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Finance, Accounting and Business Analysis 
Volume 7 Issue 1, 2025 

http://faba.bg/       
ISSN  2603-5324 

DOI: https://doi.org/10.37075/FABA.2025.1.07 

 

Portfolio Optimization Based on MPT-LSTM Neural Networks: 

A case study of Cryptocurrency Markets   
 

Habib Zouaoui1* , Meryem-Nadjat Naas2  
  
Department of Finance and Accounting, University of Relizane, Relizane, Algeria1 

Department of Management, University of Relizane, Relizane, Algeria2 

* Corresponding author 

 

Info Articles   Abstract 
 

History Article: 

Submitted  19 February 2025 

Revised   19 May 2025 

Accepted  22 May 2025 

 Purpose: This study aims to examines advanced portfolio management 

techniques using Long Short-Term Memory (LSTM) networks, the study was 
applied to investing in cryptocurrencies whose markets are characterized by 

high-frequency trading, and using behavioral finance models based on the 

concept of return-risk and deep learning based on the work of artificial neural 
networks (ANN) and long-term memory (LSTM) algorithms 

Design/Methodology/Approach: This study adopts quantitative approach. 

Moreover, A random portfolio consisting of 25 cryptocurrencies was selected 
based on the database of the website: https://finance.yahoo.com/crypto/ 

during the period 2021-2024 AD and programming the Python language. And 
an attempt to evaluate the performance of the models used in accurately 

predicting the optimal relative weights of the investment portfolio, which proved 
the relative effectiveness of deep learning models by estimating the values of the 

mean square error (MSE) at a level of 0.0218% to predict the optimal portfolio 

weights for 5 days based on training 80% and testing 20% of the study data. 

Findings: The second hypothesis of this study was accepted, which states the 

effectiveness of deep learning algorithms to predict the weights of optimal 
portfolios with a return estimated at 1.7239% and a risk of 1.1219% and a Sharpe 

index value estimated at 1.5365%, while the Markowitz return-risk model 

portfolio came with a return rate estimated at 31.15% and a risk of 39.05%. With 
no diversification of investment on all portfolio assets and a Sharpe index value 

of 0.7978%. 

Practical Implications: This study provides important insights that machine 

learning offers significant advantages in portfolio optimization, from improved 

forecasting of asset returns to dynamic rebalancing, better risk management, and 
automation. The ability to handle high-dimensional, non-linear, and non-

stationary data makes ML an ideal tool for optimizing portfolios in complex and 
fast-moving markets; especially in cryptocurrency markets. However, challenges 

like data quality, overfitting, and interpretability must be addressed to ensure 

effective deployment of ML in real-world portfolio. 

Originality/Value: This study provides an original and timely contribution to 

understanding the use of deep learning for portfolio optimization represents a 
significant advancement over traditional financial models by offering several 

original and valuable benefits. These include the ability to capture complex non-

linear relationships, dynamic rebalancing in response to real-time data, 
processing of unstructured data (like sentiment analysis), advanced risk 

management, and the integration of high-dimensional data. The combination of 
these capabilities enables more accurate, adaptive, and robust portfolio 

optimization, ultimately enhancing portfolio performance and reducing risk. 

Paper Type:  Research Paper 

 

Keywords:  

Markowitz Model,  

Deep Learning,  

Portfolio Optimization, 

Cryptocurrencies,  

LSTM Neural Networks  
 

 

JEL: C38, C4, C45, C5, 

C58, G1, G11. 

 

* Address Correspondence:   

E-mail:  habib.zouaoui@univ-relizane.dz1 

meryemnadjat.naas@univ-relizane.dz2 

 

  

http://faba.bg/
https://doi.org/10.37075/FABA.2025.1.07
mailto:meryemnadjat.naas@univ-relizane.dz
https://orcid.org/0000-0001-7694-2473
https://orcid.org/0009-0004-4018-1261


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

83 

 

INTRODUCTION 
 

Portfolio optimization is a cornerstone of modern finance, focusing on the strategic allocation of 

capital across various assets to achieve specific investment objectives, such as maximizing returns or 

minimizing risk. Traditional methods, such as the Markowitz mean-variance model, rely on statistical 

assumptions and linear relationships, which may not adequately capture the complexities of financial 

markets. Given that these markets often exhibit non-linear dynamics and intricate interdependencies, Deep 

Learning (DL) emerges as a compelling alternative for portfolio optimization. This literature review explores 

the application of DL techniques in optimizing portfolios, highlighting their capacity to model non-linear 

relationships and enhance predictive accuracy. We examine various DL architectures, including neural 

networks and reinforcement learning, and their efficacy in addressing the limitations of traditional models. 

The review also discusses practical implications, such as feature selection and data preprocessing, and 

identifies future research directions to further integrate DL methods into portfolio optimization frameworks. 

Ultimately, this review underscores the potential of deep learning to revolutionize portfolio management in 

an increasingly complex financial landscape (Zhang et al. 2025). 

    However, Artificial intelligence is one of the sciences that the world has begun to rely on in various 

areas of life due to the ability of this science to collect and analyze big data (Big Data) and make decisions 

and reach accurate results that exceed the ability of the human element. As for the field of financial markets, 

with the increasing complexities of financial globalization that have increased the conditions of future 

uncertainty and high risks, asymmetry of circulating information and problems of fear and panic among 

investors, and in light of the emergence of the Fourth Industrial Revolution, automation of financial services 

and high-frequency trading, more than 80% of daily global stock market trading has become trading done 

through artificial intelligence (AI) and algorithmic trading, as they are trading done without any human 

intervention. Artificial intelligence enables investors to trade by creating, examining and testing data and 

making investment decisions automatically through machine learning. Machine learning is programmed 

through algorithms and placing orders according to specific criteria such as average daily trading and 

comparing them with trading averages in past periods, price changes, offered quantities, market fluctuations 

as a whole, price changes in the derivatives market and the future outlook of the economy, taking into 

