Predicting Chinese stock market using XGBoost multi-objective optimization with optimal weighting

Selection of risk and return indicators

In order to provide a comprehensive depiction of both returns and risk, our study has chosen five crucial indicators. When it comes to returns, we take into account both the rate of return and the excess return. The rate of return represents the actual increase in assets, whereas the excess return indicates the proficiency in generating additional returns through a particular investment strategy. Regarding risk assessment, our attention is directed towards the metrics of standard deviation and maximal drawdown. The standard deviation measures the consistency of returns generated by the investment strategy, while the maximal drawdown indicates the asymmetry and peakedness in the distribution of returns, addressing specific shortcomings of the standard deviation. Moreover, the Sharpe ratio quantifies the additional return gained in relation to the assumed level of risk, which is of utmost importance for investors when making decisions.

The chosen indicators create a composite that better captures the various aspects of stock returns and risks compared to relying solely on return rates or stock prices. This composite is crucial as it possesses qualities that are more suitable for machine learning models. It allows for the identification of complex and nonlinear relationships among features in the market, which is both intricate and unstable. The calculation methods for the five indicators are as follows.

Let $T_i$ symbolize the net asset value (NAV) on the $i^{th}$ day, while $R_p$ signifies the cumulative return of the investment portfolio. $R_f$ stands for the risk-free rate, and ${\beta }_p$ denotes the beta value of the portfolio. We have:

Cumulative return:

(1) $$R_p=\frac{T_i-T_0}{T_0}.$$

Excess return:

(2) $$Alpha=R_p-\left(R_f-{\beta }_p\times (R_m-R_f)\right).$$

Standard deviation:

(3) $${\sigma }_p=\sqrt{\frac{1}{N}\sum _{i=1}^N{(R_i-\overline{R_p})}^2}.$$

Max Drawdown:

(4) $$MaxDrawdown=\frac{max(T_m-T_n)}{T_m},$$where $T_i$ represents the net value of the composition on day $i$.

SharpeRatio:

(5) $$SharpeRatio=\frac{R_p-R_f}{{\sigma }_p}.$$

The process employed to generate labels for the accumulated return rate and excess return rate entails assigning a value of 1 to samples that are greater than 0, while assigning a value of 0 to all other samples. Nevertheless, when it comes to standard deviation, maximal drawdown, and Sharpe ratio, samples that are lower than the sample mean are designated with a label of 1, while all other samples are assigned a label of 0.

Fusing labels based on optimal weights

The use of multi-objective optimization methods demands the simultaneous realization of profit acquisition and risk control objectives. In this way, they closely resemble the multi-label classification problems seen in the machine learning field. Researchers have proposed various approaches to address such issues, which can broadly be divided into traditional multi-label classification methods (Moyano et al., 2018; Zhang & Zhou, 2014) and deep learning-based multi-label classification methods (Kim, 2014; Vaswani et al., 2017). Yet, the aforementioned methods are subject to certain limitations when applied in the financial domain, such as the difficulties encountered when addressing label correlation and label imbalance issues, as well as the potential for low model interpretability and weak generalization capabilities.

Hence, this article introduces the concept of “fusing labels”. The construction process is as follows: (1) assign specific weights to each label; (2) sum the products of multiple labels and their weights, with the resulting sum taken as the fused label value. Using this approach, multi-labels are transformed into a single composite label that merges both return and risk information, hence the term “fusing labels”. The detailed fusion process is as follows:

To ensure optimal model performance, the best weights are filtered out for label construction. These weights possess the following characteristics: (1) the weight of a single indicator falls between 0 and 1; (2) the value of the fusing labels obtained based on the weight maps falls between 0 and 1; and (3) the minimum change unit for each weight is 0.1.

The fusing labels method based on optimal weights offers the following advantages:

• 1)

  Adjustable weights: The impact of various indicators on the model can be influenced by adjusting the weights. This allows for the theoretically effective analysis of the individual contributions of each indicator and practical adjustments to the model to be made based on economic finance theory, historical experience, or investor risk-return preferences.

• 2)

  Alleviating label correlation issues: Fusing labels can eliminate redundant information introduced by correlations, thereby enhancing the model’s prediction accuracy.

