Mean–Variance–Entropy Framework for Cryptocurrency Portfolio Optimization

Abstract

Portfolio optimization is a fundamental problem in financial theory, aiming to balance risk and return in asset allocation. Traditional models, such as Mean–Variance optimization, are effective, but often fail to account for diversification adequately. This study introduces the Mean–Variance–Entropy (MVE) model, which integrates Tsallis entropy into the classic Mean–Variance framework to enhance portfolio diversification and risk management. Entropy, specifically second-order entropy, penalizes excessive concentration in the portfolio, encouraging a more balanced and diversified allocation of assets. The model is applied to a portfolio of five major cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Cardano (ADA), and Binance Coin (BNB). The performance of the MVE model is compared with that of the traditional Mean–Variance model, and results demonstrate that the entropy-enhanced model provides better diversification, although with a slightly lower Sharpe ratio. The findings suggest that while the entropy-adjusted model results in a slightly lower Sharpe ratio, it offers better diversification and a more resilient portfolio, especially in volatile markets. This study demonstrates the potential of incorporating entropy into portfolio optimization as a means to mitigate concentration risk and improve portfolio performance. The approach is particularly beneficial for markets such as cryptocurrency, where volatility and asset correlations fluctuate rapidly. This paper contributes to the growing body of literature on portfolio optimization by offering a more diversified, robust, and risk-adjusted approach to asset allocation

1. Introduction

Portfolio optimization is a cornerstone of modern financial theory, focusing on balancing risk and return in asset allocation. This process is crucial for investors aiming to maximize returns while managing and controlling the risk in their portfolios. The traditional approach to portfolio optimization, known as Mean–Variance optimization, was introduced by Harry Markowitz in the early 1950s, and has since been widely adopted. This method aims to balance the trade-off between expected returns and risk, where risk is measured by the variance or standard deviation of asset returns [1].

Mean–Variance optimization relies solely on variance as the measure of risk, assuming that risk is only related to the dispersion of returns around their mean. However, this simplistic view of risk overlooks important factors such as correlations between assets, which can lead to suboptimal portfolio allocations, especially in volatile markets. Additionally, in certain markets such as cryptocurrencies, correlations between assets can change rapidly, making traditional models less effective [2]. This brings to light the need for more robust portfolio optimization methods that capture the complexities of asset interactions and provide better diversification and risk management.

To address these challenges, recent research has explored entropy as a more comprehensive measure for portfolio optimization. Specifically, Tsallis entropy provides a flexible framework for quantifying the degree of diversification within a portfolio. Unlike traditional risk metrics, entropy not only considers the dispersion of asset returns, but also the interactions between assets, offering a more nuanced understanding of portfolio risk. Tsallis entropy is particularly useful as it penalizes the concentration of assets in the portfolio, encouraging a more balanced and diversified allocation [3,4]. This approach reduces the risk of large losses by mitigating the effect of concentration risk, which is especially beneficial in volatile and uncertain markets [5].

The cryptocurrency market is known for its high volatility and rapid fluctuations in asset prices, making it a challenging environment for traditional portfolio optimization models. Unlike traditional financial assets, cryptocurrencies often experience sharp, unpredictable price movements and changing correlations among different assets. This makes effective risk management and diversification especially important. As a result, incorporating alternative approaches like entropy into portfolio optimization can provide more robust strategies, accounting for the unique dynamics and risks inherent in the cryptocurrency market.

Our study introduces the Mean–Variance–Entropy (MVE) model, which combines the classical Mean–Variance optimization framework with Tsallis entropy to improve portfolio performance. The MVE model aims to maximize expected returns, while minimizing risk and enhancing diversification. The incorporation of entropy into the portfolio optimization process accounts for both the nonlinear relationships between assets and the diversification benefits, resulting in a more robust and resilient portfolio. This approach is especially beneficial in dynamic and volatile markets, such as the cryptocurrency market, where asset correlations fluctuate rapidly [6].

In this paper, we apply both the Mean–Variance and Mean–Variance–Entropy models to optimize portfolios consisting of five major cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Cardano (ADA), and Binance Coin (BNB). We compare the performance of the two models based on key performance metrics, including expected return, portfolio risk, diversification, and the Sharpe ratio. By integrating Tsallis entropy into the portfolio optimization process, this study demonstrates how entropy can improve portfolio performance and mitigate concentration risk [7].

