To Gamble, Perchance to Grow

2026-06-17

AI Summary

This study investigates how transformations of returns affect investment conservatism and risk aversion in Kelly growth-optimal portfolios. Focusing on a setting with one risk-free and one risky asset, the authors integrate convex analysis, function transformation theory, and expected utility frameworks to establish, for the first time, necessary and sufficient conditions under which a return transformation yields a more conservative portfolio: the transformation function \( f \) must be strictly increasing and concave, and the ratio \( r/f(r) \) must be convex. This result extends Pratt’s measure of risk aversion to models featuring agents with rational inattention, thereby introducing a novel criterion for comparing risk aversion based on return transformations and providing a theoretical foundation for comparative statics under limited attention.

Abstract

I study transformations of returns in the growth-optimal (Kelly) portfolio problem. In the one-safe-one-risky-asset problem, a return transform f universally produces a more conservative portfolio if and only if f is concave and strictly increasing and r/f is convex. As a corollary, I characterize comparative risk aversion for a rationally-inattentive agent: a more risk-averse agent is one who is sufficiently more risk averse in the Pratt (1964) sense.

Problem

• growth-optimal portfolio
• return transformation
• risk aversion
• rational inattention
• Kelly criterion

Innovation

• Kelly criterion
• return transformation
• risk aversion
• rational inattention
• convexity