DoubleInverse ETFs and Tracking Error
(Guest Commentary By Bill Rempel – October 2, 2008)
Dear Subscribers and Readers,
For those who had wanted to learn more about individual stocks, the art of stock selection, and modelbased trading/investing, it is again time to turn to one of our regular guest commentators, Bill Rempel. Bill is a prolific writing on the financial markets and trading strategies and is the author of a very active market blog at: http://billrempel.com (“The Rempel Report”).
In this commentary, Bill is going discuss the tracking errors of leveraged ETFs (for the S&P 500 and the NASDAQ100, specifically), as well as whether these are good vehicles for trading purposes. Without further ado, following is a biography of Bill:
Bill Rempel (aka nodoodahs) is an active poster on the MarketThoughts forum as well as a few others around the web. Bill is a regular, monthly guest commentator on our website (see “The Correlations, They Are AChangin'” for his last guest commentary). Bill graduated from Caddo Magnet High School (a high school for nerds) back in 1985 and proceeded to learn the hard way when he drank his way out of a scholarship to Tulane later that year. After a few years of sweating for a living, he decided to go back to school, and graduated from LSUShreveport in 1995 with a Bachelors in Mathematics  all the while working the overnight shift stocking shelves in a grocery store.
Postcollege, Bill has been in the P&C insurance industry as an actuary, product manager, and pricing manager. Bill and his wife Millie are amateur investors with a variety of holdings, but they prefer to buy and hold value investments. In typical "value" style, they live cheap, driving old cars and preferring to save or invest instead of buying fancy "stuff."
Disclaimer: This commentary is solely meant for education purposes and is not intended as investment advice. Please note that the opinions expressed in this commentary are those of the individual author and do not necessarily represent the opinion of MarketThoughts LLC or its management.
Just about every time the market's gotten wild recently, I've heard the question asked by daytraders, "What's up with this doubleinverse ETF? It's not acting right!"
My response has been along the lines of "They have big tracking errors when the market is volatile. Oddly, that's just when you need them the most."
It also begs the question, why use the doubleinverse ETFs, or even the doublelong ETFs, for a day trade anyway? If you need exactly double the move of an index, I suggest that the futures contracts are a better choice. If you need leverage, again, the futures market has all you need and more. If all you need is a strong, unleveraged intraday mover, I suggest scanning for highvol liquid stocks and forgetting about the movement in the broad market indices.
Here I want to compare the tracking errors over the last two years for two widelyfollowed indices, and the ETFs associated with them.
The tracking errors were assembled from closetoclose daily moves over the period from September 2006 through September 2008. The SPY and QQQQ should track the move of the index in question; the QLD and SSO should track twice the move of the index in question; and the QID and SDS should track twice the inverse of the move of the index in question. For illustration, here's a concrete example:
September 28, 2008 
S&P 500 moves down 106.85 points or 8.81% from close to close.
SPY moves down 7.84% for tracking error of 0.97% and absolute error of 0.97%
SSO moves down 13.79% for tracking error of 3.82% and absolute error of 3.82%.
SDS moves up +14.68% for tracking error of 2.94% and absolute error of 2.94%
The error itself, if it is random and unbiased, should add up to zero over time, with negative errors canceling out positive errors. The absolute errors (or absolute value of the errors) are important because they'll be used to measure the average deviation from expected result, without regard to whether the error was up or down. An average error different from zero indicates bias in the tracking; a high average absolute error indicates inaccuracy; these are somewhat mutually independent concepts.
In the case of September 28th, the errors of the two leveraged ETFs Vs. the S&P 500 were large, and biased towards UNDERreaction. Neither moved far enough to make their tracking accurate.
On average, the tracking errors overall are refreshingly close to zero, which means that, on average, the tracking ETFs are unbiased. The problem arises at the extremes. When the S&P 500 makes extremely large moves (more than a percentage point or so in either direction), all of the index tracking ETFs underestimate the move  their tracking errors are biased.
The average absolute errors are pretty instructive. Not only does the error show bias when the moves in the underlying are extreme, the raw size of the errors increases. It's good to note that the SSO and SDS have pretty much double the average absolute error of the SPY, across most rows of the table; at least they're consistently bad in light of their purpose, and not far off of a double move. If you compare the average absolute error for the two extreme rows, vs. the average absolute for the remaining eight rows, then the error increases by 7080% at the extremes.
The big problem with the S&P 500 trackers is the underestimation bias combined with the error size increase on volatile days. If you don't get the full move on a wild trend day, the ETF isn't all that good a daytrading instrument, is it?
For the Nasdaq 100 tracking ETFs, the errors overall are still close to zero, which means that, on average, the tracking ETFs are unbiased. The problem again arises at the extremes. When the index makes extremely large moves (more than a percentageandahalf or so in either direction), all of the index tracking ETFs underestimate the move  their tracking errors are biased.
The average absolute errors show the same bias when the moves in the underlying are extreme, as the raw size of the errors increases. What's scary is that the errors for the leveraged ETFs (QLD and QID) are four times as large, on average, as the QQQQ errors. That level of lousyness is at least fairly consistent across all the rows, however. If you compare the average absolute error for the two extreme rows, vs. the average absolute for the remaining eight rows, then the error again increases by 7080% at the extremes.
We see the same problem with the Nasdaq 100 trackers that we saw with the S&P 500 trackers, an underestimation bias combined with the error size increase on volatile days. These instruments aren't really all that good for what they're supposedly designed to do.
I'd like to close with a nod or two to the longtermers.
Because of the impact of compounding changes, one shouldn't expect the doubleinverse ETFs to translate their "twice the inverse of daily movement" into "twice the inverse of the longerterm move." Even if they tracked perfectly accurately on a daytoday basis, the longterm impact would be more like 1.8 times the inverse of the annual move (a Monte Carlo simulation result).
However, assuming the doublelong ETFs tracked perfectly accurately on a daily basis, the tendency would be to get just barely more than twice the annual appreciation of the index. This would be much less accurate in years when the index didn't appreciate much, and more accurate when the index showed considerable annual appreciation (again, a Monte Carlo simulation result).
There are other longterm differences in absolute performance, because the doublelong and doubleinverse ETFs pay distributions based on their own transactions in the futures and swaps markets, whereas the long ETFs have distributions based on the underlying index's dividend yield, and there are possible tax treatment differences as well.
While I haven't done any rigorous analysis on the sector or industryspecific leveraged ETFs, or on the foreign market index ETFs, I expect to find the same general trends of larger tracking errors in general on leveraged products, as well as larger errors with underestimation bias when volatility is high.
For my part, I don't see these tools as very useful for a daytrader. If it were my money, and I needed or desired a leveraged move based on the indices, I would just use the futures contracts to get a more accurately measured move. If one can't afford to use those contracts, then one should be happy with what one can get, I suppose.
Assuming I wanted to day trade and just needed a fastmoving target, it would probably be better to just use individual liquid stocks or apply actual leverage to the QQQQ, since the tracking error on QLD and QID is so large.
From where I sit, I think the best use of the doublelong and doubleinverse ETFs is in the holding of markettiming positions for weeks or months, rather than daytrading. But what do I know?