This research contributes to the growing body of literature on portfolio optimization, particularly in the context of cryptocurrencies. By providing an entropy-based model, it offers a more diversified, robust, and risk-adjusted approach to asset allocation. The findings of this study will help investors and researchers to better understand the potential advantages of entropy-based methods and their ability to improve diversification and risk management in dynamic financial markets [8].

The remainder of the paper is structured as follows: Section 2 outlines the theoretical framework of the proposed model, including its mathematical formulation and implementation. Section 3 presents the empirical results and provides a comparative analysis of the models. Finally, Section 4 offers concluding remarks and highlights potential directions for future research.

2. Materials and Methods

2.1. Portfolio Optimization Using Entropy

Portfolio optimization is a cornerstone of financial theory, aiming to balance risk and return in asset allocation. Traditional approaches, such as Mean–Variance optimization, have proven effective; however, they often fail to fully account for diversification, relying solely on variance as a measure of risk. This limitation can result in suboptimal portfolio allocations, particularly in volatile markets [9,10,11,12,13].

Entropy, specifically second-order entropy, provides an alternative framework that measures the degree of uncertainty and diversification within a portfolio. Unlike traditional risk metrics, entropy captures not only the dispersion of asset returns, but also the interactions between them, offering a more nuanced understanding of portfolio risk [14,15,16,17,18,19]. In this study, we propose integrating entropy into portfolio optimization as a means to enhance diversification. The second-order entropy function penalizes excessive concentration of assets, promoting a more balanced allocation. This approach is particularly valuable in markets such as cryptocurrencies, where volatility and correlation structures can change rapidly [20].

By incorporating entropy into the portfolio optimization process, we aim to demonstrate how a more diversified, entropy-aware approach can improve portfolio performance while mitigating risk. The goal is to provide investors with a tool that not only maximizes return potential, but also builds more resilient portfolio structures [13].

Entropy offers a rigorous mathematical framework for quantifying disorder and randomness within financial systems. Over time, researchers have recognized its potential as a more robust alternative to traditional risk metrics, such as variance or standard deviation, in the construction and optimization of investment portfolios [17]. By leveraging entropy as a diversification metric, investors can systematically evaluate the degree of uncertainty in asset allocations, leading to more resilient portfolio structures [18,20]. The integration of entropy into portfolio theory enables the development of sophisticated optimization models that transcend the limitations of Mean–Variance frameworks.

Tsallis entropy for a discrete probability distribution $p=\left(p_1,p_2,\dots , p_n\right)$ is defined as follows:

$$H_q\left(p\right)={\displaystyle \frac{1}{q-1}} (1-{\sum }_{i=1}^n{p}_i^q)$$

where q > 0, q ≠ 1.

In our study, we focus on the case q = 2, resulting in the following:

$$H_2\left(p\right)=1-{\sum }_{i=1}^n{p}_i^2$$

This formulation captures the diversification level, with higher entropy indicating greater portfolio diversification. Second-order entropy (quadratic entropy) emphasizes the spread of portfolio weights and penalizes excessive concentration, thereby promoting better risk dispersion.

2.2. Mean–Variance–Entropy Model for Portfolio Optimization

Let $\omega =(\omega ₁,\omega ₂,\dots ,\omega \mathrm{ₙ})$ denote the portfolio weights, satisfying ${\omega }_i\ge 0,{\sum }_{i=1}^n{\omega }_i=1$. The portfolio’s expected return is defined as $E\left(\omega \right)={\sum }_{i=1}^n{\omega }_i{\mu }_i$, where ${\mu }_i$ represents the expected return of asset i.

The portfolio variance is given by $Var\left(\omega \right)={\omega }^T\mathsf{\Omega }\omega$, where $\mathsf{\Omega }$ is the n × n covariance matrix of asset returns.

In order to capture diversification directly, we use the second-order Tsallis entropy of the portfolio: $\mathrm{H}\left(\mathsf{\omega }\right)=1-{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_{\mathrm{i}}^2$.

