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!