The other day I was asked why I'm short SSO as opposed to just long SDS. The answer is that there is natural drag on leveraged ETF prices. Part of this is due to the decay factor in futures pricing. But the bigger factor is just the math of multiplication and time linked returns.

Bear in mind these are my personal investments, not anything I'm doing professionally.

OK, let's use a product like SSO, which is the 2x leveraged S&P 500. The way its supposed to work is that every day, you get 2x whatever percentage return is on the S&P 500. On Monday, SSO closed at $25.73. If the S&P 500 were to finish up 1% today, SSO *should* be up 2%, or $0.51 to $26.24. If its then down 1% the next day, SSO should be down 2%, or -$0.52 to $25.72.

Over time, the math of total return (in percentage) looks like this (just in general, not of these ETF's specifically):

(1 + X0) * (1 + X1) * (1 + X2) ... (1 + Xn) -1

Where X is day n's return.

Now you'll notice that if we get the exact same percentage return, but in opposite directions, on two separate days, it doesn't mean your total return will be zero. For example, say you lose 1% on day 1, but gain 1% on day 2.

(1 - 0.01) * (1+ 0.01) - 1 = -0.0001, or -1bps.

The more severe the return, the more severe the result. Say its -10% and +10%. The result would be ...

(1 - 0.1) * (1 + 0.1) - 1 = -0.01, or -1%.

It doesn't matter what order these occur in, because multiplication is commutative.

(1 + 0.1) * (1 - 0.1) - 1 = -0.01, or -1%.

It would therefore seem like there is a natural negative drift in security prices. But in the normal world, we assume security prices aren't dependent on previous percentage gains, but on some fundamental valuation. For example, if I buy a bond at $100 but it subsequently has some credit problem that results in it falling to $90, I will have lost 10%. But if the credit problem is resolved and it gets back to par, I realize a 11% gain. I'm not limited to getting back the opposite of what I lost.

But the multiplication factor of ETFs sort of mess with this. Let's say the S&P 500 drops by 2% on day 1, then rises by 2.0408% on day 2 (which puts you back to where you started), and repeats this pattern for 6 days.

Now let's say you own the double long ETF, and we'll assume the ETF works as its supposed to. On day 1, you'd lose 4% (-2% * 2) and on day 2 you'd make 4.0816% (2.0408% * 2). But do the math...

(1 - 0.04) * (1 + 0.040816) * (1 - 0.04) * (1 + 0.040816) * (1 - 0.04) * (1 + 0.040816) - 1 = -0.24%

Why? Think about the pay back formula, i.e., percentage return you need in period 2 to go back to zero given a loss in period 1. Its ...

1 / (1+x) - 1

Now if the S&P return was x, then the ETF return is going to be 2x. But notice that...

2 [1 / (1+x) -1] <> [1 / (1+2x) -1]

See? In fact, the left equation is always going to be larger than the right equation if x is negative, and always smaller than if x is positive.

Now I can't say there is a real arbitrage here, because if the market moves higher or lower decisively, that will dominate all these pretty equations. But if you are short-term trading, it seems to me you're better off shorting the opposite ETF than going long. So I'm making a bearish play by going short SSO as opposed to owning SDS.

## Tuesday, May 12, 2009

### Leveraged ETF Math: This may smell bad, kid

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## 21 comments:

Because of the rebalance and the gamma effect you don't really match the market when you oscillate. Here's a simple chart. I'm trying to cut and paste from Excel:

Day Idx Lvl @ Close etf px idx shares Expect

1 100 100.00 2.00 100

2 90 80.00 1.78 80

3 85 71.11 1.67 70

4 95 87.84 1.85 90

5 88 74.90 1.70 76

6 102 98.73 1.94 104

I'll email it to you separately. As always, great blog.

Josh:

I'm not arguing with you. This is why there is no arb by just shorting both SDS and SSO. But I think these are reasonable short-term plays.

I suspect that your math is wrong, and there's no negative convexity drift. Looking at the SDS prospectus, it appears that the ETF is a claim on a levered portfolio, essentially 2 units of S&P (partly synthetic, with futures) and -1 unit of cash (i.e., the borrowing required for leverage). If that's so, then investors get twice the S&P return, less borrowing costs and fees, over any chosen horizon. The prospectus refers in places to "approximately 200%" of the return on the index. (Also, your convexity problem goes away if returns are tracked continuously rather than at discrete intervals.)

Oren, it's that you have something like gamma at the end of the day b/c the counterparties to the swaps are re-hedging. It will allow you to have a return over time that doesn't have anything to do with the holding period for the index. This is b/c they do a daily rebal. If you want short financial exposure you're kind of screwed, but if you want genenral .SPX exposure you can at least sell futures. You may have to roll, depending on your timeframe, but that's at least going to give you delta 1 and no gamma.

