1. Introduction

The food industry, an integral part of the global economy, has always attracted widespread investor interest. The food industry provides essential consumer goods and is characterized by a large market size and relatively stable demand, and research by Morelli suggests that the food industry is characterized by relatively stable cash flows and sustained basic demand compared to other industries [1]. Other studies have found that the share prices of essential consumer goods, such as food, fall relatively little during economic downturns, e.g., Han et al. found that stocks in the food sector fell less than the broader market during the 2008 financial crisis [2]. These characteristics have attracted a certain amount of value and growth investors. Its strong cyclicality makes it a good choice for constructing value portfolios with stable returns.

Compared to investing in individual stocks, optimizing a portfolio can effectively reduce unsystematic risk and achieve the best match between the overall risk and return of the portfolio [3]. To rationally allocate food industry stocks, accurately predicting individual stock price trends becomes very important. LSTM has strong capabilities in time series modeling and has shown great potential in stock price forecasting. For instance, Nelson et al. built stock price forecasting models based on LSTM networks that outperformed traditional ARIMA models on high-dimensional stock data [4].

This paper plans to apply the LSTM methodology, using historical data of listed food and beverage companies as the basis to establish stock price forecasting models, predict future price trends of individual stocks, and provide support for optimized portfolio allocation. The experiment will collect 10 years of historical stock price sequences of companies like Nestle, Pepsi and Coca-Cola, use LSTM networks for multi-step forecasting, evaluate different model parameter settings, and analyze the accuracy of prediction results. The ultimate goal is to construct stock price expectations, optimize food and beverage stock investment portfolios, and achieve low-risk asset allocation.

2. Methodologies

2.1. Data collection

This paper selected 5 stocks to construct the portfolio: AB InBev, Heinz, Nestle, Pepsi, and Coca-Cola, which are companies that have the top market value. The data I used is the stock trend from August 14, 2013 to August 14, 2023, a time span of 10 years. In total, there are 2673 pieces of information. The source of data is accessed from Yahoo Finance by a python package “yfinance”. The data set from “yfinance” package is already clean, the processing of data is limited to putting several sets of data together to form a dataframe. Usually, stock price data set includes the following features: Open, Volume, High, Low, Closing price, Adjusted closing price. However, only the Adjusted closing price will be used in the dataset, since the adjusted closing price takes into account factors such as stock dividends and splits. Since corporate events like dividends and stock splits can impact stock prices, the adjusted closing price offers a better depiction of the stock's real performance. Also, with the help of minmaxscaler, the data is normalized in a range between 0 and 1. This not only accelerates the method, but also increases its accuracy [5]. Standard deviation will also be included in the method. Since one goal of portfolio construction is to achieve a balance between return and risk, the standard deviation provides a measure of the volatility of each asset in the portfolio, helping investors control overall risk while pursuing higher returns. By combining assets with different standard deviations, you can find the optimal balance of return and risk.

2.2. Selections of features

Monte Carlo simulation is being used, which estimates the expected performance of a portfolio by stochastically simulating possible future market scenarios, as well as changes in the prices and performance of investment assets. Such simulations can help investors better understand the various scenarios that their investment strategies may face and the trade-offs between risk and return [6].

2.3. LSTM models

Long Short-Term Memory (LSTM) is a type of recurrent neural network that is appropriate for digesting and forecasting significant events in a time series with relatively large intervals and delays [7]. The gradient disappearance issue in the RNN structure is addressed using LSTM. The overview of the structure of LSTM is shown in Fig.1.

Figure 1: Overview of LSTM

The forget gate's role is to decide how much information to discard from the cell state C_t. It uses the Sigmoid activation function, taking the current input x_t and the previous hidden state h_t-1 as inputs, and outputs a value between 0 and 1. This output is denoted as f_t, representing the amount of information to forget.