account, for example, news related to stimulus packages or any news affecting the market by decision 

makers, in order to study and analyze them and reach a final result by machine learning, which leads to the 

implementation of a specific investment order. Trading in this innovative method results in objectivity in the 

investment decision. Behavioral Finance is predominant in humans during trading, meaning that they are 

more likely to be affected behaviorally by the environment and economic changes, which leads to a change 

in the investment decision that was previously taken, and this change in behavior and investment decision 

may result in unexpected losses. One of the benefits of this innovative trading method is the speed of 

executing trading orders, as an investment opportunity is searched for, information about it is collected, and 

a huge amount of data is analyzed accurately, and the appropriate investment decision is made through 

machine learning. This entire process is completed in a matter of seconds. As the human being’s ability to 

search for a similar opportunity and implement the appropriate investment decision regarding it takes longer, 

which may lead to wasting the investment opportunity. 

Generally, our study will try to test the following hypotheses:  

H1: The DL models provides the forecasting of cryptocurrencies portfolio optimization with higher 

accuracy than MPT model.  

H2: The MPT model provides the forecasting of cryptocurrencies portfolio optimization with higher 

accuracy than DL models. 

 

LITERATURE REVIEW 

 

Table below is a summarized comparison of  results from recent studies (2020–2025) on portfolio 

optimization using deep learning models versus Harry Markowitz's Modern Portfolio Theory (MPT) model. 

These results highlight the key findings and performance metrics from the studies. 

However, MPT remains a cornerstone of  portfolio optimization due to its simplicity and effectiveness 

in diversification. However, its limitations in handling dynamic and complex markets have led to the rise of  

alternative approaches like Deep Learning (DL) models (e.g., LSTM, GRU), which offer superior 

adaptability and predictive power. Hybrid models combining MPT and DL are emerging as a promising 

direction for robust portfolio optimization 

 

 

 

 



Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

84 

 

Table 1. Reviewed Previous Studies 

Authors/Year StudyTitle Methodology Key Findings 

Heydarpour et 

al. (2024) 

Robust Portfolio 

Optimization using 

LSTM-based Stock 

And Cryptocurrency 

Price Prediction: An 

Application of 

Algorithmic 

Trading Strategies 

VLMA, FLMA, 

EMA, and SMA 

algorithms based on 

the LSTM's predicted 

price 

LSTM and RNN capture temporal 

dependencies, outperforming MPT 

in dynamic markets. 

LSTM/RNN adapt better to time-

series data; MPT assumes static 
correlations. 

Yu (2023) Mean-variance 

Portfolio 

Optimization by 

LSTM-based 

Predictions 

LSTM Model and 

based calculated the 

predicted returns on a 

rolling basis 

Hybrid model which combine the 

stock price forecasting with asset 

allocation can indeed bring excess 

returns 

Xu et al.(2022) LSTM-MPT Based 

Quantitative 

Portfolio Decision 

Model, 

Combining 

Markowitz mean-

variance model, 

Monte Carlo 

algorithm, and LSTM 

prediction price curve 

a comparative analysis with the four 

commonly used portfolio model 

strategies shows that the LSTM-

MPT decision model is valid and 

reliable for long-term investments 

Cui et al. 

(2023) 

Portfolio 

constructions in 

cryptocurrency 

market: A CVaR-

based deep 

reinforcement 

learning 

the CVaR risk 

measure and a deep 

reinforcement 

learning optimization 

unfolding that CVaR measure with 

deep learning outperforms the 

traditional portfolio construction 

technique 

Xu 

et al. (2025) 

Cryptocurrency 

Portfolio 

Optimisation Based 

on LSTM Time 

Series Forecasting 

combining Long 

Short-Term Memory 

(LSTM) time series 

forecasting with 

traditional portfolio 

optimization methods 

The results indicate that the LSTM-

enhanced portfolio optimization 

method yields higher returns and 

better risk management compared to 

traditional methods 

Zhang 

)2025(et al.  

Portfolio 

Optimization with 

Lstm-Based Return 

and Risk 

Information 

 

deep learning-based 

portfolio strategy 

with prediction-based 

return as well as 

prediction-based risk 

information 

LSTM and RNN capture temporal 

dependencies, outperforming MPT 

in dynamic markets 

Durall (2022) Asset Allocation: 

From Markowitz to 

Deep Reinforcement 

Learning 

 

MPT, and on ML 

approaches based on 

deep reinforcement 

learning 

DRL method has the potential to 

construct a promising 

investment strategy 

Sebastian et al. 

(2024) 

Deep Learning for 

Stock Price 

Prediction and 

Portfolio 

Optimization 

LSTM- MPT. Mean-

Variance 

Optimization 

The study hence concludes that 

combining forecasting theory with 

portfolio selection could 

improve portfolio returns 

Source: Authors’ analysis from literature review (2025). 

 

 Broadly, the integration of deep learning techniques into cryptocurrency portfolio optimization has 

garnered increasing attention due to the unique challenges posed by the highly volatile and non-linear nature 

https://ieeexplore.ieee.org/author/37089994028
https://www.researchgate.net/scientific-contributions/Zhihan-Xu-2310944930?_sg%5B0%5D=nSHxW2tzfs_NLqlGcz6XZaexLSzTZPHyK7yyKTUhHfGWykz-6XLi54nkLpasJN5Sk6WYSzM.cR3p79X3Re5v8zARcOqKKRfrX1_7uiuzoEkMLBwtmX5LVIQJhC_xdoLTc7pnGI-Yc9NlGkZn7xN5kQ_SawHqcQ&_sg%5B1%5D=-Qo9DNqhyd-CUdf6f7ri50_ZuzEC_IWjsNfl5Sent-1NSgnjxlZlfobVTbjWQjrRL4H9nng.i3p6F0LEbdAYWUWi5Lg2tLuf8W4DNr79JZ4gcFXdXumKJcTVcmUhj4-arT06vgjjZr3RgCuZPDdXcX4VjAHRlA&_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIiwicG9zaXRpb24iOiJwYWdlSGVhZGVyIn19
https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=5176908


