# 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

### 5 Responses

1. I realized this on my own when I was in a double inverse ETF and after a few months barely had broken even when I “should” have had a 10% gain. The interesting property this creates is that the value of the ETF is not path independent. If the underlying index goes up 10% in one day as opposed to over many days, then the values will be very different.

I don’t do this as my day job so I don’t want to spend much time on it, but it seems to me that this creates a scenario where it should be “easy” to profit because of the fact that volatility has a strong downward bias.

mikkel

January 4, 2009 at 1:01 pm

2. I’m surprised this property isn’t more widely known or publicised. These are marketed as straightforward short positions. However the trading strategy which underlies the etf results in distributional properties which I see as much more derivative-like.

krazyj

July 30, 2009 at 4:13 am

3. Your RshortETF = -3Ridex + 12 R1R2 , I believe is wrong and should be 6R1R2

Anonymous

February 13, 2011 at 9:31 am

• Thanks for the comment, but I think it is correct.

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

and

$R_{Index} = R_1 + R_2 + R_1 R_2$

so that

$R_{Short ETF} + 3 R_{Index} = 12 R_1 R_2.$

phorgyphynance

February 13, 2011 at 11:44 am

4. I stand corrected. –thanks

Anonymous

February 14, 2011 at 1:16 am