Silvia Dedu

Traditional mean–variance portfolio optimization proves inadequate for cryptocurrency markets, where extreme volatility, fat-tailed return distributions, and unstable correlation structures undermine the validity of variance as a comprehensive risk measure. To address these limitations, this paper proposes a unified entropy-based portfolio optimization framework grounded in the Maximum Entropy Principle (MaxEnt). Within this setting, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) are formally derived as particular specifications of a common constrained optimization problem solved via the method of Lagrange multipliers, ensuring analytical coherence and mathematical transparency. Moreover, the proposed MaxEnt formulation provides an information-theoretic interpretation of portfolio diversification as an inference problem under uncertainty, where optimal allocations correspond to the least informative distributions consistent with prescribed moment constraints. In this perspective, entropy acts as a structural regularizer that governs the geometry of diversification rather than as a direct proxy for risk. This interpretation strengthens the conceptual link between entropy, uncertainty quantification, and decision-making in complex financial systems, offering a robust and distribution-free alternative to classical variance-based portfolio optimization. The proposed framework is empirically illustrated using a portfolio composed of major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—based on weekly return data. The results reveal systematic differences in the diversification behavior induced by each entropy measure: Shannon entropy favors near-uniform allocations, Tsallis entropy imposes stronger penalties on concentration and enhances robustness to tail risk, while WSE enables the incorporation of asset-specific informational weights reflecting heterogeneous market characteristics. From a theoretical perspective, the paper contributes a coherent MaxEnt formulation that unifies several entropy measures within a single information-theoretic optimization framework, clarifying the role of entropy as a structural regularizer of diversification. From an applied standpoint, the results indicate that entropy-based criteria yield stable and interpretable allocations across turbulent market regimes, offering a flexible alternative to classical risk-based portfolio construction. The framework naturally extends to dynamic multi-period settings and alternative entropy formulations, providing a foundation for future research on robust portfolio optimization under uncertainty.

Traditional mean–variance portfolio optimization proves inadequate for cryptocurrency markets, where extreme volatility, fat-tailed return distributions, and unstable correlation structures undermine the validity of variance as a comprehensive risk measure. To address these limitations, this paper proposes a unified entropy-based portfolio optimization framework grounded in the Maximum Entropy Principle (MaxEnt). Within this setting, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) are formally derived as particular specifications of a common constrained optimization problem solved via the method of Lagrange multipliers, ensuring analytical coherence and mathematical transparency. Moreover, the proposed MaxEnt formulation provides an information-theoretic interpretation of portfolio diversification as an inference problem under uncertainty, where optimal allocations correspond to the least informative distributions consistent with prescribed moment constraints. In this perspective, entropy acts as a structural regularizer that governs the geometry of diversification rather than as a direct proxy for risk. This interpretation strengthens the conceptual link between entropy, uncertainty quantification, and decision-making in complex financial systems, offering a robust and distribution-free alternative to classical variance-based portfolio optimization. The proposed framework is empirically illustrated using a portfolio composed of major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—based on weekly return data. The results reveal systematic differences in the diversification behavior induced by each entropy measure: Shannon entropy favors near-uniform allocations, Tsallis entropy imposes stronger penalties on concentration and enhances robustness to tail risk, while WSE enables the incorporation of asset-specific informational weights reflecting heterogeneous market characteristics. From a theoretical perspective, the paper contributes a coherent MaxEnt formulation that unifies several entropy measures within a single information-theoretic optimization framework, clarifying the role of entropy as a structural regularizer of diversification. From an applied standpoint, the results indicate that entropy-based criteria yield stable and interpretable allocations across turbulent market regimes, offering a flexible alternative to classical risk-based portfolio construction. The framework naturally extends to dynamic multi-period settings and alternative entropy formulations, providing a foundation for future research on robust portfolio optimization under uncertainty.

