Phorgy Phynance

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.



Written by Eric

April 5, 2011 at 11:12 am

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