Maximizing Portfolio Diversification via Weighted Shannon Entropy: Application to the Cryptocurrency Market

Abstract

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.