• 3)

  Mitigating label imbalance problems: Multi-label scenarios can easily produce label imbalances. As a result, compared to single-label models, the trained model will be biased towards high-frequency labels or label combinations. Fusing labels can alleviate this, enabling the model to equally address different categories of samples and improve its prediction capabilities in relation to minority classes.

• 4)

  Enhancing model interpretability: Financial applications entail the use of models with high interpretability, necessitating an understanding and explanation of the model’s prediction results. By converting multi-label classification problems into single-label classifications, fusing labels can offer a better, more comprehensive understanding of the model’s predictive outcomes.

• 5)

  Resolving label conflict issues: When dealing with financial data, different labels might inherently conflict with each other, such as return labels and risk labels, making it more difficult for the model to understand the labels. The use of fusing labels can avoid such conflicts.

• 6)

  Improving generalization capabilities: Multi-label classification generally produces more complex models with a higher probability of overfitting. The utilization of fusing labels simplifies the model structure and suppresses the overfitting issue, enhancing the model’s generalization capabilities in the process.

OW-XGBoost

The XGBoost algorithm (Chen & Guestrin, 2016) is an optimized algorithm based on boosting tree algorithms, such as Adaptive Boosting (Adaboost) and Gradient Boosting Decision Tree (GBDT). Specifically, it learns by integrating multiple weak classifiers. Given a training set $D=(x_i,y_i)$, where D contains $n$ records and $m$ variables, and $|D|=n$, $x_i\in R^m$, and $y_i\in R$, the predicted value ${\hat{y}}_i$ for the $i^{th}$ sample, it can be represented by a model composed of $k$ decision trees, denoted as:

(6) $${\hat{y}}_i=\sum _{i=1}^K{f}_k(x_i),f_k\in F.$$

Here, $f_k$ represents the $k^{th}$ decision tree, and F is a function space representing the collection of all decision trees.

Unlike the objective function of GBDT, XGBoost incorporates an additional regularization term on top of the original objective function to reduce overfitting and enhance generalization. The objective function is as follows:

(7) $$L=\sum _{i=1}^{n}l(y_i,{\hat{y}}_i)+\sum _{k=1}^K\mathrm{\Omega }(f_k).$$

Here, function $l$ can select different loss functions, while $\mathrm{\Omega }(f_k)$ represents the penalty term for the $k^{th}$ tree. The specific formula is as follows:

(8) $$\mathrm{\Omega }(f_k)=\gamma T_k+\frac{1}{2}\lambda \sum _{j=1}^{T_k}{w}_{k,j}^2,$$where $w_{k,j}^2$ represents the weight of the $j^{th}$ leaf in the $k^{th}$ tree, T represents the number of leaf nodes, and $\gamma$ and $\lambda$ are parameters used to balance importance. By approximating the L equation using a second-order Taylor series expansion, the following formula is obtained:

(9) $$\begin{array}{l}{L}^{(t)}={\displaystyle \sum _{i=1}^{n}l(y_i,{\widehat{y}}_i)+{\displaystyle \sum _{k=1}^K\Omega (f_k)}}\\ ={\displaystyle \sum _{i=1}^{n}l\left(y_i,{\widehat{y}}_i^{(t-1)}+f_t(x_i)\right)+{\displaystyle \sum _{k=1}^K\Omega (f_k)}}\\ \approx {\displaystyle \sum _{i=1}^n\left[l\left(y_i,{\widehat{y}}_i^{(t-1)}+g_i{f}_t(x_i)+\frac{1}{2}{h}_i{f}_i^2(x_i)\right)+{\displaystyle \sum _{k=1}^K\Omega (f_k)}\right].}\end{array}$$

Here, $g_i={\mathrm{\partial }}_{{\hat{y}}_i^{(t-1)}}l(y_i,{\hat{y}}_i)$ and $h_i={\mathrm{\partial }}_{{\hat{y}}_i^{(t-1)}}^{2}l(y_i,{\hat{y}}_i)$ respectively represent the first- and second-order approximations of the loss function L with respect to $y_i^{(t-1)}$.

This model possesses the following advantages:

• The model supports parallel computation, resulting in higher computational efficiency.

• The algorithm supports column sampling, which reduces overfitting and enhances generalization ability while effectively reducing computational complexity.