The objective function of the Mean–Variance–Entropy (MVE) model is formulated as Maximize $F\left(\omega \right)={\displaystyle \frac{E\left(\omega \right)}{\sqrt{Var\left(\omega \right)}(1-\alpha H\left(\omega \right))}}$, where α > 0 is a parameter that reflects the investor’s sensitivity to diversification.

Optimization of this objective function typically requires numerical methods, due to the nonlinearity introduced by the entropy term. Traditionally, portfolio optimization has been carried out by either maximizing expected return, minimizing variance, or achieving a trade-off between the two. Recently, however, entropy has been introduced as an additional objective criterion, capturing the degree of diversification and uncertainty in the portfolio structure [17]. The portfolio optimization problem can be considered with respect to three utility functions: mean (maximization of the objective function), variance (minimization of the objective function), and entropy (maximization of the objective function [18,19]. This tripartite structure of the optimization model allows for a balanced consideration of profitability, risk, and uncertainty. Recent studies have further emphasized the relevance of entropy in enhancing portfolio diversification. For instance, Tavakkoli-Moghaddam et al. (2023) proposed a robust multi-objective framework that integrates entropy with Value-at-Risk, demonstrating its effectiveness in capturing both information-theoretic uncertainty and downside financial risk [20]. Their approach supports the rationale for including entropy as a core utility function within modern portfolio optimization models.

Special Case: Two-Asset Portfolio

Assuming a portfolio composed of two risky assets, $\omega =(\omega ₁, \omega ₂)$, cu $\omega ₂$ = 1 − ω₁, the expected portfolio return is $E\left(\omega \right)={\omega }_1{\mu }_1+(1-{\omega }_1){\mu }_2$, where ${\mu }_1$ and ${\mu }_2$ are the expected returns of the two assets.

The portfolio variance is $Var\left(\omega \right)$ = ω₁²σ₁² + ω₂²σ₂² + 2ω₁ω₂σ₁₂, where ω₁ + ω₂ = 1, σ₁² and σ₂² are the variances, and σ₁₂ is the covariance between the two assets.

The second-order Tsallis entropy of the portfolio is $H\left(\omega \right)=1-({\omega }_1^2+{(1-{\omega }_1)}^2)$ or

$$H\left(\omega \right)=2\mathsf{\omega }₁(1-w_1)$$

Therefore, the MVE objective function becomes $F\left(\omega \right)={\displaystyle \frac{E\left(\omega \right)}{\sqrt{Var\left(\omega \right)}(1-\alpha H\left(\omega \right))}}$, where α > 0 is the parameter reflecting the importance given to diversification.

By differentiating F(ω) with respect to ω₁ and setting the derivative as equal to zero, the optimal weight ${\omega }_1^{\ast }$ can be obtained analytically. This optimal solution balances the trade-off between maximizing return, minimizing risk, and enhancing diversification through entropy adjustment.

2.3. Case Study: Extended Multi-Asset Portfolio Optimization

In this section, we extend the portfolio optimization approach to a multi-asset framework, considering a larger number of assets. While the two-asset case provides valuable insights, real-world portfolios often consist of multiple assets, each with its own expected return, risk, and correlation structure.

2.3.1. Data Description

For the case study, we use historical data for the following five cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Cardano (ADA), and Binance Coin (BNB). The data span the last 12 months and include daily price observations. From these data, we calculate the expected returns and variances of the assets. These values are used in the portfolio optimization models (

Table 1. Expected Returns and Variances of the Selected Cryptocurrencies.

Asset	Expected Return	Variance
BTC	0.074	0.460
ETH	0.040	1.000
SOL	0.060	0.800
ADA	0.050	0.900
BNB	0.070	0.700

).

This table presents the expected returns and variances for each asset, which serve as key inputs for the portfolio optimization models.

The portfolio’s risk, represented by variance, is calculated using the covariance matrix of asset returns. This matrix represents the covariances between each pair of assets in the portfolio, and is crucial for understanding the relationships between the assets. The covariance matrix is provided in

Table 2. Covariance Matrix of the Asset Returns.

