1.1. Background

1.2. Research Objectives

This study aims to develop a portfolio optimization algorithm that leverages deep learning models to generate optimal investment decisions based on predicted asset returns.

1.3. Organization of the Paper

This paper consists of the following six sections. Section 1 presents the background and objectives of the study, as well as the overall structure of the paper. In the subsequent sections, key terminology is also introduced. Section 2 reviews several prior studies that are relevant to this research. Section 3 describes the architecture of the deep learning model proposed in this study and explains how its intended functions are realized. Section 4 outlines the types of datasets used, the methodology for evaluating the algorithm’s performance, and the experimental environment. Section 5 presents the experimental results and their analysis. It compares outcomes under different experimental conditions and discusses the effectiveness of the proposed method. Section 6 provides the final conclusions of the study and discusses potential directions for future research.

1.4. Evaluation Metrics for Prediction

1. 1.

   Mean absolute error (MAE) measures the average absolute difference between predicted and actual values. It is calculated by summing the absolute errors of each data point and taking the average. The formula is as follows:

   \begin{equation} \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} \left| y_i - \hat{y}_i \right| . \tag{1} \end{equation}
2. 2.

   Mean absolute percentage error (MAPE) measures the average percentage difference between predicted and actual values. The formula is as follows:

   \begin{equation} \text{MAPE} = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{y_i - \hat{y}_i}{y_i} \right| \times 100\% . \tag{2} \end{equation}
3. 3.

   Mean squared error (MSE) measures the average of the squared differences between predicted and actual values. The formula is given by:

   \begin{equation} \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} \left( y_i - \hat{y}_i \right)^2 . \tag{3} \end{equation}
4. 4.

   Root mean squared error (RMSE) is the square root of the MSE and provides the error in the same unit as the original data. It is computed as:

   \begin{equation} \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} \left( y_i - \hat{y}_i \right)^2 } . \tag{4} \end{equation}

2.1. Predict-Then-Optimize (PO)

The PO paradigm is a two-stage decision-making framework commonly used when optimization parameters must be estimated from data. Given a feature vector \(x\), a predictive model \(f_\theta(x)\) estimates the cost vector \(\hat{c}\), where \(f_\theta(\cdot): \mathcal{X} \rightarrow \mathbb{R}^d\) denotes a mapping from the feature space to the cost space. This prediction is then used in an optimization problem to obtain the decision \(w\). The PO paradigm can be formally written ¹⁵ as:

The smart PO framework ¹⁵, also referred to as decision-focused learning ¹⁶, has been increasingly explored in operations research problems such as shortest path, matching, and scheduling. These approaches aim to improve decision quality by incorporating the optimization objective into the learning process. However, its application to portfolio optimization is still relatively limited, and the existing studies remain less mature compared to other well-established domains.

2.2. Transformer-Based vs. Non-Transformer Approaches for Time Series Forecasting

Various approaches have been proposed for time series forecasting based on deep learning. In recent years, with the growing popularity of Transformer architectures, several Attention ¹⁷-based time series forecasting models have been developed ¹⁸,¹⁹. However, the study by Zeng et al. ²⁰ demonstrated that their proposed model, DLinear—a multi-layer perceptron based on time series decomposition—not only achieves better forecasting performance but also improved computational efficiency compared to Transformer-based models.

2.3. Deep Learning-Based Direct Generation of Portfolio Weights

Zhang et al. ¹⁰ proposed a network architecture that directly optimizes the SR. The output of the neural network is a vector representing the portfolio weights. However, their approach does not address a fundamental limitation of the original Markowitz model—namely, the use of historical asset returns as expected returns.

