Archive for the ‘ETF’ Category
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 
Unfortunately, that is not correct. The problem is that
only holds when 
This should have also been obvious by plugging 
![\frac{A_t}{A_0} = \left(\frac{S_t}{S_0}\right)^x \exp\left[\frac{\left(x-x^2\right)\sigma^2 t}{2}\right],](http://s0.wp.com/latex.php?latex=%5Cfrac%7BA_t%7D%7BA_0%7D+%3D+%5Cleft%28%5Cfrac%7BS_t%7D%7BS_0%7D%5Cright%29%5Ex+%5Cexp%5Cleft%5B%5Cfrac%7B%5Cleft%28x-x%5E2%5Cright%29%5Csigma%5E2+t%7D%7B2%7D%5Cright%5D%2C&bg=ffffff&fg=1c1c1c&s=0 "\frac{A_t}{A_0} = \left(\frac{S_t}{S_0}\right)^x \exp\left[\frac{\left(x-x^2\right)\sigma^2 t}{2}\right],")
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
If 
![\begin{aligned} G_{x,i-1,i} &= 1 + x \left(G_{i-1,i} - 1\right) \\ &= (1-x)\left[1+\left(\frac{x}{1-x}\right) G_{i-1,i}\right]\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+G_%7Bx%2Ci-1%2Ci%7D+%26%3D+1+%2B+x+%5Cleft%28G_%7Bi-1%2Ci%7D+-+1%5Cright%29+%5C%5C+%26%3D+%281-x%29%5Cleft%5B1%2B%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29+G_%7Bi-1%2Ci%7D%5Cright%5D%5Cend%7Baligned%7D&bg=ffffff&fg=1c1c1c&s=0 "\begin{aligned} G_{x,i-1,i} &= 1 + x \left(G_{i-1,i} - 1\right) \\ &= (1-x)\left[1+\left(\frac{x}{1-x}\right) G_{i-1,i}\right]\end{aligned}")
so that
![\begin{aligned} G_{x,0,n} &= \prod_{i=1}^n G_{x,i-1,i} \\ &= (1-x)^n\prod_{i=1}^n \left[1+\left(\frac{x}{1-x}\right) G_{i-1,i}\right].\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+G_%7Bx%2C0%2Cn%7D+%26%3D+%5Cprod_%7Bi%3D1%7D%5En+G_%7Bx%2Ci-1%2Ci%7D+%5C%5C+%26%3D+%281-x%29%5En%5Cprod_%7Bi%3D1%7D%5En+%5Cleft%5B1%2B%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29+G_%7Bi-1%2Ci%7D%5Cright%5D.%5Cend%7Baligned%7D&bg=ffffff&fg=1c1c1c&s=0 "\begin{aligned} G_{x,0,n} &= \prod_{i=1}^n G_{x,i-1,i} \\ &= (1-x)^n\prod_{i=1}^n \left[1+\left(\frac{x}{1-x}\right) G_{i-1,i}\right].\end{aligned}")
If we assume 
where 
so that
This allows us to rewrite (using the definition of the binomial coefficient)
![\begin{aligned} G_{x,0,n} &= (1-x)^n \left[1+\left(\frac{x}{1-x}\right) G \right]^n \\ &=(1-x)^n \sum_{i=0}^n \binom{n}{i}\left(\frac{x}{1-x}\right)^i G^i. \end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+G_%7Bx%2C0%2Cn%7D+%26%3D+%281-x%29%5En+%5Cleft%5B1%2B%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29+G+%5Cright%5D%5En+%5C%5C+%26%3D%281-x%29%5En+%5Csum_%7Bi%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bi%7D%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29%5Ei+G%5Ei.+%5Cend%7Baligned%7D&bg=ffffff&fg=1c1c1c&s=0 "\begin{aligned} G_{x,0,n} &= (1-x)^n \left[1+\left(\frac{x}{1-x}\right) G \right]^n \\ &=(1-x)^n \sum_{i=0}^n \binom{n}{i}\left(\frac{x}{1-x}\right)^i G^i. \end{aligned}")
Noting that
![E(G) = \exp\left[\left(\bar\mu + \frac{\sigma^2}{2}\right)\Delta t\right] = \exp\left(\mu\Delta t\right)](http://s0.wp.com/latex.php?latex=E%28G%29+%3D+%5Cexp%5Cleft%5B%5Cleft%28%5Cbar%5Cmu+%2B+%5Cfrac%7B%5Csigma%5E2%7D%7B2%7D%5Cright%29%5CDelta+t%5Cright%5D+%3D+%5Cexp%5Cleft%28%5Cmu%5CDelta+t%5Cright%29&bg=ffffff&fg=1c1c1c&s=0 "E(G) = \exp\left[\left(\bar\mu + \frac{\sigma^2}{2}\right)\Delta t\right] = \exp\left(\mu\Delta t\right)")
and
we arrive at a disappointingly simple, yet important, expression
