2.1. Portfolio formation

MPT was founded when Markowitz [2] proposed the mean-variance (MV) approach as a solution to the portfolio selection problem, using expected return to measure investment performance and variance to quantify risk. The primary concept of MV technique is to optimize anticipated return while maintaining constant variation, or to minimize variance while preserving expected return. MPT has garnered extensive acceptance and discussion by scholars [5,30]. According to Tobin [31], liquidity preference may influence the amount of wealth allocated to financial assets and creates a successful portfolio that includes both risk-free and a unique kind of risky assets. In order to simplify the computation, Sharpe [32] proposes the diagonal model, which greatly aids in the development of portfolio theory by assuming that there are no relationships between securities. Ritter and Renwick [33] present a comprehensive examination of the literature concerning asset management and investor portfolio behavior, juxtaposing contemporary real-world performance with theoretical advancements, followed by an exploration of their own perspectives on the topic. Other research encompasses Merton’s study [34], which enhanced modern portfolio theory by presenting a continuous-time model aimed at maximizing expected utility within a defined planning framework. To address a real portfolio problem without the need to solve a mathematical programming problem, Elton et al [35] studied and created decision rules. In order to compare MV and quadratic approximations that determine optimal portfolios, Gracer and Hakansson [36] introduced a discrete dynamic investment model. A broad methodology for assessing a managed portfolio’s performance was put forth by Chen and Knez [37]. In order to provide a deterministic transformation to the multi-objective stochastic programming portfolio model, Ben Abdelaziz et al [38] develop a paradigm known as chance constrained compromise programming (CCCP). Several realistic constraints were added to the Markowitz’s MV model by other researchers. Liu and Loewenstein [39] use transaction costs into stock trading methods to optimize investors’ wealth utility. In the MV model, Brown and Smith [40] take into account risk aversion, transaction cost, and portfolio limitations. They conclude that when three additional assets are introduced, portfolio optimization problems will be challenging to resolve. Additionally, several research include robust optimization strategies for portfolio management. In order to estimate uncertain parameters, Tu and Zhou [41] encompass financial objectives into Bayesian priors. They demonstrate that the Bayesian method performs better when using objective-based priors than when using alternative priors for portfolio selection. Fuzzy set theory is also being used by several academics to analyze portfolio issues. According to Li and Xu [42], the financial market frequently experiences both random and fuzzy uncertainty; for this reason, they take into account the opinions of experts and investors and building a portfolio. An expert system is created by Yunusoglu and Selim [43] to assist portfolio managers in making investment decisions. The first step in the three-stage expert method is to remove any merchandise that is deemed unsuitable. The second step involves assessing the stock by conducting in-depth literature searches and speaking with subject-matter experts. Building a portfolio using a mixed-integer linear programming model is the final step. Their findings show that ES performance does not differentiate significantly among risk choices; additionally, ES is better suited for investment periods of six, nine, and twelve months. In order to determine the ideal portfolio, Almahdi and Yang [44] first established three optimization goals: the analytical Sharpe ratio, the Sterling ratio, and the Calmar ratio. Then, they chose the method that performed the best. During the decision-making process, Larni-Fooeik et al. [45] analyzed investment scenarios and determined the quantity of regret attained in each scenario. They evaluated volatility risk measurements and employed stochastic optimization to determine the optimal scenario that optimizes investment portfolio returns, minimizes risk, and reduces resultant regret. The augmented epsilon constraint (AEC) method was employed to transform each multi-objective model into a single objective and to provide the Pareto-efficient frontier. The literature review that was done in the area of portfolio formation is summarized in Table 1 below.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. A summary of the research done in the area of portfolio optimization.

https://doi.org/10.1371/journal.pone.0335036.t001

2.2. Sentiment analysis

The growing interest and applicability of automated sentiment or polarity analysis using NLP has led to the development of various methodologies in recent years. These approaches can be broadly classified into two main categories: lexical-based methods and supervised machine learning methods [5]. Lexical approaches generally employ a predetermined lexicon of words, each linked to a particular sentiment, however they are highly contingent upon the context for which they were developed, as noted by Ribeiro et al [65]. Several instruments such the Valence Aware Dictionary and sEntiment Reasoner (VADER) [66], integrate lexical analysis with the evaluation of phrase features to detect term polarity. Recently, following the introduction of the Transformer conception in 2017 [67] and the subsequent development of models derived from it, particularly Google’s Bidirectional Transformer (BERT) model [68] and its different versions [69,70], have achieved remarkable popularity and established new benchmarks in NLP tasks within the GLUE (General Language Understanding Evaluation) Benchmark. Other models utilizing transformers consist of OpenAI-ChatGPT [71]. Typically, no one strategy consistently attains optimal predictive performance across every collection of data [72]. Consequently, regardless of the selected strategy, the model must be refined according to the particular dataset. Araci [73] observed that financial texts feature a specialized vocabulary, marked by unique terminology and a tendency toward ambiguous expressions instead of clearly defined positive or negative terms, which makes models trained on general corpora ineffective for this context. Consequently, he optimized BERT for financial data (FinBERT) and demonstrated its superiority over leading machine learning techniques for financial sentiment analysis datasets. The provided model was then applied in numerous studies and merged with other models, ultimately producing favorable results. For instance, Hiew et al [74] used a well-known pre-trained BERT model created by Google to analyze a text-based sentiment indicator, specifically for three individual stocks that are active in the Hong Kong market and have a lot of discussion on Weibo.com. They showed significant improvement in using BERT for financial sentiment analysis compared to existing models. Within the context of sentiment analysis, Wu et al [75] conducted a study examining the impact of deep learning techniques, with a particular focus on BERT models. The performance of LLM and FinBERT on news articles and company announcements was compared by Shen and Zhang [76] with respect to their applicability to FSA. Their study highlighted the advantages of quick engineering utilizing zero-shot and multi-shot methodologies to enhance sentiment categorization accuracy. To identify sentiment in financial documents, Jun Gu [77] and associates developed FinBERT, a pre-trained natural language processing model. Subsequently, this model was improved by integrating a LSTM architecture, and this novel model was named FinBERT-LSTM. The model employs news categories pertaining to the hierarchical structure of the stock market, namely those connected to the market, industry, and individual stocks, in conjunction with the prior week’s stock price status to generate predictions. Table 2 presents a summary of the literature review on sentiment analysis, organized by author, publication date, methodology used, and the data applied. Figure 1 provides an overview of the model categorizations.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. A summary of the research done in the area of sentiment analysis.

