# Long-Term Returns Estimation of Leveraged Indexes and ETFs

**Authors:** Hayden Brown  
**arXiv:** [2301.03186](https://arxiv.org/abs/2301.03186)  
**Date:** 2023-01-09  
**Summary page:** https://www.summarizepaper.com/en/paper/2301.03186v1/

## Key points

- Daily leveraged ETFs amplify gains and losses of underlying benchmark indexes on a daily basis
- Long-term effects of going long in a daily leveraged ETF for more than one day are unclear
- Bounds for log-returns of a leveraged ETF when going long for more than one day are quadratic in the daily log-returns of the underlying benchmark index
- Sufficient conditions for outperformance and underperformance of a leveraged ETF in relation to its underlying benchmark index can be found using these bounds
- Going long in a -3x leveraged ETF during a period where the underlying benchmark index drops by 10% or more 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, gives a log-return of at least 1.5 times the log-return of a short position in the underlying benchmark index, suggesting that investors may benefit from using -3x leveraged ETFs as hedges during periods of market decline
- A 2x daily leveraged S&P 500 ETF will perform at least as well as the S&P 500 index in the long run if the average annual log-return of the S&P 500 index continues to be at least .0658 (as it has been historically), and the standard deviation of daily S&P 500 log-returns is under .0125, suggesting that investors who want to increase their exposure to equities may benefit from using these types of ETFs
- Leveraged ETFs are complex financial instruments and may not be suitable for all investors; it is crucial to carefully consider individual financial goals and risk tolerance before investing in them

## Summary

In the world of finance, daily leveraged exchange traded funds (ETFs) are known to amplify both gains and losses of their underlying benchmark indexes on a daily basis. However, the long-term effects of going long in a daily leveraged ETF for more than one day have been less clear. In this study, Hayden Brown provides bounds for the log-returns of a leveraged ETF when going long for more than just one day. These bounds are quadratic in the daily log-returns of the underlying benchmark index and are used to find sufficient conditions for outperformance and underperformance of a leveraged ETF in relation to its underlying benchmark index. The results show that if the underlying benchmark index drops by 10% or more 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. This suggests that investors may be able to benefit from using -3x leveraged ETFs as hedges during periods of market decline. Additionally, Brown's research shows 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 historically), 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. This finding suggests that investors who want to increase their exposure to equities may benefit from using these types of ETFs. Overall, Brown's study provides valuable insights into how investors can use leverage through ETFs to potentially enhance returns or hedge against market downturns. However, it is important to note that leveraged ETFs are complex financial instruments and may not be suitable for all investors. As with any investment decision, it is crucial to carefully consider one's individual financial goals and risk tolerance before investing in these types of products.