3.1. Discussion on Conventional Methods

The method proposed by Zhang et al. ¹⁰ aims to optimize the SR in order to obtain an investment portfolio. Specifically, they input eight features—date, open, high, low, close, adjusted close, volume, and return—into a 64-layer long short-term memory (LSTM) ²¹ network. The loss function is defined as the SR, and the output vector is passed through a SoftMax function to obtain a portfolio allocation vector whose elements sum to one. Investments are executed based on this vector. During the backtest period, the model receives the most recent 50 days of data at the beginning of each month, outputs portfolio weights for that month, and repeats this process monthly.

Although their method directly utilizes a neural network to generate portfolio weights, it still inherits a key limitation of the MV model—namely, it uses historical returns directly as a proxy for future expected returns. Given that deep learning-based time series forecasting has been widely validated for its effectiveness, we consider a framework in which a deep learning model is first used to forecast future returns, and these predicted returns are then fed into an LSTM model to generate the final portfolio allocation.

3.2. Preliminary Evaluation of the Proposed Method

To evaluate the validity of our idea, we conducted a simple empirical test using data from January 1, 2021 to January 1, 2022. We focused on three assets: SPY, VTI, and JPXN. For each asset, we employed the DLinear model (Figs. 1 and 2) to forecast the average return for the upcoming month at the beginning of each month, using the most recent 50 days of data up to that point.

We then compared the predicted monthly returns with the actual values and computed the following evaluation metrics: MAE, MAPE, MSE, and RMSE. The results are summarized in Tables 1 and 2.

From the tables above, it can be observed that DLinear yields better forecasting performance for SPY and VTI, while for JPXN, using the simple average of the most recent 50 days appears to provide more reliable results. Overall, the predictive approach based on DLinear demonstrates effectiveness and can be considered a meaningful improvement.

3.3. Proposed Deep Learning Model

The model proposed in this study is divided into two components: a prediction step and an allocation step.

3.3.2. Allocation Step

In the allocation step, we adopt a 64-layer LSTM model as the neural network architecture (Fig. 3). The predicted returns from the prediction step are fed into the network, which outputs a vector \(w' \in \mathbb{R}^n\). The magnitude of each element in this vector represents the relative preference for investing in each of the \(n\) assets.

However, since the sum of the values in \(w'\) does not necessarily equal 1, it cannot be directly used as a valid portfolio allocation. To address this, the vector is passed through a SoftMax function to obtain the final portfolio weight vector \(w \in \mathbb{R}^n\), which satisfies the constraint that all elements sum to one.

For the loss function, we adopt the SR. Specifically, the predicted return matrix \(\hat{R}\) is multiplied by the weight vector \(w\) to obtain the predicted portfolio return for the corresponding month. The mean and standard deviation of the resulting return vector are then calculated, and the SR is maximized by minimizing its negative value as the loss function.

\begin{equation} \text{SR} = \frac{\mathbb{E}\left[\hat{R}w\right]}{\text{Std}\left(\hat{R}w\right)}, \quad \hat{R} \in \mathbb{R}^{T \times n}, \quad w \in \mathbb{R}^n . \tag{8} \end{equation}

4.1. Experimental Environment

The basic experimental environment used in this study is summarized in Table 3.

4.2. Experimental Design

The experiment in this study consists of three steps: prediction, allocation, and backtesting. In the prediction step, the DLinear model is used to generate return forecasts. These forecasts are then passed to an LSTM model to obtain the portfolio weights. Using these weights, a financial investment simulation is conducted within a selected asset universe. This simulation process is referred to as backtesting.

The method of sliding the time window in the prediction step is illustrated in Fig. 4.

4.3. Dataset Description and Backtest Framework

The backtesting strategy involves generating forecasts using deep learning models and allocating the portfolio based on the predicted returns at the beginning of each period during the backtesting window from January 1, 2021 to January 1, 2022. To examine the robustness of the proposed method across different market regimes, we additionally conduct a long-horizon backtest over the period 2018–2022 for Universe 1, which covers diverse market conditions, including bull markets, volatile periods, and extreme shocks such as the COVID-19 crisis.