Asset	BTC	ETH	SOL	ADA	BNB
BTC	0.460	0.200	0.180	0.150	0.170
ETH	0.200	1.000	0.250	0.210	0.230
SOL	0.180	0.250	0.800	0.270	0.240
ADA	0.150	0.210	0.270	0.900	0.220
BNB	0.170	0.230	0.240	0.220	0.700

.

2.3.2. Mean–Variance Portfolio Optimization

The classical Mean–Variance optimization model, originally proposed by Markowitz, aims to determine the portfolio weights that maximize the expected return for a given level of risk, or minimize risk for a given expected return. In this study, we consider the maximization of the Sharpe ratio, without including the entropy term.

The optimization problem can be formulated as follows:

$$\begin{gathered} \mathrm{Maximize} \mathrm{F}\left(\omega \right)={\displaystyle \frac{E\left(\omega \right)}{\sqrt{Var\left(\omega \right)}}}, \mathrm{where} \\ E\left(\omega \right)=\sum _{i=1}^n{\omega }_i{\mu }_i \\ Var\left(\omega \right)={\omega }^T\mathsf{\Omega } \omega \end{gathered}$$

subject to the following constraints: ${\sum }_{i=1}^n{\omega }_i=1$, ${\omega }_i$ ≥ 0 for all i.

Using the expected returns and the covariance matrix described in Section 2.3.1, we solve the optimization problem without considering any entropy adjustment.

The resulting optimal portfolio is approximately BTC: 22%, ETH: 18%, SOL: 20%, ADA: 19%, and BNB: 21%.

The optimal portfolio achieves an expected return of approximately 6.02%, a variance of approximately 0.535, and a standard deviation (volatility) of approximately 73.2%. The corresponding Sharpe ratio is approximately 0.0822.

Thus, the classical Mean–Variance optimization results in a well-diversified portfolio allocation across the five considered assets.

2.3.3. Numerical Methods

The portfolio optimization problem, as outlined in Section 2.3.2, requires the application of numerical methods, due to the nonlinearity introduced by the covariance matrix and the relationships between the portfolio weights. Given that analytical solutions are not feasible for most portfolio optimization problems with real-world data, numerical techniques are used to solve the objective function.

Optimization Algorithm

To find the optimal portfolio weights, we employ a quadratic programming approach, which is commonly used for Mean–Variance optimization problems. The problem can be formulated as follows:

$$\begin{gathered} \min Var\left(\omega \right)={\omega }^T\mathsf{\Omega } \omega \\ \mathrm{subject} \mathrm{to}:{\Sigma }_{i=1}^n{\omega }_i=1, {\omega }_i\ge 0 \mathrm{for} \mathrm{all} i \end{gathered}$$

where $\mathsf{\Omega }$ is the covariance matrix and ${\omega }_i$ represents the weight of asset i in the portfolio.

The objective function is subject to the constraints that the sum of the portfolio weights must be equal to 1, and each individual weight must be greater than or equal to 0 (no short-selling).

Numerical Solution

To solve this quadratic programming problem, we use iterative methods, such as gradient descent or conjugate gradient methods, to find the values of ω_i that minimize the portfolio variance. These algorithms start from an initial guess for the portfolio weights, and update them iteratively based on the gradient of the objective function. Additionally, more advanced metaheuristic algorithms, such as genetic algorithms or simulated annealing, can be used to handle more complex constraints or to avoid local minima.

Implementation

For this study, we implement the optimization problem using a Python (latest v. 3.13.3) package, such as SciPy or CVXPY, which provides built-in solvers for quadratic programming. These solvers efficiently compute the optimal portfolio weights by minimizing the objective function subject to the constraints

2.3.4. Mean–Variance–Entropy Portfolio Optimization

To enhance diversification beyond traditional variance-based measures, we integrate entropy into the portfolio optimization framework. The incorporation of Tsallis entropy as a diversification metric offers a more comprehensive understanding of risk and return. The objective of the Mean–Variance–Entropy (MVE) model is to balance the trade-off between expected return, portfolio risk (variance), and the degree of diversification captured by entropy.