• It incorporates a mechanism for handling missing values, enabling the model to automatically learn the splitting directions of tree nodes where there is missing data.

• Unlike GBDT, which solely employs first-order derivative information, XGBoost uses a second-order Taylor series, enabling the model to capture more granular data patterns and enhancing its accuracy.

• By incorporating L1 and L2 regularization into the loss function, the model’s generalization ability is significantly improved.

The constructed OW-XGBoost model will be backtested to demonstrate its superiority over YL-XGBoost and MLC-XGBoost, as well as highlight its robustness.

Model backtesting

The process of backtesting the model can be divided into two parts: stock selection and portfolio optimization. Given the current difficulties and limitations with short-selling stocks in the Chinese market, for present purposes, no consideration is given to shorting stocks that may decline. For the model’s specific implementation process, we adopted a sliding window research method on the basis that it retains the time series characteristics of the data and better simulates the actual investment process. It should also be noted that ST stocks and stocks listed for less than 6 months were excluded from trading. The commission fee was set to 0.03% of the transaction value for buying and 0.03% of the transaction value plus 0.1% for stamp duty for selling, with a minimum commission deduction of 5 RMB per transaction.

Data source and preprocessing

Parameter selection

Trading frequency selection

To control trading costs and ensure the present research is comparable with others, the optimal trading frequency was selected based on monthly and weekly portfolio adjustments (Table 2).

Table 2:

Backtesting data for OW-XGBoost under different trading frequencies.

	Rate of return	Annualized return	Excess return	$\alpha$	$\beta$	Volatility	Sharpe ratio	Max drawdown
Monthly rebalancing	30.09%	61.05%	22.85%	0.547	0.339	0.183	3.113	5.9%
Weekly rebalancing	18.57%	36.15%	11.97%	0.302	0.287	0.146	2.202	6.13%
CSI 300 Index	5.89%	10.93%		−0.811	1.237	0.169	0.143	8.78%

DOI: 10.7717/peerj-cs.1931/table-2

As shown in Fig. 1, the return curve of monthly adjustments using OW-XGBoost remained consistently higher than that of the weekly adjustments. In terms of return metrics, monthly adjustments with OW-XGBoost significantly outperformed weekly adjustments for the following reasons:

• 1)

  A decreased trading frequency reduces tax and trading expenses.

• 2)

  The model incorporates a substantial number of fundamental factors, thus exhibiting a high value-investing attribute. Weekly adjustments cannot adequately capture the benefits of value regression, resulting in lower returns than monthly adjustments when using OW-XGBoost.

• 3)

  Compared to weekly adjustments, monthly adjustments are better at balancing short-term market fluctuations and evaluating and adjusting portfolios over a longer time span, thereby minimizing the impact of short-term fluctuations on returns.

Regarding risk metrics, the volatility of monthly adjustments with OW-XGBoost is slightly higher than that of weekly adjustments, though the maximum drawdown is slightly lower. Both exhibited robust risk-control capabilities, though there are minor differences in risk metrics.

In conclusion, monthly adjustments with OW-XGBoost outperformed weekly adjustments in terms of return rate, excess return rate, and maximum drawdown, demonstrating superior overall market performance. On this basis, monthly adjustments were selected. Also, regardless of the trade frequency, the performance of OW-XGBoost significantly surpassed that of the CSI 300 Index, demonstrating and underscoring the efficacy of fusing labels.

Comparison with YL-XBGoost

In order to compare the differences between OW-XGBoost, the CSI 300 Index, and YL-XGBoost, the market performance of the three models was analyzed. The method of minimizing portfolio variance was used to determine the investment weights of the targets for both OW-XGBoost and YL-XGBoost. Specific backtesting data can be found in Table 3.

It can be seen from Fig. 2 that the return trend of OW-XGBoost is typically higher than that of YL-XGBoost and the CSI 300 Index. Except for October 2022, the cumulative return rate of OW-XGBoost exhibited only minor fluctuations and drawdowns, broadly maintaining an upward trajectory. Where there is minimal volatility and maximum drawdown, it consistently achieves excess returns relative to the benchmark index. These results effectively demonstrated OW-XGBoost’s ability to achieve high return rates while controlling for low volatility and a low maximum drawdown.

