Long-Term Growth Rate of Expected Utility for Leveraged ETFs: Martingale Extraction Approach
International Journal of Theoretical & Applied Finance, Vol. 20, No. 6, 2017
30 Pages Posted: 6 Dec 2016 Last revised: 18 Feb 2019
Date Written: December 3, 2016
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.
Keywords: LETF, Long-Term Growth Rate, Expected Utility
JEL Classification: C52, G13
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