https://doi.org/10.1371/journal.pone.0335036.t002

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. An overview of sentiment analysis algorithms.

https://doi.org/10.1371/journal.pone.0335036.g001

In the field of sentiment analysis, the choice of pre-training data plays a crucial role in model performance. FinBERT is a specialized pre-trained language model developed specifically for sentiment analysis in the financial domain, having been trained on a large corpus of financial texts. While previous studies have predominantly relied on news articles, social media platforms (such as X), financial websites, and corporate statements as input sources, this study takes a different approach by utilizing companies’ quarterly reports to forecast future stock price movements using the FinBERT model. This choice is motivated by the fact that financial reports offer comprehensive and reliable insights into a company’s operations and future outlook.

2.3. Stock market prediction

In the last few years, the continuous development of big data technology and artificial intelligence (AI) has prompted an increasing number of researchers to use AI as the basis for their research methods, demonstrating that AI-based algorithms more effectively address the challenges of nonlinear and non-stationary features compared to traditional statistical models [90]. Notable methods include Support Vector Machine (SVM) [91], Principal Component Analysis (PCA) [92], Random Forest (RF) [93]and Genetic Algorithm (GA) [42], and Artificial Neural Network (ANN) [26,94]. A substantial quantity of review articles exists on stock price forecasting and prediction, thoroughly investigating the application of machine learning techniques in stock market forecasting.

Stock market prediction constitutes a regression application because of the continuous nature of stock prices [95]. Di Persio and Honchar [96] employed RNN for predicting Google stock prices. RNN, LSTM, and GRU are the three most effective neural networks for processing sequential data. RNN is utilized for analyzing historical data. LSTM and GRU circumvent the vanishing gradient problem through the utilization of forget, reset, and update gates. The GRU is determined to be quicker because to its utilization of reset and update gates. Future stock market prices can be predicted attributed to significant advancements in machine learning algorithms, particularly deep learning algorithms that are particularly good at identifying intricate linkages and patterns in financial time series data.

Because it can model long-term dependencies, LSTM, a kind of RNN, has garnered a lot of interest among these algorithms [97]. The following are some other studies that have been done on that subject matter. In a study, Bao et al [98] introduced an innovative deep learning architecture that integrates wavelet transformations (WT), stacked autoencoders (SAE), and LSTM for forecasting stock prices. SAEs were utilized for the inaugural time in forecasting stock prices based on hierarchically derived deep characteristics. Gulmes [99] has evaluated multiple algorithms, including LSTM1D, LSTM2D, LSTM3D, ANN, LSTM-GA, and LSTM optimized with ARO (LSTM-ARO), using the DJIA stock dataset from 2018 to 2023 in a study. Pang et al [100] introduced a deep long short-term memory (LSTM) neural network with an embedded layer and a long-term short-term memory (LSTM) neural network utilizing an auto encoder for stock market prediction. Gill et al [101] have examined the efficacy of advanced deep learning in a study. The research additionally introduces models for short-term trend prediction utilizing daily and hourly closing prices of the S&P 500 index and the Brazilian ETF EWZ. Liu et al. [102] developed a recurrent convolutional neural kernel (RCNK) model, which learns complementing characteristics from different data sources, primarily historical price data and text data from message boards, to predict stock fluctuations. Qiu et al [103] conducted an experiment using different wavelet functions and used SNR and RMSE criteria to determine the most effective wavelet for reducing stock price noise. Table 3 provides an overview of the literature review, summarizing key studies, methodologies, and findings related to the topic. It includes details such as the authors, publication dates, approaches used, and datasets analyzed, offering a comprehensive snapshot of the existing research in the field. Based on the findings from the literature analysis about the LSTM model (as indicated in Table 2) and its outstanding outcomes, this study will employ this model to forecast the expected returns.

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. A summary of the research done in the area of stock market prediction.

https://doi.org/10.1371/journal.pone.0335036.t003

Various extensions of Markowitz’s MV model improve modern portfolio theory and provide researchers with additional perspectives for research. These developments highlight the importance of improving the MV model in portfolio management. In general, both short-term and long-term investment methods can be used to purchase stocks listed on the stock market. Long-term investment involves holding stocks for a long time, while short-term investment involves buying and selling stocks in shorter time frames, and investors aim to make a profit within a few days or weeks [102]. Traders use several trading tactics, such as short-term, long-term, and intermediate-term [103]. However, no previous research has simultaneously examined the three objective functions of risk, return, and utility (the trend prediction score obtained from sentiment analysis of text data) for a portfolio. To achieve quarterly long-term investment, we incorporate a new objective function into the original model in this study to maximize utility. This utility function takes into account the impact of environmental data (quarterly reports) in the portfolio formation model. The presented model was finally implemented on real data of index DJIA companies and then solved using the WGT method and the results were presented. The methodology will be examined in the next section.

3.1. FinBERT model

Analyzing financial sentiment is challenging due to the specialized vocabulary and the limited availability of labeled data in this domain [17]. This study utilizes the pre-trained FinBERT model to conduct sentiment analysis and predict future fluctuations in stock prices. FinBERT is specifically designed for financial text analysis, offering improved accuracy over general-purpose language models when applied to financial content. In this section, we elaborate on the model’s architecture, theoretical foundations, and its advantages for sentiment classification within the financial domain. The FinBERT model is based on the widely adopted BERT framework, but it is uniquely fine-tuned on large-scale financial corpora. This enables the model to better understand the nuanced language and context typical of financial documents. Through domain-specific training and task-oriented fine-tuning, FinBERT achieves a high level of precision in detecting sentiment signals from textual data. The detailed architecture of the FinBERT model is illustrated in Fig 3, highlighting the components tailored for financial sentiment analysis.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. FinBERT architecture following Araci [17].

https://doi.org/10.1371/journal.pone.0335036.g003

The primary function of the input layer is to accept financial text and transform it into numerical vectors suitable for model processing, utilizing techniques such as word segmentation and word embedding. The BERT encoder employs a multilayer transformer architecture that effectively collects contextual information from the text via the attention mechanism, generating a corresponding vector representation of the text. A sentiment classification layer is subsequently incorporated atop the BERT encoder, utilizing the encoder’s output as input to execute sentiment classification over a fully linked network. The sentiment categorization layer comprises several neurons, each representing a distinct sentiment category. The training procedure of the FinBERT Sentiment model is primarily segmented into two phases: pre-training and fine-tuning.

