Long-Term Growth Rate of Expected Utility for Leveraged ETFs: Martingale Extraction Approach

TL;DR

This paper introduces an analytical martingale extraction method to compute the long-term growth rate of expected utility for leveraged ETFs across various stochastic models, aiding optimal leverage decisions.

Contribution

It develops a novel analytical approach using martingale extraction to explicitly derive long-term growth rates for leveraged ETFs under multiple stochastic models.

Findings

Explicit formulas for growth rates under different models

Impact of stochastic interest rates on utility growth

Optimal leverage ratios identified for long-term investors

Abstract

This paper studies the long-term growth rate of expected utility from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. Working with the power utility function, we develop an analytical approach that employs martingale extraction and involves finding the eigenpair associated with the infinitesimal generator of a Markovian time-homogeneous diffusion. We derive explicitly the long-term growth rates under a number of models for the reference asset, including the geometric Brownian motion model, GARCH model, inverse GARCH model, extended CIR model, 3/2 model, quadratic model, as well as the Heston and 3/2 stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. We determine the optimal leverage ratio for the long-term investor and examine the effects of model parameters.