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Tuesday, April 28, 2009

via This is the Green Room by J on 4/27/09

I am very interested in the topic of leveraged ETFs (I wrote on it just last week) - and a key point I always come back to is that levered ETFs exhibit a negative drift over time. I illustrated this by plotting the inverse double levered financials ETF (SKF) vs its underlying index, the IYF. The IYF was down 66% and the SKF, which by design rises when the IYF falls, was down 16%.  That should be warning enough that leveraged ETFs do not move as common sense might expect.

One can go through the math and actually prove that

r_L = (r_U)^x\exp\left(\frac{(x-x^2)\sigma^2N}{2}\right)

where r_L and r_U are the respective ETF and underlier N-day returns, x is the ETF leverage ratio and \sigma is the N-day realized volatility. Critically, the term in the exponent must be less than one, which accounts for the negative drift. The negative drift term grows as either leverage or volatility increase, so levered ETFs are actually short volatility! As the authors of the linked piece put it:

The gross return of a leveraged or inverse ETF over a finite time period can be shown algebraically to be simply the gross return of the ETF’s underlying index over the same period raised to the power of the leveraged multiple of the ETF, multiplied by a scalar that is less than one.

continue this article here.

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