• Pre-training: The pre-training FinBERT model comprises two primary phases:
  1. Pre-training: the original BERT model is trained on extensive, general datasets (such as BookCorpus and Wikipedia) to comprehend general language and sentence structure. This phase is non-specialized and emphasizes general language understanding.
  2. Further Pre-training: the model undergoes additional training on financial datasets, including news articles, financial reports, and corporate announcements. The objective is to acquire financial terminology (e.g., “stocks,” “bonds,” “profit,” “loss”) and boost expertise in financial NLP tasks, such as sentiment analysis and named entity recognition, to achieve better accuracy and efficiency.
• Fine-tuning: during the fine-tuning process, the model is trained using dataset specific to finance. The goal is for the model to adjust its weights to process financial language more effectively while retaining the broad language understanding gained in the pre-training phase. In this phase, the model is trained on labeled data using techniques such as backpropagation and gradient descent. A reduced learning rate is typically used during fine-tuning to avoid disturbing the general language knowledge already acquired by BERT.

The FinBERT model generally has the following advantages over other models:

• Domain-Specific Training: FinBERT is trained on financial texts, giving it a deeper understanding of financial terminology and context.
• Improved Sentiment Analysis: It provides more accurate sentiment classification for financial content like earnings reports or market news.
• Better Context Understanding: FinBERT handles ambiguous financial terms (e.g., “bullish,” “liability”) more accurately than general models.
• Higher Accuracy in Financial Tasks: It outperforms general models in tasks such as financial entity recognition and document classification.
• Reduced Noise in Predictions: Because it’s specialized, FinBERT generates fewer irrelevant or incorrect outputs when analyzing financial texts.
• Time and Resource Efficiency: Using a pre-trained domain-specific model like FinBERT reduces the need for extensive retraining on financial datasets.

3.2. LSTM networks

LSTM networks were first introduced by Hochreiter and Schmidhuber [24] as a technique for capturing and learning sequential patterns. LSTM networks are a specialized form RNNs that are capable of retaining information over extended periods, making them more effective than traditional RNNs in handling long-term dependencies [62]. Graves and Schmidhuber [121] provide evidence that LSTM networks are capable of overcoming the inherent issues that were previously present and memorizing temporal patterns over an extended length of time. Consequently, in our study, we will make use of this model to forecast the prices of stocks in the future. LSTM networks are composed of an input layer, several hidden layers, and an output layer. What distinguishes LSTMs from other types of networks is the inclusion of memory cells within the hidden layers. Fig 4 shows the architecture of an LSTM memory cell, which plays a crucial role in storing and managing information over time. To effectively model time-series data such as stock prices, the LSTM model is configured with the following components:we use quarterly repor

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. The architecture of the LSTM memory cell, as outlined by Fischer and Krauss [29].

https://doi.org/10.1371/journal.pone.0335036.g004

1. Input Layer: The input layer takes in a sequence of historical financial data, such as the opening price, closing price, high, low, and trading volume. This input is typically structured into time windows (e.g., 60-time steps) to give the model temporal context. Each input vector at a time step can include multiple features.
2. LSTM Layers: The core of the model is the LSTM layer (or layers), which consists of memory blocks called cells.Each cell maintains an internal state, controlled by three gates:
   • Forget Gate (f_t): Decides what information to discard from the cell state.
   • Input Gate (i_t): Determines which new information to store in the cell state.
   • Output Gate (o_t): Controls how much of the internal state to pass to the output.

The input and hidden states are represented by and at time t. respectively, is adjusting its cell state. The input gate controls which data should be stored in the memory cell, the output gate governs what information is extracted from the memory cell, and the forget gate determines which data should be discarded. The mathematical formulation of an LSTM cell is as follows:

(1)

(2)

(3)

(4)

(5)

(6)

Where ,,,,,,and are weight matrices, , , , and are the bias vectors associated with each of the respective gates. The bias vectors are incorporated to enhance the model’s adaptability to the data. The bias vectors, , , and are initialized to zero, whereas the bias the forget gate in the LSTM is initialized to 1.0. The symbol * represents element-wise multiplication. These operations allow the model to regulate information flow dynamically and effectively manage both short-term and long-term dependencies.

1. Stacked LSTM Architecture: In many advanced configurations, multiple LSTM layers are stacked to increase the model’s representational power. The output of each LSTM layer is passed to the next layer, enabling deeper abstraction of sequential features.
2. Dropout Regularization: Dropout layers are inserted between LSTM layers to prevent overfitting by randomly deactivating a fraction of the neurons during training. This improves generalization, especially when the dataset is small or noisy.
3. Dense (Fully Connected) Layer: The final dense layer maps the high-level sequence representations to a single output value (e.g., predicted stock price). For regression tasks, the activation function is typically linear.
4. Compilation and Training: The model is compiled using a loss function such as Mean Squared Error (MSE), which is appropriate for continuous output variables. The optimizer is often Adam, which adapts learning rates and accelerates convergence. Training is conducted over multiple epochs, with early stopping or validation monitoring to prevent overfitting.

The architecture of the LSTM network is particularly advantageous for modeling financial time series data, where both short-term fluctuations and long-range temporal dependencies play a critical role in shaping future trends. In the context of stock market forecasting, price movements are rarely independent of historical behavior; rather, they are influenced by recurring patterns, investor sentiment, and macroeconomic events that unfold over varying timescales. LSTM networks are inherently designed to capture such temporal structures by maintaining a memory of past states while dynamically adjusting their focus to recent changes. This ability allows them to outperform traditional machine learning models, which typically rely on fixed-size input vectors and handcrafted features. Unlike methods such as linear regression or support vector machines that require domain-specific feature engineering to capture time dependencies, LSTM models automatically learn relevant temporal representations directly from raw sequential data. This makes them not only more adaptable to the complex dynamics of financial markets but also more scalable for real-world forecasting applications where patterns may shift over time. For more information, please refer to the article by Yu et al [124].