This paper develops a robust portfolio optimization framework that integrates Weighted Shannon Entropy (WSE) into the classical mean–variance paradigm, offering a distribution-free approach to diversification suited for volatile and heavy-tailed markets. While traditional variance-based models are highly sensitive to estimation errors and instability in covariance structures—issues that are particularly acute in cryptocurrency markets—entropy provides a structural mechanism for mitigating concentration risk and enhancing resilience under uncertainty. By incorporating informational weights that reflect asset-specific characteristics such as volatility, market capitalization, and liquidity, the WSE model generalizes classical Shannon entropy and allows for more realistic, data-driven diversification profiles. Analytical solutions derived from the maximum entropy principle and Lagrange multipliers yield exponential-form portfolio weights that balance expected return, variance, and diversification. The empirical analysis examines two case studies: a four-asset cryptocurrency portfolio (BTC, ETH, SOL, and BNB) over January–March 2025, and an extended twelve-asset portfolio over April 2024–March 2025 with rolling rebalancing and proportional transaction costs. The results show that WSE portfolios achieve systematically higher entropy scores, more balanced allocations, and improved downside protection relative to both equal-weight and classical mean–variance portfolios. Risk-adjusted metrics confirm these improvements: WSE delivers higher Sharpe ratios and less negative Conditional Value-at-Risk (CVaR), together with reduced overexposure to highly volatile assets. Overall, the findings demonstrate that Weighted Shannon Entropy offers a transparent, flexible, and robust framework for portfolio construction in environments characterized by nonlinear dependencies, structural breaks, and parameter uncertainty. Beyond its empirical performance, the WSE model provides a theoretically grounded bridge between information theory and risk management, with strong potential for applications in algorithmic allocation, index construction, and regulatory settings where diversification and stability are essential. Moreover, the integration of informational weighting schemes highlights the capacity of WSE to incorporate both statistical properties and market microstructure signals, thereby enhancing its practical relevance for real-world investment decision-making.

Traditional mean–variance portfolio optimization is ill-suited to cryptocurrency markets, where extreme volatility, fat-tailed distributions, and unstable correlations undermine variance as a risk measure. To overcome these limitations, this paper develops a unified entropy-based framework for portfolio diversification grounded in the Maximum Entropy Principle (MaxEnt). Within this formulation, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) emerge as complementary specifications derived analytically via the method of Lagrange multipliers, ensuring mathematical tractability and interpretability. Empirical validation is conducted on a portfolio of four leading cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—using weekly return data from January to March 2025. Results reveal that Shannon entropy converges to near-uniform diversification, Tsallis entropy (q = 2) penalizes concentration more strongly and enhances robustness against tail risk, while WSE integrates asset-specific informational priorities, aligning allocations with investor preferences or market characteristics. Comparative analysis confirms that all three models yield allocations more resilient and structurally balanced than variance-driven portfolios, mitigating estimation risk and concentration effects. This study provides a coherent mathematical formulation of entropy-based portfolio optimization by embedding Shannon, Tsallis, and Weighted Shannon entropies within a common Maximum Entropy (MaxEnt) optimization framework. Beyond its immediate empirical scope, this work also opens several avenues for future research. First, entropy-based portfolio construction can be extended to dynamic multi-period settings with transaction costs and liquidity frictions, which are particularly relevant in cryptocurrency markets. Second, the framework may be generalized to incorporate alternative entropy measures such as Rényi or Kaniadakis entropy, enabling more refined sensitivity to tail risks and nonlinear dependencies. The proposed framework provides a flexible foundation for future extensions toward dynamic, multi-period portfolio optimization under uncertainty.

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.

Portfolio optimization is a cornerstone of modern financial decision-making, tradition-ally 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 max-imization, mean absolute deviation (MAD) minimization, and entropy-based diversifi-cation into a unified optimization structure: the Mean–Deviation–Entropy (MDE) model. The MAD metric offers a robust alternative to variance by capturing the average mag-nitude 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 en-tropy. 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 di-versification, 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 allo-cation, and decentralized finance, especially under high-uncertainty market conditions.

In this paper, we propose a novel optimization model for portfolio selection that integrates the classical mean–variance criterion with a second-order Tsallis entropy term. This approach enables a trade-off between expected return, risk, and diversification, extending Markowitz’s theory to account for non-Gaussian characteristics and heavy-tailed distributions that are typical in financial markets—especially in cryptocurrency assets. Unlike the first-order Tsallis entropy, the second-order version amplifies the effects of distributional structure and allows for more refined penalization of portfolio concentration. We derive the analytical solution for the optimal weights under this extended framework and demonstrate its performance through a case study using real data from selected cryptocurrencies. Efficient frontiers, portfolio weights, and entropy indicators are compared across models. This novel combination may improve portfolio selection under uncertainty, especially in the context of volatile assets such as cryptocurrencies, as the proposed model can provide a more robust and diversified portfolio structure compared to conventional theories.