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85 

 

of cryptocurrency markets. This literature review systematically examines recent advancements in applying 

deep learning methods for optimizing cryptocurrency portfolios. We analyze various architectures, including 

recurrent neural networks (RNNs), convolutional neural networks (CNNs), and reinforcement learning, 

assessing their effectiveness in predicting price movements and enhancing portfolio performance. The review 

highlights key findings on the ability of deep learning models to capture complex relationships and patterns 

within cryptocurrency price data, leading to improved risk-adjusted returns. Additionally, we discuss the 

implications of feature selection, data preprocessing, and model evaluation metrics critical for successful 

implementation. The review concludes by identifying gaps in the current literature and proposing directions 

for future research, particularly in the areas of model interpretability and the incorporation of 

macroeconomic factors into deep learning frameworks for cryptocurrency portfolio optimization. (Ashy et 

al. 2024). 

 

MATERIALS AND METHODS 

 

Modern Portfolio Theory (MPT) model 

  Modern Portfolio Theory (MPT) provides a rigorous, quantitative framework for portfolio 

construction that emphasizes diversification and the tradeoff  between risk and return. While it has become 

a cornerstone of  modern investment theory, its assumptions of  normal returns and constant correlations can 

limit its practical application in volatile or non-normal market conditions. However, MPT continues to serve 

as a benchmark, and modern variations (e.g., Black-Litterman model, Dynamic MPT) have been developed 

to address some of  its shortcomings (Zouaoui and Naas 2021). 

Furthermore, We should present the MPT mathematical model, moreover, we consider a more 

general case with n risky securities Notations: for i = 1,..., n,  

 

𝐖 = (𝑤1, … , 𝑤𝑛)  : is the vector of portfolio weights. 

𝐑 = (𝑅1, … , 𝑅𝑛)  :  is the vector of asset returns. 

�̅� = (�̅�1, … , �̅�𝑛)  : is the vector ofasset returns expectations. 

𝐞 = (1, … ,1)   : is the vector with all components equal to 1. 

𝐕 = [𝜎𝑖𝑗]
1≤𝑖,𝑗≤𝑛

 : is the (n × n) variance-covariance matrix of returns. The matrix V is supposed to be 

invertible. 

Denote by W' the vector deduced from transposition of the vector W. For each given expected return, 

we have to determine the minimal variance portfolio. 

Therefore, following the Markowitz approach to determine optimal weights of portfolio, we have to 

determine the set of portfolios which minimize the variance for given expected returns 𝔼[𝑅𝑃]. This leads to 

the following quadratic optimization problem (Prigent 2007): 

minww′⋁w,  
𝑤𝑖𝑡ℎ w′. R̅′   = 𝔼[𝑅𝑃] 

w′. e  = 1 

(1) 

The first constraint corresponds to the fixed expectation level. The second constraint is simply that w 

is a vector of weights. However, short selling is allowed and no other specific constraints are introduced.        

The expected return of any portfolio P with weights w is given by: 

𝔼[𝑅𝑃] = ∑ 𝑤𝑖𝔼[𝑅𝑖]

𝑛

𝑖=1

= w. R̅′ (2) 

The variance of the return of P is equal to: 

σ2(𝑅𝑃) = w′V w = ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗

𝑛

𝑗=1

𝑛

𝑖=1

= ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗 + ∑ 𝑤𝑖
2𝜎𝑖

2

𝑛

𝑖=1

𝑛

𝑗=𝑖+1

𝑛

𝑖=1

 (3) 

 

The previous relation shows the decomposition of the variance of the portfolio return into two 

components. This relation proves that the marginal contribution of a given asset to the risk of the whole 

portfolio is not reduced to its own risk (its variance), but also takes account of its potential correlations to 

other securities. This latter property induces the diversification effect.  



Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

86 

 

From relation below, the partial derivative with respect to any weight wi is deduced: 

∂σ2(𝑅𝑃)

∂𝑤𝑖

= 2 ∑ 𝑤𝑖𝜎𝑖𝑗

𝑛

𝑗=1

 (4) 

Denote by σiP the correlation coefficient between asset i and portfolio P. Then: 

∑ 𝑤𝑗𝜎𝑖𝑗

𝑛

𝑗=1

= ∑ 𝑤𝑗

𝑛

𝑗=1

𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗) = 𝐶𝑜𝑣 (𝑅𝑖, ∑ 𝑤𝑗

𝑛

𝑗=1

. 𝑅𝑗) = 𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗) = 𝜎𝑖𝑃 (5) 

and finally: 

∂σ2(𝑅𝑃)

∂𝑤𝑖

= 2𝜎𝑖𝑃. (6) 

 

Deep learning (DL) models 
This section is devoted to briefly describe the basic principle of four Non-linear machine learning 

models or deep learning models that will be used later for cryptocurrency forecasting namely RNN, LSTM 

(Zouaoui and Naas 2023). 

 
Recurrent neural networks (RNN) 

 Forecasting with recurrent neural networks (RNNs) is a common application in time series analysis, 

where the goal is to predict future values based on past observations. RNNs are particularly well-suited for 

sequential data due to their ability to capture temporal dependencies. Here's a general guide on how to use 

RNNs for forecasting (Ibri and Slimane 2022):  

a. Data Preparation: 

o Collect and preprocess the time series data (e.g., normalize or standardize the data). 

o Create input-output pairs by sliding a window over the sequence. For example: 

 Input: (xt,xt+1,…,xt+n−1) 

 Output: xt+n (the value to predict). 

b. Model Design: 

o Choose the RNN architecture (e.g., LSTM, GRU). 

o Define the number of layers, hidden units, and activation functions. 

o Add a dense layer at the end to produce the final output (e.g., a single value for univariate 

forecasting or a vector for multivariate forecasting). 

c. Training: 

o Use a loss function like Mean Squared Error (MSE) or Mean Absolute Error (MAE) to 

measure the difference between predicted and actual values. 

o Optimize the model using backpropagation through time (BPTT) and an optimizer like Adam 

or SGD. 

d. Evaluation: 

o Evaluate the model on a test set using metrics like RMSE, MAE, or MAPE. 

o Visualize the predictions against the actual values to assess performance. 

e. Forecasting: 

o Use the trained model to predict future values by feeding it the most recent sequence of data. 

