## Monday, May 16, 2011

### Guess what, leveraged etfs don't decay!

While writing the previous posts and doing the math I still had a feeling that something in my reasoning was not quite right. I could not put a finger on it, but with the help of the author of OnlyVix blog, I seem to have figured it all out.
Let me start with an old problem of 50/50 chance of winning 1% every timestep. Here is a puzzle from E.Chans blog (and book):
"Here is a little puzzle that may stymie many a professional trader. Suppose a certain stock exhibits a true (geometric) random walk, by which I mean there is a 50-50 chance that the stock is going up 1% or down 1% every minute. If you buy this stock, are you most likely, in the long run, to make money, lose money, or be flat? Most traders will blurt out the answer “Flat!”

...And they would be right. To prove this, let's write down a binomial tree for this case. I'll use 10% step to simplify the math:

Here we start with initial 100\$. Every branch has 50% probability. Notice that the expected value at each time step is exactly 100\$. However, on the third timestep the most probable value is 99\$ .

To double-check it, I've run a Monte-Carlo simulation of the above problem. Here is the result:
The average value is 0, while the median is -0.5% for 100 steps or -0.005% for one step.

The case for 5% change per timestep looks like this:
The distribution shifts to the left, bu again, average value is zero, and median is -0.1176.

So the answer to above puzzle is indeed 'flat'.

Now returning to a leveraged pair like FAS&FAZ, here is a Monte-Carlo simulation of a leveraged pair:
Here I've used a normal distribution for returns of the underlying with sigma = 1%. Once again, the average return over 100 periods is zero, while most of the occurrences are negative.

This means that leveraged etfs don't decay over time, they just look like they do, because that is the most likely outcome.
So here we go, contrary to common belief, the leveraged etfs don't decay after all!

1. This is easiest to see at the extreme: double or nothing.

The expectation E(X) = 2*X*(p=0.5)+0*X*(p=0.5) = X

So the expectation is to be flat. However, after n rounds, you will have 2^n money with probability 2^-n, and zero otherwise. So, given enough time, you will almost surely have zero money, although the expectation is always to be flat.

2. Great post, keep 'em coming!

3. Jev, hi. Your binomial tree is fine, but I am not sure that it disproves ETF decay. Perhaps the question is decay relative to what?

So for instance if one invests 100\$ in a 2x ultrashort ETF and 200\$ short on the underlying ETF for a period of a week, the end value of the two will surely be different because of:

- the path up and down of the price at end-of-day decreases or increases the capital base for the next day's investment in the ETF (i.e. compounding)
- tracking error in the implementation of leverage either through options or borrowing costs, or both

Perhaps I misunderstood what you are showing.

4. @Tr8der: my point is that *expected* return over the long run in case of leveraged etfs is zero. However, negative result is *most probable*.
The 'double-or-nothing' case presented in the comments illustrates this perfectly.
In other words, there is no 'edge' in shorting leveraged etfs without expressing a view on the
future volatility.

5. The expectation of a price process with 0 drift and even probabilities of up/down movements is 0, yes. So you are correct in terms of long run process under these assumptions.

In fact in equities there is more often a significant bias towards positive return (i.e. higher prob of daily positive return). So with the compounded approach on an ultrashort, the capital / share base will bleed away on these days since at EOD these are closed out back into cash (in theory) and on the next day will be able to purchase fewer shares.

Negative price movements tend to be much larger, but less frequent than positive. One could have k x as many positive movements as negative but because negative larger could net out in the end to a 0 return, say, over some period. Because of the price path over the period though, the short on \$200 vs the ultrashort ETF on \$100 loses less (or 0 in this case).

So in practice this positive return bias (or bias of the price movement probabilities) plus the tracking error often make for the decay ...

But I'm sure I'm not saying anything you don't already know. Just clarifying for my purposes.

6. Absolutely! Leveraged etfs don't decay. At least at theory (or in ideal world). This even can be shown analytically.

Recently I performed Monte Carlo Simulation Model of some leveraged ETFs (based on GBM ... with Margin Call, with limitation on daily profit/loss, constant vs dinamic adjustment of leverage, etc). Results help me understand why this myth is so popular.

