*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.

Your use of log returns is incorrect.

ReplyDeleteThe 3x bull fund has an exposure of 300% to the underlying. That means a $100 position will be worth $103 after a single-day move of 1% in the underlying.

Similarly, the 3x bear fund has an exposure of -300% to the same underlying. Given the same 1% move, the $100 initial position would now be worth $97.

These funds invest in the stocks themselves and swaps, I believe. There's no reason the returns shouldn't be a linear function of the underlying return, excluding management fees, transaction fees, slippage, etc. They do not invest in an imaginary derivative based on log returns.

@Pete: you're right. Some time after writing this post I realized there is an error in my reasoning. This has been resolved in a later 'Leveraged etfs don't decay' post.

ReplyDeleteHave you read this paper:

ReplyDeletehttp://www.docstoc.com/docs/5577389/The-Dynamics-of-Leveraged-and-InverseExchange-Traded-Funds

They basically show that a daily-rebalanced ETF decays at a speed proportional to abs(2X-1) where X is the leverage (3 and -3) in your case. That explains why FAZ is so "bad". FAS should be only as bad as SKF: abs(2*3-1) = abs(2*(-2)-1) = 5

Not sure if the paper contradicts your statement of no decay in leveraged ETFs under iid assumptions.