 
Longshort-term memory (LSTM) model  

LSTM (Long Short-Term Memory) is a specialized type of Recurrent Neural Network (RNN) that is 

capable of learning long-term dependencies. It was introduced by Hochreiter & Schmidhuber in 1997 to 

address the vanishing gradient problem encountered by traditional RNNs. LSTM achieves this by using 

gates that regulate the flow of information in the network (Brown et al. 2023). 

Moreover, the LSTMs are highly effective for tasks involving sequential data, such as time series 

forecasting, natural language processing (NLP), speech recognition, and more (Zeroual et al. 2020). 

Furthermore, Figure 1 shows a complete diagram of LSTM, similar to Figure1 with RNN. The LSTM has 

four components: input gates, forget gate, cell state, and output gate. 



Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

87 

 

 

Figure 1. Schematic diagram of LSTM model 

 

o Input Gate (rt) &Candidate Cell State(dt):The input gate decides which new information to store in 

the cell state: 

𝑟𝑡 = σ(𝑊𝑓 . [ℎ𝑡−1, 𝑥𝑡]) + 𝑏𝑓 (7) 

The candidate cell state represents the new information that could be added to the cell state. 

dt = tanh(𝑊𝑑 . [ℎ𝑡−1, 𝑥𝑡]) + 𝑏𝑑   (8) 

o Forget Gate: The forget gate decides how much of the previous memory should be discarded from 

the cell state: 

ft = σ(𝑊𝑖 . [ℎ𝑡−1, 𝑥𝑡]) + 𝑏𝑖 (9) 

When: σ = Sigmoid function that outputs values between 0 and 1 (0 means “forget” and 1 means 

“retain”). 

o Cell State Update: The new cell state is updated based on the forget gate and input gate decisions: 

𝐶t = ft. 𝐶𝑡−1 + 𝑟𝑡 . 𝑑𝑡 (10) 

The forget gate Ct scales the previous cell state Ct−1, and the input gate rt scales the candidate cell 

state dt. 

o Output gate: The output gate controls what part of the cell state to output as the next hidden state ht: 

𝑜t = σ(𝑊0. [ℎ𝑡−1, 𝑥𝑡]) + 𝑏0 (11) 

ℎt = 𝑜ttanh 𝐶t (12) 

The output gate decides how much of the cell state should be passed to the next time step. 

 
Performance Metrics 

     Table below summarizing the key risk-adjusted performance metrics in portfolio optimization: Sharpe 

Ratio, Sortino Ratio, Variance (Var), and Conditional Value at Risk (CVaR). Moreover, this table highlights 

how these metrics are used in portfolio optimization to balance risk and return, such as purpose in 

optimization and interpretation of results. 

 

 



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Table 2. Key risk-adjusted performance metrics in portfolio optimization 

Metric Formula Purpose in Optimization  Interpretation 

Sharpe Ratio 
Sharpe Ratio =

𝔼[𝑅𝑃] − 𝑅𝑓

𝜎𝑃

 
Maximize risk-adjusted 

return relative to a risk-free 

rate 

 

Higher values indicate 

better risk-adjusted 

performance. 

Sortino Ratio 
Sortino Ratio =

𝔼[𝑅𝑃] − 𝑅𝑓

𝜎𝑑𝑜𝑤𝑛

 
Maximize risk-adjusted 

return, focusing only on 

downside risk. 

Higher values indicate 

better performance 

with less downside 

risk. 

Variance 

(Var) 

𝑉𝑎𝑟(𝑅𝑃)

= ∑ ∑ 𝑤𝑖𝑤𝑗

𝑛

𝑗=1

𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗)

n

𝑖=1

 

Minimize the dispersion of 

returns around the mean 

(reduce volatility). 

Lower variance 

indicates less risk and 

more stable returns. 

Conditional 

Value at Risk 

(CVaR) 
CVaR =

1

1 − c
∫ 𝑥𝑝(𝑥)𝑑𝑥

VaR

−∞

 

Minimize the average loss 

in the worst α% of cases 

(reduce tail risk). 

Lower CVaR 

indicates less exposure 

to extreme losses. 

Source: Prigent (2007). 

 

RESULTS AND ANALYSIS 

 

Data description  
This study constructs an optimal investment portfolio for high-frequency, high-risk cryptocurrency 

markets by conducting a comparative analysis between traditional Markowitz mean-variance optimization 

and advanced deep learning models. By evaluating their performance across key metrics—such as risk-

adjusted returns, volatility resilience, and scalability—we aim to identify the most effective strategy for 

algorithmic cryptocurrency trading. Our findings will provide actionable insights for quantitative investors, 
hedge funds, and automated trading systems operating in ultra-volatile digital asset environments. 

Therefore, The study was applied to real data of time series of daily prices of a financial portfolio 

consisting of twenty-five (25) cryptocurrencies with high market values and the most traded in the market, 

during the period between (08/15/2021-08/16/2024) at 1093 observations based on the database of the 

website: https://finance.yahoo.com/crypto/ and Python programming. However, the following figure 

shows the development of the returns of the portfolio assets during the study period. 