Actually many people just are confused. There is a trick in puzzles like this. I'll try to publish my thoughts on this to my blog or seekingalpha soon.

Anyway thanks for articles. I get a lot of pleasure from reading.

Have a good time ...

7. Leveraged ETFs need not decay, in terms of expectation, in your idealized iid models, but the differences between the real world and your models for many of these products imply that they do decay under a more realistic expectation.

For example, there are many leveraged ETFs where bonds are the underlying. Bond prices are mean reverting (interest rates are mean reverting), so are not realistically modeled by iid processes. This will imply a decay in expectation of these leveraged products over time.

Even equities exhibit some mean reversion that would imply that these decay. For example, try computing the variance rate per unit time of SPY taken from yahoo daily, weekly, and monthly data. Any stochastic process with uncorrelated increments (e.g. GBM) has a variance rate that is independent of the sampling window. You will find, however, that for SPY that rate strongly shrinks as the sampling window increases. This is evidence of mean-reversion. To the extent that you believe the future prices will retain this feature, then any realistic expectation for a leveraged SPY shows it leaking value due to its mean-reversion.

8. I might be wrong but I think you are looking at the wrong average. Shouldn't you be looking at the geometric average instead? Or alternatively, the mean of log returns. Notice that log(0.99) < -0.01 whereas log(1.01) < 0.01 so 1% gains and losses don't exactly match.
As a result, as sequence of 1% returns with an equal number of gains and losses will lose money: (1.01)^n * (0.99)^n < 1

9. @iggy: I've got fooled by the geometric returns at one moment, but take a second look at the binomial tree. It is all about the expected return, which stays the same, no matter what kind of leverage is employed. Still not convinced? Just run a Monte-Carlo sim...

10. Yes, you are right. Expected returns are indeed the same under your assumptions.

What others have said is true though. There is a clear decay in real market environments. Shorting a leveraged ETF does have an edge but it also requires more capital since the position can ago against you (plus the borrowing costs).

11. Ummm there IS decay, if you compare a leveraged ETF vs the underlying. Compare going through the SAME branches of the binary tree for a 10% vs a 5% return. The 10% does NOT yield 2x!!! It decays when there is any volatility!

12. There is decay relative to an underlying index. You are just showing the averages for each binomial tree and only using a two period case (a critical error of extrapolation is then made). Find the average value for a 2x vs an index for each simulation, not independently over many nodes. This average will be negative, ie.there is expected decay. This decay is based on the fact returns are both positive and negative over time. Even theoretically, you almost always get the middle modes of your binomial tree. If the returns are always up, or always down, like in the two cases of your two period binomial there is actually positive "decay." This is offsetting the negative decay you find in the middle node. With a bigger tree, your negative decay outcomes will far exceed your positive and the average will not be zero.

13. J.Douglas.wood@gmail.comMarch 13, 2012 at 8:10 AM

question: if one was to short svxy or xiv as a hedge on if the market crumbles, does con tango/ backwardation / decay help as a tailwind? I'm researching this topic of whether inverse etn/etf's have this advantage and I'm getting different directions from multiple websites. I would think there is some sort of decay. Case in point: chart 1year vxx vs xiv and 1 year chart xiv vs spy. Please help me understand this

14. The analysis is correct based on the assumption that daily price movements are either UP or DOWN by a fixed percentage (e.g. 0.5% per time step). Financial markets do not follow this rule. Some days the index remains unchanged.

One option is to download daily price movements from Yahoo. Another is to just use a random number generator and scale the output distribution:

N=200;
dY_percent = randn(N,1)./100; % one year of pct changes

Now when you run the Monte Carlo simulution, use dY_percent for the price change:

values=nan*ones(N,1);
values(1)=1.00;
for i=2:N;
values(i)=values(i-1).*(1+dY_percent(i));
end

The numbers is 'values' will most always trend down in time, however sometimes they can go up. Put all the code in a second for loop and repeat a few hundred times. You can now calculate the expected probability that the leveraged ETF will be above 1.0 or below 1.0 after N trading days... or just plot the median for a quick idea if the trend is up or down. As number of trading days (N) increases, the probability of the ETF being DOWN is greater than the probability of it being UP.

