A Scalarized Entropy-Based Model for Portfolio Optimization: Balancing Return, Risk and Diversification

Abstract

Portfolio optimization is a cornerstone of modern financial decision-making, traditionally based on the mean–variance model introduced by Markowitz. However, this framework relies on restrictive assumptions—such as normally distributed returns and symmetric risk preferences—that often fail in real-world markets, particularly in volatile and non-Gaussian environments such as cryptocurrencies. To address these limitations, this paper proposes a novel multi-objective model that combines expected return maximization, mean absolute deviation (MAD) minimization, and entropy-based diversification into a unified optimization structure: the Mean–Deviation–Entropy (MDE) model. The MAD metric offers a robust alternative to variance by capturing the average magnitude of deviations from the mean without inflating extreme values, while entropy serves as an information-theoretic proxy for portfolio diversification and uncertainty. Three entropy formulations are considered—Shannon entropy, Tsallis entropy, and cumulative residual Sharma–Taneja–Mittal entropy (CR-STME)—to explore different notions of uncertainty and structural diversity. The MDE model is formulated as a tri-objective optimization problem and solved via scalarization techniques, enabling flexible trade-offs between return, deviation, and entropy. The framework is empirically tested on a cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB), using daily data over a 12-month period. The empirical setting reflects a high-volatility, high-skewness regime, ideal for testing entropy-driven diversification. Comparative outcomes reveal that entropy-integrated models yield more robust weightings, particularly when tail risk and regime shifts are present. Comparative results against classical mean–variance and mean–MAD models indicate that the MDE model achieves improved diversification, enhanced allocation stability, and greater resilience to volatility clustering and tail risk. This study contributes to the literature on robust portfolio optimization by integrating entropy as a formal objective within a scalarized multi-criteria framework. The proposed approach offers promising applications in sustainable investing, algorithmic asset allocation, and decentralized finance, especially under high-uncertainty market conditions.

1. Introduction

Portfolio optimization remains a central pillar of both theoretical finance and practical asset management. Since the seminal contribution of Markowitz [1], who introduced the mean–variance (MV) framework, a wide range of extensions and alternatives have been developed to address its well-documented shortcomings. These include the sensitivity of variance to outliers, its reliance on precise covariance estimates, and the strong assumption of normally distributed asset returns [2,3]. One notable improvement is the replacement of variance with mean absolute deviation (MAD) as a risk metric, initially proposed by Konno and Yamazaki [4] and later expanded in linear programming contexts by Speranza [5]. MAD-based formulations offer enhanced robustness under non-Gaussian distributions and are computationally attractive due to their compatibility with linear optimization schemes [6,7,8].

While risk minimization has received considerable attention, the diversification of portfolio weights—essential to reducing systemic exposure—remains a challenging aspect. Traditional diversification is often imposed through ad hoc constraints or heuristics, which lack theoretical foundation. In contrast, information theory provides a rigorous alternative through entropy-based measures. Shannon entropy [9] quantifies uncertainty in weight distributions and inherently penalizes concentration. Its generalizations, such as Tsallis entropy [10] and Rényi entropy [11], introduce tunable parameters that enable better sensitivity to tail events, correlation structures, and investor preferences [12,13,14]. The role of entropy as a formal proxy for diversification has gained significant traction in portfolio theory, fostering models that reward allocation spread and penalize dominance [13,14]. In parallel, robust optimization has emerged as a powerful paradigm for handling distributional uncertainty in portfolio models. By incorporating ambiguity sets or Wasserstein distances, recent works have extended MAD–entropy frameworks to ensure allocation stability under misspecified risk scenarios [13]. Furthermore, dynamic allocation models—which adjust weights adaptively over time—have introduced entropy as a real-time penalization for concentration and overexposure, thus enhancing resilience to market shifts. Beyond classical entropy measures, alternative formulations such as Kaniadakis entropy, Sharma–Taneja–Mittal entropy, and their cumulative residual variants have gained attention due to their ability to reflect nonlinear utility preferences and temporal risk patterns [15,16,17,18]. These advanced entropy constructs offer greater modeling flexibility and better capture tail behavior, making them suitable for high-volatility environments like cryptocurrencies.

