameter changes on the terminal wealth of investment portfolios. It can be concluded that the terminal wealth and weight parameters change in the opposite direction.
Considering realistic constraints, a multi-period mean-variance investment portfolio model is proposed and its optimal time-consistent investment strategy is solved by a discrete iteration method. On the one hand, it enriches the research of modern financial decision-making theory and provides a new idea for exploring dynamic risk control technologies. On the other hand, it helps investors adjust their investment strategies in the dynamic investment and enhances the ability of individual and institutional investors to adapt to the investment environment. This is better for investors making scientific and rational decision strategies. However, due to the existing management costs in the financial market, investors cannot invest in too many assets, and there is a minimum purchase volume for each stock. Therefore, it can be extended by considering cardinal constraints and minimum trading volume.

Key words: multiperiod portfolio selection, mean-variance, transaction costs, time consistent strategy, discrete iteration method

摘要： 从动态规划的角度分析,方差算子的不可分离性导致标准的多阶段均值-方差模型的最优投资策略不满足时间一致性。文章采用条件期望映射的方法,构建了一个具有交易成本、借贷约束和阈值约束的多阶段M-V投资组合模型。由于考虑了交易成本,该模型是一个具有路径依赖性的动态优化问题。为了获得其时间一致性投资策略,文章将该问题近似地转化为连续性动态规划模型,证明最优解的近似度,并运用离散迭代算法求解。最后,使用上海证券交易所的部分历史数据验证了模型和算法的有效性。

关键词: 多阶段投资组合, 均值-方差, 交易成本, 时间一致性策略, 离散迭代法