## 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 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 Solution

The correct way to look at this is to let

and

If is 1 day, then

so that

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

so that

This allows us to rewrite (using the definition of the binomial coefficient)

Noting that

and

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…

interesting to see where this goes. i hadn’t spotted the mistake but their conclusion no longer holds. maybe this will tell us why.

filthyMay 6, 2009 at 6:29 am

Good to see you filthy.

I’ll post what I found when I get a few extra minutes.

phorgyphynanceMay 6, 2009 at 8:03 am

come on eric. i’m hoping you can shed some light on a possibly profitable pattern.

filthyMay 20, 2009 at 11:22 am

Do you have Mathematica?

The next step is to just perform a series expansion. I did it on a napkin, but wanted to check my work before posting. Then life intervened 🙂

The basic idea is to note that

E(G_x) = 1 – x + x E(G).

Setting

E(G_x) = exp(x mu’ dt)

we have

mu’ = log[1 – x + x exp(mu dt)]/(x dt).

With Mathematica, you could expand this in terms of mu dt symbolically.

If I recall correctly, you get something like

mu’ = x mu – mu (x^2 – x) + O[(mu dt)^2].

So you do see the negative drift there, but it does not depend on sigma.

I still hope to post more on this, but am recovering from a string of 17+ hour workdays and am suffering from TLA syndrome, i.e. trying to add some more three-letter acronyms to my business card).

phorgyphynanceMay 20, 2009 at 11:43 am

thanks. when i get time i’m going to rework the paper myself.

filthyMay 21, 2009 at 12:03 pm

Hey! We all have a bit of Mathematica now via Wolfram Alpha.

Here is the correct expansion:

http://www88.wolframalpha.com/input/?i=series+log(1+-+a+%2B+a+exp(x))%2Fa

mu’ = x mu – .5 (x^2 – x) mu^2 + O(mu^3)

where I made a slight change to

E(G_x) = exp(mu’)

and set dt = 1.

phorgyphynanceMay 21, 2009 at 12:08 pm

Thanks for an excellent post. You should really continue work on this and publish either here or elsewhere.

Anyway, another indication that the Cheng and Madhavan [not Madhaven] final result must be wrong is that it depends on the final value of the underlying asset (the non-leveraged ETF) only and not on the trajectory. This is obviously not the case, as the following classic example shows:

Let’s take two days (t = 2) in which the underlying asset does not move in total,

S_2 = S_0.

This might arise if it didn’t move at all also in the middle, S_1 = S_0, in which case there is nothing to leverage and obviously A_2 = A_0 also.

On the other hand, S could have fallen by 20% and then rose by 25%, S_1 = 0.8 S_0; in this case, taking for example a two-fold leverage, x = 2, would give A_1 = 0.6 S_0 and finally A_2 = 1.5 A_1 = 0.9 S_0.

Ehud SchreiberDecember 31, 2009 at 4:37 am

Sorry, I typo’ed S instead of A twice in the final line, but the idea should be clear.

Ehud SchreiberDecember 31, 2009 at 4:41 am

Hi Ehud,

Thanks for your comments. If you have any thoughts on where to go next with this, I’d be happy to explore it. It has been a while and I’ve been distracted with other things lately.

phorgyphynanceDecember 31, 2009 at 9:14 pm

[…] might find the three articles by here, the previous post and here, respectively. Possibly related posts: (automatically generated)Mixed Bag – getting […]

Study on leveraged and inverse exchanged-traded funds « Financial Quant's BlogMarch 6, 2010 at 5:24 am

Talking practically: If we assume the difference between tn and tn-1 is one day then the subsequent maths does hold. However, shouldn’t the volatility term x*sigma actually be |x|*sigma as volatility can’t be negative?

Joe

AnonymousSeptember 11, 2011 at 9:43 pm

Hi Joe. Thanks for the comment, but no, the subsequent maths is not correct. The problem is that the expression

is false. From this point, everything else is incorrect, including any relationship between the underlying vol and the leveraged vol.

phorgyphynanceSeptember 13, 2011 at 4:20 pm

Hi, can you also check 3.2 Order flow and prices? it is substituting and rearranging (12) and (13) with no reason and I can’t understand where the error is.

PasqualeOctober 13, 2011 at 12:29 am

Hi Pasquale,

It looks like they have a typo in (12). It should be

phorgyphynanceOctober 15, 2011 at 5:49 pm