Traditional portfolio optimization models, rooted in the mean–variance framework of Markowitz, rely heavily on variance as a risk measure. Although theoretically elegant, this approach becomes fragile in volatile and structurally unstable markets such as cryptocurrencies, where return distributions deviate significantly from normality, cor-relations are unstable, and concentration risk emerges. These limitations have motivated the search for alternative frameworks capable of capturing uncertainty in a more flexible and distribution-free manner. Entropy, originally introduced by Shannon as a measure of information, has gradually been recognized in the financial literature as a suitable proxy for diversification and systemic uncertainty. To address the shortcomings of variance-based models, this paper introduces the Weighted Shannon Entropy (WSE) model as a diversification-oriented alternative. By extending the classical Shannon entropy with asset-specific informational weights, the WSE framework provides additional flexibility for modeling heterogeneous asset char-acteristics, such as liquidity, informational value, or perceived reliability. Using the principle of maximum entropy and the method of Lagrange multipliers, we derive ex-ponential-form solutions for portfolio weights that naturally discourage concentration, ensure balanced allocations, and remain analytically tractable. The methodology is validated empirically on a portfolio of four leading cryptocurren-cies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—using market data from January to March 2025. The results demonstrate that the entropy-based optimization framework produces well-diversified portfolios, robust to volatility and structural instability, and provides a distribution-free alternative to the classical mean–variance model. Beyond its empirical performance, the WSE formulation highlights the conceptual advantage of entropy in integrating return, risk, and diversification into a single unified framework. The paper contributes both theoretically and practically: it strengthens the mathematical foundation of entropy-based portfolio selection, extends its applicability to digital asset markets, and illustrates how weighting schemes can enrich the classical Shannon measure. Future research may extend this approach to multi-period optimization, gen-eralized entropies such as Tsallis and Kaniadakis, or integration with machine learning models for dynamic portfolio management.

In this paper, we propose a novel optimization model for portfolio selection that integrates the classical mean–variance criterion with a second-order Tsallis entropy term. This approach enables a trade-off between expected return, risk, and diversification, extending Markowitz’s theory to account for non-Gaussian characteristics and heavy-tailed distributions that are typical in financial markets—especially in cryptocurrency assets. Unlike the first-order Tsallis entropy, the second-order version amplifies the effects of distributional structure and allows for more refined penalization of portfolio concentration. We derive the analytical solution for the optimal weights under this extended framework and demonstrate its performance through a case study using real data from selected cryptocurrencies. Efficient frontiers, portfolio weights, and entropy indicators are compared across models. This novel combination may improve portfolio selection under uncertainty, especially in the context of volatile assets such as cryptocurrencies, as the proposed model can provide a more robust and diversified portfolio structure compared to conventional theories.

Sustainable economic decision making increasingly requires robust methodologies capable of withstanding deep uncertainty, particularly in volatile financial and resource-constrained environments. This paper introduces a unified optimization framework based on nonlinear scalarizing functionals, designed to support resilient planning under structural ambiguity. By integrating performance objectives with risk boundaries, the proposed model generalizes classical robustness paradigms—such as strict and reliable robustness—into a single tractable and economically interpretable formulation. A key innovation lies in translating scenario-based uncertainty into a directional performance index, aligned with stakeholder-defined sustainability criteria and encoded via a preference vector k. This scalarization approach supports behaviorally consistent and computationally efficient decision-making even in the absence of complete probabilistic information. A case study in multi-scenario portfolio allocation demonstrates the model’s capacity to maintain return stability while respecting predefined risk tolerances. Computational benchmarks confirm the framework’s scalability to larger problem instances, validating its practical applicability. Beyond financial applications, the model also holds promise for sustainable policy design, infrastructure planning, and resource allocation under deep uncertainty. This work contributes to bridging the gap between abstract optimization theory and applied sustainability challenges, offering a robust and adaptive decision-support tool for real-world implementation.