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.

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