\( {f_{t}}= σ({W_{\lbrace xf\rbrace }}\cdot {x_{t}}+ {W_{\lbrace hf\rbrace }}\cdot {h_{\lbrace t-1\rbrace }}+ {b_{f}}) \) (1)

Figure 2: Forget Gate

The input gate's job is to decide which new data should be included in the cell state C_t. With the current input x_t and the prior hidden state h_t-1 as inputs, it uses Sigmoid activation and gives back a value from 0 to 1. How much fresh information should be retained is indicated by the output i_t.

\( {i_{t}}= σ({W_{\lbrace xi\rbrace }}\cdot {x_{t}}+ {W_{\lbrace hi\rbrace }}\cdot {h_{\lbrace t-1\rbrace }}+ {b_{i}}) \) (2)

Figure 3: Input Gate

The cell state, which is utilized to store and spread long-term information, is an essential part of the LSTM. It is calculated by taking the input gate, forget gate, and the prior cell state C_t-1 into account.

\( {C_{t}}= {f_{t}}\cdot {C_{t-1}}+ {i_{t}}\cdot tanh({W_{xc}}\cdot {x_{t}}+ {W_{hc}}\cdot {h_{t-1}}+ {b_{c}}) \) (3)

Figure 4: Cell State

The output gate chooses which data from the cell state will be output as the hidden state h_t. o_t indicates how much data from the cell state should be output. The hyperbolic tangent activation function is simultaneously used to process the candidate cell state values.

\( {o_{t}}= σ({W_{\lbrace xo\rbrace }}\cdot {x_{t}}+ {W_{\lbrace ho\rbrace }}\cdot {h_{\lbrace t-1\rbrace }}+ {b_{o}}) \) (4)

\( {h_{t}}= {o_{t}}\cdot tanh({C_{t}}) \) (5)

Figure 5: Output Gate

2.4. Evaluation indicators

In order to concretize the credibility of this approach, Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) is decided to be utilized here.

\( MAE=\frac{1}{n}\sum _{i=1}^{n}|{y_{i}}-\hat{{y_{i}}}| \) (6)

Where n is the quantity of observations; y_i is Actual values (observed data); ŷ_i is the Predicted values.

\( RMSE=\sqrt[]{\frac{1}{n}\sum _{i=1}^{n}{({y_{i}}-\hat{{y_{i}}})^{2}}} \) (7)

Where n is the quantity of observations; y_i is Actual values (observed data); ŷ_i is the Predicted values.

3. Results and discussion

3.1. Portfolio construction

In the initial portfolio allocation, the proportion of weights of stocks are: NESN.SW 15%; PEP 10%; KO 20%; BUD 15%; KHC 40%. This paper reduced the weights of NESN.SW and PEP because these companies seem to have higher volatility than other stocks. KHC’s weight is the largest, since it has the second lowest standard deviation, and its mean is larger than KO. This allocation was aimed at reducing risk to a certain extent.

Table 1: Descriptive Statistics of Stock Prices

Based on the calculation of daily return and covariance, an annual return of 7% and volatility of 17% is obtained for this portfolio. To explore more possibilities, this paper carried out a Monte Carlo simulation with 25,000 iterations and obtained a scatter plot. Based on Fig.6, the portfolio with the highest Sharpe ratio had an annual return of 10.04% and volatility of 15.33%, which is the red star in Fig.6. The weights for the five companies in this portfolio were as follows: NESN.SW: 41.15%, PEP: 41.62%, KO: 15.60%; BUD 0.961%; KHC 0.668%. However, the portfolio with the lowest standard deviation also had a good annual return of 8.575% and volatility of 14.42%, which is the yellow star in Fig.6. In this portfolio, NESN.SW made up approximately 58.47%, KO: 17.30%, but PEP dropped to only 18.92%. BUD and KHC increased to 1.921% and 3.389%. The change in the weight of these stocks in the portfolio is presented in the Fig.7. These data proved that Pepsi has a high volatility, but the reason why Nestle increases is still unclear. Also, this paper has to consider whether it is necessary to add AB InBev and Heinz to the portfolio, since they make up such a small portion of the portfolio. The adjustment of their proportions in the portfolio based on performance is decided.

