## 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!

## Sunday, May 15, 2011

### The problem with shorting leveraged etfs

As I've described in my previous post, inverse etfs decay relative to each other. After looking at their charts it is not difficult to imagine earning 'easy mony' by shorting a pair of leveraged etfs. Max Dama has done this and there are more people doing this according to Google, but I think this type of strategy is pretty risky.

Here is an example of 'easy money'. I've simulated a random walk (upper chart in blue) that has 50/50% chance of going up or down every day. And it always moves exactly by 3% (log, so it should be flat over the long run ). From this reference I've created two leveraged trackers, with +2x and -2x leverage, I'll call them 'up' and 'down'.

Suppose we start a 100 day period with a pair  consisting of equal amounts of capital in 'up' and 'down', both equal to 100\$. So the pair value on day 1 is \$200. Pair value is plotted in the lower chart.  Without a trend in the reference, the pair value decays at a constant rate exp(0.5log(leverage*(1+daily_delta))+0.5log(leverage*(1-daily_delta))).  If we short both 'up' and 'down' the lower chart will flip upside down, producing pretty good looking pnl.
But before you short sell  every available leveraged etf out there, take a look at the next chart:
All the parameters here are the same, only the underlying has two brief periods of consecutive wins or losses. This results in two heavy spikes in the pair value. If we would have shorted both 'up' and 'down' the result would be a pretty heavy drawdown. No easy money here...

## Saturday, May 14, 2011

### FAS vs FAZ - inverse etf behavior

There has been a lot of talk about leveraged etf (under) performance . In general, these etfs seem to underperform their benchmark. A google search for 'leveraged etf decay' will provide a couple of hours worth of reading material, so  I will try to limit information redundancy to a minimum. I'll limit myself to a single sentence introduction: 'leveraged and inverse etfs are based on the arithmetic returns of their benchmark, which introduces a negative tracking error'.
If you have little idea about what I'm talking about, take a look here for an explanation of the difference between the arithmetic and geometric returns.
So I'll continue the examination of inverse etf dynamics from what is already known: underperformance.

Let's first take a look at the relation between FAS and FAZ. Both are 3x leveraged versions of the same underlying index, FAZ being the inverse one.
Here can be seen clearly that while the etfs move in the opposite directions, FAS in the long run outperforms FAZ.
Their daily arithmetic returns however are performing exactly as advertised:

However, anybody holding a position for longer than one time period (being a day) should be only interested in geometric returns, or log returns.
When log returns of these two are examined, the picture changes:
Instead of following a straight line, the returns are skewed in favor of FAS. The green line here is a theoretical estimation of inverse relation based on algebraic returns.
For example: FAS gains 10% on a given day and FAZ follows with a 10% decline. In log returns this would translate to FAS: log(1.1) = 0.0953   FAZ: log(0.9)=-0.1054.   The log returns are not equal (duh!) but skewed in favor of FAS . When the position is held for a longer time and the pair moves 10% every day (no matter in which direction), we loose approx 0.5% per day of the total position.
Please take a note that this 'skew' is not about leverage, but inverse algebraic relationship. Leverage only provides more daily movement, exaggerating the skew.
A handy chart below shows the under performance of inverse etf as a function of its underlying daily change. One can see that the error is relatively small for <1% moves, but increases rapidly with bigger moves.

The difference between geometric and algebraic returns has been explained by E.Chan on his blog (and in his book) . However, he made a mistake in the calculation of average loss per time period stating it to be -0.5%.
When we have a 50/50 chance of winning or loosing 1% , in fact the expected return per minute is exp(0.5*log(1.01)+0.5*log(0.99)), which translates to -.005 % per minute, which is equivalent to -7%  in 24 hours ;-).

There are a couple of very interesting strategies that can be derived from this asymmetry, if one can handle the  math and rebalancing logic.

## Wednesday, May 11, 2011

### In trading no one should ever be absolutely sure

...and the one who is will get punished sooner or later. This has happened countless times, to traders of all skill levels. Once you think you've got a trade on your hands that can't possibly go wrong, you're brewing a recipe for disaster. Everybody knows this, me included, however I just barely escaped such a situation by plain luck.

A couple of days ago I ran my spread scanner and the good old GLD-GDX spread caught my eye. It was so streched up, that I decided to trade it. In fact it was at an extreme that has not happened for a very verly long time. This spread is a 'classic' for spread traders and it has been mean reverting for years now. I'm was sure that it would mean  revert again, and quickly got on board. The only thing that prevented me from betting big was the shortage of free cash and I did not want to close any other positions. So I ended up with a usual size of a spread bet, that I don't allow to produce more than 2% of portfolio volatility.
What happened next is a short story. The spread moved further against me, knocking 2% off my portfolio. A pity, but not a disaster.
The good part is that I can still handle this situation with calmness and make reasonable decisions about keeping this spread or taking a loss. Things could be different if my position was larger and I lost months worth of work on a single trade.
This reminded me again that trading should not be convinced with gambling or one will end up the as >90% of amateur day traders- with a blown account, but more on this later...