Long-term returns estimation of leveraged indexes and ETFs

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 daily 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 daily leveraged ETF in relation to its underlying benchmark index. Of note, results 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.