关键词: 均值-方差投资组合, 机器学习, 超参数优化, 股票预测, 萤火虫算法

Abstract: The returns and risks of assets in the securities market are uncertain. The core problem faced with by investors is how to optimize the allocation of wealth in an uncertain environment so that the risks are minimized or the returns are maximized. To achieve this goal, the artificial intelligence and optimization methods are used by the investors studying portfolio selection problems, which provide investors with new theoretical frameworks and investment strategies. And numerous widely accepted empirical researches suggest that stock prices and returns are predictable to some extent. In this case, it is necessary to observe the change and volatility of financial data over a long time in the past so as to make a good preparation for future trend forecast and investment decisions. Up to now, existing studies have mainly focused on: (1)Statistical methods. They aim at prediction by analyzing the past price characteristics, such as linear regression, autoregressive conditional heteroscedasticity, autoregressive integrated moving average, and generalized autoregressive conditional heteroscedasticity model. And (2)machine learning methods. They are K-Nearest Neighbor, support vector regression, eXtreme Gradient Boosting, deep neural network, long-short term memory, convolutional neural networks, etc. Some comparative studies emphasize that machine learning has stronger ability to deal with non-linear and non-stationary problems than statistical models.
In this paper, a portfolio selection model is proposed using machine learning with hyperparameter optimization for the stock prediction and Mean-variance model for portfolio selection. To be specific, two stages are involved in this model: the stock prediction and portfolio optimization. In the first stage, a hybrid model combining XGBoost with a modified firefly algorithm based on specific probability (pFA) is proposed to predict stock prices for the next period, and compares its predictive ability with XGBoost, LSTM and SVR algorithms. The pFA is developed to optimize the hyperparameters of the XGBoost. In the second stage, the stock selection (stocks with higher predicted returns are selected) and asset allocation (spreading funds across selected stocks). Considering the constraints such as transaction costs and threshold constraints, the mean-variance model and equally weighted model are employed for the portfolio selection. The MV model aims to make a trade-off between maximizing returns and minimizing risks, which is expressed by a typical multi-objective optimization formula, and introduce the risk aversion coefficient to change the multi-objective formula into the mono-objective formulation.
At the same time, using China Securities 300 Index component stocks as study sample, we give a numerical example to demonstrate the designed algorithm’s performance and the proposed model’s application. We specifically and randomly select 48 stocks in the CSI300 as candidate assets, large enough for individual investors to choose stocks before forming portfolios. The sample interval is from January 2013 to December 2021. The history data is divided into six periods, every period containing four-year data divided into a training set and a testing set as a ratio of 8∶2. The training set is used to train the model and adjust the hyperparameters to get a good generalization, and the test set to evaluate the performance of the final model.
In this paper we select 19 indicators as the input of the stock prediction, and the MV model is used to conduct the portfolio’s asset allocation based on the selected high-quality asset. We use the pivoting algorithm to solve the MV model without short sales and to obtain the optimal portfolio strategy. In addition to the MV model, an equal-weight portfolio is also studied. To investigate the accuracy of stock prediction methods, four indexes, namely mean square error, root mean square error, mean absolute error, and hit ratio are used. From the experimental results, we have several important findings: (1)to improve the FA’s optimization ability, the pFA is developed, and after comparing the pFA with FA, PSO, AFSA, GA, and DE, the advantage of pFA is verified by a set of unimodal and multimodal test functions; (2)the empirical results demonstrate that the pFAXGBoost+MV model achieves better results than its counterparts and the market index in terms of return and return-risk metrics.
Considering realistic constraints, a portfolio selection model is proposed using machine learning with hyperparameter optimization for the stock prediction and Mean-variance model for the portfolio selection. And it is a convex quadratic programming problem with equality and linear inequality constraints, which is solved by a novel improved pivoting algorithm. On the one hand, it enriches the research into the modern financial decision-making theory and provides a new idea based on machine learning for the stock prediction. On the other hand, it helps investors adjust their investment strategies in the quantitative investment, and enhances the ability of individual and institutional investors to adapt to the investment environment. However, there are some limitations to this study. This paper can further be improved and extended from the following aspects: (1)Semi-variance, VaR, and skewness can be used for portfolio selection. (2)Cardinal constraint and minimum trading volume can be considered.

Key words: mean-variance portfolio selection, machine learning, hyperparameter optimization, stock prediction, firefly algorithm