 

 

 
Source: based on python code github/https://github.com/dimasthoriq/DL-portfolio 

optimization/blob/main/experiment.ipynb/ Yahoo, https://finance.yahoo.com/markets/crypto/all 

Figure 2. The returns of cryptocurrency portfolio assets 

 

The cryptocurrency market witnessed many fluctuations during the study period, especially with the 

beginning of the Covid-19 crisis, which was a difficult and harsh year, until 2022, when cryptocurrencies 

https://www.amazon.com/Jean-Luc-Prigent/e/B001JOENCM/ref=dp_byline_cont_book_1
https://finance.yahoo.com/crypto/


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

89 

 

faced more than one dilemma that caused them to face the largest wave of losses since the Corona crisis. 

The losses are not only due to the return of central banks around the world to warn against this market, 

which has no controls yet. But with central banks moving to tighten monetary policy and raise US interest 

rates to high levels, 2022 witnessed a mass wave of investors fleeing risky asset markets, including "crypto" 

and stocks, to hold the US dollar. Due to the waves of mass exodus of investors, some platforms failed to 

return customers' dues, which is what happened with the "FTX" platform, which declared bankruptcy and 

is currently being investigated, and its CEO is scheduled to appear before an investigation committee in the 

US Congress. In terms of trading during 2022, the combined market value of cryptocurrencies fell by 614 

percent, losing about $1,340 billion after their total value fell from $2,182.5 billion at the beginning of the 

year’s trading to about $842.5 billion at the end. During 2023, the market value of cryptocurrencies witnessed 

mixed developments, as the total market value of cryptocurrencies increased from about $800 billion at the 

beginning of the year to nearly $1.1 trillion by the end of 2023, an increase of 37.5%. Bitcoin (BTC) 

maintained its position as the largest cryptocurrency in terms of market value, reaching $450 billion by the 

end of the year, followed by Ethereum (ETH), which came in second place with a market value of $300 

billion. Thus, the gap between Bitcoin and Ethereum narrowed during the year, as it was about $250 billion 

at the beginning of 2023 and decreased to $150 billion by the end of the year. Other cryptocurrencies such 

as Ripple (XRP) and Coin Lite (LTC) recorded growth in their market value, but at lower rates than Bitcoin 

and Ethereum. In general, the cryptocurrency market witnessed a remarkable growth in total market value 

during 2023, with Bitcoin and Ethereum dominating the sector. The analysis of these fluctuations is linked 

to several reasons, including the occurrence of many fraud operations through cryptocurrencies, which led 

to tarnishing their reputation. 

Moreover, the occurrence of more than one hacking operation on cryptocurrency platforms. The 

declaration of bankruptcy by some platforms, the most famous of which was the largest platform, FTX, 

which was declared bankrupt. 

Furthermore, the exit of cryptocurrency platforms and their cessation in some major countries due to 

tightening restrictions and their move to other less powerful markets. The rules and laws began to be 

tightened more in some countries, perhaps the most prominent of which is the United States of America, 

which was the opposite of the year 2022. Some political unrest and geopolitical factors around the world 

also boosted the rise in the price of Bitcoin in particular, which recorded record increases exceeding 100 

percent (but it has not reached peak levels yet), which analysts attributed to technical and economic effects, 

in addition to the impact of recent geopolitical tensions (the Russian-Ukrainian war - and the war on Gaza), 

so that its price exceeded forty thousand dollars. In 2024, cryptocurrency markets are witnessing a noticeable 

decline during today's trading, with the prices of many major digital currencies declining. Bitcoin, the largest 

cryptocurrency by market value, recorded a 0.83% decline to reach $64.8 thousand, after approaching the 

$70,000 barrier in the past weeks. Ethereum was not immune to these declines, as it fell by 0.80% to reach 

$3,514, and other currencies such as Tether, BNB, and Solana also declined by varying percentages, as these 

currencies fell by 0.04%, 0.49%, and 0.68% respectively, reflecting a general downward trend in the market 

until August. 

Broadly, the period from 2021 to 2024 has been marked by extreme volatility and varying returns in 

the cryptocurrency market. Understanding these dynamics is crucial for effective portfolio management and 

optimization strategies. Investors should continuously monitor market conditions and adjust their strategies 

accordingly. 

 

Application of the MPT model 

Before applying the Mean-variance model and finding the optimal portfolio weights as well as the 

return and risk of the investment portfolio, we calculated the model inputs for the cryptocurrency prices to 

be invested in, including calculating the returns and standard deviation and extracting the variance-

covariance matrix that gives us an idea of the effectiveness of diversification in maximizing the objective 

function as shown in the tables below: 

o Download study data using Jupyter Notebook: 

  Using yfinance in Jupyter Notebook, you can easily download financial data from Yahoo Finance 

for analysis. This approach is particularly useful for portfolio optimization studies, where you need historical 

price data, returns, and other financial metrics. By automating the data download process and organizing 

your data efficiently, you can streamline your analysis workflow: 

  



Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

90 

 

#Import Libraries 
import math 

importnumpy as np 

import pandas as pd 

importtensorflow as tf 

fromtensorflow import keras 

fromkeras.models import Sequential 

fromkeras.layers import Dense 

fromkeras.layers import LSTM 

importmatplotlib as mpl 

importmatplotlib.pyplot as plt 

importseaborn as sns 

importplotly.express as px 

importplotly.graph_objects as go 

import plotly.io as pio 

importrandom 

importos 

importyfinance as yf 

importdatetime 

data = yf.download(dji_stocks, start=start_date, end=end_date) 

stocks_df = data['Close'] 

[*********************100%%**********************]  25 of 25 completed 

Source: based on python code github/https://github.com/dimasthoriq/DL-portfolio-

optimization/blob/main/experiment.ipynb 

 

o Heatmap Correlation Matrix 

A heatmap of the correlation matrix (appendix 1) is a valuable tool for portfolio optimization. It helps 

investors identify diversification opportunities, manage risk, and make informed decisions about asset 

allocation. By combining this visualization with tools like Modern Portfolio Theory (MPT) or Deep 

Learning (DL), you can build robust and efficient portfolios (Figure 6). 

