Minimum Rényi Entropy Portfolios

• Minimum Rényi Entropy Portfolios are a robust optimization framework that minimizes the exponential Rényi entropy, integrating higher-order moments and tail risks.
• The methodology employs nonparametric m‑spacings estimation and numerical global optimization to effectively address non-Gaussian features in asset returns.
• Empirical results demonstrate that MRE portfolios achieve superior Sharpe ratios compared to minimum variance portfolios, despite modestly higher turnover.

Minimum Rényi Entropy Portfolios provide a robust framework for portfolio optimization under non-normal asset return distributions by minimizing the "amount of randomness" in portfolio returns, measured via Rényi entropy. Unlike classical approaches that often rely on variance or normality assumptions, this methodology incorporates higher-order moments and tail risk contributions through an information-theoretic lens. The core objective is to identify portfolio weights that minimize the exponential Rényi entropy of the portfolio return distribution, yielding improved risk-return profiles relative to minimum variance portfolios (Lassance et al., 2017).

1. Rényi Entropy in Portfolio Context

Rényi entropy is a generalization of Shannon entropy for continuous random variables, quantifying uncertainty by accounting for distributional characteristics beyond variance. For a continuous portfolio return random variable $P$ with density $f_P$, Rényi entropy of order $\alpha > 0,\,\alpha \neq 1$ is defined as:

$$H_\alpha(P) = \frac{1}{1-\alpha} \ln\left[\mathbb{E}(f_P(P)^{\alpha-1})\right] = \frac{1}{1-\alpha} \ln \int (f_P(x))^\alpha\,dx$$

As $\alpha \to 1$,

$$\lim_{\alpha \to 1} H_\alpha(P) = -\mathbb{E}[\ln f_P(P)] = H(P)$$

The exponential Rényi entropy is given by:

$$H_\alpha^{\exp}(P) = \exp(H_\alpha(P)) = \left(\int (f_P(x))^\alpha dx\right)^{1/(1-\alpha)}$$

The tuning parameter $\alpha$ modulates the notion of uncertainty: small $\alpha$ emphasizes tail behavior (fat tails), large $\alpha$ focuses near the mode. This flexibility allows practitioners to account for non-Gaussian features present in financial returns.

2. Minimum Rényi Entropy Portfolio Optimization

Given asset returns $X=(X_1,\dots,X_n)'$ and weights $w \in \mathcal{W}$ (e.g., $\sum_i w_i = 1$, $w_i \ge 0$), portfolio return is

$P = w'X$

The minimum Rényi entropy (MRE) portfolio is the solution to:

$$w^*_\alpha = \arg\min_{w \in \mathcal{W}} H_\alpha^{\exp}(w'X)$$

With a minimum expected return constraint $w'\mu \ge \mu_0$, the Lagrangian formulation is:

$$\mathcal{L}(w, \lambda, \gamma) = H_\alpha(w'X) - \lambda(\mathbf{1}'w - 1) - \gamma(w'\mu - \mu_0)$$

Numerical methods are required to solve the first-order conditions due to the non-convexity introduced by higher-order moments.

3. Sensitivity to Higher-Order Moments and Tail Behavior

Through a truncated Gram–Charlier expansion, Rényi entropy is shown to capture not only variance but also skewness and kurtosis:

$$H_\alpha(X) \approx H_\alpha\big[N(0,\sqrt{\mathrm{Var}(X)})\big] + k_1(\alpha)\,\mathrm{Kurt}(X) + k_2(\alpha)\,\mathrm{Skew}(X)^2 + k_3(\alpha)\,\mathrm{Kurt}(X)^2$$

With coefficients:

$$k_1(\alpha) = \frac{1-\alpha}{8\,\alpha}, \quad k_2(\alpha) = -\frac{3\alpha^2-6\alpha+5}{24\,\alpha^{3/2}}, \quad k_3(\alpha) = -\frac{3\alpha^4-12\alpha^3+42\alpha^2-60\alpha+35}{384\,\alpha^{5/2}}$$

For $\alpha < 1$, $k_1(\alpha)>0$, so higher kurtosis increases entropy—thus portfolios minimizing Rényi entropy penalize fat-tailed distributions, addressing extreme return risks beyond simple variance minimization.

4. Nonparametric Estimation via $m$‑Spacings

Practical computation employs a robust sample-spacings estimator:

$$\widehat{H}_\alpha^{\exp}(m, T) = \left[\frac{1}{T-m}\sum_{i=1}^{T-m} \left(\frac{T+1}{m}(P^{(i+m)}-P^{(i)})\right)^{1-\alpha}\right]^{1/(1-\alpha)}$$

where $\{P_t\}$ are sorted sample returns over $T$ periods, and $m$ is a tuning parameter. As $\alpha \to 1$, this reduces to the Learned–Miller & Fisher estimator for Shannon entropy. This estimator is robust to non-normality and is recalculated each re-optimization cycle.

5. Numerical Optimization and Regularization

The mapping $w \mapsto H_\alpha^{\exp}(w'X)$ is generally non-convex due to higher moments, necessitating global optimization algorithms such as Nelder–Mead-based scatter search. Constraints typically include:

• Full investment: $\sum_i w_i = 1$
• No short sales: $w_i \ge 0$
• Global variance-based regularization (Levy & Levy 2014): $$\sum_i(w_i - 1/n)^2\,\sigma_i/\bar{\sigma} \le \delta$$

This last constraint promotes weight shrinkage toward equal weighting in proportion to asset volatility, improving out-of-sample stability. In practice, objective gradients are approximated by finite differences due to estimator complexity.

6. Empirical Performance and Comparison

Empirical analysis on six diversified datasets, monthly rebalancing with a 120‑month window and variance-based constraint ($\delta=25\%$), yields the following out-of-sample averages:

Portfolio	Sharpe Ratio (SR)	Adjusted Sharpe (ASR)	Turnover
MRE ($\alpha=0.3$)	0.911	0.903	0.36
MRE ($\alpha=0.5$)	0.914	0.906	0.38
MRE ($\alpha=0.7$)	0.913	0.904	0.43
MRE ($\alpha=1$)	0.915	0.906	0.54
Minimum Variance (MV)	0.890	0.884	0.32

Across datasets, MRE portfolios outperform minimum-variance portfolios with respect to both Sharpe and Adjusted Sharpe ratios, at the cost of a modestly higher turnover. The best overall trade-off is observed around $\alpha \simeq 0.3$, suggesting enhanced risk-return efficiency through entropy-based optimization in the region where tail sensitivity is heightened.

7. Connections, Implications, and Significance

Minimum Rényi entropy portfolios generalize classical portfolio theory by incorporating non-Gaussian features inherent in asset returns distributions. The methodology addresses robust risk assessment for heavy-tailed and skewed data, offering flexibility through the tunable entropy parameter $\alpha$. The adoption of nonparametric spacings-based estimators and suitable regularization mitigates estimation risk and promotes stability. Empirical evidence supports the superiority of the MRE framework over conventional variance-based approaches, particularly in environments with pronounced higher-order moment effects (Lassance et al., 2017).

A plausible implication is that further research may extend these entropy-based frameworks to structured asset classes, alternative risk measures, or dynamic re-allocation under regime shifts where distributional assumptions are complex or evolving.