Monday, February 23, 2009

Measuring ETF Decay

First visit - More On ETF Decay.

So how much do levered ETFs actually decay? I ran a few examples on paper, then computed the formula.

Say we have an underlying A, that is trading at 27.00 while it's double short Z is trading at 100.

Consider the example of a two day move in A where it drops 2% and then the next day retraces completely back to 27.00.

On that move, Z, the double short, will first rise to 104.00. Then the following day, it should theoretically close at 99.755.

So, a 2% drop and retrace (note the up move is greater than 2%) will *decay* the levered short by .25% - which isn't too bad.

Let's now consider a larger drop-and-retrace. A 10% drop on A would bring its price down to 24.30 - and obviously the retrace would bring it back to 27.00.

In that case, Z, the double short, would rise to 120.00 on day one - and then fall to 93.33 on the underlying's retrace. So after a two days of net-flat trading, the double short has just gotten hammered, losing 6.66% of its value.

Here's the formula:

Lemma - Given an underlying A and its daily compounding double inverse Z, a drop of R-percent which retraces fully the following day will DECAY Z by:

6*R2 / (1 - R)

when .3333 > R > 0

In other words, the initial drop can't be greater than 33%. Taylor will explain why in the comments.

Let's test my examples.

6*(.022) / (1 - .02) = .0024 which correlates to the 99.75 price we *retraced* to.

6*(.102) / (1 - .10) = .0666 which correlates to the 93.66 price the double inverse Z *retraced* to the second example.

Conclusion - The last thing a levered ETF holder wants is rapid, large price retracements!

Time for homeschool to open. This discussion will be continued later.


Anonymous said...

that explains, as you were saying earlier, why ETFs aren't suitable for volatile markets. Is there some way you might be able to work the VIX into the calculation as well? I was thinking that that might be able to tell you when the volatility will do you in. I just got up so if this doesn't make sense, I'll try again later.

Further, I saw a similar conclusion on ETFs from Goldman Sachs in either the Financial Times, WSJ or NYTimes...can't remember which but I've been trying to track down a link to post.

auntulna said...

My broker told me there were about 1000 shares of SDS and SSO available to short. With such a small number, the trade could be closed out without warning.

CaptiousNut said...


They have PLENTY of shares to short. They are partially lying.

Ask them how much you have to pay (10% interest) to borrow some shares. What they are doing is lending to the big boys; they are lending to whomever pays the most margin interest.

Or, buy a deep put like I did with XHB - they may let you stay short after expiration. Or sell a deep call. You won't have *pay up* too much.

Taylor Conant said...

Not sure, I did some math and a drop of 34% for the original stock seemed to work out okay. However, if the original drop is 50%, then the retrace is 100% (a doubling) so the inverse would have to lose -200% and that isn't possible.

What am I missing here? I'm sure the answer is obvious, but alas...

There's gotta be some bound whereby the retracement results in a 100% or greater loss. But 33% didn't seem like it was it.

CaptiousNut said...

Your math is wrong. A stock drops from 3 to 2, a drop of 33%.

Its retrace, from 2 to 3 represents a rise of X %.

Solve for X.

Then reconsider.

Taylor Conant said...

Rise from 2 to 3 is a rise of 50%. The inverse goes down by 100%. Okay, I guess I get it now. I don't know why I didn't get that at first. I'm telling you, I'm a mathtard.

Kevin said...

Here is yet another article describing leveraged decay: