# Alpay Kaya, CFA

La vérité existe. On n'invente que le mensonge. —Braque (Truth exists. One only invents falsehood.)

# Don’t be Victimized by the Semantics of LETFs

Since shortly after leveraged ETFs appeared in 2006, the investing public has on & off discussed a phenomenon termed ‘negative drift’ or ‘decay’. Typically described by example in flat markets, it refers to a loss in LETF value over a period the tracked index (and 1x-leveraged ‘standard’ ETF) is flat. This is actually just an example of negative drift but serves the purpose of a working definition.

Recently I came upon a finance blog posting declaring that such discussions are misguided. The author assumes the future expected value of an index equals its current value,

$\mathbf{E} \bigl[ x (t > t_0) \bigr] =\;\! \mathbf{E} \bigl[ x (t_0) \bigr]$

and under this assumption, the future expected value of a leveraged ETF tracking that index equals its current value. The conclusion is correct; unfortunately, laypeople will be misled by it.

Although the assumption effects a loss of generality, it is not the issue. The author’s conclusion is consistent with the general case, that the average LETF growth factor (price ratio) equals

${\displaystyle \mathbf{E} \biggl[ \frac{\, l (t > t_0)}{l (t_0)} \;\! \biggr] } = {\displaystyle \Biggl( \! \mathbf{E} \biggl[ \frac{\;\! x (t > t_0)}{x (t_0)} \;\! \biggr] \Biggr)^{\textstyle \! \beta} }$

the average index growth factor exponentiated by leverage. So the author’s assumption is consistent with reality…then what is the issue?

In statistical parlance, drift is the percent change in value associated with a variable’s future expected value. Two completely unrelated variables can have the same drift. Therein lies the issue, the general investing public was not commenting on statistics, but on its perception of tracking. Even if their expectations were ill-conceived and their terminology unsound, the general concern regarding why LETFs trade the way they do is nonetheless valid.

EXAMPLE  Consider the S&P 500 and Nasdaq-100 indices. I’ve chosen these indices because of the popularity of the +3x/-3x-leveraged ETFs tracking them: UPRO, SPXU for S&P 500 and TQQQ, SQQQ for Nasdaq-100.

For the 250 days leading up to the end of March 2013 (3/29/12 to 3/28/13), the two indices’ daily log return statistics are shown here.

$\begin{array}{lccc} \mbox{} &\mathtt{Mean,}\ \mu &\mathtt{Volatility,}\ \sigma &\mathtt{Implied\ Mean,}\ \zeta \\[0.75ex] \mathtt{S\&P\,500} & 0.0441\% & 0.8097\% & 0.0473\% \\ \mathtt{N-100} & 0.0068\% & 0.9850\% & 0.0117\% \end{array}$

The implied mean return, $\zeta = \mu + 0.5 \sigma^{2}$, is the daily log return implied by the expected value. It is used to calculate expected value, $\exp(n\zeta)$, n days into the future. From LETFs’ stated daily return objectives, their implied means are as shown here,

$\begin{array}{ccc} \mathtt{Leverage,}\ \beta & & \mathtt{Implied\ Mean,}\ \zeta \\[0.75ex] +3 & & \zeta_\mathrm{l} \, (+3) = +3 \, \zeta_\mathrm{x} \\ -3 & & \zeta_\mathrm{l} \, (-3) = -3 \, \zeta_\mathrm{x} \end{array}$

which could be used to calculate the expected value of a leveraged ETF in the future. This relationship between $\zeta_\mathrm{l}$ and $\zeta_\mathrm{x}$ is consistent with the continuous rebalancing assumption and is derived and shown to be accurate in LEVERAGED ETFs, my book on this topic.

While the statistics of the LETFs and indices are related in a straightforward manner, the same cannot be said for the relationship between the two for a specific price trajectory. For the same 250-day period, consider idealized LETFs (zero financing costs, expenses) with the same initial adjusted prices as the aforementioned LETFs, with perfect-tracking-model final prices italicized.

$\begin{array}{lccc} \mbox{} & \mathtt{3/28/12} & \mathtt{3/28/13} & \mathtt{Daily\ Return} \\[0.75ex] \mathtt{S\&P\,500} & 1405.54 & 1569.19 & 0.0441\% \\ \mathit{Model\ } \mathtt{UPRO} & 84.43 & \mathit{111.87} & 0.113\% < 0.132\% \\ \mathit{Model\ } \mathtt{SPXU} & 45.60 & \mathit{29.69} & -0.172\% < -0.132\% \\ \mathtt{Nasdaq-100} & 2770.97 & 2818.69 & 0.0068\% \\ \mathit{Model\ } \mathtt{TQQQ} & 60.87 & \mathit{59.60} & -0.008\% < 0.020\% \\ \mathit{Model\ } \mathtt{SQQQ} & 42.40 & \mathit{34.81} & -0.079\% < -0.020\% \end{array}$

As can be seen from this chart, the daily log returns of even idealized LETFs are less than those of the index scaled by leverage…and here less means smaller-magnitude if positive and larger-magnitude if negative. Note that LETFs are path-dependent; even with the indices’ initial/final prices kept constant, the model LETFs’ final ‘prices’ will change with the indices’ price trajectories.

LETF decay as perceived by investors is real. These funds leverage an index’s percent change, which is a log-normal distribution’s output. Whether or not investors realized it, many thought the leveraged parameter was the log-normal distribution’s input.

Leveraged ETFs represent a class of financial instrument in that they have a unique risk/reward profile. Investors who understand this profile can make use of them without being surprised at the results.