# Phorgy Phynance

## Leveraged ETFs: Selling vs Hedging

In this brief note, we’ll compare two similar leveraged ETF strategies. We begin by assuming a portfolio consists of an $x$-times leveraged bull ETF with daily return given by

$R_{\text{Long}} = x R_{\text{Index}} - R_{\text{Fee}},$

where $R_{\text{Fee}}$ is the fee charged by the manager and some cash equivalent with daily return $R_{\text{Cash}}$. The daily portfolio return is given by

\begin{aligned} R_{\text{Portfolio}} &= w_{\text{Long}} R_{\text{Long}} + w_{\text{Cash}} R_{\text{Cash}} \\ &= w_{\text{Long}} \left(x R_{\text{Index}} - R_{\text{Fee}}\right) + w_{\text{Cash}} R_{\text{Cash}}.\end{aligned}

We wish to reduce our exposure to the index.

### Strategy 1

An obvious thing to do to reduce exposure is to sell some shares of the leveraged ETF. In this case, the weight of the ETF is reduced by $\Delta w$ and the weight of cash increases by $\Delta w$. The daily portfolio return is then

$R_{\text{Strategy 1}} = R_{\text{Portfolio}} + \Delta w \left(-x R_{\text{Index}} + R_{\text{Fee}} + R_{\text{Cash}}\right).$

### Strategy 2

Another way to reduce exposure is to buy shares in the leveraged bear ETF. The daily return of the bear ETF is

$R_{\text{Short}} = -x R_{\text{Index}} - R_{\text{Fee}}.$

The daily return of this strategy is

$R_{\text{Strategy 2}} = R_{\text{Portfolio}} + \Delta w \left(-x R_{\text{Index}} - R_{\text{Fee}} - R_{\text{Cash}}\right).$

### Comparison

For most, I think it should be fairly obvious that Strategy 1 is preferred. However, I occasionally come across people with positions in both the bear and bull ETFs. The difference in the daily return of the two strategies is given by

$\Delta R = 2\left(R_{\text{Fee}} + R_{\text{Cash}}\right).$

In other words, if you reduce exposure by buying the bull ETF, you’ll get hit both by fees as well as lost return on your cash equivalent.

Unless you’ve got some interesting derivatives strategy (I’d love to hear about), I recommend not holding both the bear and bull ETFs simultaneously.

Note: I remain long BGU (which is now SPXL) at a cost of US$36 as a long-term investment – despite experts warning against holding these things. It closed yesterday at US$90.92.

Written by Eric

October 2, 2012 at 3:24 pm

## 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 , ,