 

fig = px.imshow(returns_df.iloc[:,0:25].corr(), text_auto=True, aspect="auto",title='Correlation Heatmap') 

fig.write_image("correlation_Heatmap.png") 

fig.write_html("correlation_Heatmap.html") 

fig.show() 

 

The table below summarizes the statistical characteristics of the MPT framework allows for effective 

risk-return trade-offs, enabling investors to construct optimized portfolios tailored to their investment goals. 

Understanding these metrics is crucial for effective portfolio management and decision-making. 

 

 

Table 3. Characteristics of Statistics for MPT optimization 

STOCKS 
P-Weights 

(Cvar) 

P-Weights 

(Sortino) 

P-Weights 

(Variance) 

P-Weights 

(Sharpe) 

ADA-USD 0 0 0 0 

BNB-USD 0 0 0 0 

BTC-USD 0 0 0 0 

DAI-USD 0 0 0 0 

DOGE-USD 0 0 0 0 

ETH-USD 0 0 0 0 

LINK-USD 0 0 24.09 4.21 

LTC-USD 0 0 0 0 

USDT-USD 0 0 0 0 

XRP-USD 0 0 0 0 

XLM-USD 0 0.67 0.08 40.03 

BCH-USD 0 0 0 0 

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.researchgate.net/figure/Descriptive-statistics-for-cryptocurrencies-portfolios_tbl3_343861353&ved=2ahUKEwi93IeJtcGLAxXf2wIHHZfUHoUQFnoECBwQAQ&usg=AOvVaw20zz_uzNgMqFUkcawmj1zW


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

91 

 

STOCKS 
P-Weights 

(Cvar) 

P-Weights 

(Sortino) 

P-Weights 

(Variance) 

P-Weights 

(Sharpe) 

WETH-USD 0 0 0 0 

WBTC-USD 0 0 0 0 

AVAX-USD 0 0.2 0 16.39 

SHIB-USD 0 0 0 22.9 

DOT-USD 0 0 0 0 

LEO-USD 15.38 0.86 25 4.78 

SOL-USD 43.79 0 25.4 4.69 

USDC-USD 0.24 0.2 0 0 

STETH-USD 40.58 98.07 25.42 7.01 

AAVE-USD 0 0 0 0 

VUSDT-USD 0 0 0 0 

RARE11294-USD 0 0 0 0 

VBTC-USD 0 0 0 0 

Annualized Return 0.39 1.26 0.27 31.15 

Annualized 

Volatility 
0.47 0.63 0.96 39.05 

Skewness -16.88 -29.22 -245.2 65.63 

Kurtosis 2905.05 3066.74 20780.99 678.6 

Max Drawdown -0.34 -0.55 -1.19 -59.42 

Count Data   1093.00 1093.00 1093 1093 

Sharpe Ratio 0,8331 2.0032 0.2781 0.7978 

CVaR 1.16 2.29 1.84 95.38 

Sortino Ratio 100.42 257.00 29.22 122.77 

Variance 0.00 0 0.01 15.25 

Source: based on python code github/ https://github.com/dimasthoriq/DL-portfolio 

optimization/blob/main/experiment.ipynb  

 
This study investigates the optimal relative weights derived from the Markowitz Mean-Variance 

model, focusing on a comprehensive summary of statistical characteristics for each optimal portfolio. We 

implemented various strategies to enhance the objective function, utilizing performance evaluation 

indicators such as the Sharpe Ratio, Sortino Ratio, variance, and Conditional Value at Risk (CVaR). The 

analysis highlights the benefits of Markowitz diversification in managing risk through an examination of the 

correlation matrix between the returns of the portfolio's assets (Zaki 2021). Furthermore, we compare the 

performance of these traditional optimization approaches with those derived from deep learning algorithms 

(DL). By assessing the strengths and weaknesses of both methodologies, this research aims to provide 

insights into the effectiveness of deep learning in enhancing portfolio optimization, ultimately contributing 

to more robust investment strategies in an increasingly complex financial environment (Yifu et al. 2024). 

 

Application of the deep learning (DL) models 

In order to create this type of portfolio, which consists of assets with different risk levels, a 

comprehensive analysis was conducted. Unlike previous portfolios that were created based on historical data 

only, this portfolio will leverage the predictions of the LSTM model to predict the optimal combination of 

assets that will generate the highest returns over 5, 10, 15, and 30 days. The process followed to build this 

diversified portfolio involves a series of sequential steps. The figure below illustrates and summarizes the 

general approach taken to create the model (Li and Liu 2023). 
 

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.researchgate.net/figure/Descriptive-statistics-for-cryptocurrencies-portfolios_tbl3_343861353&ved=2ahUKEwi93IeJtcGLAxXf2wIHHZfUHoUQFnoECBwQAQ&usg=AOvVaw20zz_uzNgMqFUkcawmj1zW
https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.researchgate.net/figure/Descriptive-statistics-for-cryptocurrencies-portfolios_tbl3_343861353&ved=2ahUKEwi93IeJtcGLAxXf2wIHHZfUHoUQFnoECBwQAQ&usg=AOvVaw20zz_uzNgMqFUkcawmj1zW


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

92 

 

 
Figure 3. Portfolio Construction Process 

 

Therefore, splitting the data into training and testing or splitting it into training, validation, and testing 

is common in supervised machine learning projects (Espiga-Fernández et al. 2025) The training set is used 

to fit and train the model while the test set is used to evaluate the trained model to get a better idea of how 

well the model will perform on new data and how it will behave in a production environment. Therefore, 

the test data should be similar to what is expected to be seen in a production environment. Another common 

splitting technique is splitting the data into three datasets: training, validation, and testing. The validation 

dataset will be used to choose the best hyperparameters for each model (Ketkar et al. 2021): 

 

n = len(returns_df) 

# Split the data 
train_data = returns_df[:int(0.8*len(returns_df))] 

val_data = returns_df[int(0.8*len(returns_df)):int(0.2*len(returns_df))] 

test_data = returns_df[int(0.2*len(returns_df)):] 

train_data.shape, val_data.shape, test_data.shape 

((874, 25), (0, 25), (875, 25)) 

