## Dec 12, 2012

### Leveraged ETFs Decay And Symmetric vs Skewed Distributions

It has been already widely discussed that LEFT do not decay in the conventional sense like for example options. As a simple thought experiment consider an options that has non-zero Θ even when underlying is not changing in price. Unlike options, LEFTs will not move if underlying ETF or index does not move. The reason why we usually observe downward drift in leveraged ETFs is due to mean-median divergence, and I will try to provide an intuitive explanation of this phenomenon without unnecessary mathematical formulas.

The only working assumption that I will use is that the asset price distribution is "fair", i.e. its expected value tomorrow is it's price today. Also, purely for illustration purposes I made up an ETF with leverage = 1.5 to better show how properties change as leverage factor is increased.

Many people have a mental model of ETF prices changes as a fair process - something like 50% chance it will go up a dollar tomorrow, 50% chance it will fall down a dollar - or some kind of symmetric price distribution.

As leverage and volatility increase, distribution becomes wider, as greater range of prices is possible. However any symmetric distribution of prices has one major problem - they allow negative prices. This is particularly obvious with higher volatility charts - the probability at zero price are greater than zero. A quick fix would be to truncate such distribution at zero, but that would break the symmetry. Practically, symmetric distributions of prices do not have economic sense.

Breaking the symmetry of price distribution however requires an adjustment - consider the case of truncated distribution: now the probability mass to the right of center (median) will be greater than the left, and will "create" expected price higher than what we started with. To make it fair we would have to lower the center (lower the median) and extend the right tail.

The expected price (mean) in all charts is the same, however the center (median) of the distribution decreases as we increase leverage. In simple words we are more likely to see lower prices as leverage is increased. That does not mean that there is a negative drift - the increased likelihood of lower prices is compensated with larger right tail.

The distribution that I used in the last chart - lognormal distribution - has been widely criticized as a poor model because it does not empirically match the likelihood of rare but extreme events, such as crashes. However the arguments that I made above do not depend on assumption of lognormality; they hold for other fat-tailed distributions as well.