LeverageAfter recent discussions on the site about the levered index ETFs, I became curious as to how well these products are tracking their targets. So, using daily data for 9 Nov 2006 through 9 Nov 2007, each 1-day, 2-day, 3- , 4- , 5- , 10- , 15- and 20-day % change was calculated for both the relevant index (either S&P 500 or Nasdaq 100) and the positive and inverse ETFs.

Then the ratio of "ETF % move / index % move" was calculated. For the positive ETFs, the ratio should be ideally 2, and -2 for the inverse products. (The only tricky part is that if the index move is close to zero, the ratio can go to infinity.  So, included were only x-day periods where the absolute value of the index move was at least 0.5%.)

Means and sd's were calculated for all the ratios in each x-day period. Below are the results for each of four ETFs:

length in days | mean ETF/index ratio | sd of ratios

Ticker SSO
1d   1.96   0.40
2d   1.96   0.38
3d   1.97   0.35
4d   1.99   0.28
5d   1.95   0.33
10d  1.94   0.31
15d  1.98   0.29
20d  1.97   0.30

Ticker SDS
1d   -1.98   0.42
2d   -1.95   0.45
3d   -1.92   0.39
4d   -1.97   0.35
5d   -1.91   0.34
10d  -1.87   0.43
15d  -1.84   0.40
20d  -1.81   0.42

Ticker QLD
1d   1.89   0.59
2d   1.91   0.45
3d   1.94   0.42
4d   1.98   0.36
5d   1.95   0.36
10d  1.96   0.38
15d  1.98   0.37
20d  1.99   0.53

Ticker QID
1d   -1.90   0.55
2d   -1.91   0.45
3d   -1.89   0.47
4d   -1.93   0.38
5d   -1.89   0.40
10d  -1.94   0.41
15d  -1.95   0.32
20d  -1.93   0.36    

Adi Schnytzer suggests:

Looks fairly good, but a more revealing test might be to regress daily % change in the relevant index (Y) on change in the relevant ETF (X). So we have Y=a+bX and the test would be not only b=0.5 (which is what you have done) but also the joint F test, a=0 and b=1. Why? Because if a is not zero, then there is a bias in the tracking, i.e. either there is an over/under-reaction to large changes in the index or to small changes in the index depending upon the sign of a.

Kim Zussman writes:

While waiting for this week's bombs to start flying, here is regression of (daily return = [c2/c1]-1 )SSO vs SPY since inception of SSO June 2006 (including dividends):

Regression Analysis: SSO versus SPY

The regression equation is
SSO = - 0.000217 + 2.00 SPY

Predictor        Coef    SE Coef       T      P
Constant   -0.00022  0.00013   -1.64  0.101
SPY          1.99573   0.01630  122.44  0.000

S = 0.00245947   R-Sq = 97.7%   R-Sq(adj) = 97.7%

Analysis of Variance

Source              DF        SS        MS         F           P
Regression        1  0.090682  0.090682  14991.23  0.000
Residual Error  348  0.002105  0.000006
Total                349  0.092787

Obviously a significant slope coefficient, with beta of 2. Notice however that the intercept is almost significantly negative (alpha), suggesting the ETF manufacturer is skimming something every day (probably in the prospectus). Recall that SPY is the SP500 ETF which levies its own (tiny) fee, so you are paying more for the leveraged ETF and might rather consider futures (unless you treasure your sanity).

Adi Schnytzer explains:

Adi SchnytzerWhat matters (in the way you have run it) is the joint F test a=0 and b=2, and I have no doubt you will be unable to reject it at any reasonable level of significance. Note also that 0.00022 is a teensy number. So it would seem that these are a good buy if one is bullish medium term and doesn't mind staying in the market. Mind you, there are those of us who got into the market just before the latest crash and so, mind or no mind, are in there till the recovery. That's the trouble with futures, unless you can pick your closing date far enough down the track.

Gordon Haave remarks:

Theoretically speaking, levered ETFs work in directional markets. That is, the constant leverage results in buying on up days and selling on down days. So, in certain market periods they work out just fine and are good short term trading vehicles.

Phil McDonnell summarizes:

There are three ways investors in leveraged ETFs incur costs. First, management fees, which are usually lower in non-leveraged ETFs, presumably because there is less juggling to do. Second, the leveraged half of the fund must pay interest at the going margin rate. Even if the fund uses futures or options the interest is implicitly built into the price of the derivative. Third, he constant leverage trap. The 2x funds are designed to give returns which are twice the daily return of the underlying. They rebalance daily, which means they sell low and buy high. In choppy markets and over multiple days this leads to slight under performance relative to the 2x benchmark. Mr. Mabry's study looked at multiple days and found this slight underperformance. In contrast, Dr. Zussman's study found a perfect 2.00 multiplier on a daily basis. That is exactly what they promise, 2x returns for the day. The negative alpha is due to the sum of all three costs above. Not to quibble with Prof. Schnytzer, but .00022 is about 5.5% per year in costs. Most of this is because of the leverage. It is either real or implied interest which must be paid.


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