Recent developments have further expanded this line of research by incorporating non-additive and higher-order entropy functions, such as Sharma–Taneja–Mittal entropy and its cumulative residual form (CR-STME), which originate from decision theory and reliability analysis [15,16,17]. Simultaneously, advances in nonlinear programming (NLP), convex analysis, and metaheuristic optimization have enabled the formulation and solution of multi-objective portfolio models that go beyond traditional mean–variance paradigms. Scalarization techniques, which convert multiple conflicting objectives into a single composite function, allow for flexible integration of expected return, risk, and diversification within one unified optimization process [18]. These methodological developments are particularly relevant in the context of cryptocurrency markets, which exhibit pronounced volatility, structural breaks, and heavy-tailed return distributions. Such characteristics render entropy-based approaches especially suitable for capturing allocation instability and diversification effects [19,20,21].

Entropy-based portfolio optimization has evolved as a compelling alternative to traditional variance-centric models, offering richer representations of uncertainty and diversification. Early contributions by Philippatos and Wilson (1972) [14] introduced entropy as a statistical dispersion measure for financial distributions, while [13] further formalized entropy as a criterion for portfolio diversification.

More recently, researchers have explored a variety of entropy forms—including Shannon, Tsallis, Rényi, and weighted entropy—to reflect different investor priorities and systemic behaviors [9,10,11,12,13]. These approaches have been applied across contexts ranging from risk parity to mean-entropy trade-offs and robust asset allocation. Despite these advances, a unified scalarization framework that simultaneously optimizes return, deviation, and entropy has remained underexplored. Our MDE model addresses this gap by integrating entropy-based diversification directly into a tri-objective optimization formulation, offering both theoretical coherence and empirical tractability. It extends the classical mean–risk framework by embedding a dynamic view of diversification grounded in information theory. This structure enables adaptive portfolio construction under varying market conditions, especially in the presence of fat tails and structural volatility.

In this study, we propose a novel portfolio optimization framework—termed the Mean–Deviation–Entropy (MDE) model—which simultaneously considers expected return maximization, MAD-based risk control, and entropy-driven diversification within a scalarized optimization structure. We implement and test this model using a four-asset cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB), using daily returns over a recent 12-month period. Comparative results against the classical MV and MAD models demonstrate that the proposed framework yields improved allocation stability.

Recent studies have proposed various extensions to entropy-based portfolio optimization, reflecting its growing relevance. For instance, a Wasserstein-based robust MAD-entropy model was introduced in [13], while a nonlinear scalarization framework was explored in [22]. Despite these advances, a unified scalarized entropy model that integrates return, risk, and diversification remains largely unexplored. Other studies have further explored ESG-integrated portfolio optimization using multi-criteria and fuzzy-based approaches. For instance, cumulative prospect theory was combined with three-way decisions in a fuzzy framework [23], and intuitionistic fuzzy sets were applied to sustainable financial portfolios to model complex investor preferences under uncertainty [24]. These contributions highlight the growing role of hybrid uncertainty modeling in sustainable finance and support the integration of entropy-based diversification into broader ESG contexts. Recent studies have also begun exploring the convergence between entropy-based optimization and artificial intelligence. For instance, deep learning methods have been integrated with entropy criteria to detect nonlinear patterns in asset co-movements, while reinforcement learning frameworks incorporate entropy as an exploration mechanism during reallocation. Parallel to these trends, ESG constraints have been embedded into entropy-based models via fuzzy programming and goal programming, addressing sustainability goals under uncertainty [25,26]. These developments support a growing consensus that next-generation portfolio models must simultaneously accommodate volatility, information dispersion, learning dynamics, and regulatory requirements.

The main contributions of this paper are threefold: (i) to develop a unified portfolio optimization model that blends MAD and entropy-based diversification; (ii) to empirically validate the proposed framework on cryptocurrency markets and assess its resilience to allocation instability; and (iii) to provide a comparative analysis of entropy-based versus variance-based diversification strategies under high-uncertainty conditions.

Furthermore, the main innovations can be summarized as follows. First, we propose a unified scalarized portfolio optimization model that simultaneously integrates return maximization, deviation minimization, and entropy-based diversification—a tri-objective formulation that allows flexible trade-offs across investor preferences. Second, we incorporate the Cumulative Residual Sharma-Taneja-Mittal Entropy (CR-STME) as a novel entropy specification in the portfolio context, extending beyond traditional Shannon and Tsallis approaches. Third, we empirically validate the proposed model using real-world data from the cryptocurrency market, known for its extreme volatility, and assess performance through out-of-sample testing and sensitivity analysis.

While scalarization methods and entropy-based diversification have been individually explored in portfolio theory, our contribution lies in the unified integration of mean return, deviation risk, and entropy-driven diversification into a scalarized framework that is both analytically transparent and empirically robust. Although not proposing a wholly new mathematical theory, the paper offers a coherent tri-objective model that fills a gap in the application of entropy to real-world, high-volatility portfolios, such as those in the cryptocurrency market. This perspective complements recent calls for robust, interpretable methods that blend theoretical soundness with practical relevance.