4.4. Experimental Settings

Table 4 summarizes the hyperparameter settings for both the prediction and allocation models.

5.1. Results of the Prediction Step

The deep learning model used in the prediction step is DLinear. For comparison, we also implemented a forecasting model using XGBoost under the same rolling window scheme. The prediction performance was evaluated using four comprehensive metrics: MAE, MAPE, MSE, and RMSE.

The results of the prediction step for Universes 1 and 2 are summarized in Tables 5–8 and Fig. 5.

As shown in the tables, DLinear outperforms XGBoost across all four evaluation metrics. In particular, XGBoost appears to suffer from underfitting on the dataset used in this study. This is likely due to the lack of a time-series decomposition mechanism in XGBoost, which makes it less effective at capturing temporal structures inherent in financial return data.

5.2. Backtest Result

As shown in Table 9, the turnover levels remain moderate across all asset universes.

5.2.1. Result of Universe 1

The blue line represents the backtest performance of the proposed algorithm, the orange line corresponds to SPY, the green line to the equally weighted strategy, and the red line to the MV model (Figs. 6 and 7). These results indicate that the two-stage deep learning-based prediction and allocation strategy delivers higher returns than SPY, the equally weighted approach, and the MV model.

For the four investment portfolios, the annualized return, SR, Sortino ratio, and maximum drawdown were calculated in Table 10. While the proposed algorithm achieved the highest return among all strategies, it showed slightly lower stability compared to SPY in terms of volatility and drawdown. Nevertheless, it outperformed the other portfolios across multiple performance indicators, demonstrating a strong overall balance between return and risk.

Table 10. Backtest performance comparison on Universe 1.

Portfolio	Return	Volatility	SR	Sortino ratio	Max drawdown
MyPortfolio	0.4008	0.1875	2.0841	3.1567	0.0877
SPY	0.3205	0.1324	2.3454	3.3289	0.0542
Equal-weighted	0.1820	0.1163	1.4795	2.0350	0.0481
Markowitz	\(-0.0125\)	0.0642	\(-0.3513\)	\(-0.3688\)	0.0523

To evaluate robustness under different market regimes, we conduct an extended backtest on Universe 1 from 2018 to 2022 (Fig. 8 and Table 11). The proposed model achieves competitive performance, with a SR (0.79) comparable to SPY, while outperforming the equally weighted and Markowitz portfolios in both return and risk-adjusted metrics. Although slightly underperforming SPY in annualized return, it demonstrates a better balance between return and risk than the Markowitz model and improved stability compared to naive allocation. Overall, the results indicate that the proposed method maintains stable and robust performance across diverse market conditions.

Table 11. Backtest performance comparison on Universe 1 (2018–2022, annualized).

Portfolio	Return	Volatility	SR	Sortino ratio	Max drawdown
MyPortfolio	0.1502	0.1772	0.7914	0.8489	0.2950
SPY	0.1752	0.2080	0.7942	0.8908	0.3372
Equal-weighted	0.1134	0.1623	0.6369	0.7086	0.2725
Markowitz	0.0498	0.0532	0.7492	0.7217	0.1165

5.2.2. Result of Universe 2

The blue line represents the backtest performance of the proposed algorithm, the orange line corresponds to SPY, the green line to the equally weighted strategy, and the red line to the MV model (Fig. 9). When the universe is selected randomly, the model’s performance slightly declines; however, it still outperforms both the equally weighted strategy and the MV model.

For the four investment portfolios, the annualized return, SR, Sortino ratio, and maximum drawdown were calculated in Table 12. The proposed algorithm achieved a return of 26%, but exhibited lower stability compared to SPY and the equally weighted strategy. However, it demonstrated significantly better performance than the MV model.

Table 12. Backtest performance comparison on Universe 2.