3.3. Mean-variance (MV) model

MV model was established by Markowitz [2] in order to address the process of choosing the optimal portfolio problem. This model is the foundation upon which MPT is built. Both the return on investment and the risk associated with it are defined in this approach by the expected return and the variance, respectively. According to Santos and Tessari [125], The key factor in portfolio selection for investors is identifying the portfolio that best suits their preferences, considering the associated risk level and expected returns. Consequently, rational investors tend to prefer portfolios with lower risk and stable expected returns, or those with higher expected returns but consistent risk levels. To address this issue, a set of optimal alternatives is created, referred to as the efficient investment frontier. The model is defined by the following set of equations:

(7)

(8)

Subject to:

(9)

(10)

The objective function aimed at minimizing risk is expressed in Equation 7, while the expected return of the portfolio comprising the chosen stocks is determined using Equation 8., and and denote the percentage of the total investment that is allocated to assets 𝑖 and j, respectively. The is used to specify the covariance between assets i and asset j and describes the expected return on ith asset. Equation 9 specifies that the proportion of budget distributed between different stocks must sum to 1. Equation 10 states that the investment % per share can never be negative. The following section provides a detailed explanation of the underlying mathematical modeling of the proposed approach, including the formulation of key equations, the assumptions made, and the rationale behind the chosen solution technique.

4.1. A modification of the fundamental model

By incorporating a third objective function into the basic model that was discussed in Section 3.4, we further develop the model in this paper. A representation of the extended mathematical model can be shown following.

(11)

(12)

(13)

Subject to:

(14)

(15)

Additionally, represents the degree of bullish trend for the ith asset in the upcoming quarter, as determined by the FinBERT model. Section 3.5 provides clarification on additional indicators and marks. In the next section, the proposed solution method is discussed, and finally, the final modified model is presented.

4.2. Proposed approach for solving multi-objective models

In this research, we apply the Goal Programming (GP) approach to tackle multi-objective optimization problems within the proposed model. GP was first developed by Charnes et al. [124] and Charnes and Cooper [125]. Romero [126], Jones and Tamiz [127], and Ignizio [128] all contributed to its continued development. Romero states that the achievement function, which gauges the extent to which the undesirable deviational variables of the model’s objectives have been minimized, is a crucial component of a GP model. This section presents a concise overview of the GP methods and normalization techniques employed in this paper. For a detailed overview of GP, including its various types and the normalization techniques employed, the reader is directed to the work of Jones and Tamiz [127].

4.2.1. The theory of the Weighted Goal Programming (WGP) approach.

WGP is generally considered a deterministic approach that provides effective solutions for addressing multi-objective challenges in technological and economic contexts, operating within the scope of multi-criteria decision-making analysis [129]. WGP is a form of goal programming where each objective is allocated a distinct weight to signify its relative significance in the decision-making procedure. The purpose is to reduce discrepancies from the target objectives by taking into account varying priorities or the relative significance of each goal. The general WGP model:

(16)

Subject to:

(17)(18)(19)

Note that the normalization constant for deviant variables is , the negative deviant variable i is , the positive deviant variable i is , the weight assigned to positive factors deviant variable i is , and the weighting factor for the negative deviant variable is . Is an optional set of hard limitations, is the ith target value, is the ith objective function, and x is the vector containing the decision variables. The WGP facilitates direct trade-offs among all unwanted deviational characteristics by including them into a weighted, normalized singular achievement function that is minimized. Each goal, , is assigned a target value, , that must be attained. The positive deviational variable, , indicates overachievement, whereas the negative deviational variable, , indicates how much the aim was underachieved. The achievement function is then used to minimize the unwanted deviation from the objective value. The decision maker’s preferences are reflected in the values of α and β. If attaining the profit aim of k is deemed twice as significant as meeting the capacity target of , then the associated deviational variables are assigned weights of = 2 and = 1 in the achievement function. The weights in GP plays two roles: (a) they normalize the unit and measurement scales, and (b) they value the decision maker’s preferences. The preferences of the decision maker can be incorporated into Goal Programming (GP) models by:

• Weights and penalties for the undesirable deviational variables;
• Choosing the target values for each objective.

A normalization approach is used in GP to quantify deviations from each aim using a consistent unit of measurement. Prominent normalizing approaches include Euclidean, percentage, and zero-one methods. Euclidean normalization is especially advantageous when the goal values are near zero. Thus, the Euclidean mean of the objective function’s technical coefficients is the normalization constant. . According to Jones and Tamiz [127], Euclidean normalization is a computationally robust technique that is applicable to all goals, even those with minimal or zero target values, and does not need optimization or intricate computations to get the normalization constants. For the following goal i, the normalization constant, denoted by. , is as follows:

(20)

Is calculated as:

(21)

4.2.2. Final model.

Building upon the explanations provided in the previous sections, the proposed model has been systematically reformulated to incorporate the identified modifications and improvements. This revised formulation reflects the underlying assumptions, structural adjustments, and methodological choices made during the development process. The updated model is presented as follows to clearly illustrate these enhancements and their role in addressing the problem more effectively.

(22)

Subject to:

(23)(24)(25)(26)(27)

Where ,, represents the negative deviation variables and ,,represents the positive deviation variables. , and represent the ideal values of the objective functions obtained by solving the single-objective model while accounting for additional constraints. In the next section, the proposed model is implemented on real data of the companies that make up the Dow Jones Index and its results are presented.

5.1. Data

One of the primary challenges in financial forecasting lies in identifying meaningful relationships between historical and future events within time-series data [37]. In this study, we rely on two main categories of data: numerical historical data and unstructured textual data. The focus of our case study is on the companies listed in the DJIA, which consists of 30 major publicly traded firms. Accordingly, both historical and textual information have been collected for each of these 30 companies to support our analysis.