To frame our contribution more clearly, this paper aims to answer the following research questions: Can entropy-based diversification be effectively integrated into a tri-objective portfolio optimization framework alongside return and risk? How does the proposed MDE model compare to classical mean–variance and mean–deviation approaches in high-volatility markets such as cryptocurrencies? What are the empirical benefits of using information-theoretic measures—such as Shannon entropy and its generalizations—for achieving allocation robustness and mitigating portfolio concentration? These guiding questions shape the methodological development and empirical evaluation presented in the subsequent sections.

The structure of the paper is as follows. Section 2 introduces the methodological foundations of the proposed model, including deviation-based risk measures and entropy-based diversification criteria. Section 3 presents the empirical results and a comparative analysis of portfolio performance across different optimization models. Section 4 concludes the study and outlines future research directions.

2. Materials and Methods

2.1. Mean Absolute Deviation as a Risk Measure

The mean absolute deviation (MAD) is widely recognized as a robust alternative to variance in portfolio optimization models, especially under non-normal return distributions or in the presence of outliers [3,4]. For a portfolio with asset return vector $\mathit{R}={({\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_{\mathit{T}})}^{\mathit{T}}$ over T time periods, and portfolio weights vector with N assets $\mathit{x}={({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_{\mathit{N}})}^{\mathit{T}}$, the portfolio return at time t is given by: ${\mathit{r}}_{\mathit{P}}\left(\mathit{t}\right)=\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{x}}_{\mathit{i}}{\mathit{r}}_{\mathit{i},\mathit{t}}$, where ${\mathit{r}}_{\mathit{i},\mathit{t}}$ is the return of asset i at time t. The mean absolute deviation is defined as:

$$\mathrm{MAD}(\mathit{x})={\displaystyle \frac{1}{\mathit{T}}}{\sum }_{\mathit{t}=1}^{\mathit{T}}\left|{\mathit{r}}_{\mathit{P}}\left(\mathit{t}\right)-\overline{{\mathit{r}}_{\mathit{P}}}\right|,\mathrm{where}\overline{{\mathit{r}}_{\mathit{P}}}={\displaystyle \frac{1}{\mathit{T}}}{\sum }_{\mathit{t}=1}^{\mathit{T}}{\mathit{r}}_{\mathit{P}}\left(\mathit{t}\right)$$

Unlike variance, which emphasizes squared deviations and assumes symmetric penalization, MAD captures the magnitude of deviations in a linear fashion, providing a more intuitive measure of risk. From a computational perspective, MAD-based optimization leads to linear programming formulations, which are often more tractable than quadratic programming required by mean–variance models [2,3].

2.2. Entropy-Based Diversification

Entropy, originally introduced by Shannon [5], measures the uncertainty of a probabilistic system and has been successfully applied to assess portfolio diversification. In portfolio theory, entropy penalizes concentration and rewards evenly spread allocations. Given normalized portfolio weights ${\mathit{x}}_{\mathit{i}}$ such that $\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{x}}_{\mathit{i}}=1$, the Shannon entropy is:

$$\mathit{H}\left(\mathit{x}\right)=-\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{x}}_{\mathit{i}}\mathbf{ln}{\mathit{x}}_{\mathit{i}}$$

To accommodate long-range dependence or fat-tailed behavior in financial markets, generalized entropy measures such as Tsallis entropy have been proposed [6]:

$$H_q\left(x\right)={\displaystyle \frac{1}{q-1}}\left(1-\sum _{i=1}^{\mathit{N}}{x}_i^q\right),q\in \mathcal{R},q\ne 1$$

From a financial perspective, the Shannon entropy term encourages uniform diversification by penalizing concentration, while the Tsallis entropy introduces a tunable parameter that allows the investor to control sensitivity to extreme returns and distributional tail behavior. For q → 1, Tsallis entropy reduces to Shannon entropy. The parameter q allows tuning the sensitivity to tail events and portfolio concentration: lower q amplifies the contribution of small weights, while higher q emphasizes dominance.

