## Archive for **May 2009**

## Barclays quants error on leveraged ETFs

In a recent article, Cheng and Madhaven from Barclays Global Investors published a good article on leveraged ETFs

**The Dynamics of Leveraged and Inverse Exchange-Traded Funds**

April 8, 2009

Check it out.

### The Error

They begin from a fairly standard starting point

However, they proceed to state that since

“holds for any period”, then it follows that

where is the ETF NAV and is the leverage factor.

Unfortunately, that is not correct. The problem is that

only holds when is 1 day. Otherwise, we could let be 1 year and this would say that the 1-year return of the ETF is times the 1-year return of the index, which we already know is not true.

This should have also been obvious by plugging into their final expression

which violates the relation defining leveraged ETFs they started with. As a result of this error, their discussion of return dynamics in Section 4 must be re-examined

### The Solution

The correct way to look at this is to let

and

If is 1 day, then

so that

If we assume is a geometric Brownian motion (as they do), then

where . With a slight abuse of notation, we can drop the indices and let

so that

This allows us to rewrite (using the definition of the binomial coefficient)

Noting that

and

we arrive at a disappointingly simple, yet important, expression

The expression above governing leveraged ETFs is the starting point for further analysis. We will come back to this in a subsequent post.

To be continued…