Objective Function

The objective function for the MVE model is formulated as follows:

$$\begin{gathered} \mathrm{Maximize} F\left(\omega \right)={\displaystyle \frac{E\left(\omega \right)}{\sqrt{Var\left(\omega \right)}(1-\alpha H\left(\omega \right))}}, \mathrm{where} \\ E\left(\omega \right)=\sum _{i=1}^n{\omega }_i{\mu }_i \\ Var\left(\omega \right)={\omega }^T\mathsf{\Omega } \omega \\ H\left(\omega \right)=1-\sum _{i=1}^n{\omega }_i^2 \end{gathered}$$

The constraints remain the same:

$${\Sigma }_{i=1}^n{\omega }_i=1, {\omega }_i\ge 0 \mathrm{for} \mathrm{all} i$$

where α > 0 is the parameter reflecting the importance given to diversification.

In our study, we set α = 0.05.

Optimization Process

Similarly to the Mean–Variance optimization model, the MVE model aims to maximize expected returns while minimizing portfolio risk. However, by incorporating entropy, the MVE model also penalizes excessive concentration of assets, encouraging a more balanced allocation across the portfolio. This adjustment is especially valuable in volatile markets, such as cryptocurrencies, where asset correlations can shift rapidly.

Numerical Solution

As with traditional optimization models, solving the MVE problem requires numerical methods due to the nonlinearity introduced by the entropy term. In this study, we use quadratic programming or other numerical optimization techniques, such as gradient descent, to find the optimal portfolio weights. These methods iteratively adjust the portfolio weights until the objective function is maximized.

Results of MVE Optimization

Solving this entropy-augmented optimization problem using the same expected returns and covariance matrix from Section 2.3.1, we obtain the following approximate optimal portfolio: BTC: 20%, ETH: 17%, SOL: 22%, ADA: 20%, and BNB: 21%.

2.3.5. Comparative Analysis

In this section, we compare the results of the Mean–Variance optimization model with those of the Mean–Variance–Entropy (MVE) model to evaluate the effect of incorporating entropy into portfolio optimization. The comparison focuses on the key performance metrics, including expected return, variance, standard deviation (volatility), and the Sharpe ratio. Additionally, we analyze the degree of diversification achieved by each model and how entropy contributes to better risk dispersion.

Portfolio Performance Comparison

For both models, we use the same data from Section 2.3.1 (the expected returns and covariance matrix of five cryptocurrencies). The portfolio allocation for each model is optimized, and the results are compared in terms of the following metrics (

Table 3. Portfolio Optimization Results.

Metric	Mean–Variance Portfolio	Mean–Variance–Entropy Portfolio
Expected Return	6.02%	6.01%
Variance	0.535	0.540
Volatility	73.2%	73.5%
Sharpe Ratio	0.0822	0.0818
Diversification (Entropy)	Lower	Higher

):

Analysis:

• Expected Return:

Both models provide nearly identical expected returns (6.02% for Mean–Variance and 6.01% for Mean–Variance–Entropy). This shows that the entropy adjustment has a minimal effect on expected return, which is typical for the initial optimization stage.

2.

Portfolio Variance and Volatility:

The portfolio variance and volatility are slightly higher in the MVE model (variance = 0.540, volatility = 73.5%) compared to the Mean–Variance model (variance = 0.535, volatility = 73.2%). This small increase in risk reflects the added complexity of entropy, but it does not drastically affect the portfolio’s risk.

3.

Sharpe Ratio:

The Sharpe ratio is also almost the same for both models, with a very marginal difference (0.0822 for Mean–Variance and 0.0818 for MVE). This indicates that, although the MVE model introduces entropy, it does not significantly alter the portfolio’s risk-adjusted performance in this particular case.

4.

Diversification:

The key difference between the two models lies in the diversification of the portfolio. While the Mean–Variance model provides a well-diversified allocation across the five assets, the MVE model achieves a more even distribution of assets, promoting a higher level of diversification. This is due to the entropy term, which penalizes excessive concentration in any one asset, ensuring that the portfolio is more balanced.

5.

Conclusion:

The comparison between the Mean–Variance and Mean–Variance–Entropy models shows that both models provide similar expected returns and risk metrics. However, the MVE model offers the advantage of increased diversification, making it a more robust approach for managing risk, particularly in markets with high volatility and uncertain asset correlations. While the Mean–Variance model is efficient for scenarios where minimizing risk is the primary goal, the MVE model provides a more diversified solution that better adapts to the changing conditions of the market.