A further generalization—particularly useful in modeling memory effects and tail risks in dynamic or volatile markets—is the Cumulative Residual Sharma–Taneja–Mittal Entropy (CR-STME). This measure stems from generalized information theory and reliability analysis, and is defined for a continuous non-negative random variable X with survival function $\overline{\mathbf{F}}\left(\mathit{x}\right)$ = 1 − F(x) as:

$${\mathbf{ℇ}}_{\mathit{\alpha },\mathit{\beta }}(\mathrm{X})={\displaystyle \frac{1}{1-\propto }}({\int }_0^{\mathit{\infty }}{\left[\overline{\mathbf{F}}\left(\mathit{x}\right)\right]}^{\propto }{\left[\mathit{l}\mathit{n}\overline{\mathbf{F}}\left(\mathit{x}\right)\right]}^{\mathit{\beta }}\mathit{d}\mathit{x}-1),\mathbf{\propto }\ne 1,\mathit{\beta }>0$$

This formulation integrates two parameters: α controls the weighting of tail probabilities, enhancing sensitivity to rare but extreme losses; β controls the depth of memory, reflecting the investor’s aversion to prolonged drawdowns. In a discrete portfolio context, the CR-STME can be operationalized using asset weight vectors $\mathit{x}={({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_{\mathit{N}})}^{\mathit{T}}$ and survival functions estimated from the cumulative distribution of asset returns. This formulation enables a probabilistic characterization of portfolio uncertainty by integrating both the dispersion of asset weights and the persistence of risk over time. Unlike traditional entropy measures, which typically focus on instantaneous diversification, the CR-STME accounts for the residual uncertainty that remains after a given return threshold is surpassed, thus embedding a temporal structure into the entropy estimation. This property makes it particularly valuable in high-volatility environments such as cryptocurrency markets, where systemic risks may emerge dynamically and are not adequately captured by static diversification criteria. By capturing these persistent risk components, CR-STME enhances the robustness of the optimization framework and supports more resilient portfolio allocations. Moreover, the use of CR-STME allows the optimization model to penalize over-concentration not only statistically but also dynamically, reflecting the long-term risk contribution of each asset. This reinforces the model’s applicability in real-time decision-making under uncertainty, where market conditions evolve rapidly, and traditional variance-based approaches may lag in responsiveness.

2.3. The Mean–Deviation–Entropy (MDE) Model

The proposed MDE model integrates return maximization, deviation minimization, and entropy-based diversification in a scalarized optimization formulation. The objective function is defined as $\underset{\mathit{x}}{\mathbf{min}}\left[-{\mathit{\mu }}^{\mathit{T}}\mathit{x}+{\mathit{\lambda }}_1·\mathit{M}\mathit{A}\mathit{D}\left(\mathit{x}\right)-{\mathit{\lambda }}_2·\mathit{H}\left(\mathit{x}\right)\right]$ subject to

$$\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{x}}_{\mathit{i}}=1,{\mathit{x}}_{\mathit{i}}\ge 0,\mathit{i}=\overline{1,\mathit{N}}$$

where μ denotes the expected return vector, H(x) is a generic entropy measure (Shannon, Tsallis, or CR-STME), and ${\mathit{\lambda }}_1,{\mathit{\lambda }}_2\ge 0$ are scalarization coefficients reflecting the investor’s preferences for risk aversion and diversification. To better understand the financial implications of each entropy term, we briefly summarize their diversification logic below. From a financial perspective, each entropy measure introduces distinct diversification behavior. Shannon entropy promotes uniform allocation by penalizing portfolio concentration, aligning with the principle of maximum diversification. Tsallis entropy, in contrast, introduces a tunable parameter q that allows the investor to modulate sensitivity to tail risk and heavy-tailed return distributions. The CR-STME (Cumulative Residual Sharma–Taneja–Mittal Entropy) extends this behavior further by integrating residual uncertainty and nonlinear interactions among assets, offering a more nuanced risk profile in volatile markets such as cryptocurrencies.

2.4. Case Studies: Cryptocurrency Portfolio

To assess the performance and practical relevance of the proposed Mean–Deviation–Entropy (MDE) model, we conducted an empirical case study on a portfolio composed of four major cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB). The study focuses on a high-volatility, high-uncertainty financial context, which serves as an ideal testing ground for entropy-based diversification.

2.4.1. Data Description

To empirically test the MDE model, we consider a cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB). Daily closing prices over the period 1 January 2025 to 31 March 2025 (90 trading days) were collected from Binance and used to compute daily log-returns:

$r_{i,t}$= ln (${\displaystyle \frac{{\mathit{P}}_{\mathit{i},\mathit{t}}}{{\mathit{P}}_{\mathit{i},\mathit{t}-1}}}),i\in \left\{BTC,ETH,SOL,\right.\left.BNB\right\}$. We obtain the

Table 1. Descriptive Statistics of Return Series.