In terms of return rate, OW-XGBoost achieved a return rate of 30.09% during the backtesting period, with an annualized return of 61.05% (approximately five times the benchmark return). This performance is significantly higher than both YL-XGBoost and the CSI 300 Index, yielding an excess return rate of 22.85% during the backtesting period. Looking at the entire 7 months of backtesting, 4 months witnessed considerable return growth, with increases of more than 4%. The highest increase was in October, with 15.80%. Contrastingly, the 3 months with declining returns saw reductions of less than 1.5%, two of which experienced drops of less than 0.7%. It is noteworthy that in 3 of the 4 months when OW-XGBoost’s returns increased, the CSI 300 Index experienced falling returns. This illustrates that OW-XGBoost remains able to effectively identify potential growth opportunities and achieve significant return growth in a bearish market. Meanwhile, YL-XGBoost’s return fluctuations were strongly correlated with index movements, showing a greater vulnerability to market volatility.

Regarding volatility, YL-XGBoost showed the highest volatility, followed by OW-XGBoost. This indicates that YL-XGBoost’s risk is further elevated, which imposes certain limitations. The overall volatility of OW-XGBoost is slightly higher than that of the CSI 300 Index, though this can largely be attributed to the rapid return increase in October 2023. As returns and volatility are inherently interconnected, a consistently rising return curve can still generate relatively high volatility, even where high returns are achieved in the short term. This is the case with OW-XGBoost: when examining monthly volatility, OW-XGBoost’s volatility is lower than that of the index for all months except October, revealing its effective ability to withstand market risks once fusing labels are introduced.

In terms of maximum drawdown, OW-XGBoost was significantly lower than both the CSI 300 Index and YL-XGBoost. Moreover, it demonstrated effective drawdown risk control in five of the monthly drawdowns, where it was notably lower than YL-XGBoost. An examination of OW-XGBoost’s market performance reveals that its maximum drawdown, highest return, and highest volatility all occurred in October. This suggests that OW-XGBoost sacrifices some risk-control capacity in the pursuit of high returns.

Furthermore, OW-XGBoost’s beta value is 0.339, significantly lower than 1, reflecting its diminished susceptibility to market impact and increased resistance to market shocks. In comparison, YL-XGBoost’s beta value is 0.458, indicating a higher vulnerability to market fluctuations. Meanwhile, OW-XGBoost’s alpha is 0.131, reflecting a strong capacity to generate excess returns, and its Sharpe ratio stands at 3.113, indicating that for every unit of risk, 3.113 units of return can be achieved, which is an exceptional risk-reward ratio. Additionally, OW-XGBoost has a win rate and profit-loss ratio of 0.667 and 11.773, respectively; both values are significantly higher than those of YL-XGBoost. This suggests that OW-XGBoost has a greater capacity for risk control and profit-making compared to using the return rate alone as a benchmark.

Comparison with MLC-XGBoost

In order to assess the superiority of OW-XGBoost over MLC-XGBoost, the backtesting results for both models were compared. The multi-label classification algorithm also used XGBoost, as shown in Fig. 3, to maintain comparability. After obtaining the classification results using the multi-classification algorithm, the score for each target was calculated based on the optimal weight. Ultimately, only the targets that achieved high scores surpassing the predetermined threshold were chosen to be included in the investment portfolio.

The overall trends exhibited by OW-XGBoost and MLC-XGBoost are somewhat similar, though the cumulative return curve of OW-XGBoost remained higher than that of MLC-XGBoost. Table 4 shows that both the return and excess return of OW-XGBoost are higher than those of MLC-XGBoost, though its volatility and maximum drawdown are lower. These results demonstrate that the OW-XGBoost-based model significantly outperforms the MLC-XGBoost model in terms of earning returns and controlling risk.

For present purposes, the OW-XGBoost model offers the following advantages: First, it can comprehensively consider the interrelated returns and risk indicators in the market, meaning it is better able to capture the correlation patterns among different indexes, avoid the loss of related information, and provide a holistic assessment of the investment targets. Second, the model makes the multi-label problem easier to solve by turning it into a single-label classification problem based on optimal weights. This simplifies the model structure, lowers the risk of overfitting, improves generalization and stability, and makes the problem easier to solve. Third, utilizing fusing labels can better leverage the feature extraction capabilities of machine learning models, making the trained targets exhibit superior consistency and overall coordination, which in turn elevates the model’s stability and risk resistance.

