# Phorgy Phynance

## Leveraged ETF Math

There seems to be some confusion out there regarding leveraged ETFs and their ability to track the underlying index.

The key thing to keep in mind is that ETFs attempt to track the DAILY returns. This leads to some nonintuitive, but perfectly natural, behavior when looking at cumulative returns. For example, consider the return of an index over a two-day period:

$R_{\text{Index}} = (1+R_1)(1+R_2)-1 = R_1 + R_2 + R_1 R_2$

Now consider the return of a triple-leveraged ETF over the same two-day period:

$R_{\text{ETF}} = (1+3 R_1)(1+3 R_2)-1 = 3 (R_1+R_2) + 9 R_1 R_2$

In other words, err… symbols

$R_{\text{ETF}} = 3 R_{\text{Index}} + 6 R_1 R_2$

After just two days, you can see that the ETF return will naturally deviate from the index return by a factor of $6 R_1 R_2$ even if the ETF is perfectly tracking the index.

The same logic extends to ultra-short ETFs:

$R_{\text{Short ETF}} = (1-3 R_1)(1-3 R_2)-1 = -3 (R_1+R_2) + 9 R_1 R_2$

Or

$R_{\text{Short ETF}} = -3 R_{\text{Index}} + 12 R_1 R_2$

You can now see that the deviation is not symmetric since the short ETF deviates by a factor of 12 as opposed to 6 for the long ETF. As a result, if you were to plot the cumulative returns for ultra-long and ultra-short ETFs versus their index, things may begin to look screwy over time.

THIS IS NOTHING MAGICAL. It doesn’t mean the ETF is not doing its job. It is just a perfectly natural consequence ETF math.

Written by Eric

December 3, 2008 at 11:58 pm

Posted in ETF, Leverage