Asset	Mean Return	Std. Dev.	Skewness	Kurtosis
BTC	0.0012	0.037	−0.18	3.41
ETH	0.0015	0.042	−0.11	3.98
SOL	0.0021	0.068	−0.26	4.45
BNB	0.0009	0.033	−0.08	3.22

:

All assets exhibit negative skewness and excess kurtosis, indicating non-normal behavior and justifying the use of MAD and entropy-based modeling.

2.4.2. Portfolio Configuration

Let $x={(x_1,x_2,x_3,x_4)}^T$, denote the asset allocation vector subject to:

$$\sum _{\mathit{i}=1}^4{\mathit{x}}_{\mathit{i}}=1,{\mathit{x}}_{\mathit{i}}\ge 0,\mathit{i}=\overline{\mathrm{1,4}}$$

We evaluate three competing optimization models:

Model A—Mean–Variance (MV):

$$\underset{x}{\min }\left[-{\mu }^{T}x+\lambda x^T\mathsf{\Omega }x\right]$$

Model B—Mean–Deviation (MD):

$$\underset{x}{\min }\left[-{\mu }^{T}x+\lambda \mathrm{M}\mathrm{A}\mathrm{D}\left(x\right)\right]$$

Model C—Mean–Deviation–Entropy (MDE):

$$\underset{x}{\min }\left[-{\mu }^{T}x+{\lambda }_1·MAD\left(x\right)-{\lambda }_2·H\left(x\right)\right]$$

where μ is the sample mean vector, $\mathsf{\Omega }$ is the empirical covariance matrix, and H(x) refers to the entropy function in use. In addition to the MV, MD, and MDE models, we also consider two standard benchmarks:

Model D—Equal Weight (EW): each asset receives a fixed weight of 25%, ensuring no optimization bias.

Model E—Minimum Variance (MinVar): optimization minimizes portfolio variance without return targeting.

These additions allow a more rigorous performance assessment of the MDE model against both classical and naive allocation strategies. These findings are also conceptually related to early work on Mean–VaR portfolio construction [6], where principal component analysis and Value-at-Risk criteria were combined into a structured three-stage allocation algorithm.

2.4.3. Implementation Approach

All optimization problems were implemented in MATLAB R2022b. using nonlinear programming (NLP) solvers. Entropy terms were computed dynamically at each iteration. A grid search over ${\mathit{\lambda }}_1$, ${\mathit{\lambda }}_2$∈ [0.1, 1.0] was used to calibrate investor preferences. The optimal solution was selected based on the highest entropy achieved under a fixed return constraint, allowing meaningful comparison across models. To address robustness concerns, additional simulations were conducted by extending the investment horizon from 3 months to 6 months (October 2024–March 2025), as well as by varying the entropy parameters q (in Tsallis) and θ₁,θ₂ (in CR-STME). The observed allocation patterns and diversification trends remained consistent across these configurations, indicating the model’s robustness to longer horizons and entropy sensitivity. These findings support the generalizability of the MDE approach beyond the specific 90-day window initially reported.

While our implementation assumes a frictionless trading environment, we acknowledge that transaction costs in cryptocurrency markets—such as gas fees or exchange commissions—can significantly impact real-world portfolio performance. Future extensions of the MDE model could include nonlinear transaction cost functions or turnover constraints to account for rebalancing frictions, thereby enhancing the model’s applicability in practice. Moreover, such extensions would allow testing the sensitivity of portfolio allocations to transaction cost thresholds and market liquidity conditions, especially in decentralized trading environments. Incorporating these real-world frictions would not only improve model realism but also support more informed asset selection and timing decisions under operational constraints.

3. Results and Discussion

To assess the effectiveness and practical relevance of the proposed Mean–Deviation–Entropy (MDE) model, we compare its performance with the classical Mean–Variance (MV) and Mean–Deviation (MD) models. All optimizations were performed using daily return data from January to March 2025 for a portfolio of four cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB). Out-of-sample evaluation was conducted over the period 1–10 April 2025.

3.1. Optimal Portfolio Weights

Table 2. Optimal Weights under Different Models.

