There are many leveraged exchange-traded funds (ETF) designed to track multiples of short-term (daily) changes in popular indexes. Over longer holding periods, these ETFs tend to veer off track. The cumulative tracking error can be large. How well do leveraged ETFs track benchmarks over a multi-year period? What return metric drives the degree to which they fail to achieve targeted leverage? To investigate, we consider two sets of the oldest leveraged ETFs:

- 34 ProShares +2X and -2X leveraged equity index ETFs (17 matched long-short pairs), with start date 3/14/07 (limited by the youngest fund), which track U.S. broad market and sector indexes.
- 10 ProShares +3X and -3X leveraged equity index ETFs (five matched long-short pairs), with start date 2/11/10, which track U.S. broad stock market indexes only.

We measure actual average daily tracking by comparing the average daily return of each leveraged ETF to the average daily return of a +1X ETF that tracks the same index. We measure longer-term (monthly) tracking by comparing the monthly Sharpe ratio of each leveraged ETF to that of a +1X ETF that tracks the same index. Using daily and monthly adjusted closing prices for the above funds and +1X counterparts through May 2015 and the contemporaneous monthly U.S. Treasury bill yield as the risk-free rate for Sharpe ratio calculations, *we find that:*

The following table summarizes average daily leverage outcomes for 17 +2X and 17 -2X ETFs over the common March 2007 through May 2015 sample period. All 17 +2x funds track a little short of targeted leverage, while nine of 17 -2X overachieve and track more than targeted leverage. Some results are affected by thin trading, with zero volume on some days. An extreme example is SZK (ProShares UltraShort Consumer Goods), which exhibits no close-to-close price changes for 274 of 2,072 (13%) of daily returns.

Among +2X funds, the correlation between average daily leverages and daily volatilities (standard deviations of daily returns) of corresponding +1X funds is -0.68, suggesting that higher volatility of the target index translates to lower +2X average achieved leverage. Among -2X funds including (excluding) SZK, the correlation between average daily leverages and daily volatilities of corresponding +1X funds is -0.14 (-0.59), suggesting that higher volatility of the target index translates to higher (in magnitude) -2X average achieved leverage.

A reasonable extrapolation of these findings is that actual +2X (-2X) ETF leverage tends to be lower (higher, in magnitude) than 2X when market volatility is elevated.

How do these daily behaviors translate to monthly tracking?

The following chart summarizes monthly Sharpe ratios for broad index +1X ETFs and corresponding relatively liquid +2X and -2X leveraged ETFs over the 99-month sample period, which includes the 2008 U.S. stock market crash. Over this period:

+2X fund Sharpe ratios are consistently lower than those of corresponding +1X funds, by 11% to 26%.

-2X fund Sharpe ratios are consistently lower than minus those of the corresponding +1X funds, 38% to 91%.

In other words, tracking at a monthly horizon is unfavorable for fund holders, and tracking error is worse for -2X than +2X funds.

What about the +3X and -3X ETFs?

The next table summarizes average daily leverages for the five +3X and five -3X ETFs during February 2010 through May 2015. All five +3x funds tend to track a little short of targeted leverage, while all five -3X funds tend to overachieve and track more than targeted leverage.

How do these daily behaviors translate to monthly tracking?

The final chart summarizes monthly Sharpe ratios for broad index +1X ETFs and corresponding +3X and -3X leveraged ETFs over the available 64-month sample period. Over this period:

+3X fund Sharpe ratios are consistently lower than those of correponding +1X funds, by 9% to 13%.

-3X fund Sharpe ratios are consistently lower than minus those of the corresponding +1X funds, 22% to 11%.

Again, tracking at a monthly horizon is unfavorable for fund holders, and tracking error is worse for -3X than +3X funds.

Sharpe ratio deterioration is not as severe here as for the +2X/-2X funds above, because the sample period here does not include the 2008 market crash. High market volatility during the crash amplifies leveraged ETF tracking errors.

In summary, *evidence from simple tests shows that risk-adjusted performances (Sharpe ratios) of leveraged ETFs over extended holding periods are materially lower than those of corresponding unleveraged funds.*

Alternatives for achieving leverage over extended periods involve borrowing to buy or short +1X ETFs. For these alternatives, broker fees/interest charges would reduce returns and lower Sharpe ratios (and there could be margin calls).

Cautions regarding findings include:

- The sample period is short in terms of number of U.S. stock market regimes, especially for the 3X and -3X funds.
- Sharpe ratio is one way to measure risk-adjusted performance. Other measures may produce different degrees of leveraged ETF tracking errors.