Guess What: Leveraged ETFs Don't Decay After All

That said, most traders will still tell you that leveraged etfs don't decay after all. They'll say "flat" or something similar, but the truth is much more complex.

On May 16, 2011, onlyvix blogger E.Chans posed a puzzle to us: 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. 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:

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