Abstract: Modern portfolio theory has been inspired by Markowitz's pioneering work, which firstly used variance to measure the risk of portfolio and originally initiated the mean variance method. The mean-variance model has been widely used and extended in the research of portfolio optimization. Based on the modern portfolio theory, most mean-variance models were proposed, which assumed that security returns are random variables and the mean of asset returns to measure returns is used.
However, in the increasingly dynamic and complex financial market, historical data is insufficient to accurately estimate the probability distribution of asset returns. The historical data of asset returns alone cannot correctly reflect its future performance, and the probability distribution under this assumption is partially effective. Relying solely on historical data on an asset's return we cannot accurately predict its future performance, so the probability distribution under this assumption is partially valid. In the realistic investment process, there are a large number of non-probability factors. Fuzzy set theory is widely applied in risk management to solve the uncertainty problem. Many scholars have studied the problem of portfolio optimization with asset returns as fuzzy numbers. Actually, the investors may encounter the uncertainty of both randomness and fuzziness simultaneously when handling the practical portfolio selection problem. Asset returns reflect not only the probability distribution of asset returns which may be partially known, but also vague information estimated by experts on the basis historical data and empirical knowledge. Taking the mixed uncertainty of asset returns into account, random fuzzy variables are considered to describe the random fuzzy phenomenon.
Considering the asset return as a trapezoidal fuzzy number, we first define the variance of random fuzzy variable and then employ it as risk measure. In the investment process, investors may face many realistic and objective constraints. In this paper, a new mean variance random fuzzy portfolio selection model with the transaction costs, borrowing constraints and threshold constraints is proposed. Based on the random fuzzy theories, the model is transformed into a convex quadratic programming problem with linear equalities and linear inequalities constraints. The KKT conditions for the proposed model can be obtained. To find the optimal solution, we present a novel improved pivoting algorithm which solves the linear part while maintaining the complementarity conditions in the computational process. The remarkable feature of the algorithm is that many variables are deleted from the KKT conditions, and it is extremely easy to implement.
A numerical example for synthetic data from China securities market is presented to illustrate the validity of the method and algorithm. Assume that aninvestor randomly chooses 20 stocks from Shanghai Stock Exchange for his investment. We collect historical data of them from April 2006 to June 2018 and set every three months as a period to handle the historical data so that the trapezoidal possibility distributions of the return rates of assets can be obtained. We analyze the variation trend of variance when the target return takes different values, and apply the rolling-window method to compare the out-of-sample investment performance of the random fuzzy mean-variance portfolio model and the equally weighted proportion.
In the in-sample analysis it can be seen that the investment proportion of risk-free asset will increase when the minimum target value of the total future profit increases. The out-of-sample analysis shows that the Sharpe ratio of the random fuzzy mean-variance model is higher than the equally weighted portfolio. Compared with the previous research, the random fuzzy portfolio model proposed in this paper is more consistent with the realistic portfolio. The improved pivoting algorithm can solve the optimal investment problem quickly and effectively, which has strong operability and practical value. In the further study, we will research the multi-period portfolio optimization problem under the random fuzzy environment.

Key words: uncertainty modelling, mean-variance portfolio optimization model, random fuzzy variable, threshold constraints, an improved pivoting algorithm

摘要： Markowitz首先采用方差度量风险,并应用于投资组合优化中,大多数的均值方差模型仅对随机投资组合优化或模糊投资组合优化进行研究,然而,实际投资组合优化问题既包含随机信息也包含模糊信息。本文首先定义随机模糊变量的方差,并用其度量风险,提出了具有交易成本、借贷约束和阀值约束的均值-方差随机模糊投资组合优化模型。基于随机模糊理论,将上述模型转化为具有线性等式和线性不等式约束的凸二次规划问题,并得到其KKT条件。本文还提出改进的旋转算法求解上述模型,该算法消掉KKT条件中部分变量,减少计算量。最后,采用中国证券市场的实际数据进行样本内分析和样本外分析,验证了上述模型和算法的有效性。

关键词: 不确定性建模, 均值-方差投资组合优化模型, 随机模糊变量, 阀值约束, 改进旋转算法