Long-Term Returns Estimation of Leveraged Indexes and ETFs

TL;DR

This paper derives bounds for long-term log-returns of leveraged ETFs based on daily benchmark returns, providing conditions under which leveraged ETFs outperform or match the underlying index over extended periods.

Contribution

It introduces quadratic bounds for long-term returns of leveraged ETFs and identifies conditions for their outperformance relative to benchmarks.

Findings

If the benchmark drops over 10% in 63 days, a -3x ETF yields at least 1.5 times the log-return of shorting the index.

A 2x leveraged S&P 500 ETF can outperform the index if the average annual return exceeds 0.0658 and daily volatility is below 0.0125.

The bounds help assess long-term performance of leveraged ETFs based on daily index behavior.

Abstract

Daily leveraged exchange traded funds amplify gains and losses of their underlying benchmark indexes on a daily basis. The result of going long in a daily leveraged ETF for more than one day is less clear. Here, bounds are given for the log-returns of a leveraged ETF when going long for more than just one day. The bounds are quadratic in the daily log-returns of the underlying benchmark index, and they are used to find sufficient conditions for outperformance and underperformance of a leveraged ETF in relation to its underlying benchmark index. Results show that if the underlying benchmark index drops 10+\% over the course of 63 consecutive trading days, and the standard deviation of the benchmark index's daily log-returns is no more than .015, then going long in a -3x leveraged ETF during that period gives a log-return of at least 1.5 times the log-return of a short position in the underlying benchmark index. Results also show promise for a 2x daily leveraged S&P 500 ETF. If the average annual log-return of the S&P 500 index continues to be at least .0658, as it has been in the past, and the standard deviation of daily S&P 500 log-returns is under .0125, then a 2x daily leveraged S&P 500 ETF will perform at least as well as the S&P 500 index in the long-run.