15. Shouldn't you have used risk-neutral probabilities when constructing the binomial model above?

I love articles like this - they take money from the suckers, and put it into the hands of those who can think for themselves, and actually do the math for themselves.

17. Reexamine your binomial tree. When the price rises 10% and falls 10% you loose one percent. When the opposite happens you also loose one percent. Sure you gain 1% if there is 2 consecutive 10% gains or losses. The disconect between your simulation and reality lies here. You assume that there is an equal chance of two consecutive gains or losses, or alternating gains and losses. In reality regression towards the mean occurs and you experience the 1% loss more often than you recieve the one percent gain for this reason alone. Even if there was an equal likelihood of either results two consecutive moves in either direction only increases the descrepancy between "current" and "regular" pricing setting the stage for an adjustment which will eliminate what you earned times 2 (or 3 for 3x leveraged etfs). In all your math you loose the psychology of the buyers and sellers, which is where the irrationality inhearant in the market lies, and while well documented cases of reality time and time again contrast with your utopian equations.

18. Think in terms of half as much being invested in a 2x ETF, half deposited in a cash deposit account, let's say \$50 invested, \$50 cash deposit, compared to \$100 being invested in the 1x (non leveraged).

The 2x ETF manager takes the \$50 you invested with them, and borrows another \$50 in order to invest \$100 in the underlying asset.

If the return on the cash deposit is the same as what the ETF pays to borrow, there's no drag. More typically it costs more to borrow than the interest that is paid on cash, but by investing cash for a slightly longer term (3 months perhaps), the daily cost of borrowing can compare to the return received on 'cash'.

If there's no difference on a individual daily basis, there will be no difference over the longer term either.

Where apparent drag becomes apparent is that comparisons are made of the 1x non leveraged with that of having twice as much (same amount invested in 1x and 2x), which are two totally different investments (one has twice the volatility as the other for a starter) and compound out differently over time.

19. Hi

I am not sure if I understand E Chan's question.
Geometric random walk means Geometric Brownian motion? If yes, then

dS/S = mu dt + sigma dW

E(S(T)) = S0 exp ( mu T)?

it is all depending on whether the trader knows the "true" drift rate from a crystal ball when he enters to the position, and ideally the vol is constant etc.

I think random walk doesnt mean we can't make money assuming that we are SOOO smart knowing all the true parameter values

Paul

1. Hi Paul,

Making money from a random walk??? Wow, how??? (provided that a random walk has no mean bias)

20. Hi sjev,

Trevor is correct in that your example already will cause the ETF decay over time without any leverage.

Let's take your example of an ETF priced at \$100. If this stock goes up 10% one day, then drops 10% the next, the end price will be 99, as your example shows.

The demonstration of leveraged ETF decay should be shown by the original unleveraged stock maintaining its value. For this, you should take an ETF that starts at 100, and either goes up 10% or -9.0909%. Assuming it continues this trend daily, the price will stay at 100. 100*1.1*0.90909 = 1.

Now take my example and make it 2x leveraged. So instead of going up 10% and going down -9.0909%, it would go up 20% and go down 18.1818%.

If you multiply 100*1.2*0.81818, you will get 98.18, showing that levered ETFs decay over time.

