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One-Day Returns
On a one-day basis, the funds did a very good job tracking their benchmarks, as shown in Figure 2. In the chart, a “positive” tracking error indicates that the fund outperformed its expected return, and a negative tracking error indicates that the fund underperformed its benchmark.
The largest tracking error on both an average and absolute basis was in the leveraged-inverse product, DXD. The smallest average error was in the simple inverse product, DOG.
Interestingly, a plurality of tracking errors was positive, particularly for the inverse funds. This may be because of the structure of the funds. These funds achieve their exposure in the market by buying options, futures or swaps. In each case, they only have to put up a portion of the value of their positions; the remaining cash can be invested in Treasuries, earning interest. As a result, they have a small but positive daily return. All else being equal, this will give them positive tracking error against the benchmarks used in this study.
The tracking error throughout this study tended to have fat tails; i.e., most days produced little or no tracking error, while a few anomalous days had high errors. In Figure 3, the colored boxes indicate where 95 percent of all tracking error results landed. Outlier results are indicated by associated black lines.
Figure 3 emphasizes that the vast majority of tracking error for all three funds was very small: less than 0.13 percent on an absolute basis. To put these figures in perspective, consider that 95 percent of the one-day returns for the Dow Jones industrial average during the period studied ranged from 3.1 percent to -3.2 percent.
In sum, these funds follow through on their promise of delivering 200%, -100% and -200% of the daily return of their benchmark indexes.
One-Week Returns
As expected, tracking errors widen as you move to longer time windows. The results of the tracking error study for a one-week holding period are shown in Figure 4. Although the vast majority of results are still close to their target, the tails are fatter, with DXD in particular posting a handful of unusual results, including one that was more than 8 percent off its benchmark index. As a reminder, this does not mean the product was working incorrectly; it’s simply due to the nature of leveraged and inverse returns when measured for periods longer than one day.
But while DXD—and to a lesser extent, DDM and DOG—experienced a few “bad” results, in most cases, the funds continued to perform well. As is the case for all longer time periods, DOG and DXD exhibit higher tracking error than DDM, suggesting it is harder to deliver accurate inverse returns than it is to deliver accurate leveraged returns.
The larger tracking errors shown in Figure 5 should be measured against the larger average move in the Dow over the one-week time frames. Ninety-five percent of the one-week returns for the Dow during the period studied fell between 4.39 percent and -5.70 percent. The largest one-week return was 16.91 percent, with the largest decline being -18.16 percent.
Given those results, the tracking errors on both DDM and DOG seem quite acceptable. The vast majority of returns for these two funds fell within 1 percent of their expected result. DXD had more trouble, with returns straying as much as 2.78 percent from the linear result. But even DXD’s tracking bands were tolerable, given the relatively large moves in the Dow itself.
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