I would interpret no mean-reversion to mean >=0% autocorrelation

Having thought about the math a bit, I take this back. Mean reversion is a strict requirement for these things to decay for a leveraged ETF pair that perfectly tracks the underlying.

It's pretty easy to see that the long ETF grows forever at a daily expected rate of 3x the underlying expected return, while the short ETF does the opposite. But if we take a position and hold forever the position size in the 3x long ETF grows over time while the 3x short ETF position shrinks, so as long as the expected return on the underlying stays positive (ie. no mean reversion!) the position in the long ETF grows forever and the short ETF tends to zero.

So if you short both and hold forever, and the tracking is perfect, and the expected return on the underlying is always positive, you eventually go broke every time.

Make any distribution of daily returns for the underlying that 1) isn't volatile enough to wipe out the 3x (or the shorts obviously win eventually), and 2) is 0EV, and you're going to win. Obviously if you're shorting something with a +EV underlying at low leverage, like shorting SPY at 1.05 leverage, you'll get killed, but shorting SPYx6 wins and at x7+ is a virtual doubleup.

Nothing anybody's going to pair-short for leverage decay purposes can reasonably have a sufficient ROI long-term. An underlying with a 1% SD needs a 2.5% annualized ROI (on the underlying) to make shorting the +3x breakeven and a 16% annualized ROI to make the pair breakeven over the next year. An underlying with a 2% SD needs a 10% annualized ROI (on the underlying) to make shorting the +3x breakeven and a 34% annualized ROI to make the pair breakeven over the next year. An underlying with a 3% daily SD needs a 56% annualized ROI to make the pair breakeven over the next year.

Obviously on any particular underlying, the annualized ROI can far exceed that (and/or the volatility can drop) for periods of time and bury you (or just be inherently awful like naked permashorting 3x spy because the ROI is too high for the low variance), but I don't see how you can expect ROIs anywhere near necessary to persist long-term for the 2%+ daily SD stuff.

I was taking no mean reversion to mean that the expectation of day T+1 close is the day T close in a walk process, or for +EV underlyings, that the distribution of percentage daily growth is the same except rightshifted by a constant factor representing the annualized growth ^ (1/250).

This isn't true though. Let's say you have a stupid stock that starts at 100 and every day either goes up 25% or down 20%. The median on every even day is obviously 100 forever. If you short this thing, it'll bust any finite bankroll eventually because that's how random walks roll.

Now 3x it and you go up 75% or down 60% each day and the thing ****s off 30% median every 2 days. It certainly CAN bust any finite bankroll, but you're a huge favorite to survive starting with a 10k roll. Making the underlying marginally +EG (say 76% or -60%) doesn't change that.

Shorting a leveraged +EV underlying is stupid and -EV, but you do get away with it a lot (like a -EV martingale). It takes quite a bit of underlying +EV (see 2 posts up) to bust you almost surely.

The overleveraged +EV ETFs go to 0 almost surely while growing unbounded in EV without mean reversion.

I don't think there's any disagreement on these points. I'm not sure what you mean when you say "This isn't true though." When I say "grows forever" I mean it has +EV at all points in time. I do agree that the median or even the vast majority of results can fall on the opposite side of breakeven than the EV.

This looks good too.


I don't think these numbers are right unless I'm misunderstanding terminology. What is your definition of breakeven over the next year? Are you referring to medians?

Yeah, the median of the 3x (or the 3x/-3x combo when I say that) goes down if the returns don't exceed those numbers. It takes strong mean reversion to get the actual EV of leveraged +EV stuff to go negative.

I don't think there's any disagreement aside from terminology then. When you said "expected decay" earlier I took that to mean decay in EV terms rather than the median result.

Absolutely. I'd tried estimating actual mean reversion in various forms and simulating strategies that had defined risk (e.g. bail if you've lost x% on the pair, call it a win if you've made y%) and the borrow rates were fairly in line with what I was getting as possible returns. I have no idea how much of that was a halfway decent yet still ****ty model and how much was lucky GIGO, but nothing made me want to play any pairs at anything near current borrows. I don't even know why I was really looking- if it matches, the market's right, and if by chance something looked really good, I'd just assume it was because my model was dumb lol.

Ok, so I misremembered the data I have in the spreadsheet because I haven't updated it to include any timeframes more recent than performance since the start of 2014. The model I have is slightly different from just shorting both as it includes a long holding in the underlying.

It also only includes 2x levered ETFs and not 3x ones and I don't plan to spend time adding those.

That said it includes Market Cap ETFs, Sectors, Styles, Commodities, and International.

The trade I have modeled is long 1 unit of underlying and short .5 units of both 2x ETFs.

