[Paper Review] Entropy and Optimization of Portfolios

Krzysztof Urbanowicz|arXiv (Cornell University)|Aug 1, 2014

Complex Systems and Time Series Analysis9 references3 citations

TL;DR

This paper proposes integrating Shannon entropy of asset return time series into Markowitz portfolio optimization to enhance risk assessment and improve portfolio performance. By modifying the utility function to include entropic corrections, the method shifts focus from average returns to correlations and distributional uncertainty, resulting in higher realized profits and greater robustness, especially in 'cool' (low-temperature) markets with lower volatility.

ABSTRACT

We briefly review the approach to optimization of portfolios according to the theory of Markowitz and propose a further modification that can improve the outcome of the optimization process. The modification takes account of the entropic contribution from the time series used to compute the parameters in the Markowitz method.

Motivation & Objective

• To address the limitations of traditional Markowitz optimization, which relies solely on variance and covariance and may be biased by historical noise.
• To explore whether entropy of return time series provides additional, non-redundant information beyond variance for portfolio optimization.
• To develop a modified optimization framework that incorporates entropy as a natural extension of Markowitz theory, improving diversification and predictive accuracy.
• To empirically evaluate the impact of entropic corrections on portfolio performance using a 27-stock portfolio from the Warsaw Stock Exchange (2001–2007).
• To investigate the role of temperature (Lagrange multiplier for entropy) in controlling risk and return dynamics, identifying optimal operating regimes.

Proposed method

• The paper extends the Markowitz utility function by adding an entropic correction term based on the Shannon entropy of individual asset returns.
• The modified utility function is expressed as $ F = D_p - \lambda m_p + \gamma \sum p_i + \alpha \sum p_i \log p_i $, where the last term introduces entropy-based regularization.
• The Lagrange multiplier $ \alpha $ controls the temperature of the system, with negative $ \alpha $ values emphasizing correlation-based diversification over return maximization.
• The optimization is performed using historical return data to compute variances, correlations, and entropies, with entropy calculated from normalized return probabilities.
• The method is applied to a 27-stock portfolio on the Warsaw Stock Exchange from 2001 to 2007, comparing results with equal-weighted and standard Markowitz portfolios.
• The performance is evaluated using the quality ratio (QR), realized annual profit, and variance of returns across different temperature regimes.

Experimental results

Research questions

• RQ1 Does incorporating the Shannon entropy of return time series improve the robustness and performance of Markowitz portfolio optimization?
• RQ2 How does the inclusion of entropy alter the balance between risk and return in portfolio optimization, especially in comparison to variance-only models?
• RQ3 What is the optimal temperature (Lagrange multiplier for entropy) that maximizes portfolio predictability and minimizes volatility?
• RQ4 Can entropy-based optimization outperform standard Markowitz and equal-weighted portfolios in terms of realized profit and risk-adjusted returns?
• RQ5 How does the non-monotonic relationship between expected return and realized profit manifest under entropy-informed optimization?

Key findings

• The inclusion of entropy in the optimization process leads to a significant increase in realized portfolio profits compared to standard Markowitz and equal-weighted benchmarks.
• Portfolios optimized with entropic corrections show higher quality ratios (QR) in 'cool' markets (low temperature), indicating better predictability and lower risk.
• The realized annual profit is a non-monotonic function of expected return, peaking at approximately 60% expected return—well above the market portfolio’s 20% return—demonstrating the potential for higher performance under entropy-informed optimization.
• Negative values of the Lagrange multiplier $ \alpha $ shift the optimization focus from maximizing expected return to minimizing risk through correlation-based diversification.
• High-temperature regimes lead to increased volatility and abrupt drops in profit, indicating 'overheating' effects, while low-temperature regimes yield smoother, more stable performance.
• The entropy-informed optimization outperforms both the standard Markowitz model and equal-weighted portfolios, as shown by compound value trajectories in Figure 6, confirming the added value of entropic information.