A mathematical study of the excess growth rate

TL;DR

This work studies the excess growth rate $\Gamma(\boldsymbol{\pi}, \mathbf{R}) = \log\left(\sum_{i\in\mathrm{supp}(\boldsymbol{\pi})} \pi_i R_i\right) - \sum_{i\in \mathrm{supp}(\boldsymbol{\pi})} \pi_i \log R_i$ from information-theoretic perspectives, revealing deep connections to relative entropy, logarithmic divergences, Helmholtz free energy, Campbell's measure, Rényi divergences, and large deviations. It develops three axiomatic characterizations of $\Gamma$: via relative entropy, via Jensen's gap, and via logarithmic divergence (cross-entropy), demonstrating that $\Gamma$ is uniquely determined (up to a constant) by natural invariances and analytic properties. The paper further analyzes optimization of $\Gamma$, providing a closed-form solution for the deterministic maximization (two-point support on the extreme returns) and a variational framework that links to penalized and constrained dual problems; it also derives a first-order condition for maximizing the expected excess growth rate and contrasts it with the growth-optimal portfolio. Through these results, the authors establish new connections between information theory and quantitative finance, and discuss potential extensions to dynamic settings, transaction costs, and market-diversity measures. The work thus advances theoretical understanding of rebalancing premiums and offers tools for robust portfolio design grounded in information-theoretic principles.

Abstract

We study the excess growth rate -- a fundamental logarithmic functional arising in portfolio theory -- from the perspective of information theory. We show that the excess growth rate can be connected to the Rényi and cross entropies, the Helmholtz free energy, L. Campbell's measure of average code length and large deviations. Our main results consist of three axiomatic characterization theorems of the excess growth rate, in terms of (i) the relative entropy, (ii) the gap in Jensen's inequality, and (iii) the logarithmic divergence that generalizes the Bregman divergence. Furthermore, we study maximization of the excess growth rate and compare it with the growth optimal portfolio. Our results not only provide theoretical justifications of the significance of the excess growth rate, but also establish new connections between information theory and quantitative finance.

Figures (2)

• Figure 1: Excess growth rates of the largest $500$ stocks, over consecutive $20$-day periods and equally weighted, of the US stock market from $1962$ to $2024$. We show both the per period excess growth rate and its aggregate through time.
• Figure 2: A path $\mathbf{p}(t)$ (in black) on $\Delta_n^{\circ}$ and its perturbation $\mathbf{q}(t) = \mathbf{p}(t) \oplus \mathbf{h}$ (in grey).

Theorems & Definitions (91)

• Definition 1.1: Excess growth rate
• Remark 1.2
• Definition 1.3: Excess growth rate as a divergence
• Remark 1.4: Information theory and quantitative finance
• Proposition 2.1: Permutation invariance
• Proposition 2.2: Dependence on support
• Proposition 2.3: Numéraire invariance
• proof
• Proposition 2.4: Chain rule (first version)
• proof : Proof of Proposition \ref{['prop:chain.rule.1']}