I have been writing about a great ETF of our times called SVXY, and its ETN clone XIV, for a while now. I wouldn't go into the details of why these are great ETFs. Readers can check out my earlier article for that viz. The Best ETF Of Our Time (AKA How To Become A Billionaire Without Really Trying).

One of the main complaints that came out of that article is that these are inverse ETFs, and as such, they must erode value in a sideways market.

Well, actually not. I see way too many articles stating this, and these are mathematically wrong. Yes, leveraged inverse ETFs do lose value from volatility drag, but 1x inverse ETFs based on futures don't. This is why.

The common belief is that 1x inverse ETFs return the negative of the gain of the underlying in a day with daily rebalancing. I do not fault people for having this belief as that's exactly how the funds are advertised. But a reading of the fine print shows that this is close, but alas, no cigar.

Let's first understand how futures work. They are daily cash settled. Let's say the future is at 100 at the start of the day, and investor A is long while investor B is short. Let's say the future ends at 110 at the end of the day. Then B will give 10 dollars to A. If the future ends at 90, then A will give 10 dollars to B. This looks surprising like A gained 10% and B lost 10% when the future went up, and just the reverse when the future went down. Actually, that is not the case.

What, in effect is happening that A is long the market at 100 and B is short the market at 100. So, if the future goes up, A buys at 100 and sells at 110, gaining 10, on an investment of 10. Of course, all these steps are not actually executed, as it is unnecessary. Just the cash settlement of the difference is enough. But what about B? B shorted at 100 and covered at 110. So B bought at 110 and sold at 110, instead of buying at 100 and selling at 90. The cash difference is the same, but the percentage return is not. A gained 10%, sure, but B lost not 10/100, or 10% but 10/110, or ~9%.

Just the reverse happens when the future goes down. A buys at 100 and sells at 90, for a 10% loss. But B buys at 90 and sells at 100, for a ~11% gain. So, the returns are not merely the negative of each other. The mathematical relationship is different.

So what is the mathematical relationship?

Let's say the underlying changes by x% in a day, where x can be negative or positive. The inverse will change by 1/(1+x) times, or in percentage terms by -x/(1+x)% in the day. Let's see if this math holds for the above cases.

Case 1: x = 10%, and the inverse should change by (per the above formula) -10(1+10%)%, or (-10/1.1)%, or about 9%.

Case 2: x = -10%, and the inverse should change by (per the above formula) 10(1-10%)%, or (10/0.9)%, or about 11%.

So these formulas hold. But why then people believe that if the underlying changes by x% the inverse would change by -x%? Well, when x is very small, 1+x is almost equal to 1, and then the formula -x(1+x)% is almost equal to -x%.

This is a small difference, but is crucial when calculating the volatility drag. Let's say the underlying goes from 100 to 110 to 90 to 100 to 110 and cycles back and forth for a month (20 trading days) finally ending at 100. This is how the inverse ETF calculated using the accurate formula i.e., -x/(1+x)% change for the inverse for each x% change in the underlying will behave in comparison to the ETF calculated using the plain negative i.e., -x% for each x%, which I call the approximate formula.

The approximate formula would indicate a lot of erosion. But the accurate formula wouldn't. As the underlying returns to 100, it returns to 100 as well. There is no volatility drag. Perhaps a graphical representation will make this even more clear.

So, there you have it. Futures based 1x inverse ETFs by definition (i.e., how futures work), should have no volatility drag. Leveraged ETFs, long or short, are a whole different issue. They do have significant volatility drag. But don't confuse them with plain vanilla inverse 1x ETFs. Those have no volatility drag.

By definition.

Now, just to double check my methodology with actual market data, I compared the performance of the S&P500 ETF (NYSEARCA:SPY) and its inverse SH (NYSEARCA:SH). Since 6/21/2006 when SH debuted, SH has returned (annualized) -9.3%. If you use the approximate formula on daily returns of SPY, SH should have returned, annualized, -11.0%. Using the accurate formula, it should have returned -6.1%. So, the approximate formula clearly overestimates the loss. But why is the accurate formula yielding a result so much higher than the actual return of SH?

Well, it is easy. SH has to pay the S&P500 ~2.1% dividend yield being a short fund, and it has a load of ~0.8% above that of SPY. Subtract that from the -6.1% annualized return for the simulated SH calculated using the accurate formula, and you get -9.0%, compared to the actual return of -9.3%. (The difference is market randomness where ETFs don't always trade at NAV.) The approximate formula would give you -13.9% compared to the actual return of -9.3%.

Q.E.D. I hope that people will now stop talking about the volatility drag for 1x inverse ETFs. There is none.

**Disclosure: **I am long XIV, SVXY, UPRO. 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.