Portfolio	Return	Volatility	SR	Sortino ratio	Max drawdown
MyPortfolio	0.2618	0.1343	1.8755	2.8204	0.0522
SPY	0.3205	0.1324	2.3454	3.3289	0.0542
Equal-weighted	0.2001	0.1316	1.4446	1.9880	0.0581
Markowitz	\(-0.0876\)	0.2255	\(-0.4327\)	\(-0.4170\)	0.1657

5.2.3. Result of Universe 3

The blue line represents the backtest performance of the proposed algorithm, the orange line corresponds to SPY, the green line to the equally weighted strategy, and the red line to the MV model (Fig. 10).

As the number of assets increased to 266, the model increasingly tended toward an equally weighted portfolio during training, and gradient explosion was observed. To address these issues, L2 regularization was introduced and a Dropout layer was added after the LSTM layer. In addition, to reduce computational cost given the large volume of data, an early stopping strategy was also employed.

The results from Universe 3 (Table 13) indicate that as the number of assets increases, the model tends to shift toward an equally weighted portfolio, failing to effectively implement stock selection. While this behavior achieves a certain level of diversification, it suggests that the model does not fully leverage the capabilities of artificial intelligence for asset selection, highlighting an area for potential improvement in future research.

Table 13. Backtest performance comparison on Universe 3.

Portfolio	Return	Volatility	SR	Sortino ratio	Max drawdown
MyPortfolio	0.1504	0.1228	1.1439	1.5737	0.0670
SPY	0.3205	0.1324	2.3454	3.3289	0.0542
Equal-weighted	0.1189	0.1069	1.0187	1.4441	0.0539
Markowitz	\(-0.0486\)	0.0636	\(-0.9212\)	\(-0.8024\)	0.0529

6.1. Conclusion

This study proposes a two-stage portfolio optimization framework integrating return prediction and allocation learning. Empirical results across multiple asset universes demonstrate that the proposed method consistently outperforms traditional baselines, including equal-weighted and mean-variance portfolios, in terms of risk-adjusted performance.

While the model does not always exceed the SPY benchmark in raw returns and may exhibit lower stability in certain scenarios, it achieves competitive SRs and improved downside resilience, indicating a more balanced trade-off between return and risk. Notably, in large-scale universes, the method maintains robustness across different market regimes, as evidenced by the extended backtest over 2018–2022.

However, two limitations remain. First, the predictive performance of DLinear degrades for highly volatile assets. Second, as the asset universe expands, the allocation model tends to converge toward near-equal weighting, suggesting limited differentiation in high-dimensional settings.

Overall, the proposed framework provides a stable and scalable approach to portfolio optimization, offering consistent improvements over classical methods while highlighting key directions for future refinement.

6.2. Future Work

Future research could explore several promising directions. First, replacing the dataset with one that represents a broader market scope, such as the SPDR S&P 500 ETF, may allow for performance evaluation under more diverse asset environments. Alternatively, using data from policy-sensitive markets, such as China, could help assess the model’s robustness under highly volatile and government-influenced conditions.

In terms of feature selection, this study relied on default indicators provided by the yfinance library. Future work could enhance model inputs by incorporating additional financial factors, including moving averages of ETF prices, PBR, PER, and return on equity. Furthermore, applying sentiment analysis to economic news and using sentiment scores as input features offer a promising way to integrate textual data into the modeling pipeline.

For the prediction model, adopting time-series forecasting techniques specifically designed for financial data may improve accuracy. On the optimization side, the objective function could be extended with penalty terms that better capture the trade-off between return and risk, offering more flexible and realistic portfolio allocation.

With regard to investment strategy design, additional constraints could be introduced, such as filtering out assets with low expected returns. Moreover, a preliminary stock selection step using machine learning techniques could help identify low-risk assets for constructing a more robust investment universe.

Finally, the current model required over one hour to process a universe of 266 assets. As the dataset scales, computational cost may become a bottleneck. Thus, future work could consider algorithmic simplification or the use of distributed computing to improve scalability and efficiency in large-scale applications.