Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

TL;DR

This paper derives explicit formulas for the robust long-term growth rate of expected utility for leveraged ETFs under model uncertainty, analyzing worst-case scenarios across various stochastic models and determining optimal leverage ratios.

Contribution

It introduces a method to compute the robust long-term growth rate of leveraged ETFs considering model uncertainty, including stochastic interest rates, and identifies optimal leverage strategies.

Findings

Explicit growth rate formulas under multiple models

Optimal leverage ratios derived for worst-case scenarios

Impact of stochastic interest rates on growth rates

Abstract

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). 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 martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox--Ingersoll--Ross (CIR), 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 \citet{Leung2017}.