 

We proceeded to build an LSTM model algorithm. To build the RNN to accurately predict the returns 

of cryptocurrencies and from there build an optimal portfolio based on return and risk and evaluate its 

performance using the Sharpe ratio, some modules had to be imported from Keras. After that, an LSTM 

layer and other dropout layers were added. Regarding the LSTM layer, the dimensions of the output space 

were set to 50 units. 20% of the layer was selected to be dropped and a dense layer with a single unit output 

was added. Adam was chosen as the optimizer for the model clustering and the loss was set to be the mean 

square error (MSE). After that, the model was fit to 100 epochs, a batch size of 32, a learning rate of 0.001, 

and a neural network of 200 cells (Nafia et al.2023): 

 

 #Model 1 (5-day window) 

model1 = tf.keras.models.Sequential([ 

LSTM(64, return_sequences=True, input_shape=(5, stocks_df.shape[1])), 

LSTM(32, return_sequences=True), 

Dense(units=stocks_df.shape[1])]) 

history1 = compile_and_fit(model1, window1.train_ds, window1.val_ds) 

perf_v=}{ 

perf=   }{  

perf_v['5 days'] = model1.evaluate(window1.val_ds) 

perf['5 days'] = model1.evaluate(window1.test_ds, verbose=0) 

  



Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

93 

 

Table 4.The effectiveness of LSTM models for predicting optimal portfolio returns  
Validation Performance Test Performance Time 

Date Loss MAE Loss MAE Epochs/Time 

5 days 0,0016 0,0230 0,0012 0,0218 14/14--1s 29ms/step 

10 days 0,0016 0,0233 0,0011 0,0219 13/13--1s 42ms/step 

15 days 0,0017 0,0246 0,0012 0,0222 12/12--2s 110ms/step 

30 days 0,0018 0,0255 0,0012 0,0228 10/10--0s 4ms/step 

Source: based on python code github/ https://github.com/dimasthoriq/DL-portfolio-

optimization/blob/main/experiment.ipynb 

 

We noted from the outputs of the application of the LSTM model to predict the returns of the 

cryptocurrencies that make up the portfolio assets, (Junhuan et al. 2024) where we note the superiority of 

the LSTM model in predicting for 5 days to the last day in the series of subsequent returns with the lowest 

mean square error rate estimated at 0.0218 for the test (Test_loss) and 0.0230 for the verification (Val_loss). 

Thus, the algorithm of this model can be adopted to estimate the optimal portfolio and extract the ratios for 

financial investments in the cryptocurrency assets that make up the investment portfolio, and then compare 

them to the space of previous solutions for the Markowitz model and trying to compare them based on the 

Sharpe index to evaluate the quality of the performance of the extracted portfolio (Das et al. 2024). 

 

o The Efficient Frontier of cryptocurrency optimal portfolio 

# Plot efficient forntier and our portfolio 
fig = go.Figure() 

fig.add_trace(go.Scatter(x=portfolio_volatilities,y=portfolio_returns,mode='markers',marker=dict(size=5,c

olor='blue',opacity=0.5),name='Random Portfolios')) 

fig.add_trace(go.Scatter(x=[sigma],y=[portfolio_return],mode='markers',marker=dict(size=6,color='red',s

ymbol='x'),name='Calculated Portfolio')) 

fig.update_layout(title='EfficientFrontier',xaxis_title='Risk',yaxis_title='Return',showlegend=True,hoverm

ode='closest') 

fig.write_image("efficient_forntier.png") 

fig.write_html("efficient_forntier.html") 

fig.show() 
 

 
Source: based on python code github/ https://github.com/dimasthoriq/DL-portfolio-

optimization/blob/main/experiment.ipynb  

Figure 4. Efficient frontier 

 

RESULTS AND DISCUSSION 
 

Through the previous simulation of time series of cryptocurrency returns using one of the traditional 

behavioral finance models (Markowitz model) and deep learning models using the LSTM algorithm based 

on the Sharpe index in evaluating the quality of the solution (optimal portfolio performance), (Tamuly et al. 

2024).the outputs of the comparative study were summarized in the following table: 

 

https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

94 

 

Table 5. The performance comparison of MPT-LSTM Models 

STOCKS P-Weights(Sharpe) P-Weights(LSTM) 

ADA-USD 0 0.014606 

BNB-USD 0 -0.05918 

BTC-USD 0 -1.44428 

DAI-USD 0 -4.74505 

DOGE-USD 0 4.70012e-05 

ETH-USD 0 2.26891 

LINK-USD 4.21 0.0689542 

LTC-USD 0 0.0518116 

USDT-USD 0 0.57744 

XRP-USD 0 0.0210249 

XLM-USD 40.03 -0.0330404 

BCH-USD 0 -0.035509 

WETH-USD 0 -2.70253 

WBTC-USD 0 1.41132 

AVAX-USD 16.39 0.00331112 

SHIB-USD 22.9 0.00729518 

DOT-USD 0 -0.0759262 

LEO-USD 4.78 0.0481682 

SOL-USD 4.69 0.033132 

USDC-USD 0 -0.315186 

STETH-USD 7.01 0.443353 

AAVE-USD 0 0.0234812 

VUSDT-USD 0 5.36088 

RARE11294-USD 0 0.00904898 

VBTC-USD 0 0.0679007 

Annualized Return 31.15 0.017239 

Annualized Volatility 39.05 0.011219 

Count Data   1093 1093 

Sharpe Ratio 0.7978 1.5365 

Source: based on python code github/ https://github.com/dimasthoriq/DL-portfolio-

optimization/blob/main/experiment.ipynb 

 
Source: based on python code github/ https://github.com/dimasthoriq/DL-portfolio-

optimization/blob/main/experiment.ipynb  

Figure 5. Cumulative returns comparison of crypto10 vs LSTM 

 

https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb
https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