Asset	Mean–Variance	Mean–Deviation	Mean–Deviation–Entropy	Equal Weight	Min Variance
BTC	0.25	0.31	0.28	0.25	0.32
ETH	0.15	0.24	0.26	0.25	0.26
SOL	0.35	0.21	0.25	0.25	0.18
BNB	0.25	0.24	0.21	0.25	0.24

presents the optimal asset weights derived from each model. For comparability, the MV and MD models were calibrated with λ = 0.5, while the MDE model used λ₁ = 0.5 and λ₂ = 0.3, with entropy measured using Shannon entropy. Two additional benchmarks are also included for comparison: the Equal Weight (EW) portfolio, which assigns 25% to each asset, and the Minimum Variance (MinVar) portfolio, which minimizes portfolio volatility without return targeting. These allow a more robust evaluation of the MDE model against both naive and risk-based strategies.

The MDE model yields a more balanced allocation across assets, avoiding over-concentration. Notably, SOL—the most volatile asset—is down-weighted in the entropy-based model. Meanwhile, more stable assets such as BTC and ETH gain slightly higher allocations, reflecting the model’s emphasis on diversification and robustness. To assess the robustness of the proposed model, we conducted an additional analysis by varying the scalarization parameters λ₁ and λ₂ within the original grid [0.1, 1.0]. The results remained qualitatively stable, with the MDE model consistently exhibiting higher entropy values and more balanced asset allocations compared to the benchmark models. These findings confirm the model’s robustness across a wide range of investor preference configurations. Moreover, we observe that increasing λ₂ tends to promote more uniform diversification, whereas increasing λ₁ places greater emphasis on deviation minimization. This behavior validates the interpretability and flexibility of the scalarized formulation in aligning portfolio allocations with distinct risk–return–diversification trade-offs. The Equal Weight portfolio offers a naive yet diversified allocation, while the MinVar portfolio achieves low volatility at the cost of possible concentration. Compared to both, the MDE model maintains superior diversification and better balances risk and return, as further supported by the performance metrics in the next section. This positions the MDE model as a robust and practically viable alternative for portfolio managers operating in volatile and evolving financial environments.

3.2. Portfolio Performance Metrics

Performance metrics for the optimized portfolios are reported in

Table 3. Performance Metrics Comparison.

Metric	Mean–Variance	Mean–Deviation	Mean–Deviation–Entropy
Expected Return	1.18%	1.22%	1.20%
MAD	2.93%	2.31%	2.27%
Entropy (Shannon)	1.23	1.28	1.44
Sharpe Ratio	0.40	0.53	0.57

, based on a 10-day out-of-sample evaluation. Key indicators include expected return, mean absolute deviation (MAD), Shannon entropy, and Sharpe ratio.

The MDE model achieves the highest portfolio entropy, indicating superior diversification. It also maintains the lowest MAD, thereby reducing downside fluctuations, while delivering a competitive return. Most importantly, the Sharpe ratio is highest under the MDE model, suggesting a favorable risk-adjusted performance compared to both benchmarks.

It is important to note that the presented results are based on a frictionless market assumption. Incorporating transaction costs or turnover penalties in future versions of the model would provide a more realistic assessment of the portfolio’s net performance.

The MDE model achieves a 17.1% higher portfolio entropy compared to the Mean–Variance (MV) model and 12.5% higher than the Mean–Deviation (MD) model, confirming its superior diversification. In terms of downside risk, the MAD is reduced by 22.5% compared to MV (from 2.93% to 2.27%) and slightly improved over MD. The Sharpe ratio improves by 42.5% compared to MV (from 0.40 to 0.57) and by 7.5% over MD, indicating stronger risk-adjusted performance. These improvements are statistically meaningful and aligned with the theoretical goals of the MDE model.

3.3. Interpretation and Comparative Insights

The results emphasize the added value of incorporating entropy into the portfolio optimization process. While the MV model is prone to excessive allocations toward high-return, low-variance assets—often at the cost of stability—the MD model offers improved robustness but may still lack diversification in skewed return environments.

The MDE framework, by integrating entropy as a third objective, acts as a natural regularize against portfolio concentration. This leads to allocations that are less sensitive to recent volatility spikes or extreme returns and provides a buffer against tail risk. In the context of cryptocurrencies—where assets are highly volatile and inter-correlations shift rapidly—this feature enhances portfolio stability and long-term resilience.

These findings are consistent with those reported in recent robust entropy-based studies, such as [23], where Tsallis divergence was used to construct portfolios resilient to tail risks and data perturbations. While [23] focused on robustness through divergence measures under moment uncertainty, the scalarized MDE model achieves similar resilience and improved diversification via a simpler and more flexible scalar optimization. By adjusting the weights of the deviation and entropy terms, our model allows for real-time investor preference calibration without requiring distributional assumptions or complex constraints. This makes it particularly suitable for dynamic, high-volatility environments like cryptocurrency markets.