Out of the 24 ETFs that have had an underlying, 2x short, and 2x long ETF available since 1/1/10 all 24 trades have been profitable (absent borrow costs and absent dividends) since then.

If you assume a 2% borrow off the base of 1 short unit (not bothering to do a daily borrow calc as net short exposure moves) then 1 of the trades turns negative. That would be Consumer Services (IYC, SCC, UCC). If you assume a 4% borrow then 2 additional trades are negative. Those are Health Care (IYH, RXD, RXL) and Consumer Goods (IYK, SZK, UGE) though the latter is profitable if you include dividends.

These trades make more money across the board if you go back to 1/1/07. I don't have anything before that because not enough of these levered ETFs existed to make it worth tracking previously.


Basically, the math is too strong over time to not be willing to hold this and starting with long exposure to the underlying keeps it from running you out of the position in a trend.

I will just leave this here for you guys:

LOL @ concluding that it's a good trade because "the math is too strong" when the only reason it made money is because you held a long position in a yuuuuuge bull market to offset your losses.

It's all beta. Your alpha is negative.

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You jump from results to attribution without making a convincing argument for why it is attributed to 'the math'. It seems trivial that under your strategy if the underlying went down your strategy would lose money.

How many of the names in your data went down over this period?

Of course. His position is similar to selling put options. Why wouldn't it make money in a strong bull market?

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Also SOXL/SOXS was in a huge bull and still lost with this trade. The underlying went up ~150% (+~1.5 u), SOXS died (+~.5u on the short), SOXL went up around 550% (-2.75u on the short) for a clear net loss. His trade was supposed to lock it in if he got one direction of movement, it moved that way, and the leveraged still outran the "hedge".

some of these positions have losses in the underlying. Most of then are winners without the underlying contributions.

Crude Oil (USO, SCO, UCO), FTSE China 25 (FXI, FXP, XPP), and MSCI Brazil (EWZ, BZQ, UBR) all had the underlying down but the trade outlined above is up.

Markets do tend to go up over time so there is less data for down markets even in timeframes of 5-10 years and almost none when you zoom out and look at it over the course of decades.

You guys are looking at too short of timeframes. The math always wins over time. These might not go down over the next 5 days/months/years, but they will go down over the next 5 decades. It doesn't matter if you're looking at the long or short side, nor what specific market/sector/style, nor how levered it is.

You have to trade in the present though- even with some mean reversion, the things can go on a tear and wipe you out on small percentage stakes before they eventually **** off. If you want capped risk, you can design bets where you bail after losing x%, and those don't appear to outrun the borrow, or you can buy long-dated puts, but again, NUGT is 10.93 now and the Jan 19 10.6 puts are more expensive than the 10.6 calls, so you're obviously paying a huge premium to be short instead of long if you go about it that way.

Just for funsies, I checked the borrow rate at IB on the ETFs mentioned:

SCC 4.91%
UCC 17.36%

RXD 3.41%
RXL 10.85%

SZK 3.96%
UGE 11.95%

SCO 11.89%
UCO 2.17%

FXP 5.38%
XPP 16.74%

BZQ 9.36%
UBR 7.62%

Yeah, usually. The median result is good if you have unrealistic assumptions about borrowing costs and an unlimited bankroll. Your EV is terrible though, as it is with any -EV martingale bet.

I wouldn't short any of these at those rates, but rates are negotiable at least by institutions. You say they're unrealistic assumptions, but its really not unrealistic to expect stable borrows under 3% and sometimes even under 1% on a few of these securities.

So your best case is you get cheap borrow on a few of these ETFs, so you only get to make the trade on the ones nobody else wants to short.

And that still doesn't change the fact that you get all the EV (minus small t-costs) with none of the martingale stupidity if you short both ETFs and rebalance daily.

Why don't you post the results of the ETFs you looked at so we can see what the best case is? Then everyone can evaluate for themselves if those returns are good enough to justify the large risk of blowing up.

1. you can often get a cheap and stable borrow (read: significantly below market) on some hard to borrow securities through your broker if you are willing to hold for extended periods

2. the martingale aspect is a feature not a bug, no market has ever gone up or down EVERY single day in the history of markets and thus with long holding periods (read: many years) you will profit

3. can't do this because I did it for work and not in my spare time, but the data is out there if anyone wants to do it on their own time

If you want to make huge mean reversion bets after the market trends for a significant period of time you can do it in far more efficient and controlled ways than letting an exponentially growing runaway short position inflate to astronomical risk levels.

This kind of trade is only a good idea if you're okay with running a martingale scam disguised as an alpha generator and don't care about blowing up because you're trading other people's money.