Robust long-term growth rate of expected utility for leveraged ETFs

Abstract

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund. When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the Hansen–Scheinkman decomposition. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion, Cox–Ingersoll–Ross, 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of Leung and Park (Int J Theor Appl Finance 20(6):1750037, 2017).