There are lots of levered ETFs out there but they all function in more or less the same way - they try to provide +/-2x the percentage daily move of whatever its benchmark index is. Just to take one example, this is from the ProShares prospectus (page 9):

*Ultra ProShares are designed to correspond to a multiple of the*

The Funds

**daily performance**of an underlying index. Short ProShares are designed to correspond to the inverse of the daily performance or twice (200%) the inverse of the daily performance of an underlying index.The Funds

**do not seek to achieve their stated investment objective over a period of time**greater than one day.(emphasis mine)

These are very important caveats to bear in mind and they are in the prospectus to protect ProShares from real-world behaviour vs. uninformed investor expectations. So how can these ETFs be used effectively, given their path dependence? The post on DXO/DTO already showed that there are occasions when the ultra or levered ETFs perform outstandingly well - basically when the underlying benchmark moves in a straight line with little deviation. So what of the flip-side? I skirted this issue with an earlier post called "The Chop!" where it was noted that SPX had been exceedingly volatile on an intraday basis for the previous couple of months. I'll try to give a look at both possibilities.

First, you have to have a basis for modeling or in this case, generating random but similar paths. I took 100 days of SPX data ending on 1/16/09 and noted the open-close price percentage change in the index. Here's that daily price change distribution in a nice histogram. This isn't exactly a normal distribution and for the stats nuts, here is the description: mean=-0.308, std error=0.359, std dev=3.59, kurtosis=0.839, skew=0.178. Unfortunately, I'm somewhat limited by knowledge of JMP and thus confined more or less to what Excel has on offer. So I took a short-cut and decided to pretend this was a normal distribution anyway. I'll accept any help to improve on this assumption so critique away.

To ensure that the normal distribution assumption wasn't wildly off, I decided to randomly generate 5 sets of n=100 based on the mean and std. dev. noted above using Excel. Then these sets were compared to the actual data set. These results are to the left in the JMP graphic. (I apologize for the squashed look but it's a tall graphic.) They aren't dead-on, but statistically speaking they are pretty similar to the original set. (Set #1 in this case is the original SPX data.) Not great, but not terrible.

Moving ahead from this point seemed acceptable and so I decided to model the returns (before expenses) according to the daily goals outlined in the prospectus of 1x, +2x and -2x funds in this environment. I ran this 200 times with some interesting results.

The 1x is the basis for daily calculation of the levered models. But the table on the right is the final return from day 1 to day 100. For the use of these Ultra-ETFs to be ideal, one would hope to see a high probability that the performance would be beyond the 2x daily expectation. Looking just at the mean, one might be tempted to think that these funds are excellent performers but when the median performance is looked at, a different story emerges. Better than 50% (about 55%) of the time, the -2x fund "underperforms" in a choppy market. In this instance, "underperform" simply means less than +/-2x the target benchmark - it says nothing of its performance to prospectus specifications. For reference the actual performances during the period under analysis were SPY= -32.8%, SSO= -60.73%, SDS = 19.14%.

So that's the examination of a relatively choppy market. What about the performance under circumstances like the summer to fall period in crude oil?

You'll have to just grant me the same assumptions as the charts illustrated above because I don't want to further clutter up the space. Briefly, the sets generated are even more similar statistically than in the first case. If you would like to see the numbers from above let me know but I'll summarize with this: the data in question is USO from June 16 until December 24 or 135 sessions. The average intraday change was -0.64% with a std. dev. of 2.65. This is a couple of weeks before the peak, and seems a little more realistic in terms of market timing in that allows some wiggle room for being off.

The results are to the right in the same table. In this environment, the -2x modelled fund performed well even on a comparison of medians and in the simulated runs actually beat a 2x performance of the underlying benchmark 81% of the time.

So what can be drawn from this overly-long post? First, it should be remembered that this simulating is totally ideal - no expense ratio (generally >0.8%) and no slippage/remainder that seems to occur in terms of performance vs. stated goal on a daily basis. My hunch is that on significant volume/volatile days the NAV of these ETFs gets away from share price due to in/out-flows and eventually must be reconciled. But in terms of applicable trading, I suppose if you believe the benchmark that you are targeting is going to enter a period of strong directional movement without much volatility, then a levered ETF might be a decent wager. But beware the chop and churn that, combined with expenses and remainders, can systematically erode returns.

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