95 

 

Through the results shown above and according to the distribution of optimal weights for the 

investor's expected investments during the year 2024 AD for the next 5 days, we note the effectiveness of the 

deep learning model using the LSTM algorithm (Alzaman 2024) with an expected rate of return for the 

optimal portfolio estimated at: 1.7239%, the highest and lowest risk level estimated at 1.1219% compared to 

the outputs of the Markowitz model, which estimated the rate of return for the optimal portfolio at 31.15%, 

and a relatively higher risk rate estimated at 39.05%. The results confirm the validity of the second 

hypothesis, which states that deep learning algorithms can benefit better from the investment diversification 

method. As for the Markowitz portfolio weights, investment in most cryptocurrencies was not employed 

and only seven (7) were used, namely LINK-USD, XLM-USD, AVAX-USD, LEO-USD, SOL-USD, 

STETH-USD, due to the limited operation of the model-based on the correlation coefficient and generating 

a limited number of possible portfolios, which achieved an average Sharpe index that is not motivating for 

investment according to the selected portfolio, estimated at 0.7978%. In contrast, the deep learning model 

that trains a large number of predicted portfolios gave an excellent Sharpe index estimated at 1.5365% for 

the performance of the selected optimal portfolio. This means that this may be an ideal investment decision. 

While adding a new asset class to the portfolio increases risk, the sharply higher ratio indicates that it is a 

risk worth taking. The high risk of cryptocurrency price fluctuations can also be explained by the global 

conditions that the world has gone through, especially since the study period coincided with the 

repercussions of the Corona pandemic (Covid-19) and political factors (the US elections) and geopolitical 

factors (the Ukraine war, the war on Gaza), where cryptocurrency trading platforms experienced several 

collapses due to investors' fears for their assets. However, the results of the study remain relative, especially 

in the field of deep learning, which raises the issue of the transparency of big data that depends on its training 

in the field of making investment decisions in financial markets, with the possibility of changing the basic 

parameters in building the LSTM model to reduce errors and even the possibility of adding GRU and Bilstm 

algorithms in future studies to increase the power of deep learning in the investment process in high-risk 

markets (the cryptocurrency market) while increasing the number of observations to become big data that 

helps in training and testing (Xu et al. 2025). 

 

CONCLUSION 

 
This study has highlighted the transformative potential of deep learning techniques in the realm of 

cryptocurrency portfolio optimization. Traditional methods often fall short in capturing the inherent 

complexities and non-linear dynamics of cryptocurrency markets. Deep learning approaches, including 

neural networks and reinforcement learning, demonstrate a superior ability to model intricate relationships 

among assets, leading to enhanced predictive accuracy and improved risk-adjusted returns. 

The review also underscored the importance of data preprocessing, feature selection, and the 

evaluation of model performance metrics, which are critical for effective implementation. Despite the 

promising results, several gaps remain in the literature, particularly concerning the interpretability of deep 

learning models and their adaptability to changing market conditions. 

Therefore, future research should focus on integrating macroeconomic factors and exploring hybrid 

models (Das et al. 2024) that combine traditional financial theories with advanced machine learning 

techniques. Additionally, empirical studies assessing the long-term performance of deep learning-based 

portfolios could provide valuable insights. Ultimately, leveraging deep learning in cryptocurrency portfolio 

optimization represents a significant advancement in investment strategies, offering a more nuanced 

approach to navigating the complexities of modern financial markets 

However, the integration of Long Short-Term Memory (LSTM) neural networks with MPT for 

cryptocurrency portfolio optimization presents a compelling solution to these challenges. LSTM, a deep 

learning model designed for sequential data, can effectively capture the time-series dependencies inherent in 

cryptocurrency prices, which fluctuate based on factors like market sentiment, regulatory news, and 

macroeconomic trends. When combined with MPT, LSTM can address many of the limitations of 

traditional portfolio optimization methods by incorporating dynamic forecasting, non-linear relationships, 

and the ability to adapt to changing market conditions. 

Furthermore, based on current results, future research can anticipate substantial performance 

enhancements by further optimizing the parameters of deep learning models. Fine-tuning these parameters 

is expected to increase the accuracy of predictions and overall portfolio performance. Additionally, exploring 

hybrid approaches that combine deep learning with traditional financial models could yield new insights 

and strategies for effective portfolio management. 

Finally, as the cryptocurrency landscape continues to evolve, continued investment in developing 

robust, data-driven optimization techniques will be essential for navigating its inherent volatility and 

complexity. This ongoing exploration promises to advance both theoretical frameworks and practical 

applications in the field of finance. 

https://www.researchgate.net/scientific-contributions/Zhihan-Xu-2310944930?_sg%5B0%5D=nSHxW2tzfs_NLqlGcz6XZaexLSzTZPHyK7yyKTUhHfGWykz-6XLi54nkLpasJN5Sk6WYSzM.cR3p79X3Re5v8zARcOqKKRfrX1_7uiuzoEkMLBwtmX5LVIQJhC_xdoLTc7pnGI-Yc9NlGkZn7xN5kQ_SawHqcQ&_sg%5B1%5D=-Qo9DNqhyd-CUdf6f7ri50_ZuzEC_IWjsNfl5Sent-1NSgnjxlZlfobVTbjWQjrRL4H9nng.i3p6F0LEbdAYWUWi5Lg2tLuf8W4DNr79JZ4gcFXdXumKJcTVcmUhj4-arT06vgjjZr3RgCuZPDdXcX4VjAHRlA&_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIiwicG9zaXRpb24iOiJwYWdlSGVhZGVyIn19


Habib Zouaoui, Meryem-Nadjat Naas / Finance, Accounting and Business Analysis, Volume 7, Issue 1, 2025 

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APPENDIX 1 

Source: based on python code github/https://github.com/dimasthoriq/DL-portfolio-optimization/blob/main/experiment.ipynb 

Figure 6.Heatmap of the correlations matrix 

 
 