Bingo! Which is why you would want to be short ultras which have high vol. Just look at what happened to FAZ and SRS over the last few months. That is, if you can stomach the upswings.

But I'm curious, how did you get a borrow on the shares to short? I've heard you can't get them.

My math is simplistic for a number of reasons:

1) You don't get

exactly200% of the index because fees, any oddities of futures/index trading differentials, timing leverage cost, etc.2) The real market has a direction to it, so the tiny negative drift is going to be overwhelmed by the market direction. Put another way, if you knew for a fact the S&P was going to rise 25% in a month, you'd almost assuredly make money by just holding SSO.

I think what Josh is trying to convey is that even if you knew the index was going to rise 25% and you bought SSO, you likely wouldn't get a 50% return. It would be something less than that and possibly a lot less.

More importantly, if you have a long-bias portfolio and you want a generic hedge, SDS isn't a good choice.

There is arbitrage, well, close to an arb available when they price of a 2x is the same as a -2x... short them both.

AI:

I'm trying to point out that even if you were 100% right that the .SPX would be down/up x% and tried to implement that with a fund that uses any kind of leverage you are not guaranteed and kind of return. You might get a return of -x%.

Just go to google and compare UYG and IYG. They should be inverse, but they have negative returns.

Hate the game, not the playa'.

-Josh

Leverage ETFs are just the latest fad product from the fund industry. Out there only for day traders...there is NO reason to hold these things overnight so why even worry about the math. They are designed to take you anyway.

I wonder the effect price slippage has on them or if there is any because it is done base on a fixed value EOD.

Barclays has a pretty good paper out about the mechanics of leveraged ETF's and market implications- I keep this one on my desk at work for whenever my excel decides to crash and reboot (i never seem to get the memo that having 60+ excel files open is a bad idea)

http://www.barclaysglobal.com/secure/repository/publications/usa/ResearchPapers/Leveraged_ETF.pdf

the leveraged ETF's are capital grinders to the nth degree- there is never, ever any excuse for holding them longer than a few days to catch a trend.

if that link isn't showing up...

yahoo.com and search "barclays leveraged etf", first link

paper is called

"The Dyanmics of Leveraged and Inverse Exchange-Traded Funds"

It is always good to talk about problems, but I am surprised to see you guys have trouble with this.

There is no negative "convexity" : at each period, you payout is a strictly linear function.... !

Just to remind you, the linear performance ( = SF / SI -1 )is additive in composition, that is the perf of a basket of weighted stock will be the weighted perf of each stock.

And as you discovered.. this additivity property won't hold on subperiods.

Conversely, the Log perf (= log SF / SI) is additive in time (do the math for 2 periods) but not in composition... (Log being concave, Jensen inequality tells you so...)

Would you qualify this mismatch as a "convexity adjustment" ?

The problem is not one of convexity or time decay, but your measure instrument.

by the way, there is indeed a time convexity in those ETF, but not where you think it is.

Imagine what happens if your underlying goes at -60pct in 1 day.

Then your pefr is .. -120 pct : you owe money to the bank, and the bank is then at risk of not being paid.

To prevent that, I guess the ETF buys options, strike = 100%( 1 - 1 / leverage) = 50 pct for a *2 , 66 for *3 etc... that restrike each day.

Those are called daily clicquet option, or krach put options.

As option they are convex indeed, and have a time decay..

For a scary visualization of how terrible these instruments are, check out the performance of XLF (1x long financials), UYG (2x long), and SKF (2x short). Very roughly, XLF is down about 50% since the beginning of '08. UYG is down about 90%. And SKF is DOWN over 50%. Imagine getting "short banks" at the beginning of 2008 and being down 50%... Granted these don't all track the same financial index, and it was a period of unprecedented volatility, but the results are quite astonishing.

One other thing to note for long term holders, these things not only are virtually guaranteed to lose value over time but also are highly tax inefficient. Check out some of the massive distributions that were pumped out by the 2x short ETFs last year. Some were like 50% of NAV as a short term capital gain. Trading sardines only, and likely to be banned by the SEC within 12 months if I had to guess.

I don't agree that you can't own these things overnight. You just have to know that the level of your bet changes slightly every day.

I played with an options strategy involving short puts on both SSO and SDS but couldn't make it work.

Your math is right. Just look at what happened to SKF and FAZ.

Here's something else:

VIX ETNs (ETNs are cousins of ETFs but are treated differently for tax purposes). Could be a nice play setting up if you choose to take advantage of the decay. Volume is a bit light though.

http://debtsofanation.blogspot.com/

2009/05/debts-of-spenders-vix-etns-

and.html

you can absolutely short both Ultras - the risk is that you get a big ONE WAY rally - which will hurt you (assuming on day 1 you short $100k of each, for example, and don't rebalance).

also, it's nearly impossible to find a "pair" to borrow in order to short.

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