# Phorgy Phynance

## 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.

### The Error

They begin from a fairly standard starting point

$dS_t = \mu S_t dt + \sigma S_t dW_t$

However, they proceed to state that since

$\frac{A_{t_i}-A_{t_{i-1}}}{A_{t_{i-1}}} = x\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}$

“holds for any period”, then it follows that

$\frac{dA_t}{A_t} = x\frac{dS_t}{S_t}$

where $A_t$ is the ETF NAV and $x$ is the leverage factor.

Unfortunately, that is not correct. The problem is that

$\frac{A_{t_i}-A_{t_{i-1}}}{A_{t_{i-1}}} = x\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}.$

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

This should have also been obvious by plugging $t=1$ into their final expression

$\frac{A_t}{A_0} = \left(\frac{S_t}{S_0}\right)^x \exp\left[\frac{\left(x-x^2\right)\sigma^2 t}{2}\right],$

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

$G_{i-1,i} =\frac{S_{t_i}}{S_{t_{i-1}}}$ and $G_{x,i-1,i} = \frac{A_{t_i}}{A_{t_{i-1}}}.$

If $\Delta t$ is 1 day, then

\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}

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

$G_{i-1,i} = \exp\left(\bar\mu \Delta t + \sigma \sqrt{\Delta t} W_{\Delta t}\right),$

where $\bar\mu = \mu - \frac{\sigma^2}{2}$. With a slight abuse of notation, we can drop the indices and let

$G =\exp\left(\bar\mu \Delta t + \sigma\sqrt{\Delta t} W_{\Delta t}\right)$

so that

$G^i =\exp\left(\bar\mu i \Delta t + \sigma\sqrt{i\Delta t}W_{i \Delta t}\right).$

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}

Noting that

$E(G) = \exp\left[\left(\bar\mu + \frac{\sigma^2}{2}\right)\Delta t\right] = \exp\left(\mu\Delta t\right)$

and

$E(G^i) = \exp\left(\mu i\Delta t\right) = E(G)^i.$

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}

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…

Written by Eric

May 4, 2009 at 7:38 pm

## 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.

Written by Eric

April 11, 2009 at 11:05 am

Posted in ETF, Leverage, Leveraged ETF

Tagged with , ,

## 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:

$R_{\text{Index}} = (1+R_1)(1+R_2)-1 = R_1 + R_2 + R_1 R_2$

Now consider the return of a triple-leveraged ETF over the same two-day period:

$R_{\text{ETF}} = (1+3 R_1)(1+3 R_2)-1 = 3 (R_1+R_2) + 9 R_1 R_2$

In other words, err… symbols

$R_{\text{ETF}} = 3 R_{\text{Index}} + 6 R_1 R_2$

After just two days, you can see that the ETF return will naturally deviate from the index return by a factor of $6 R_1 R_2$ even if the ETF is perfectly tracking the index.

The same logic extends to ultra-short ETFs:

$R_{\text{Short ETF}} = (1-3 R_1)(1-3 R_2)-1 = -3 (R_1+R_2) + 9 R_1 R_2$

Or

$R_{\text{Short ETF}} = -3 R_{\text{Index}} + 12 R_1 R_2$

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.

Written by Eric

December 3, 2008 at 11:58 pm

Posted in ETF, Leverage