A mathematical study of the excess growth rate
A new study offers a detailed mathematical analysis of the excess growth rate, a measure of extra returns from portfolio diversification and rebalancing. The authors connect this rate to information theory, geometry, and statistical physics, introducing three theorems that uniquely define it. They show that maximizing excess growth rate can outperform standard benchmarks by capturing market volatility. The research uses US stock data and highlights links to concepts like relative entropy and Helmholtz free energy.
What it examines
This paper studies the excess growth rate, a key concept in portfolio theory, using ideas from information theory, geometry, and probability. The authors aim to show its connections to entropy, free energy, and large deviations, and provide new mathematical characterizations and optimization results for portfolio selection.
What it concludes
The study reveals deep links between finance and information theory, offering new ways to measure and optimize portfolio performance. Applications include smarter investment strategies, risk management, and understanding market diversity. Future research may extend these ideas to dynamic markets, transaction costs, and more complex financial models.
Evidence objects
A groundbreaking study redefines the 'excess growth rate' in portfolio theory, linking it to information theory, geometry, and statistical physics through concepts like relative entropy, Jensen's inequality, and logarithmic divergence.
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The authors introduce three axiomatic theorems uniquely characterizing the excess growth rate, and reinterpret it as a divergence, connecting it to Helmholtz free energy and large deviations, with new terminology and mathematical insights.
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Empirical analysis using US stock data shows that maximizing the excess growth rate systematically harvests market volatility, often outperforming benchmarks, though the study focuses on one-period models and leaves dynamic settings for future research.
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This paper rigorously analyzes the excess growth rate in portfolio theory, introducing three novel axiomatic characterizations and uncovering deep links to information theory, entropy, and statistical physics, including the Helmholtz free energy. Its originality lies in these fresh theoretical connections, offering compelling insights for quantitative finance, despite limited direct practical applications.
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Raw abstract and provenance
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… ▽ More 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. △ Less
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