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Leverage Causes Fat Tails and Clustered Volatility

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Doyne Farmer is awesome. I first ran into him back in 2001 (or maybe 2002) at the University of Chicago where he was giving a talk on order book dynamics with some awesome videos from the order book of the London Stock Exchange. He has another recent paper that also looks very interesting:

Leverage Causes Fat Tails and Clustered Volatility
Stefan Thurner, J. Doyne Farmer, John Geanakoplos
(Submitted on 11 Aug 2009 (v1), last revised 10 Jan 2010 (this version, v2))

We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called “value investing”, i.e. systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are normally distributed and uncorrelated across time. All this changes when the funds are allowed to leverage, i.e. borrow from a bank, to purchase more assets than their wealth would otherwise permit. During good times competition drives investors to funds that use more leverage, because they have higher profits. As leverage increases price fluctuations become heavy tailed and display clustered volatility, similar to what is observed in real markets. Previous explanations of fat tails and clustered volatility depended on “irrational behavior”, such as trend following. Here instead this comes from the fact that leverage limits cause funds to sell into a falling market: A prudent bank makes itself locally safer by putting a limit to leverage, so when a fund exceeds its leverage limit, it must partially repay its loan by selling the asset. Unfortunately this sometimes happens to all the funds simultaneously when the price is already falling. The resulting nonlinear feedback amplifies large downward price movements. At the extreme this causes crashes, but the effect is seen at every time scale, producing a power law of price disturbances. A standard (supposedly more sophisticated) risk control policy in which individual banks base leverage limits on volatility causes leverage to rise during periods of low volatility, and to contract more quickly when volatility gets high, making these extreme fluctuations even worse.

I completely agree with this idea. In fact, I discussed this concept with Francis Longstaff at the last advisory board meeting of UCLA’s financial engineering program. Back in December, I spent the majority of a flight back to Hong Kong from Europe doodling a bunch of math trying to express the idea in formulas, but didn’t come up with anything worth writing home about. But it seems like they make some good progress in this paper.

Basically, financial firms of all stripes have performance targets. In a period of decreasing volatility (as we were in preceding the crisis), asset returns tend to decrease as well. To compensate, firms tend to move out further along the risk spectrum and/or increase leverage to maintain a given return level. The dynamics here is that leverage tends to increase as volatility decreases. However, the increased leverage increases the chance of a tail event occurring as we experienced.

On first glance, this paper captures a lot of the dynamics I’ve been wanting to see written down somewhere. Hopefully this gets some attention.

 

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Written by Eric

April 5, 2011 at 11:12 am

Barclays quants error on leveraged ETFs

with 14 comments

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

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

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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:

Source: Bloomberg, Yahoo! Finance

Here is the cumulative performance of FAZ versus -3x the daily return of the Russell 1000 Finance Services index:

Source: Bloomberg, Yahoo! Finance

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:

Source: Bloomberg

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

with 5 comments

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

The word is out… it’s NOT about subprime

with 4 comments

Whenever I read about “subprime contagion”, I feel frustrated. When you get the flu, does the runny nose cause the muscle aches? No. Just because the runny nose came first doesn’t mean the flu can be described as “runny nose contagion”. The subprime mess was just the first symptom to appear in the bursting of a general credit bubble. The corporate high yield market saw very similar aggressive loan covenants. Commercial real estate. Emerging markets. You name it. We’ve been in the midst of a general fixed-income bubble since our friends at the Fed decided to keep rates far too low for far too long.

Think about this. Back in 2005, we were worried about the CDS market reaching $17 TRILLION notional. That is a HUGE number. But the notional amount is not indicative of overall exposure because of hedging, right?

If you want to see a perfect hedge, visit a Zen garden.

What is that number today? More like $45 TRILLION. That is insane. An entire new insurance industry has basically assumed that corporate defaults do not exist anymore. Not only that, a “hedge” can turn into naked exposure at the flip of a switch, i.e. what happens when the entity you bought insurance from no longer exists?

