Barclays quants error on leveraged ETFs
In a recent article, Cheng and Madhaven from Barclays Global Investors published a good article on leveraged ETFs
Check it out.
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 correct way to look at this is to let
If is 1 day, then
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
This allows us to rewrite (using the definition of the binomial coefficient)
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…