Comparison with baseline models

To further analyze OW-XGBoost’s performance in the stock market, five baseline models were picked for comparison with the XGBoost model: random forest (OW-RF), decision tree (OW-DT), support vector machine (OW-SVM), logistic regression (OW-LR), and ordinary least squares linear regression (OW-OLS). The results are shown in Fig. 4. Specific backtesting data can be found in Table 5.

OW-XGBoost’s returns significantly surpassed those of the other models. The remaining five baselines exhibited remarkably high trend similarity, as can be seen in how the support vector machine, random forest, and decision tree baselines nearly overlap. These five baselines generally aligned with the market trend in the initial stage, though the fluctuation became less significant during the rising and falling phases, illustrating the efficacy of models using fusing labels in controlling risk. The cumulative return of these baselines gradually increased in the later stage, all of which surpassed the index return. The OLS regression model had the lowest return curve for most of the study period, with the lowest gain among the six models, just exceeding the CSI 300 Index (Fig. 4). Moreover, compared to other baselines, its return took around 20 days longer to turn from negative to positive.

The support vector machine model generated the highest profits, with a back-testing return of 17.35% and an annualized return of 33.62%. The two baselines with the lowest returns are OLS regression (13.44%) and logistic regression (13.21%). With the exception of OW-XGBoost in October, the monthly returns of other models fluctuated considerably less than the CSI 300 Index, demonstrating their capacity to remain profitable during market declines.

The overall volatility and maximum drawdown control abilities of the baselines did not vary significantly, and all were notably less than the CSI 300 index. Among the five baselines, the largest drawdown was OLS regression (6.62%), while the smallest was decision tree and support vector machine (2.99%). Most baselines maintained a maximum drawdown level of 3–4%, which is less than half of the CSI 300 Index’s ultimate drawdown of 8.87%. In terms of overall volatility, the five baselines closely align, with OLS regression and logistic regression exhibiting the highest (0.103) and lowest (0.091) values, respectively. The remaining baselines’ volatility fell within the 0.094–0.099 range, slightly exceeding half of the CSI 300 Index’s volatility.

In summary, fusing labels based on optimal weights showed stable performance across various machine learning models, achieving superior returns compared to the market as well as lower market volatility and drawdown, thus demonstrating their effectiveness. It was discovered that nonlinear machine learning models did better than linear OLS regression and generalized linear logistic regression models in terms of return and risk control. This suggests that there are nonlinear relationships between factors and labels that linear models had trouble detecting. When it comes to linear models, generalized linear machine learning models do better than traditional linear OLS regression models. This is because machine learning models are better at finding linear relationships between factors. Finally, all models, even the traditional OLS regression, did better than the CSI 300 Index. This shows that combining labels based on optimal weights is a good way to show how return and risk work in the stock market, which in turn makes model investments perform better.

Adjusting the backtesting period

To further investigate the robustness of OW-XGBoost, a longer duration characterized by a vibrant stock market, namely, 2013–2015, was selected to explore OW-XGBoost’s performance in the context of complex market conditions. Specific backtesting data can be found in Table 6.

As can be seen in Fig. 5, from 2013 to mid-2014, OW-XGBoost was consistently higher than the Shanghai and Shenzhen 300 Index overall, with both less volatility and drawdown. During the bull market phase from July 2014 to June 2015, OW-XGBoost largely charted the same trend as the index. In the initial phase, OW-XGBoost’s cumulative return remained above the index, though the gap gradually narrowed, and it was eventually overtaken by the index in March 2015. During the stock price crash from June to September 2015, OW-XGBoost’s drawdown was significantly less than the Shanghai and Shenzhen 300 Index, at around one-third of the maximum index drawdown. Later, during the bounce at the end of 2015, OW-XGBoost also showed superior performance. However, during the bull market, although OW-XGBoost’s risk control performance significantly surpassed the market, the ultimate return was no better than that of the broader market. This could be due to the generally high stock price evaluation during the bull market: when OW-XGBoost needed to balance risk and return, its ability to unearth excessive returns in the stock selection process may have been negatively impacted.

