Just about a year ago, I analyzed some leveraged ETFs and compared their performance to their underlying index. See here.

Leveraged ETFs attempt to double, or even triple, the performance of broad indicators like the Dow Industrials or S&P 500.

I addressed two concerns. One was *tracking error.* This is a legitimate concern for **any** ETF owner. Suppose the S&P 500 advances 7 percent in 3 months. Then we would expect the SPDR 500 Index Trust ETF (NYSEARCA:SPY) to also go up 7 percent. Furthermore, we would expect it to track the zigs and zags of the broad market quite faithfully.

Look below. You have real sharp vision if you can even distinguish between the S&P500 and SPY on this chart. No tracking error here.

source: www.bigcharts.com

What about the ProShares Ultra S&P500 ETF (NYSEARCA:SSO)? Their Yahoo profile says the fund is designed to track "*two times the daily performance of the S&P500 ."*

Pay attention to the words *daily performance.* Leveraged ETFs such as SSO rebalance their portfolio each day, using futures, derivatives and stock holdings, in order to achieve this 2x performance. This can lead to losses caused by volatility drag. Let me turn to an illustration of volatility drag, using an extreme example to make the discussion and mathematics brief.

Suppose you bought a stock for $20 a share. The first day it rises to $25; the second, it falls to $15, and the third day it went back up to $20. A buy-and-hold investor has broken even. What happens to an ETF adjusted daily designed to "track" this one stock?

OOPS! Not so good, especially a double index. On the first day the stock gained 25 percent, doubled this is 50 percent. On the second day it fell 40 percent; doubled, this is an 80 percent loss; and on the third day it rose about 33 percent, for a 66 percent doubled gain. What is your overall result if you rebalance each of the three days? Turning the percentages into decimals, and multiplying we have

- (1.00 + .50) for day 1;
- (1.00 - .80) for day 2; and
- (1.00 + .66) for day 3. The final result is

(1.5)x(.2)x(1.66) or approximately .50.

Returning to percentages (isn't math fun!) you have 50 percent of your investment remaining. You have lost half your money! Savvy investors can see that since you lost so **much** on the second day, it was almost impossible for the recovery on day three to make much of a difference.

Now we can look at the data. The chart below compares SSO against the S&P500 for the past twelve months. There seems to be very little problem with tracking error, and as you will see in the calculations beneath it, no volatility drag, either.

Over the last year the S&P500 has gained 18 percent. If you double this, 36 percent, it appears that SSO has "overperformed" a bit, since it is up almost 40 percent.

This is not quite true. The trick lies in the leveraged ETF mathematics and arithmetic. For very small percentage changes, such as 1 percent or half a percent, typical of day to day fluctuations in the market, you can just "double the percentage" and you will get SSO performance on that date accurately enough.

For larger percentage changes typical of several months, a year, or longer, you must use a different calculation. You do not "double" the performance, you *square it.*

Thus after one year you gained 18 percent, or you had

(1.00 + 0.18) = (1.18)

times your original holdings. For a 2x leveraged ETF, when you **square** this number, you get

(1.18)*(1.18) = 1.39, or a 39 percent increase in your holdings.

And that is almost *exactly* how much SOO gained in the last twelve months.

Why has there been no volatility drag? In the extreme example above, daily fluctuations were very large. For the smaller percentage changes typical of a broad index like the S&P 500 each day the drag is barely noticeable. Mid June's 1 percent drop in the S&P 500 was the biggest daily change over the past year. On most days changes have been far less than that. So drag is minimal.

We do see a bit of "overperformance" in the ProShares Ultra Dow 30 ETF (NYSEARCA:DDM), though. This ETF is designed to double the returns on the Dow Jones Industrial Average.

source: www.bigcharts.com

The Dow has gained about 14 percent in the last year; using the calculations illustrated above we would expect DDM to gain

(1.14)*(1.14) = 1.30 or thirty percent.

The ETF actually gained 32 percent in the last year...A tad better than expected.

Thus, despite the concern about tracking error and volatility drag for broad index based ETFs over long periods of time, I find little evidence of either.

**Disclosure: **I am long SPY, IYT, XLV. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.