• Textual data: For the purpose of gathering textual data, we have utilized the quarterly reports that were released by firms (those that are included in the DJIA) during the first quarter of each year 2024. The reports in question were obtained from the Yahoo Finance website, and the Sider AI was utilized in order to extract the company’s outlook for the upcoming quarter. As an illustration of the extracted outlook, Table 4 provides an example.
• Historical data: Historical data is needed as input for the LSTM model. To do this, we have gathered historical data (daily closing prices) for the firms comprising the DJIA utilizing the yfinance library. Yfinance is an open-source library for acquiring and retrieving financial data, including business calendars, historical data, etc. In the Python programming language. We will utilize historical data from 2015 through the conclusion of the first quarter of 2024 as input for the LSTM model.

Download:

• PNG
  larger image
• TIFF
  original image

Table 4. Sample of company quarterly report outlooks.

https://doi.org/10.1371/journal.pone.0335036.t004

The constituents of the DJIA comprise “UnitedHealth Group, Inc.” (UNH), “The Goldman Sachs Group, Inc.” (GS), “Microsoft Corporation” (MSFT), “The Home Depot, Inc.” (HD), “Caterpillar, Inc.” (CAT), “Amgen Inc.” (AMGN), “McDonald’s Corporation” (MCD), “Salesforce, Inc.” (CRM), “Visa, Inc.” (V), “American Express Company” (AXP), “The Travelers Companies, Inc.” (TRV), “Apple, Inc.” (AAPL), “International Business Machines Corporation” (IBM), “JPMorgan Chase & Co.” (JPM), “Honeywell International, Inc.” (HON), “Amazon.com, Inc.” (AMZN), “The Procter & Gamble Company” (PG), “Johnson & Johnson” (JNJ), “The Boeing Company” (BA), “Chevron Corporation” (CVX), “3M Company” (MMM), “Merck & Co., Inc.” (MRK), “The Walt Disney Company” (DIS), “NIKE, Inc.” (NKE), “Walmart, Inc.” (WMT), “The Coca-Cola Company” (KO), “Cisco System, Inc.” (CSCO), “Dow Inc.” (DOW), “Verizon Communication, Inc.” (VZ) and “Intel Corporation” (INTC).

In this study, the choice of DJIA data was driven by several factors:

1. Availability of high-quality, continuous data: DJIA companies provide comprehensive and reliable datasets, allowing us to rigorously evaluate the model.
2. Focus on a specific market to control variability: Concentrating on a single, well-defined index helps reduce complexity arising from structural differences across various markets.
3. Time and resource constraints: Access to a broader and more diverse dataset was limited during the research period.

5.2. Details on usage of the FinBERT model

As mentioned in Section 5.1, we use quarterly reports to predict future price outlooks. We collect companies’ outlooks for the second quarter after the first quarterly reports are received in 2024 and use this text data as input to the FinBERT model. Also, this paper uses the Python Transformer package and the ProsusAI/finbert pipeline for the model input. The model output findings are presented in Table 5. The process of extracting the score for company future outlooks is as follows: first, companies’ quarterly reports are extracted, and then insights and information from these reports are extracted using the Sider AI tool. two of the outlooks (future trends) extracted with the help of the Sider artificial intelligence tool are shown as examples in Table 4, and the complete data has been uploaded to GitHub and its link is as follows https://github.com/esipour93/DJIA_Textual_Data-1.git. The extracted outlooks are then fed to the FinBERT model and sentiment analysis is performed on them, and then the FinBERT model outputs are converted into scores using the AHP method. This process is graphically shown in Fig 5.

Download:

• PNG
  larger image
• TIFF
  original image

Table 5. Demonstrating the results of the FinBERT model.

https://doi.org/10.1371/journal.pone.0335036.t005

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. The process of extracting corporate Outlook scores.

https://doi.org/10.1371/journal.pone.0335036.g005

We computed using the Analytic Hierarchical Process (AHP) approach and outputs FinBERT model. The AHP approach is a multi-criteria decision-making strategy that assists in choosing the optimal choice from severalAlso, using the AHP method explained options based on various factors. This approach works well in unique circumstances where making decisions necessitates weighing a variety of intricate attributes. Saaty [130] was the first to suggest this approach in 1980. Saaty’s AHP is a widely used technique for prioritizing the basic challenges in a complicated situation [131]. We determined the weights of the three criteria: positive, negative, and neutral, utilizing this procedure, and subsequently calculated the value by multiplying this vector with the output of the FeinBERT model. We have evaluated the value of preferences according to expert opinion, with positive to negative equal to 7, positive to neutral equal to 4, and neutral to negative equal to 3. The findings of this method are as follows:

Table 6 presents the decision matrix constructed as input for the AHP. This matrix captures the pairwise comparisons of the decision criteria, allowing for the quantification of subjective judgments in a structured format. It serves as the foundational step in deriving the normalized weights of the objective functions, which reflect the relative importance assigned to each criterion by the decision-maker. These weights are critical in guiding the optimization process and ensuring alignment with specific investment goals or risk preferences. Following this, Table 7 reports the outcomes of the AHP-based analysis. It presents the calculated weights, the corresponding scores for each portfolio alternative, and the resulting optimal portfolio configurations. The results are disaggregated by investor profile, enabling a nuanced understanding of how varying preferences such as risk tolerance or return expectations impact portfolio composition. This comprehensive presentation of results highlights the effectiveness of the proposed model in integrating multi-criteria decision-making with portfolio optimization.

Download:

• PNG
  larger image
• TIFF
  original image

Table 6. Decision matrix.

https://doi.org/10.1371/journal.pone.0335036.t006

Download:

• PNG
  larger image
• TIFF
  original image

Table 7. Demonstrating the resultant weights.

https://doi.org/10.1371/journal.pone.0335036.t007

It is important to highlight that the derived weight vector for the positive, neutral, and negative criteria is (0.705, 0.211, 0.084), respectively. Additionally, the Consistency Ratio (CR) obtained for the pairwise comparison matrix is 3.4%. In the context of the Analytic Hierarchy Process (AHP), the CR serves as an essential measure to evaluate the logical consistency of the judgments made during the pairwise comparison process. Given the subjective nature of human assessments, some degree of inconsistency is expected. However, a CR value below the widely accepted threshold of 10% indicates that the comparisons are sufficiently consistent and reliable for decision-making purposes. Therefore, with a CR of 3.4%, we can be confident in the robustness and credibility of the resulting weight vector. Letting SAᵢ denote the sentiment vector for the ith asset and Q the weight vector, the outlook score is calculated using the element-wise product: = SAᵢ ⊙ Q. An illustrative example follows:

5.3. Details on usage LSTM model

In this study, a LSTM model is employed to forecast two critical financial indicators: the expected return and the associated risk for each asset. These predicted values are represented within the proposed model as (expected return) and (risk or variance-covariance elements between assets). The LSTM model is particularly well-suited for this task due to its ability to capture temporal dependencies and nonlinear patterns in financial time-series data. The specific architecture and configuration details of the implemented LSTM model, including the number of layers, units, dropout rates, and other training parameters, are summarized in Table 8.

