关键词: 投资组合, 均值—下偏差, 收益预测, 深度学习, 旋转算法

Abstract: The purpose of investment portfolio is to allocate an amount of fund to multiple financial assets more effectively. The Mean Variance (MV) model, proposed by Markowitz, is an important foundation for modern portfolio theory. However, the data in stock market is complex, and it is difficult to accurately describe the returns and risks of assets using the mean and variance of historical returns only. Therefore, based on the MV model, numerous studies have proposed various extended models to optimize the decision-making effectiveness of portfolios. There are two main directions of portfolio optimization: the first one is to predict the assets’ return, which can be used to represent the return of the portfolio, the second one is to change the measurement method of risk.
In the process of portfolio optimization the prediction of return is a crucial factor. Traditional statistical methods, such as the Autoregressive Integrated Moving Average (ARIMA) model, are mainly based on the hypotheses of linearity and normal distribution, and these assumptions may not be satisfied in stock return series. To overcome this problem, deep learning models are considered to predict return in portfolio optimization, which can deal with complex, multi-dimensional and noisy time-series data, and have shown better performance than traditional statistical models. With the superiority of deep learning models in stock market prediction, it is meaningful to investigate the combination of deep learning models’ return prediction with classic portfolio optimization models.
As a candidate of measuring risk, variance has been highly criticized in academics as well as in practice. The primary reason is that the variance punishes both upside deviation and downside deviation of returns in the same way. Thus, an alternative promising candidate, Lower Partial Deviation (LPD) is used as the measure of risk, which penalizes only those outcomes that fall below the target return. In addition, in the process of making investment decisions, the needs and risk preferences of investors also can be taken into account. Therefore, this paper proposes a novel risk measurement: LPD_α, which considers investors’ demand and risk preference by combining target returns with investors’ demands and risk preferences.
In conclusion, it is very significant to measure the returns and risks of an investment portfolio reasonably and accurately. Integrating return prediction in portfolio formation can improve the performance of original portfolio optimization model. In this paper, comprehensively considering various indicators and factors, three deep learning methods including Long Short-Term Memory (LSTM), Convolutional Neural Network (CNN) and Deep Neural Network (DNN) are used to predict stock returns, and the mean absolute error, mean squared error, root mean squared error and hit ratio are selected to evaluate the performance of the three deep learning models. Since LSTM performs better than CNN and DNN in forecasting, the predicted returns based on LSTM are introduced to portfolio optimization model. Considering the demands and preferences of investors, the transaction costs, the threshold constraints and the borrowing constraints, we construct a M-LPD_α portfolio model. This model can be solved using a sequence of quadratic programming method and pivoting algorithm. Finally, in-sample tests and out-of-sample tests are conducted using historical data of SSE50 obtained from Tonghuashun. The in-sample test results indicate that when investors have higher expectations of target returns or higher risk preferences, the effective frontier of the M-LPD_α based on LSTM predicting will be lower. In addition, with the transaction costs and borrowing constraints decreasing, the effective frontier of the M-LPD_α based on LSTM predicting will become higher. And the effective frontier of the M-LPD_α based on LSTM predicting will become higher when the threshold constraints increase. The out-of-sample test results show that the M-LPD_α based on LSTM predicting proposed in this paper performs better than the Equal Weight (EW) model and MV model.
Using the deep learning to analyze the stock market data can improve the ability of individual and institutional investors to deal with complex financial data, and provide technical support for scientific and reasonable investment strategies. Moreover, considering the impact of investor preferences on investment decisions can help individual and institutional investors make more personalized investment decisions. In future study, we will use some methods, including text mining, data augmentation, and feature engineering, to improve the indicator factor system, which can enhance the predictive performance and stability of deep learning.

Key words: portfolio optimization, mean-lower partial deviation, return prediction, deep learning, pivoting algorithm