When I get a chance, I hope to start posting some more mathematical analysis of what’s going on (since that’s what I’m good at). For example, a primer on Leverage Mathematics 101 (which is partially complete) followed by Hedge Mathematics 101 would be a good start. In risk management, “hedging” basically means “let’s buy(sell) some similar securities so that we have more capital to buy other stuff.” In other words, hedging allows you to leverage yourself more.

Here is an article that may help spread the word, i.e. it’s not about subprime:

Straight Talk on the Mortgage Mess from an Insider

However, I would go even further. The truth is the current mess is not even about mortgages. Here is the word we should all be thinking about, “Fixed Income Bubble”.

Written by Eric

December 9, 2007 at 10:16 pm

Still talking about subprime

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Occasionally, I like to poke my nose in and see what my old comrades are up to over on NP. It seems the infamous thread is still alive and kicking.

However, it appears they are still talking about subprime. When are they going to realize that subprime was merely the first symptom of a massive global credit/asset bubble to surface? In economic terms, the dollar amount involved in any Treasury Dept bailout will be insignificant. However, when you multiply that amount by the insane leverage financial institutions have in, mostly off balance sheet, exposure, then things make more sense. Don’t be fooled. No policy maker cares about subprime borrowers. They are desperately trying to keep one, if not several major banks, afloat. Sorry Citigroup. The music has stopped.

Financial markets have a long way to go down still. The US economy will get whacked in a historical fashion, but we won’t be knocked out by any means. There are still strong sectors in our economy that will benefit from the coming “onshoring“. Pain is a good teacher and I, for one, will welcome the exorcism of complacency that is eminent.

Written by Eric

December 4, 2007 at 11:14 pm

Another word for hedged… leveraged

with 2 comments

Market turmoil is still quite fascinating to me and I still believe the current environment will be one for the history books and I’m still trying to take as much as I can from this learning experience.

My professional work experience is in fixed income. For two years, I was very “plugged in” to the markets and was meeting regularly with some of the greatest thinkers out there, but now I’m more of a pure “quant” and most of my news comes from blogs, web news, etc. Unfortunately, I’m not yet spending as much time with the traders as I’d like (but that should change soon I hope). Most of the major news sources, e.g. Bloomberg, seem to concentrate more on equity markets than credit and fixed income. I pay more attention to the Dow now than ever before. That is why I am so perplexed by the stock market. I thought stocks were supposed to be easier than bonds, i.e all the smart guys are in fixed income, right?

So while the credit markets seem to be imploding, stocks are doing just dandy. Maybe people are taking cues from the market cheat sheet?

Anyway, I’ve blabbered quite a bit on this blog (and at my former employer) expressing my opinion of CDOs. I even managed to upset quite a few people while expressing my opinions as well. No regrets though. I’m happy to have this hugely public diary, both here and on NP, to later look back and see how I did in regards to thinking events through. Occasionally, I still like to poke my nose in over at NP and I see kr is still giving out the occasional nugget. Here is one of his latest:

A few thoughts:
– If you took all the writedowns at a single med-to-large bank rather than seeing them across the street, you could have reduced that entity’s equity to ZERO. For instance, MER has only something like USD54bn of mkt cap and USD39bn of book equity.
– If the view is that there will be another round of writedowns in the same amount as Q3 then you will have banks desperate to raise equity (i.e. it is not just the monolines). Who would buy that equity right now? Prince Alwaleed for example has floated his own holdings so I see him more as seller than buyer for example. I don’t see guys like JC Flowers or Cerberus well positioned for this job – in retrospect, even Barclays/RBS have not been with respect to ABN, as can be seen by the action in their share price and cost of jr capital.
– Another possibility would be the downgrade to BBB like the Japanese banks, with all the implications that brings with it. I.e. serious change in business model. That has contagion and macro effects. One example is that flow trading of financials has cost people a lot.
– I think investors will call foul on the FAS157 Level-3 assets, and it will hit guys like GS seriously as their L3 reporteds are a big multiple of their mkt cap.
– There was a funny comment in this month’s BBG mag about “nobody really knows how desks are hedging the CDO assets.” That is bull – the answer is that most people were NOT HEDGING AT ALL, BECAUSE THEY COULDN’T. Stuff was originated to sell, and the exit has vanished, or, it was originated to live forever on a trading book even though people tried to avoid saying that, and there is no decent MTM approach so instead banks are showing huge volatility, mostly to the downside.
– Implications of SIV / CDO / CP demise are pretty vast. There seem to be an increasing amount of trade receivables on the market, b/c there are no conduits to fund them… means corp cost of cap is going up in unexpected areas.

