关键词: 不确定理论, 均值—方差—熵模型, 国际投资组合, 不确定汇率, 不确定税率

Abstract: In order to get better investment returns, more and more investors pay attention to international portfolios. This paper studies how to make investment decisions in the face of uncertain securities returns, exchange rates and tax rates in a complex international investment environment.
Since the mean-variance model was proposed, the mean-variance model has been the foundation of the modern portfolio theory and become a widespread criterion to help investors make their investment decisions. However, studies concerned only discuss the home portfolio selection problem. In fact, with the rapid economic development and liberalization of capital flows, international portfolio investment has become more common and has attracted more and more the attention of practitioners and academic scholars. Some scholars indicate that an international portfolio is meaningful for investors because it can bring more benefits to investors than only domestic investment. Therefore, exchange rate risk has a significant impact on international portfolios and cannot be ignored. Furthermore, apart from exchange rate risk, some countries levy capital gain tax on investment gain and tax risk should also be a concern. There are papers that provide insightful findings about international portfolio selection or portfolio selection with tax risk. But they all assume that asset returns, exchange rate risk and tax rate are considered as random variables, which are based on the assumption that the future frequencies can be precisely reflected by the historical data. However, in practice, it is difficult to obtain precise probability distributions due to the fast-changing environment or the occurrence of unexpected events like the spread of COVID-19. Then we cannot treat the indeterminate quantities as random variables, so we need to explore to use a new tool other than probability theory to make investment decisions.
Using the uncertainty theory, we propose a mean-variance-entropy model for international portfolio selection with tax consideration under the assumption that the return rates of risky assets, exchange rate and tax rate are uncertain variables. Specifically, we follow the uncertain mean-variance modeling idea and apply a proportion entropy to gain the required investment diversification. The proportion entropy has been used to describe portfolio selection problems’ degree of diversification. For further discussion, we will deduce the general deterministic equivalent form of the model and the deterministic equivalent form when the returns of risky assets, uncertain exchange rate and tax rate take normal uncertainty distributions.
On the basis of the model and its equivalent forms, an empirical study is carried out with a portfolio of fifteen stocks from the Nasdaq and New York Stock Exchanges and a risk-free asset. Based on the data, we give the optimal investment allocation on these stocks and risk-free asset. And we also analyze the questions of whether foreign exchange forward contracts should be used to hedge foreign exchange risks in international investment portfolios, and the necessity of considering uncertain tax rates in investment decisions. We find that when the risk tolerance level of investors is low, they should use forward contracts to hedge exchange risk, but when the risk tolerance level of investors is high, they should choose a portfolio without forward contracts. In addition, the experiment also shows that uncertain tax rates must be addressed in portfolio decision-making, and a more accurate distribution of uncertain tax rates should be given as far as possible. We also do sample validation by comparing the optimal portfolio produced by the mean-variance-entropy model with the equal-weighted portfolio. The results show that the optimal portfolio obtained by the model performs better than the equal-weight portfolio. The model proposed in this paper is effective.

Key words: uncertain theory, mean-variance-entropy model, international portfolio selection, uncertain exchange rate, uncertain tax