Download:

• PNG
  larger image
• TIFF
  original image

Table 8. LSTM model summary.

https://doi.org/10.1371/journal.pone.0335036.t008

The developed model consists of an input layer, an output layer, and two hidden layers. Furthermore, two dropout layers have been added to mitigate the potential for overfitting. Here, a dropout rate of 20% is assumed that it is applied to the hidden layers’ input and functions by randomly deactivating 20% of the input units. In addition, a rate of 20% was used to divide the data into two categories: train and test data. The proposed model has shown acceptable performance in price prediction. Fig 6 displays the results of the proposed model’s performance as an example of all DJIA companies. The hyperparameters of the LSTM model, including the number of layers, the number of units per layer, dropout rates, batch size, and learning rate, were determined through a manual trial-and-error approach. Various configurations were tested based on values commonly used in related literature and prior experience with time-series forecasting tasks. The performance of each configuration was evaluated using validation data and standard metrics such as RMSE and MAE. The final set of hyperparameters was selected based on the best overall performance, balancing accuracy, model complexity, and overfitting prevention.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. Showing examples of the LSTM model performance and comparing actual price with predicted price for three companies UNH, BA, DOW.

https://doi.org/10.1371/journal.pone.0335036.g006

5.4. Implementation modified model

Given that most companies’ financial reports are published in late April or early May, in this study, the three-month period of May, June, and July has been considered for investment. Therefore, using the LSTM model, the price has been predicted for these three months, and finally the expected return and covariance matrix have been obtained. The expected return is calculated using the following formula.

(28)

Where represents the return on the asset i in period t. Also, represents the price of asset i in period t and represents the price of asset i in period t-1. The average expected return obtained for DJIA companies for the first quarter of 2024 is shown in Table 9. The covariance matrix is also given in Table 10.

Download:

• PNG
  larger image
• TIFF
  original image

Table 10. Covariance matrix.

https://doi.org/10.1371/journal.pone.0335036.t010

Download:

• PNG
  larger image
• TIFF
  original image

Table 9. Average expected return.

https://doi.org/10.1371/journal.pone.0335036.t009

Using equation 20, the values for the three objective functions of risk, return, and trend are obtained as , respectively. Furthermore, to calculate , the proposed model solution with a single objective function is used. This approach involves simplifying the multi-objective problem by focusing on one objective at a time, which allows for a simpler optimization process. The value of 𝐵 is then used as a reference point or benchmark to evaluate and compare the performance of the multi-objective solution in subsequent analysis. The outcomes are displayed in Table 11.

Download:

• PNG
  larger image
• TIFF
  original image

Table 11. Ideal values for each objective.

https://doi.org/10.1371/journal.pone.0335036.t011

Also, using the AHP method explained in Section 5.1, the weighted values of the objective functions for the different investor states of risk-averse, risk-taking, and trend-based with the expert opinion are shown in Table 12. The AHP process begins with defining the objective, identifying the criteria and sub criteria, and constructing a hierarchy. Next, pairwise comparisons are made between the criteria using a relative importance scale, which typically ranges from 1 (equal importance) to 9 (extreme importance). These comparisons are used to create a pairwise comparison matrix from which the weights of the criteria are calculated, often by normalizing the matrix and calculating the average of each row. Finally, a consistency ratio is calculated to ensure that the judgments are consistent. If this ratio is acceptable (usually less than 0.1), the weights obtained can be used in the decision-making process. Here, we weighted the three objectives according to expert opinion and formed optimal portfolios based on these results.

Download:

• PNG
  larger image
• TIFF
  original image

Table 12. Objective function weights in the goal programming technique for different types of investors.

https://doi.org/10.1371/journal.pone.0335036.t012

As shown in Table 11, the CR value for all three cases was less than 10%, so the calculated weights are acceptable. To calculate the values, we solved the initial model with each of the objective functions alone, the results of which are shown in Table 12.

6.1. Review and discussion of the results

Beta: William Sharpe [132] proposed that systematic risk can be quantified using the beta coefficient (β). Beta acts as a measure of a security’s responsiveness to changes in the broader market’s performance. It quantifies the extent to which an asset’s returns correlate with those of the market portfolio. Specifically, the beta coefficient assesses how an asset’s returns react to variations in the market’s returns, calculated by analyzing the relationship between the asset’s return sensitivity and the market’s return sensitivity. Mathematically, beta is defined as the ratio of the covariance of the security’s returns and the market portfolio’s returns to the variance of the market portfolio’s returns, represented as follows:

(29)

The portfolio beta is calculated from the following formula:

(30)

Where n represents the number of stocks in the portfolio and represents the weight of the ith asset.

Sharp ratio: The Sharpe Ratio (SR), introduced by Sharpe in 1964 [132], serves as a standardized and comprehensive metric for evaluating fund performance. It effectively accounts for both returns and associated risks, mitigating the impact of risk factors on performance assessments. A higher SR value indicates a superior risk-adjusted return. The Sharpe Ratio is computed using the following formula:

(31)

Where E(R) represents the expected return, σ(R) denotes the standard deviation of the returns, and is the risk-free rate.

