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Friends and acquaintances have been mentioning how they have been using levered exchange traded funds (ETFs) in the stock market. Some use these ETFs for the overall general market, financials, or commodities. Regardless of the actual ETFs, they all behave in similar manner. Whenever someone mentions using a leveraged ETF, I always ask, "Are you using this ETF for intraday movements or holding this ETF for several days, possibly months?" Invariably, the person responds that he has an outlook and wants to capture an outsized return by holding for many days, possibly months. For example, an acquaintance at the gym mentioned that he is bullish on oil longer term and was using an ETF with double leverage. I advised him to sell. Why did I do that?

While I share his positive longer term outlook for oil, I advised him to get out of his levered oil ETF because of his exposure to volatility over a longer term. A good way to understand this material is to look at a hypothetical example.

Hypothetical Example Of Four Levered ETFs
Day Actual Positive 2X Negative 2X Positive 3X Negative 3X
0 100.00 100.00 100.00 100.00 100.00
1 125.00 150.00 50.00 175.00 25.00
2 100.00 90.00 70.00 70.00 40.00
3 125.00 135.00 35.00 122.50 10.00
4 100.00 81.00 49.00 49.00 16.00
5 125.00 121.50 24.50 85.75 4.00
6 100.00 72.90 34.30 34.30 6.40
7 125.00 109.35 17.15 60.03 1.60
8 100.00 65.61 24.01 24.01 2.56
9 125.00 98.42 12.01 42.02 0.64
10 100.00 59.05 16.81 16.81 1.02
Source: My Fictitious Sample Data

In looking at the table above, what do you find surprising? I find two things: first, Positive 2X's values equals Negative 3X's values on even numbered days; and second, Positive 2X's values plus Negative 2X's values don't equal 100 and Positive 3X's values plus Negative 3X's values don't equal 100. Let's investigate how these numbers were determined to understand these seemingly odd results.

On Day 0, we start with a stock or commodity at a base value of 100.00. The next day, it appreciated 25% to 125.00. The following day, it depreciated 20% to fall back to 100.00. The daily absolute change in the underlying stock or commodity is a positive or negative 25, the percentage change is positive 25% and negative 20% respectively. It is this difference between appreciation and depreciation rates that eats away at the levered ETF's results. As we'll see, the greater the volatility and the longer the time held, the greater the erosion of the expected result.

For the Positive 2X ETF, Day 1 yielded a result of 150.00, which is a 50% appreciation from 100.00. Day 2, it fell 40% to 90.00. Day 3, it rose 50% to 135.00. As we see, these base values are drifting further and further away from the true base values of 100 and 125. Finally, on Day 10, the Positive 2X ETF value is only 59.05, even though the base value remains at 100.00.

In a similar manner for the Positive 2X ETF, Day 1 yielded a result of 50.00, which is a 50% depreciation from 100.00. Day 2, it rose 40% to 70.00. Day 3, it fell 50% to 35.00. As we see these base values are drifting further and further away from the true base values of 100 and 125. Finally, on Day 10, the Negative 2X ETF value is only 16.81, even though the base value remains at 100.00.

Intuitively, many expect that a Positive 2X ETF plus a Negative 2X ETF should balance out to zero change from the base value. As we see from our example, that doesn't hold true.

Let's look at Day 2 for the Negative 2X and Positive 3X results. Negative 2X is 40% rise (-2X times -20%) from 50.00 to equal 70.00. Positive 3X is a 60% fall (+3X times -20%) from 175.00 to equal 70.00. So we see that math for Negative 2X and Positive 3X yields the same results for even numbered days.

Okay, so you're thinking, that's fine for this hypothetical example, where I have massive daily changes of either 25% or 20%. In real life, the results should look much more reasonable. Shouldn't they?

Let's use Yahoo! Finance to examine two ProShares ETFs: Ultra S&P500 (NYSEARCA:SSO), a Positive 2X ETF; and UltraShort S&P500 (NYSEARCA:SDS), a Negative 2X ETF.

Yahoo Graph comparing ProShares ETFs SSO and SDS with S&P 500

Please note that you can click through the above graph to see a full sized version.

The green line in the middle represents the S&P 500, and over the past year it is down by about 40%. As an aside, please note that I am eyeballing the percentages, so these figures are not exact. Intuitively we should expect SDS to yield about positive 80%. Yet, we see that SDS only yielded a positive 40%. The remaining 40% is lost. Similarly, intuitively we should expect SSO to yield about negative 80%. Yet, it is only down by 70%. Again, we should expect that the cumulative return of SSO (+2X) and SDS (-2X) should yield zero. However, in this example, negative 70 plus positive 40 yields negative 30. This negative 30, once again, demonstrates the erosion in value.

The reasons for these ETFs to differ from our intuitive expectations are primarily the volatility of the S&P 500 itself and the duration of one year, a time arbitrarily chosen. There are two other lesser reasons: tracking error and ETF fees.

As a complete aside, notice the huge increase in volume after the October meltdown. With the increased volatility in the markets, speculators and investors are using these ETFs to gain even larger swings.

I want to be very clear in that I am not picking on ProShares. I am using their ETFs as convenient examples. In fact, ProShares states the following information on its homepage:

ProShares ETFs Target Daily Returns – Each ETF targets a return that is either 200%, -100%, or -200% of the return of an index or other benchmark for a single day ("daily target"). Due to the compounding of daily returns, ProShares' returns over periods other than one day may be more or less than the daily target. For more on correlation, leverage and other risks, please read the prospectus.

The link under the word compounding in the above quote provides a somewhat similar example to my hypothetical example above. Its examples and explanation are worth reviewing. And in ProShares's FAQs on Performance and Pricing, it states the following:

Assess on daily performance, not long-term performance. ProShares ETFs seek to achieve their objectives on a daily basis (before fees and expenses).

If you look every day, you can see the daily performance of the NAV (before fees and expenses) and the daily performance of the benchmark.

Investors considering ProShares and other leveraged or short funds with daily objectives should not expect them to achieve their objectives over periods longer than one day, however. There are several reasons for this, but the most significant one is index volatility and its effect on fund compounding. In general, periods of high index volatility will cause the effect of compounding to be more pronounced, while lower index volatility will produce a more muted effect. For more details, see Understanding Long-Term Performance of Leveraged and Short Funds.

I agree with the ProShares message. Given that ProShares highlights to its customers that its leveraged ETFs are best used for meeting daily, not longer term objectives, I believe ProShares is acting in a fair and reasonable manner.

The numerical examples, SSO and SDS charts, and ProShares' quotations should provide you with sufficient knowledge to understand what happens when you buy and hold a leveraged ETF.

Source: Leveraged ETFs: Handle with Care