Figure 6: Portfolios with Monte Carlo simulation

Figure 7: Change in Portfolio weights

The removal of AB InBev and Heinz from the portfolio is being carried out and reran the Monte Carlo simulation. The resulting highest Sharpe ratio portfolio had an annual return of 10.19% and a volatility of 15.17%, which is the red star in Fig.8. The volatility is lower and the annual return is higher. This means we would have a higher return together with a lower risk, showing that it is better to remove BUD and KHC from the portfolio. The portfolio with the lowest standard deviation, which is the yellow star in Fig.8, had an annual return of 9.177% and a volatility of 14.40%, similar to the result of the highest Sharpe ratio portfolio. Therefore, the decision is made to remove AB InBev and Heinz from the portfolio.

Figure 8: Portfolios without AB InBev and Heinz

Since Nestle have high risk and return, this paper considered forecasting its general trend to decide whether to increase its proportion in the portfolio. This paper used an LSTM model for this purpose.

3.2. Model performance analysis

This paper imported Nestle's daily closing prices over the past ten years for training and built a prediction model using a Python script. The Adaptive Moment Estimation is being used as the optimization parameter, since it’s the most effective one theoretically [8], and Mean Squared Error as the loss function. This paper also added a dense layer to enhance the nonlinear analysis capability of the LSTM model.

This paper took 61 days of data as a validation set and then predicted Nestle's stock price for the next 15 days. For the training set x, each row represents the data from day 61 to the last day with 60 columns each day, and each column represents i days back from that day. For the training set y, each row represents the closing price of the day with only one column. The predictions and the actual adjusted close prices are obtained, as shown in Fig.9 and Fig.10. The difference between Actual price and predicted price are quite small. The MAE is equal to 1.09, and RMSE is equal to 0.89. Both metrics used to assess the performance of the predictive model show very low values, indicating that the model is plausible [9].

Figure 9: The overview of the prediction of price

Figure 10: The close-look of the prediction of price

The training and validation sets fit well, and the 30-day forecast results is obtained, as shown in Fig.11. The results show the possibility of upward price appreciation for Nestle's stock price, which means we need to increase its proportion in the portfolio. This explains the increase of the weight of Nestle stock despite its high volatility when constructing a portfolio with the lowest standard deviation.

Figure 11: Forecast results of Nestle’s price

3.3. Drawbacks and potential improvements

However, according to the observation of Nestle's stock price, the possibility of such a small fluctuation within 30 trading days is quite low. Therefore, this model may not be accurate enough. Some limitations of LSTM may cause this inaccuracy: for the LSTM model to create an accurate model, a lot of historical data is required. Also, other factors, like real-world policies may affect the price that may not be caught by data [10].

One way to improve this is to add error rectification method, such as including the past 3 days' average difference of stock price and use them to correct the result [11]. Discover technical indicators and external factors, such as emotional factors, that strongly affect the data used for forecasting to improve the accuracy of prediction.

4. Conclusion

This paper explores predictive modeling of food and beverage stocks using LSTM. For data, this paper collected 10 years of historical daily closing prices of 5 stocks including Nestle, Pepsi and Coca-Cola as training data. For methodology, this paper built a recurrent neural network model with LSTM and fully connected layers, using MSE as the loss function and Adam as the optimizer. The research process first evaluated weight allocation of different stocks in the portfolio via Monte Carlo simulation, then selected Nestle, a relatively high-risk stock, for individual stock price prediction. The results showed a slight upward trend for Nestle’s stock price, thus advising moderately increasing its portfolio allocation. However, prediction errors remained, inconsistent with actual volatility. The major conclusion is that while LSTM models can reasonably fit historical price time series and provide general judgments on future trends, prediction accuracy still has room for improvement, likely due to the incomplete consideration of influencing factors and short of data. Future research could enhance predictive accuracy by collecting more data and enriching model features, or introduce attention mechanisms to focus on more key factors. This preliminary study offers basic analysis of predictive modeling for food and beverage stocks, but further optimization of model design, expanded sample size and feature scope is needed to obtain more reliable risk control and investment decision support.