Archive for the ‘Leveraged ETF’ Category
Felix Salmon being hysterical about leveraged ETFs
I am a big fan of leveraged ETFs and have written about them here several times. I am not a daytrader. On the contrary, my upbringing in traditional asset management taught me the importance of investing for the long term. That is why I have happily held my position in BGU since soon after it was listed and haven’t traded it in over a year. My cost is little over $36 per share. It is now trading over $90 per share and has paid some very generous dividends in the mean time.
I first pointed out the issue of daily compounding in a December 2008 post here:
I thought the mystery would disappear, but I was mistaken. Five months later, people (bloggers in particular) still seemed confused so in April 2009 I wrote an article:
In that article, I demonstrated these leveraged ETFs are doing their jobs very well by looking at the performance of the ETF compared to the performance of a fund that exactly tracked three times the daily return of the index. The theoretical leveraged ETF and the actual leveraged ETFs were right on top of each other with negligible tracking error.
Barclay’s didn’t help things much when they published a horribly flawed analysis of leveraged ETFs which I pointed out in a May 2009 post:
Here we are, more than one year later and bloggers are still missing it on leveraged ETFs.
I like Felix Salmon. He touches on many interesting topics and I subscribe to his RSS feed to keep up with things. He is a very good filter. However, like many filters, he is not an expert in any area he touches. As a result, it is easy for him to upset people. There is no problem with this as long as it leads to interesting discussions, which it often does. One thing that does irritate me is when he takes himself too seriously and tries to step out of “filter mode” and into “expert mode”. This is such a case in his most recent posts on leveraged ETFs:
In these articles, in what could be described as crazed hysteria, Felix screeches about leveraged ETFs and how they should never ever be held longer than over night. Huh? I have happily held my leveraged ETFs for more than a year. What is the problem? I think it is fair to say I understand them.
So why do I “buy and hold” leveraged ETFs?
First, I have strong conviction in my view of the direction of the markets. I believe active management can add value, but not that much, and I do not feel like picking stocks myself. ETFs are a natural choice for me. I like the Russell 1000 because it is a highly diversified index based on large cap US stocks (which also tend to be large multinationals). As long as Bernanke remains at the helm of the Federal Reserve, I will happily stick with this leveraged ETF.
I think I’m not alone either. To quote Felix:
There are 173 million shares of TBT outstanding, which at a price of $35.65 apiece, means that more than $6 billion is tied up in TBT shares. But average daily volume is just 10.7 million shares — which means that the overwhelming majority of TBT shares are not traded on any given day.
Imagine that. Maybe they are long term investors like me. I also own TMV (3x 30-year treasury bear). That hasn’t worked out as well so far, but I have never been so confident in an investment and would happily add to this investment with more capital.
But why leveraged ETFs rather than just using margin?
I will never ever ever get a margin call for buying BGU. No matter how wrong I am. Or, as I prefer to think of it, no matter how the market behaves in the short term which I’m frankly not interested in.
Leveraged ETFs are a great way to obtain leverage without the use of a margin account.
Plus, bogus Barclay’s math aside, the thing that hurts leveraged ETFs are flat yet volatile markets, i.e. no trend. In a trending market, the leveraged ETF will outperform its margin-ed cousin significantly.
But the volatility!
Yikes. Yes, leveraged ETFs are crazily volatile, but fortunately, I do not need to write investment letters to myself and I can stomach the volatility. My own research convincingly assures me to remain happily invested.
As long as investors understand the daily compounding and can stomach the volatility, I hope these listed leveraged ETFs remain available for my portfolio.
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.