Treynor ratio: The Treynor Ratio is a prominent and important performance metric in financial literature. Commonly known as a “reward-to-volatility” metric, it evaluates the return produced by a portfolio in relation to the risk assumed, particularly emphasizing systematic risk, denoted as beta (β). This ratio is a one-dimensional metric that accounts for both residual return and the beta coefficient. Beta, denoted by the Greek letter β\beta, is a measure of the sensitivity of a portfolio’s returns to fluctuations in the overall market. It quantifies the degree to which the portfolio’s performance is affected by systematic market risk. Beta is typically calculated using a market index or a benchmark portfolio as a reference [133]. To calculate the Treynor Ratio, the residual return is divided by the portfolio’s beta, as shown in the following formula:

(32)

Sortino ratio: The Sortino ratio is a variation of the Sharpe ratio, but it replaces standard deviation with downside deviation as the measure of risk. Unlike the Sharpe ratio, which considers all fluctuations in return, the Sortino ratio focuses solely on returns that fall below a user-defined target or required rate of return, treating only those negative deviations as risky [134]. The formula for calculating the Sortino ratio is as follows:

(33)

Where represents the downward deviation of the portfolio. And it is calculated from the following formula:

(34)

For each of the proposed portfolios listed in Table 12, the following performance measures have been calculated to evaluate them. These calculations provide a detailed and comprehensive assessment of the risk-return profile for each portfolio and provide key insights into their potential performance under different market conditions. The performance measures include expected return, portfolio variance, Sharpe ratio, beta index, Treynor ratio, and Sortino ratio, all of which serve as critical indicators for assessing the profitability and risk associated with each portfolio. A summary of the results, for further analysis and comparison, is provided in Table 13.

Download:

• PNG
  larger image
• TIFF
  original image

Table 13. Performance measures results for portfolios.

https://doi.org/10.1371/journal.pone.0335036.t013

According to the results presented in Table 13, various strategies can be customized to suit investors with diverse risk appetites. For instance, if return is the only concern, portfolio number 9 stands out as the top choice, offering the highest return but also carrying a significant level of risk. However, a deeper analysis of the results reveals the following key insights:

• The beta index is a critical metric for evaluating a portfolio’s systematic risk, as it reflects the portfolio’s sensitivity to overall market fluctuations. A lower beta value is generally more favorable for risk-averse investors, indicating reduced exposure to market volatility. Among the proposed portfolios, portfolios 4 and 6 demonstrated the lowest beta ratios, making them particularly appealing from a risk-management perspective. However, there is a notable trade-off between risk and return. While portfolio 4 achieved the lowest beta index, it also yielded the lowest return, potentially limiting its appeal for investors focused on maximizing profitability. In contrast, portfolio 6 offered a more favorable balance, combining a low beta ratio with a comparatively higher return, thereby emerging as a stronger option for those seeking moderate risk alongside improved returns. Furthermore, during the optimization process, these portfolios were assigned a higher weight under objective number three, which prioritizes the desirability of future price trends. This allocation highlights the model’s emphasis on trend-following performance and reinforces the suitability of Portfolio 6 for trend-based investment strategies. These findings highlight the importance of carefully balancing risk and return according to investor preferences, as well as the significant influence of trend desirability in determining optimal portfolio configurations.
• The Sharpe ratio, a well-known indicator of risk-adjusted return, offers important insights into the performance of the proposed portfolios. Based on the calculated Sharpe ratios, Portfolio 3 exhibits the highest value, indicating that it delivers a superior return relative to its level of risk. This makes Portfolio 3 an attractive option for investors seeking to maximize returns while maintaining a favorable risk-reward balance. Conversely, Portfolio 4 displays the lowest Sharpe ratio, suggesting that its return is comparatively lower relative to the risk undertaken. While this portfolio may appeal to highly risk-averse investors prioritizing minimal exposure to volatility, its limited return potential may not suit those seeking significant gains. These findings underline the importance of tailoring portfolio selection to align with an investor’s specific risk tolerance and investment objectives. For risk-tolerant investors, portfolios with higher Sharpe ratios, such as Portfolios 2, 10 and 3, may be more appealing as they emphasize higher returns for a given level of risk. On the other hand, risk-averse investors might prioritize portfolios with lower risk, even if the risk-adjusted returns are less favorable. This analysis highlights the trade-offs inherent in portfolio selection and underscores the versatility of the proposed model in accommodating diverse investor profiles. By providing a range of portfolio options with varying risk-return characteristics, the model supports informed decision-making tailored to individual preferences and market conditions.
• The Treynor ratio is an important performance metric that evaluates a portfolio’s excess return in relation to its systematic risk, or market risk. It is calculated using the portfolio’s beta, which measures the portfolio’s sensitivity to overall market movements. A higher Treynor ratio suggests that the portfolio is generating greater returns for each unit of systematic risk, making it a crucial measure of risk-adjusted performance. In the proposed portfolios, Portfolio 4 stands out with the highest Treynor ratio, signaling that it delivers the most return relative to its systematic risk. However, it is crucial to emphasize that Portfolio 4 also has the lowest overall return, which suggests that while it is optimized for managing market risk, it may not provide the level of returns sought by more aggressive investors. This makes Portfolio 4 particularly suitable for risk-averse investors, who are focused on minimizing exposure to market fluctuations while choosing reduced returns in return for enhanced stability. Portfolio 6, on the other hand, demonstrates a relatively high Treynor ratio as well, indicating a favorable return in relation to systematic risk. In addition, Portfolio 6 offers a higher return compared to Portfolio 4, this makes it a more attractive choice for investors looking for a balance between risk and reward. This portfolio may be considered ideal for normal investors who are comfortable with some level of market risk but still prioritize a reasonable level of return. Finally, Portfolio 9 is also recommended for risk-taking investors due to its relatively high level of systematic risk and its relatively high return.
• The Sortino ratio is an essential measure for assessing portfolio performance, particularly when the primary concern is managing downside risk—the potential for negative returns that could erode capital. This metric is particularly useful for investors aiming to limit losses while maximizing returns, especially in volatile market environments where shielding against substantial drawdowns is critical. Among the proposed portfolios, Portfolios 3 and 7 exhibit higher Sortino ratios compared to the other options, signaling better control over downside risk. However, despite their relatively favorable risk profiles, these portfolios do not offer the highest returns, making them less appealing for investors prioritizing return potential. In contrast, Portfolio 9 stands out as a more balanced option. It achieves a relatively high Sortino ratio, suggesting effective downside protection, while also delivering more favorable returns than Portfolios 3 and 7. This combination of strong risk management and higher returns makes Portfolio 9 a more attractive choice for investors who aim to optimize both performance and downside risk.