In conclusion, based on OW-XGBoost’s ability to achieve better return levels while controlling for volatility and a maximum drawdown smaller than the broader market in stable, turbulent, and drastic falling stages, it can be seen to possess a high level of robustness.

Changing benchmark index

In order to assess the roburstness of OW-XGBoost across various stock pools, its effectiveness was additionally evaluated using stock pools consisting of constituent stocks from the CSI 100 and CSI 800 indices, as commonly done in literature. Specific backtesting data can be found in Tables 7 and 8, respectively.

CSI 100 index

As shown in Fig. 6, the returns of OW-XGBoost consistently surpassed the index returns, with the gap between them rapidly widening early in the backtesting period. The backtest yield for OW-XGBoost stands at 30.96%, with an annualized yield of 63% and an excess yield rate of 24.63%. During the backtesting phase, the overall volatility of OW-XGBoost was slightly higher than that of the CSI 100 index; however, once October was excluded, the overall volatility of OW-XGBoost fell below the index. In terms of monthly volatility, OW-XGBoost’s volatility was lower than the index’s volatility for 4 months. Meanwhile, the maximum drawdown for OW-XGBoost during the backtesting period was 5.9%, which was lower than the 10.08% maximum drawdown for the CSI 100 index.

CSI 800 index

As shown in Fig. 7, the OW-XGBoost and CSI 800 index return curves show a significant degree of overlap. However, in comparison to the fluctuating index, OW-XGBoost maintained an overall upward trajectory. Additionally, it displayed a largely stable or improved performance throughout the three significant market fluctuations that occurred in October 2022, December 2022 to January 2023, and March to April 2023, without notably suffering from these fluctuations.

During the backtesting period, OW-XGBoost’s returns were approximately twice that of the index, with a volatility of 0.901, which is about three-fifths of the index. Furthermore, the maximum drawdown was only 2.79%, which is significantly lower than the index’s maximum drawdown.

Comparing OW-XGBoost’s performance in relation to the CSI 300, 100, and 800 indices, it can be found that OW-XGBoost performs better in backtesting when using high-market capitalization stocks as the stock pool. There are several reasons for this: first, studies have shown that in addition to the value of the company itself, small and medium-sized stocks in the Chinese stock market also possess a certain “shell value.” However, neither fundamental nor technical factors are capable of reflecting this part of the value. As a result, OW-XGBoost struggles to accurately predict the ups and downs of small and medium-sized stock prices based on existing factors. Second, compared to large-cap companies, smaller-cap companies have more limited information disclosure, and the probability of irregularities in the disclosure process is higher, resulting in relatively lower information quality. Consequently, the characteristic factors that form on the basis of this information are also less likely to adequately predict trends. Third, large-cap stocks in the Chinese stock market have higher liquidity, resulting in relatively lower price fluctuations and less susceptibility to the daily limit system, which facilitates the OW-XGBoost fully realizing its potential.

OW-XGBoost has been tested in the past using stocks from the CSI 100 and CSI 800 indexes as stock pools. This shows that it is very good at finding extra returns and managing risks, and there is a lot of evidence to support this.

Adjusting training set duration

To further examine the robustness of OW-XGBoost under different training set durations, model training was carried out using training sets with durations of 2, 3, and 6 months, respectively. Specific backtesting data can be found in Table 9.

As shown in Fig. 8, the trends of OW-XGBoost in the three scenarios are relatively similar, all achieving higher returns than the Shanghai and Shenzhen 300 Index and exhibiting lower volatility and maximum drawdown than the index. Hence, it can be concluded that OW-XGBoost can maintain excellent performance in terms of generating profits and controlling risks under different training set durations, indicating its robustness.

As the training set duration lengthens, OW-XGBoost’s overall performance steadily declines for the following reasons: First, extending the length of the training set may cause OW-XGBoost to pay too much attention to noise or specific historical patterns in the historical data, thereby increasing complexity and overfitting while decreasing its generalization ability. Therefore, its performance in backtesting is relatively poor. Second, over time, the effectiveness of feature factors may change due to the influence of market conditions and investor behavior. Longer training sets will give OW-XGBoost more early data, but they will not look into enough of the factors that work best in new market conditions or how these factors relate to each other. This means that OW-XGBoost does not work as well as it could.