A mathematical study of the excess growth rate

TL;DR

This paper explores the excess growth rate in portfolio theory through the lens of information theory, establishing new theoretical links and characterizations that deepen understanding of financial growth measures.

Contribution

It provides three axiomatic characterizations of the excess growth rate using information-theoretic measures and compares its maximization with the growth optimal portfolio.

Findings

Connected excess growth rate to Rényi and cross entropies

Established axiomatic characterizations using relative entropy and divergence measures

Compared maximization of excess growth rate with growth optimal portfolio

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 R\'{e}nyi 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.