In a bear market, conservative investors attempt to protect the value of their portfolios by going into cash or, perhaps, seeking refuge in low beta, high dividend paying blue chips like AT&T (NYSE:T), or Procter & Gamble (NYSE:PG). More aggressive investors may attempt to sell short.

In recent years, caffeine-addicted speculators have been drawn to "inverse ETFs." They are so called because when the target index goes down, these funds increase in value. Of course, the opposite is also true. Even more intriguing are the 2x reverse funds: if the target were to fall by 10%, a reverse ETF tracking the broad market would *rise by 20%!*

It is not hard to understand the appeal of such funds to investors whose portfolios have been ravaged by several bear markets since the turn of the century. In addition, they save investors the hassles and margin involved in short selling, are traded continuously during market hours, and unlike short selling, "cap" your losses at 100%. Not that 100% is all that attractive a cap, of course; but it is better than the possible 200% or 300% losses suffered by shorts in, say, Apple (NASDAQ:AAPL) or Amazon (NASDAQ:AMZN).

In this article, we look at one popular 2x inverse ETF: Proshares Ultrashort S&P 500 (NYSEARCA:SDS). Yahoo's fund summary states that "the fund invests in derivatives which are designed to show *daily* return characteristics which are -2x that of the Standard and Poor's 500." On most days, more than 10 million shares change hands, so liquidity is not a problem. We will examine a few charts to answer the following questions, which should be a concern of all investors for *any* ETF:

- How does the inverse ETF behave when the market declines? If the S&P 500 drops 20%, does this ETF
*really*gain 40%? That is, is the ETF a good tracker? - If the market moves against you...
**up, in this case...**how do the ETF shares behave? (*hint: it involves some arithmetic).* - What happens in flat markets? Do trading costs and administrative fees dig into returns?

Most investors pay attention to the first; close their eyes and pray during the second; and completely overlook the third. In fact all of them deserve your utmost attention.

Especially that last one: for there is a curve ball in leveraged ETFs that would make Sandy Koufax proud. Called "volatility drag," it is a direct consequence of the daily rebalancing by which leveraged ETFs obtain their 2x and 3x daily returns. With the passage of time - many weeks or more - this could lead to substantial underperformance compared to expectations.

Let's show it in just 3 days by a bit of exaggeration. Suppose today you bought the S&P 500 index ETF, SPY, at 20. Tomorrow it goes up to $25 a share; the 2nd day it falls to $15 a share; the third day it rises back to $20 a share. A buy and hold investor has broken even.

What about the -2x leveraged ETF? The first day the SPY gained 25%; so the -2x ETF would fall 50% in value. The 2nd day, when the SPY fell 40%, you expect the -2x ETF would soar 80%; and the third day, when the SPY gained 33%, your double ETF would fall 66%. The ETF after three days will be worth:

(1-.50): decline in value after the first day, SPY went up 25%, you fall 50% or .50 as a decimal calculation; times

(1 + .80): increase in value after the second day: SPY went down 40%, you gain 80% or .80 as a decimal; times

(1 - .66): decline in value on the third day, SPY gained 33%, you fall 66% or .66 as a decimal. So the final result is

(1 -.50)*(1 + .80)*(1 -.66) = .306. Only 30% of your funds are remaining, even though a buy and hold investor broke even.

Your -2x ETF would show nearly a 70% loss after just 3 days of trading even though its "bogey" stayed even over this time period!

But wait. Mr. Market and Mr. Koufax have some more "chin music" for you. That capital gain you enjoyed on day #2 is fully taxable as ordinary income, since the leveraged fund closes its position at the end of every day.

That is fine as an exaggerated case. How bad is the volatility drag over intermediate time periods - say, a couple months or more - but with more realistic daily percentage changes?

SDS began to trade summer of 2006, so our examples will be from markets of the last five years. As a comparison for performance, we will use the S&P 500 Spyder (NYSEARCA:SPY).

First, we can look at some periods of declining markets. In the fall of 2007 to early Spring of 2008, SPY fell by nearly 20%. We didn't know it at the time, but this was the first down move of the 2008 crash. As you can see, SDS tracked this down move very well: zigs by one index are zags in the other.

The bottom line is that SPY fell by about 20% and SDS went **up 40%,** which looks like -2x in my book. There does not appear to be any major drag on performance, even though we are looking at almost 6 months of market action.

Much the same can be said for that dreadful summer of 2008:

So hungry were investors to jump on the short bandwagon that the SDS briefly rose more than the 90% or so you might have expected!

Similar results can be shown for the two summer selloffs since the new bull market began in March of 2009. Look at summer 2011 as an example:

A 16% drop in the SPY turned into a 32% gain for the double short.

Perhaps the drag is more evident in flat markets? Returning to 2008, by late Spring, stocks fought their way back to nearly even for the year:

The double short ETF was nearly flat for the year as well. *This means that over an intermediate time period,* trading and management fees for SDS were not enough to substantially detract from expected returns. The same can be said for the summer of 2011:

Notice despite the roller coaster ride, the SDS moved opposite the market and showed no *loss* when the market showed no *gain.*

Thus, I conclude *unlike other double and triple ETFs with monthly 2% and 3% trading, management, and slippage costs chewing into returns,* the SDS is a useful tool for investors seeking to aggressively short the broad market. There appears to be no significant volatility drag.

Finally, let's look at the 2nd case, advancing markets. One thing makes short sellers cringe at cocktail parties: while everyone is trumpeting their big gains, shorts hide in the corner with their big losses. Some analysts think a new bear market began late last summer, and the entire rally since has been a charade, based on cheap money and statistical hanky-panky with GDP, the consumer price index, and every other economic indicator. What if these analysts stuck to their guns and bought SDS on October 1st 2011 and were still holding? *It ain't pretty:*

A 32% gain for longs turned into a 50% loss for holders of SDS.

Not be sadistic and kick a short when he or she is down, but why didn't the SDS fall by 64%? The quirky algebra of inverse ETFs comes into play: since the most the SDS share price can fall is 100% (to zero), the losses are not "double the gains" when SPY advances. For you quants out there, (the rest can close their eyes) the value of your shares is calculated as follows. 32% doubled, is 64%, which is .64 as a decimal. Then:

(1.0)/(1 + .64), = .609 rounding off to about .61. So your shares should fall to about 61% of their original price.

This is a 39% loss; not 50% as actually occurred. So the inverse fund was punished more than it should have been over the last year.

For bearish day traders and speculators looking forward a few months or so, SDS is an efficient way to short the market without the margin and other risks associated with selling stocks short. Relatively few short sellers trade for periods longer than this.

There are double ETFs on gold, oil, and treasury bonds as well. In future articles, I'll show how well they (inversely) track their indexes.

**Disclosure: **I am long SPY. 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.

**Additional disclosure:** ETFs provide invaluable, low cost diversification. Double and triple ETFs use derivatives, options, futures, and other financial wizardry to amplify returns. These derivatives are expensive and involve daily rebalancing. These costs can be prohibitive when the 2x/3x ETF is held for long periods, though this is not usually how traders use them. Tax implications of daily rebalancing must also be considered.