21. The expected return is very misleading. From your binomial tree, at the 3rd timestep, there is a 75% chance that you've lost money. At the 4th timestep you're back to 50% > 100 and 50% < 100. At t=5, 68.75% of the returns are less than 100 (you've lost money). Here's the breakdown at t between 1 and 25. I'm not sure what happens as t becomes very large, but after t=20 it looks like there is consistently a greater than 50% probability that you lose money (despite the fact that the expected value is 100).

t=1; avg=100.0; % < 100 = 0.0
t=2; avg=100.0; % < 100 = 0.5
t=3; avg=100.0; % < 100 = 0.75
t=4; avg=100.0; % < 100 = 0.5
t=5; avg=100.0; % < 100 = 0.6875
t=6; avg=100.0; % < 100 = 0.5
t=7; avg=100.0; % < 100 = 0.6563
t=8; avg=100.0; % < 100 = 0.5
t=9; avg=100.0; % < 100 = 0.6367
t=10; avg=100.0; % < 100 = 0.5
t=11; avg=100.0; % < 100 = 0.623
t=12; avg=100.0; % < 100 = 0.5
t=13; avg=100.0; % < 100 = 0.6128
t=14; avg=100.0; % < 100 = 0.5
t=15; avg=100.0; % < 100 = 0.6047
t=16; avg=100.0; % < 100 = 0.5
t=17; avg=100.0; % < 100 = 0.5982
t=18; avg=100.0; % < 100 = 0.5
t=19; avg=100.0; % < 100 = 0.5927
t=20; avg=100.0; % < 100 = 0.5
t=21; avg=100.0; % < 100 = 0.5881
t=22; avg=100.0; % < 100 = 0.6682
t=23; avg=100.0; % < 100 = 0.5841
t=24; avg=100.0; % < 100 = 0.6612
t=25; avg=100.0; % < 100 = 0.5806

22. Most leveraged ETF re-balance daily. Re-balance is the cause of decay.

23. John Jiang or someone else, can you explain that further? It seems most people here don't understand the concept since everyone disagrees with another. Very interesting!

24. There is decay because leveraged ETFs double or triple the PERCENTAGE change of the non-leveraged ETF, not the monetary price. So if today the price of USO is \$10 and it increases by \$1 to \$11, that is an increase of 10%. If UCO (the 2x leveraged ETF) was also \$10, it would increase by 20%, meaning it would go to \$12. However, if USO were to fall tomorrow by \$1 from \$11 back to \$10, a decrease of 9.09%, UCO would decrease 18.18%, which is actually a \$2.18 decrease, meaning that then the price of UCO would be \$9.82 compared to USO still at \$10.

It's the same thing if the prices fall first then rise: USO falls \$1 from \$10 to \$9, a loss of 10%, which would mean UCO would drop 20%, from \$10 to \$8. If USO then regained \$1 the next day to \$10, that is a gain of 11.11%, which means UCO would rise 22.22% from \$8, which only gets it back to \$9.77.

So yes, if the price hovers around the exact same spot, you will slowly lose money, which completely depends on the volatility of the market. It also greatly increases your losses or gains if the price trends up or down over a period of time. If USO is \$10 and rises \$1, or 10% in one day, it's price would go to \$11, whereas UCO would gain 20% to \$12. However, if USO gained that same \$1 over 2 days...observe:

USO day 1: gains \$0.50 to \$10.50 (gain of 5%)
UCO day 1: gains \$1.00 to \$11.00 (gain of 10%)
USO day 2: gains \$0.50 to \$11.00 (gain of 4.762%)
UCO day 2: GAINS \$1.04764 to \$12.04764 (gain of 9.524%)

USO gained the exact same monetary amount, but split over two days, it resulted in UCO gaining \$2.04764 gain instead of a \$2 gain.

So, you can still make money in leveraged-ETFs (and it doesn't always have to be day trading), but you need a clear picture of timelines and the effects of the decay. You're doubling down but giving the house a 49-51 advantage. This is not a game you play a lot, it's only for when you are supremely confident your chosen investment is about to trend upward within a carefully thought out timeline. You cannot think "I'll just buy it and forget about it for 10 years, and hope it's doing good then". This is a "for very good reason I think that within x amount of time it will rise x amount, which supersedes the small, constant penalty I will pay".

Another way to think about it is that you are buying your investments with a credit card that you are paying interest on. If you're paying 10% APR on your card but use the money to make 18% on real estate tax liens, great, but if put that money into a CD making 2%...not so much.