My hunch is that the fed cuts on the 11th b/c liquidity is dropping again, especially with year-end. It is out of control – specifically Ben’s control. It looks like political support for the various subprime fixes has stalled. What I think is that liquidity of all things financial (i.e. non-corporate) is going to get weaker and cause a full-on crisis for a market-traded institution. The talk about Citi cutting their div is one tremor, trading activity in Barclays is another, and the fact that even AFTER all the reported loss numbers, people still don’t feel comfortable, is yet another.

I think vols are still cheap, maybe looking to buy some.

All the while I was complaining about CDOs, I was coming at it from the angle of a “quant”, i.e. thinking about how to model CDOs and how those models are used in risk management, asset allocation, etc. Too bad I didn’t understand more about the legal/accounting aspects of CDOs. The term everyone has now heard of is SIV. I was blabbering about off balance sheet leverage and fair value accounting, but didn’t realize that the entire CDO market was (to a jaded eye) a big play on accounting in addition to the obvious play on ratings agencies. If I had known about SIVs, I might have been able to do more to help some who may have now lost a lot of money. Maybe not. That’s all in hindsight. But what am I missing now? Where is the next weakest link? How are corporations hiding off balance sheet debt? Has anyone looked at “Level 3” assets in corporate, i.e. non-financial, balance sheets? Are they as scary as the big banks?

I’ll say it again… this is not a subprime issue. Subprime contagion does not explain the current environment. Subprime was just the first to blow. We are experiencing the blowup of a global fixed income bubble. In fact, some would say we’re experiencing a general global asset bubble.

Who’s going to get hurt? Financial institutions for sure. Anyone who depends directly on the value of paper assets.

Who’s going to win? People whose wealth depends on physical assets.

I’ve already lost all hope in Bernanke. He is not going to let his monicker “Helicopter Ben” go by the wayside in a “time of need”. Bernanke is going to lower rates and weaken the USD until oil exporters are forced to break the peg to the USD and inflation skyrockets. I predict that all these gloom mongers about home prices dropping by 30% will turn out to be wrong in nominal terms even if they are correct in real terms. In other words, home owners are going to be saved by the dropping value of the USD. All those on Wall Street who were so gleeful every time rates dropped are suddenly going to feel the pain when the value of their paper securities go up in smoke.

Watch out for the “happy stage of inflation”, i.e. wage increases. It will be interesting to see what the world will look like when oil is priced in EUR and the USD is no longer the world currency. Fortunately, I still have faith that we’ll come out of the current mess stronger as a country, but there will certainly be pain felt at the higher end of the wealth spectrum.

I’m actually ironically optimistic about the outlook for suburban and rural economic development. A weaker dollar will make outsourcing less attractive. That will bring manufacturing jobs back home. I can imagine a boon in suburban and rural development. Just imagine if communities developed decent broadband via fiber-to-the-home/business. Suddenly, there will be attractive jobs and living standards in affordable places.

Maybe a weak dollar is what this country needs, i.e. a good kick in the pants. Pain is the best teacher, right?

[Edit: PS, the title of the post was inspired by a great article on Financial Armageddon, but I never got around to explaining why, but have a look and it might be obvious.]