A couple points:

If we grant no mean reversion, every daily return must be i.i.d. If so by the basic properties of expected value the more leveraged you are the higher the expected value. I believe this means Tom must be incorrect in his more extreme point, even if it would be a very risky bet.

If there's an intrinsic decay to compounded daily returns, why would 1x daily returns be the optimal ratio? Why not capitalize on decay by only buying an ETF that tracks half the daily return? At what point do decay proponents conclude that we have achieved optimal leverage?

So comparing a double leveraged asset and an unleveraged asset, with no drift, no mean reversion. Vol at .01 daily for the underlying. Starting at $1000, holding with no adjustment for a 10 year holding period.

50k sims

underlying
mean - 1003.00743034
std - 536.829459263




leveraged
mean - 994.43460712
std - 1303.72193855




Completely trendless but I don't see the decay. I'll try adjusting drift and mean reversion next to see the impact

Yup

Exactly. And this is why I've mentioned several times that there's much better ways to express a mean reversion bet than to let your position grow to an arbitrary size with a short leveraged ETF position. You can always make the bet with a controlled size directly with the underlying or with options. It's probably rare that you'd want to do it with the leveraged ETF, but if you decide that's a good instrument to do it with, you'd want to adjust your position sizes over time rather than just shorting an holding forever and hoping you don't blow up.

But as I mentioned before, if the borrowing cost is within reason, there are potentially other reasons to short these things related to the funds hemorrhaging capital that have nothing to do with "the math".

These graphs might be more fun if you plot them on a linear axis so you can see the yuuuuuuuuge positive skew.

When people see that the majority of results will fall below the breakeven point but they're offset by massive gains a small percentage of the time they might think twice about the conclusions they draw from having moderate success shorting and holding them over the past 10 years. The idea that they always go to zero in the long run is very dangerous to one's presumably finite finances.

I love how everyone seems to think that the daily compounding of a +3x fund equals the daily compounding of a -3x fund.

I'm not sure why you'd think that. Basically the whole point of the majority of this conversation has been to point out that "the math" isn't the reason for "decay". There's other reasons the daily compounding isn't perfectly mirrored (ie. there's real decay, not this phantom "math" bullcrap).

I agree that you have been the most vocal proponent against math and reality, and that if posting the most times was a measure of being correct, then you must be correct.

LOL. In your last post you finally touched on the reason these things suck - that they don't perfectly track the underlying (ie. they bleed money because rebalancing isn't free or easy). I thought you might have actually gone and learned something. Bummer for you.

You aren't doing a daily readjustment to keep the percent leverage constant.

I buy $2000 worth of X , pay $1000 + borrow $1000. That is 200% leveraged.

After doubling, I own $4000 worth of x, and I still have only borrowed $1000 since I have not rebalanced. I am not not at 200% leveraged anymore.

The leveraged (and inverse) funds maintain their leverage over time.

No. I showed the math on why they suck. You continue to ignore the math.

I will post another spreadsheet picture tomorrow.

The only "math" you've shown is that these things lose money when the underlying starts at $100, fluctuates a little, and returns to $100. Nobody is ignoring it the implications of this. We all understand it and it helps to illustrate why leveraged ETFs with strong mean reverting underlyings would decay. But your little chart also shows that trending leveraged ETFs outperform their underlying (x leverage), which you chose to ignore completely.

sp,

What is "the math" that people are citing?

Basically, when it's up and down, not trending, you have less capital on the positive days and more capital on the negative days. So you gain less and lose more than the underlying. But when the underlying goes one direction for a while, the compound returns blow up.

I captured the leverage in volatility. Since the volatility is 2x on every day, we are always 200% leveraged.

There are two elements to the daily borrowing readjustment. Doubling the drift/vol (which is the 'math is too strong' argument) and the costs associated with being so active (sp argument - that I tend to agree with).

1% daily vol:

mean - 999
std - 534
max - 6011



2% daily vol:

mean - 994
std - 1278
max - 26194





Fascinating! Makes perfect sense though, normal returns will result in a lognormal price. Worth noting that the vol I chose was random so its a happy coincidence that the the difference between the two is so drastic but it is still illustrative.

Ya that seems very easy to understand. What I'm not understanding is why there is such intense debate over something that can be trivially modeled by anyone who cares. Make some reasonable assumptions and just run a monte carlo and see if a levered product makes sense for your use case.

Or just guess? I dunno. Seems kinda confusing to me since I don't really understand what either side is arguing about. Except I did have a good lol over using a 150y sample (1870 is totally gonna be like 2050) to predict a 40y sample (NEVER HAPPENING), that was amusing.

Good article on mean reversion: http://www.osfi-bsif.gc.ca/Eng/fi-if...ages/mnrv.aspx

It will be regime-based, but I can't think of how to easily model that so I'll assume a static mean reversion.

Can someone run the math on the 3x levered daily DOW return from 2/11/10 through yesterday? I am running the hypothetical return (based on 3x daily DOW moves) and coming up with an initial investment of $13.2333 (where the ETF UDOW was on day 1 on a split adjusted basis) resulting in a value of $72.26 at yesterday's close. This contrasts with the actual instrument closing at $108.59.

Pete or others, do you believe that the drift from rebalancing is always down or can it be up in a trending market? It doesn't make any intuitive sense to me. The fund is trading within .02 of NAV fwiw so its not a fund premium issue.

Here are the index* results for +3x and -3x with high realized volatility and no realized drift.



*index = absolutely no costs included.

(you did show that $ gains/losses aren't symmetrical with % gains/losses, which is cool)

I'm not sure what you think this shows? It's just a single possible path, and doesn't make a point for or against leveraged indexes, beyond showing that in this one instance they lost.

I'll run this at some point, but we've established that the median result for the 3x will likely be below the 1x. That is not in dispute. Can you respond to my previous point? If it is all about decay, wouldn't you prefer a <1x investment?

I believe a major part of the rebalancing cost comes from slippage, which will (likely) hurt you regardless of if the market is trending.

Yes, I would prefer a <1x investment if my goal was to make money with the highest probability (rather than make the most money in positive cases). A .5x daily levered ETF will come out positive more often than a levered one will. That doesn't mean it will always make more money though, just that the distribution shifts positive.

Pete keeps arguing about skew which I have never disputed. Its just that the more leverage and longer the timeframe you look at the higher the likelihood of these ending up at 0 are. If you add enough leverage all possible levered (regular or inverse) ETFs always go to 0 over time. This is mathematical and due to issues of leverage and compounding rather than slippage or skew.

This is no different from saying that if you repeatedly bet all in on a fair coin flip getting 2 to 1 on your money your bankroll always goes to zero over time.

Of course its true, but it doesn't mean taking the opposite side of the bet is a good financial decision.

Sent from my Nexus 6P using Tapatalk

Which goes back to my original point of the math being what causes you to eventually lose money if you are betting on a highly leveraged daily ETF. Leverage works against you on the downside more than it can ever work for you on the upside if your leverage is high enough and your time horizon long enough. Of course all bets are off in the short term.