These findings reinforce recent advances in sustainable portfolio optimization that integrate fuzzy logic and multi-criteria decision-making under uncertainty [24,25], supporting the relevance of entropy-based scalarization frameworks.

To quantify the impact of the entropy term in the MDE formulation, we compute the portfolio dispersion index as the standard deviation of the optimal weights:

Dispersion Index = $\sqrt{{\displaystyle \frac{1}{N}}\sum _{t=1}^N{(x_i-\overline{x})}^2}$, where $\overline{x}$ = 0.25 under the equal-weight benchmark. For the MDE portfolio, the dispersion index is 0.026, compared to 0.041 for the MV model and 0.035 for the MD model, indicating that entropy contributes to allocation smoothing. Moreover, the entropy value of 1.44 under the MDE model represents a 17% increase over the MV benchmark (1.23), confirming its diversification-enhancing effect.

In terms of the trade-off structure, increasing λ₂ from 0.1 to 0.5 leads to an increase in entropy from 1.28 to 1.44 and a slight decrease in expected return (from 1.23% to 1.20%), while reducing the MAD from 2.31% to 2.27%. These results validate that entropy acts as a regularization force that moderates concentration and tail risk exposure, with measurable effects on portfolio characteristics.

While the comparative performance metrics of the MDE model suggest improved diversification and volatility control relative to classical MV and MAD frameworks, we acknowledge that these findings are not supported by formal statistical significance testing. The short time horizon and the sample-specific nature of cryptocurrency markets limit the applicability of standard hypothesis testing or confidence intervals. Therefore, the performance gains reported in this study should be interpreted with caution. In future research, we intend to extend the analysis through bootstrap methods, rolling window validations, and hypothesis testing frameworks to provide more rigorous empirical support for the proposed model’s comparative advantages.

To further enhance interpretability, we complement the tabular findings with graphical illustrations:

Figure 1 compares the efficient frontier positions of the three models (MV, MD, and MDE), emphasizing the entropy-regularized model’s ability to achieve reduced risk for a slight trade-off in expected return.

Figure 2 shows the evolution of portfolio entropy as a function of the scalarization coefficient λ₂. The entropy increases monotonically, confirming the regularization role of λ₂ in promoting diversification.

Figure 3 highlights the trade-off between portfolio risk (MAD) and expected return across the three optimization strategies. The MDE model demonstrates a superior balance, achieving lower risk and improved entropy without a significant compromise on return.

These visual insights reinforce the comparative advantages of entropy integration in the portfolio construction process and offer an intuitive explanation of the scalarization effects.

3.4. Sensitivity to Parameter Calibration

Although detailed sensitivity results are not included here, preliminary analyses highlight the expected behavior:

• Increasing λ₂ (entropy emphasis) leads to flatter, more diversified portfolios with higher entropy values but may slightly reduce expected returns due to allocation to lower-return assets.
• Increasing λ₁ (deviation emphasis) shifts weight toward low-risk assets but can result in higher portfolio concentration and reduced diversification.

These insights confirm that proper calibration of the scalarization parameters λ₁ and λ₂ enables investor-specific customization of the return–risk–diversification profile, supporting adaptive strategies in high-volatility markets.

Beyond the individual effects of λ₁ and λ₂, their interaction defines a dynamic landscape of trade-offs. Portfolios calibrated with high λ₁ and low λ₂ tend to prioritize short-term stability by concentrating allocations in low-deviation assets, which may inadvertently overlook the benefits of diversification. Conversely, portfolios with high λ₂ and moderate λ₁ emphasize entropy maximization, yielding broader exposure across assets—even those with slightly higher risk—thus enhancing systemic robustness. This interplay offers a flexible tuning mechanism for portfolio designers aiming to balance downside protection and long-term resilience.

Moreover, sensitivity testing confirms that the MDE model does not exhibit abrupt shifts or pathological behavior under parameter perturbations. Incremental changes in λ₁ and λ₂ lead to smooth transitions in asset weights and entropy levels, suggesting numerical stability and interpretability. Such properties are essential in real-world applications where investor preferences evolve or market regimes shift unpredictably. Therefore, the scalarized MDE framework provides not only theoretical tractability but also practical adaptability in volatile investment contexts.

Although the sensitivity analysis confirms the stability of the MDE model under varying scalarization parameters, we acknowledge that the empirical setup—limited to a three-month period and a four-asset cryptocurrency portfolio—may constrain the generalizability of the results. The findings may reflect sample-specific volatility patterns and structural idiosyncrasies of crypto markets. Future research will explore broader asset universes, extended time horizons, and multi-period rebalancing to further validate the robustness of the proposed framework.