The choice of an optimal portfolio for investors is shaped by several factors, with the final decision largely based on the individual preferences and objectives of the investor. One of the most critical factors in portfolio selection is the amount of risk an investor is willing to take, which can vary significantly across individuals. Risk tolerance is a personal characteristic that shapes how much volatility or potential loss an investor is prepared to endure in pursuit of higher returns. As a general principle, investors who accept higher levels of risk are often rewarded with the possibility of greater returns, although this comes with the potential for larger fluctuations in the portfolio’s value. Another important factor considered in this study is the appropriate trend forecast of future price movements, which can provide valuable insight into how a portfolio will perform under prevailing conditions. Investors who can identify favorable trends may be better positioned to capitalize on market dynamics. To support investors in making better-informed decisions, this study analyzed the suitability of various portfolios in relation to future price movements. The findings of this analysis are presented in Table 12, where the performance of various portfolios is evaluated with respect to the consideration of price movement trend forecasts. By considering risk tolerance, expected returns, and trend desirability, investors can more effectively align their portfolio choices with investment objectives and market expectations. This three-pronged approach allows for a more comprehensive assessment and helps investors strike a balance between optimizing potential returns and controlling risk exposure. Ultimately, the decision-making process becomes more tailored, ensuring that investors select portfolios that align not only with their personal risk preferences but also with the broader company’s outlook.

6.2. Comparison with other portfolio optimization model

In this paper, we tried to present a new model in which we used new neural network models and artificial intelligence tools. In this model, we tried to add a new criterion as the outlook of the possible future price movement to improve the model results. Our proposed model included two types of text and price data, which we calculated the price data using the LSTM model, and for text data, we collected the quarterly reports of DJIA companies from the Yahoo Finance website and extracted the companies’ outlook for the next quarter using the Cider artificial intelligence tool. To convert this text data into a score, we used the FinBERT model and performed sentiment analysis on the outlooks, and considering that the output of the FinBERT model was a vector (positive, neutral, negative), we used the AHP method to convert this vector into a score and finally extracted the outlook score, this process can be seen in Fig 5. In fact, we tried to present a State-of-the-Art model. The topic of sentiment analysis in most previous models has been that first, sentiment analysis was performed on Twitter data, websites, and social networks, and second, the results were used as one of the features for price prediction. While in our study, we have used data from companies’ quarterly reports with an innovative approach. Among the previous studies, the following can be mentioned:

Colasanto et al. [135] In one study, sentiment scores, particularly polarity values, were incorporated into stock price forecasting by introducing them as an additional “view” in the Black-Letterman framework. These scores were extracted from textual analysis of Financial Times articles and were associated with various events, both favorable and unfavorable, that affected specific stocks. The key distinction of the present study is in the application of the FinBERT model, which is used not only as a predictive feature but also to extract insights into future performance from corporate disclosures. Unlike previous approaches that used sentiment as a direct input for price forecasting, our method uses FinBERT to assess the outlook of companies, thereby informing investment decisions with a more interpretative and qualitative layer. Leow et al. [136] introduced two novel models SAW and SMPT which incorporate real-time market sentiment derived from Twitter data, analyzed using BERT. These models adjust portfolio allocation weights based on the extracted sentiment signals. To optimize their performance, genetic algorithms are employed with objectives such as maximizing cumulative returns and minimizing volatility. In contrast, the present study relies on sentiment extracted from corporate quarterly reports to assess future outlooks, offering a more fundamental and document-based approach to sentiment analysis. Day and Li [137] examined the influence of various financial information sources on investment decisions and explored how deep learning techniques can enhance the accuracy of financial news classification. Their empirical findings demonstrated that the type of financial source significantly affects investor behavior and decision-making outcomes, and that classification performance benefits from the application of deep learning models. However, unlike the current study, their research did not address portfolio optimization; instead, it focused solely on analyzing the effects of information quality and classification on investment behavior. Fatouros et al. [138] conducted a pioneering study exploring the capabilities of large language models specifically ChatGPT-3.5 in the context of financial sentiment analysis, with a focus on the FOREX market. Employing a goal-free prompting strategy, they evaluated various ChatGPT prompts on a substantial dataset of FOREX related news headlines. Model performance was assessed using standard evaluation metrics, including precision, recall, F1-score, and MAE for sentiment classification. Notably, the study does not propose a structured or model-driven investment framework; rather, it centers on assessing the viability of LLMs for sentiment detection. Seshakagari et al. [139] employed three advanced models FinBERT, GPT-4, and T5 for the task of financial sentiment classification. By evaluating these models using precision, recall, and F1 score, the study contributes to the field by offering comparative insights that guide the selection of the most suitable model depending on specific application needs in financial sentiment analysis. However, similar to other works in this domain, their study does not extend to portfolio construction or optimization, focusing solely on the classification task.

Generally, the proposed model distinguishes itself through two notable and innovative features. First, it integrates qualitative news data extracted from companies’ quarterly reports, allowing the model to capture sentiment, forward-looking statements, and performance insights that are not typically reflected in historical price data alone. This incorporation of textual and contextual information enhances the depth of the analysis and supports more informed investment decisions. Second, the model introduces a three-objective optimization framework, expanding beyond the traditional focus on return maximization and risk minimization. The third objective specifically evaluates the future outlook of companies, derived from financial reports and qualitative assessments. This added dimension allows the model to account for anticipated growth, market positioning, and management guidance factors that are increasingly relevant in dynamic and uncertain market environments. The effectiveness of this approach is demonstrated in Table 12, which presents the performance metrics of the optimized portfolios. Notably, the Sharpe ratios of portfolios that assign greater weight to the third objective the forward-looking outlook show superior risk-adjusted returns. This indicates that incorporating qualitative future expectations alongside quantitative financial data can lead to more robust and strategically balanced investment outcomes.

Limitations

While the selection of DJIA companies provides a stable and well-documented dataset for model development and validation, it also introduces a limitation regarding the generalizability of the results. The DJIA includes only 30 large-cap, U.S.-based blue-chip companies, which may not fully represent the broader equity market, including small-cap stocks, emerging industries, or firms operating in different regulatory or economic environments. As such, the findings and model performance observed in this study may not directly translate to other market segments or international contexts. Future research should consider applying and testing the model on more diverse datasets, encompassing a wider range of firms across different indices, sectors, and regions