Optimal Betting: Beyond the Long-Term Growth
Levon Hakobyan, Sergey Lototsky
TL;DR
This work addresses the risk inherent in Kelly-style growth by quantifying fluctuations of the log-wealth through the asymptotic variance $\upsilon_r(f)$ and by introducing two risk metrics: the asymptotic Sharpe ratio $\mathrm{SR}_r(f)$ and the ridge coefficient $\mathrm{Ri}_r(f,\gamma)$. It develops a unified framework that covers discrete-time (The Standard Model and Beyond), high-frequency compounding, and continuous-time compounding via semimartingale returns, deriving LLN- and CLT-type results for long-run growth $g_r(f)$ and its variance $\upsilon_r(f)$, and identifying optimal fractional Kelly strategies $f\in(0,f^*)$ and their risk-control variants. In continuous time, the paper formulates $g_R(f)= f\mu - \tfrac{f^2\sigma^2}{2}$ and $\upsilon_R(f)$ under practical models, yielding closed-form-like expressions for $\mathrm{SR}_R(f)$ and a continuous-time ridge optimization, with connections to power utility via $f^{\eta}$. Overall, the results provide computationally efficient tools for constructing fractional Kelly strategies and highlight practical implications for risk management, portfolio choice, and connections to ergodic and diffusion models in finance.
Abstract
While the Kelly portfolio has many desirable properties, including optimal long-term growth rate, the resulting investment strategy is rather aggressive. In this paper, we suggest a unified approach to the risk assessment of the Kelly criterion in both discrete and continuous time by introducing and analyzing the asymptotic variance that describes fluctuations of the portfolio growth, and use the results to propose two new measures for quantifying risk.