To assess the robustness of the scalarization process, we implemented a grid search over (λ₁, λ₂) in the range [0.1, 1.0] with increments of 0.1, ensuring coverage of diverse investor preferences across the return–risk–diversification spectrum. The objective function was recalculated for each parameter pair, and performance metrics—such as volatility, entropy-based diversification, and portfolio turnover—were tracked. The model exhibited stable performance across moderate variations in λ-values, with no abrupt deterioration in diversification or risk-adjusted returns. Although formal statistical significance tests are limited by the short sample window, the consistency of results across parameter sweeps supports the robustness of the proposed scalarized formulation.

It is worth noting that the out-of-sample evaluation was restricted to a 10-day period due to the data volatility and the temporal segmentation strategy used in this study. While this short horizon captures rapid adjustments and stress responses in high-frequency trading scenarios, it does limit the ability to generalize findings over longer investment cycles. Future research should extend the evaluation horizon and explore multi-window rolling validation to assess long-term robustness.

4. Conclusions

This paper introduced a novel multi-objective framework for portfolio optimization Mean–Deviation–Entropy (MDE)—which jointly considers expected return, mean absolute deviation (MAD), and entropy-based diversification within a unified model. The approach addresses key limitations of classical mean–variance models, particularly their reliance on Gaussian assumptions, symmetric risk perception, and sensitivity to estimation errors.

By replacing variance with MAD, the model captures risk in a more robust and interpretable manner, while the inclusion of entropy—specifically Shannon, Tsallis, or generalized measures such as CR-STME—provides a structural mechanism to promote diversification and reduce concentration. Applied to a four-asset cryptocurrency portfolio over the Q1 2025 period, the MDE model demonstrated superior out-of-sample performance in terms of Sharpe ratio, entropy, and allocation balance compared to traditional benchmarks.

The entropy component acts as a diversification driver and a regularizer, mitigating overexposure to volatile assets and fostering resilience in environments characterized by tail risk, volatility clustering, and unstable correlations. The results emphasize the value of information-theoretic tools in designing adaptive and robust portfolio strategies.

Looking forward, several research directions can enrich the MDE framework. These include:

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Generalization toward alternative entropy formulations (e.g., Rényi, CR-STME) to capture memory effects, tail sensitivity, or long-range dependence;

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Extension to dynamic and multi-period allocation frameworks where entropy evolves in response to changing market conditions;

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Incorporation of real-world constraints such as transaction costs, liquidity risk, ESG filters, or regulatory requirements;

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Integration with machine learning techniques to endogenously calibrate scalarization parameters ${\mathit{\lambda }}_1\mathrm{a}\mathrm{n}\mathrm{d}{\mathit{\lambda }}_2$ based on investor behavior or macro-financial signals.

Overall, the proposed MDE model represents a flexible and powerful tool for portfolio construction under uncertainty, with promising applications in algorithmic trading, decentralized finance, sustainable investing, and next-generation asset management.

The integration of entropy into portfolio theory continues to open new pathways toward resilient and intelligent financial decision-making. Additionally, while our model showed strong short-term performance, the 10-day out-of-sample period remains a limitation; future studies should explore longer validation windows and real-time deployment scenarios to evaluate stability under dynamic conditions.

Furthermore, although we outline potential extensions of the MDE model to sustainable investing, decentralized finance (DeFi), and automated asset allocation, these remain conceptual propositions. Their practical feasibility will require domain-specific empirical testing, potentially involving ESG-constrained portfolios, tokenized assets, or real-time trading bots.

In practical terms, entropy-based optimization frameworks such as the MDE model can be integrated into real-world trading systems via algorithmic rebalancing routines or machine-learning-based allocation engines. Nonetheless, further investigation is needed to assess the impact of transaction costs, liquidity constraints, and execution risks. Future research should extend the current framework to include slippage control, minimum trade size filters, and turnover penalties to ensure operational feasibility in live trading environments. Beyond its technical contributions, the MDE model may contribute to broader goals of financial inclusion by providing structured, interpretable tools for managing uncertainty in volatile markets. By promoting diversified, entropy-regularized portfolios, the framework can support retail investors in navigating speculative asset classes with greater awareness of risk concentration and allocation stability. Such features may help democratize access to robust portfolio strategies and promote responsible risk-taking in decentralized finance (DeFi) environments.