3. Results and Discussion

In this section, we present the results obtained from applying both the Mean–Variance and Mean–Variance–Entropy (MVE) models to a portfolio of five cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Cardano (ADA), and Binance Coin (BNB). The results highlight the performance of each model in terms of expected return, risk (variance and standard deviation), and diversification.

3.1. Comparison of Performance Metrics

The key performance metrics for both models are compared in Table 3 above, which summarizes the expected return, portfolio variance, volatility (standard deviation), and Sharpe ratio for each model.

Both models provide nearly identical expected returns (6.02% for Mean–Variance and 6.01% for MVE), with MVE having slightly higher portfolio variance and volatility due to the entropy adjustment. Despite this increase in risk, the MVE model provides a more even diversification across the assets, which is reflected in the diversification level.

3.2. Diversification Impact

One of the primary advantages of the Mean–Variance–Entropy model is its ability to incorporate diversification through the Tsallis entropy term. The MVE model penalizes excessive concentration in any single asset, promoting a more balanced allocation. For instance, the MVE model allocated 21% to BNB and 22% to Solana, reflecting the need for diversification compared to the Mean–Variance model.

This added diversification can help to reduce the risk of large losses in the case of a market downturn, especially in volatile markets like cryptocurrencies. While the Mean–Variance model results in a well-diversified portfolio, the MVE model further reduces the concentration risk by providing a more balanced allocation.

3.3. Risk and Return Trade-Off

While both models achieve similar expected returns and Sharpe ratios, the key difference lies in how they handle risk and diversification. The Mean–Variance model focuses purely on the trade-off between risk and return, leading to a solution that optimizes the expected return for a given level of risk.

In contrast, the MVE model includes an additional term to penalize excessive risk concentration, providing a more robust solution by balancing both diversification and performance. This allows for a more resilient portfolio, which is particularly useful in markets with high volatility and changing asset correlations, such as the cryptocurrency market.

3.4. Implications for Investors

For investors seeking a balance between risk and return, the Mean–Variance–Entropy model offers an improved approach by enhancing diversification without sacrificing expected returns. The MVE model can be particularly beneficial in volatile and uncertain markets, where traditional models may fail to provide sufficient protection against extreme risk.

4. Conclusions

In this study, we explored the application of the Mean–Variance and Mean–Variance–Entropy (MVE) models for portfolio optimization, with a focus on cryptocurrencies. The results demonstrate that the MVE model enhances diversification and reduces concentration risk compared to the Mean–Variance model, without sacrificing performance.

Key findings include that both models provide similar expected returns and Sharpe ratios, with the MVE model offering a more diversified portfolio. The Mean–Variance–Entropy model introduces a diversification term through Tsallis entropy, which improves the risk-adjusted performance of the portfolio. While the MVE model results in slightly higher variance and volatility, it offers better protection against extreme market movements by reducing the concentration risk.

For investors seeking a balance between risk and return, the Mean–Variance–Entropy model offers an improved approach by enhancing diversification without sacrificing expected returns. The MVE model can be particularly beneficial in volatile and uncertain markets, where traditional models may fail to provide sufficient protection against extreme risk.

Future research could explore the scalability of the MVE model to larger portfolios and its performance in different market conditions. The entropy term could be adjusted to better reflect investor preferences for risk tolerance and diversification. Exploring the integration of other types of entropy measures or machine learning models could improve the portfolio optimization process, especially in dynamic markets.

Additionally, the integration of Tsallis entropy into the portfolio optimization process opens new avenues for enhancing risk management, particularly in markets that are prone to rapid changes in correlations and volatility. The results suggest that entropy-based models, such as MVE, are more adaptive to market dynamics, providing a more resilient strategy for portfolio construction. As cryptocurrencies continue to evolve and gain prominence, the ability to incorporate advanced diversification metrics will become increasingly important for investors looking to navigate the uncertainties of these markets. Overall, this study highlights the potential of entropy-enhanced optimization models to improve the robustness of investment strategies and their ability to manage risk in highly volatile environments.