![\begin{aligned} E(G_{x,0,n}) &=(1-x)^n \sum_{i=0}^n \binom{n}{i}\left(\frac{x}{1-x}\right)^i E(G)^i \\ &= (1-x)^n \left[1+\left(\frac{x}{1-x}\right) E(G) \right]^n \\ &= \left[1-x+x E(G)\right]^n. \end{aligned}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+E%28G_%7Bx%2C0%2Cn%7D%29+%26%3D%281-x%29%5En+%5Csum_%7Bi%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bi%7D%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29%5Ei+E%28G%29%5Ei+%5C%5C+%26%3D+%281-x%29%5En+%5Cleft%5B1%2B%5Cleft%28%5Cfrac%7Bx%7D%7B1-x%7D%5Cright%29+E%28G%29+%5Cright%5D%5En+%5C%5C+%26%3D+%5Cleft%5B1-x%2Bx+E%28G%29%5Cright%5D%5En.+%5Cend%7Baligned%7D+&bg=ffffff&fg=1c1c1c&s=0 "\begin{aligned} E(G_{x,0,n}) &=(1-x)^n \sum_{i=0}^n \binom{n}{i}\left(\frac{x}{1-x}\right)^i E(G)^i \\ &= (1-x)^n \left[1+\left(\frac{x}{1-x}\right) E(G) \right]^n \\ &= \left[1-x+x E(G)\right]^n. \end{aligned}")
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…
Bloggers Missing it on Leveraged ETFs
Back in December, I noticed several bloggers coming out against leverage ETFs. In response, I wrote
Leveraged ETF Math
to try to dispell some of the misunderstandings out there. Last week, Bespoke Investment Group, whom I generally admire, came out with an article:
Direxion 3x Financial ETFs Go Certifiably Crazy
The volume on Direxionshares’ 3x leveraged bull and bear financial ETFs shows that traders love the product. However, the ETFs have returned some crazy numbers this year. The 3x ETFs provide 3 times the daily change of the underlying index, and year to date, the financial index that FAS (long) and FAZ (short) track is down 14%. However, the 3x long ETF (FAS) is down 68% year to date, but the 3x short ETF (FAZ) is down 65%! And since the lows on March 9th, these things have returned some whopping numbers. FAS is up 195%, while FAZ is down $102.78 (or 87%). Rest assured that a lot of people have gotten burned with these leveraged ETFs, and even though they’re meant to track daily performance, their crazy longer-term returns won’t go unnoticed forever.
There is nothing crazy about the long-term returns of FAS and FAZ. The proper way to compare their performance is versus an index whose daily returns are exactly three times the unleveraged index. This is easy to do once you have the daily returns of the index. Here is the cumulative performance of FAS vs 3x the daily return of the Russell 1000 Financial Services index:
Here is the cumulative performance of FAZ versus -3x the daily return of the Russell 1000 Finance Services index:
I don’t think anyone can look at these charts and suggest Direxion is not tracking the indices well. Instead of spreading misinformation, perhaps it would be better if bloggers tried to explain these ETFs rather than set up strawman charts indicating how different cumulative returns of FAS and FAZ versus the cumulative return of the index. Of course with daily returns of 40%, the difference between cumulating 3x daily returns can deviate significantly from 3x the cumulative returns. This is perfectly normal and anyone investing in leverage ETFs should understand this. There is nothing “certifiably crazy” about it.
Direxion had the misfortune of introducing these ETFs during a financial crisis. Here is what the hypothetical cumulative performance of FAS would have looked like if it was around since 1995:
During bull markets, these ETFs suddenly do not seem so unattractive over long periods.
Disclosure: I own shares of BGU as a long-term investment. Let’s see where it is trading 3 years from now.
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:
Now consider the return of a triple-leveraged ETF over the same two-day period:
In other words, err… symbols
After just two days, you can see that the ETF return will naturally deviate from the index return by a factor of 
The same logic extends to ultra-short ETFs:
Or
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.
