Finance's Gaussian Copulas: The New Frankenstein Monster

A former student of mine recently sent me the link to this Wired article: "Recipe for Disaster: The Formula that Killed Wall Street," by Felix Salmon (February 23, 2009) - I love it when graduates keep in touch, and I love it even more when they email me interesting material for my blog! The article describes the groundbreaking work by a quant named David X. Li, who pioneered the use of Gaussian copula models (more on what that means below) to estimate correlation between two random events, and better quantify the probability of these events occurring simultaneously. The events of interest here are mortgage defaults.

Correlation captures how much random quantities evolve together; for instance, the sale of umbrellas and sunscreen lotions on any given day is negatively correlated - if you need a lot of one, you probably won't need much of the other. In finance, quants used to struggle with the pricing of mortgage pools, popular financial instruments because mortgages were bundled together and homeowners weren't supposed to all default at the same time: a few would, but most of them would continue repaying their loans, and investors would continue receiving their money. Investors wanted to know the likelihood of two homeowners in the pool defaulting, to make sure they were properly compensated for the risks they took. Such likelihood was linked to the correlation between the two events, which was not easy to estimate, since there weren't a lot of data points (there were relatively few defaults during the real estate boom).

Li's contribution was to model default correlation in mortgage pools using available market data for credit default swaps instead. CDS basically provide insurance against defaults and are traded much more frequently than the bonds they insure, giving quants more historical data to feed their models. Moreover (and that turned out to be a problem, but it motivated the model's huge popularity at the time), Li obtained "one clean, simple, all-sufficient figure that summed up everything:" the correlation number in his copula model.

Copula is one of those fashionable buzzwords (Value-at-Risk is another one), which almost no one in the financial world had heard of a few years ago, and which have now become omnipresent. Copulas are used to transform general multivariate distributions into related multivariate uniform distributions, in order to study the dependence of the modified (uniform) random variables. For some reason, this makes the analysis simpler. There are several families of copulas, used to model different types of dependencies, but the Gaussian one is by far the most popular because of its widespread adoption by the financial industry.

Li's model allowed derivatives to be rated simply on that one correlation number. An article in The Economist states ("In Plato's Cave -- Mathematical models are a powerful way of predicting financial markets. But they are fallible" -- January 22, 2009): "the [ratings] agencies’ models were even less sophisticated than the issuers’," for instance not distinguishing between a BBB rated CDO [credit debt obligation] and a BBB rated corporate bond, despite their different risk profile. The issuers were even able to "build securities with any risk profile they chose, including those made up from lower-quality ingredients that would nevertheless win AAA ratings" because they knew (thanks to third-party companies) the models the rating agencies used to rate financial instruments.

From Wired: "Just about anything could be bundled and turned into a triple-A bond—corporate bonds, bank loans, mortgage-backed securities, whatever you liked," simply by creating a derivative with the "right" correlation number. The deceptive simplicity of the model attracted quants and, more dangerously, their non-quant bosses like moths to a flame, with the results we see today.

A key issue, emphasized in the Wired article, was that the copula model assumed correlation to be constant, although in practice it varies with time and is very sensitive to small changes in the inputs. Another problem was that the historical data for the credit default swaps (CDS) covered a very narrow range of market conditions - the period during which CDS were traded witnessed soaring house prices, so models only knew a world where there was a real-estate bubble. A similar comment was made in The Economist: "there was no guarantee that the future would be like the past, if only because the American housing market had never before been buoyed up by a frenzy of CDOs."

Interestingly, Li had cautioned against putting too much trust in his model as early as 2005 -- see "How a Formula Ignited Market That Burned Some Big Investors", by Mark Whitehouse, Wall Street Journal, September 12, 2005; the article appears eerily prescient in hindsight. Unfortunately, Li was ignored by the Wall Street crowds eager to apply his formula no matter what, and make money off it. (Investors had been reluctant to buy financial instruments when they did not have a good understanding of risk. Li's formula solved that problem and created a very lucrative industry.) Li recently returned to China and, according to a statement issued by the company where he now works, is "no longer doing the kind of work he did in his previous job and, therefore, would not be speaking to the media".

But if he follows the current developments, Li must feel he has created the financial equivalent of Frankenstein: his creature overpowered him and destroyed his world. According to Wikipedia.org, the Frankenstein book, first published in 1818, was supposed to be a "warning against the 'over-reaching' of modern man and the Industrial Revolution, alluded to in the novel's subtitle, The Modern Prometheus." Sounds familiar?

Comments

[Rod Carvalho]

Prof. Thiele,

You might be interested in taking a look at this article too:
http://www.iht.com/articles/2009/03/10/business/10quant.php

Though I am quasi-certain you've read it already (it's on today's NYTimes, after all).

[A Reader]

Nice synopsis and explanation of how we got here, Dr. Thiele. In "Frankenstein," the monster killed the scientist and then disappeared to commit suicide. I'd say that the oversimplification of Li's model helped to kill Li's career, but do you know if his model still used today?

[Rod Carvalho]

Quant-bashing seems to be a new fad. There's something interesting about these articles on Wired and NYTimes: though my friends in academia enjoy them, the ones who work in the financial services industry greatly despise them.

Finance is complex. Almost all journalism on the topic is mediocre. Same goes for Science journalism: it does not matter how amazing one's writing skills are... it just isn't possible to explain stuff like Quantum Chromodynamics in a non-technical manner (IMHO, that is).

[Aurelie]

I'm not sure if it's quant-bashing. The articles I've read seemed to blame the quants' bosses, who couldn't understand the limitations of the models (or refused to), and applied them anyway. It's very tempting, when the model is simple and elegant, to convince yourself the assumptions always hold. I remember doing that in high school on some physics exams, when there was a theorem vaguely related to the question asked. (Let's just say physics wasn't my strong suit.) I think finance companies realize they still need the quants' models, but they also need top people who understand math and aren't afraid of speaking up about models' flaws. The quants' bosses should be closer to being quants themselves. You have to remember that financial engineering only became a discipline in the 1990s - Wall Street hired physics PhDs because it didn't have a choice; there weren't too many financial engineering programs back in the days when Emanuel Derman joined Goldman Sachs. As time goes by, more quants will have the seniority necessary to be promoted to higher, more influential positions, and hopefully prevent another fiasco.

[Rod Carvalho]

The article on Wired was a bit of "quant-bashing". The one on NYTimes was much more balanced and accurate. These days, the fad seems to be to bash and blame anyone who works in Finance (not just quants). Most people don't know the difference between a trader, a broker, a quant, or an investment banker. It's understandable. Finance is now much more specialized than what it was some decades ago when we only had vanilla instruments such as stocks and bonds.

I wholeheartedly agree with you that people in positions of power in banks should be more "quantish", and understand the limitations of their models better. However, I am not sure this will ever happen. Wall Street always had a sort of "frat house" culture. Until not long ago, Goldman Sachs bankers would shmooze with their clients at strip clubs. Bankers doing cocaine and spending money on prostitutes is not unheard of. Traders insulting, harassing and terrorizing women and "over-educated" PhD's is tolerated. I know a guy who has a PhD in Math from UC Berkeley and works for a major French bank (you easily guess which), and his boss is a former US Marine Corps fighter pilot who knows little to nothing of financial mathematics. There's a culture of the "alpha male", and quants are viewed as a necessary evil. In such testosterone-driven environments, one cannot find much rationality.

The bright side is that a new Wall Street will likely emerge from this crisis. More intellectual firepower in top positions, and less irrational monkeys with giant egos in positions of power. Hopefully that would prevent future fiascos of such doomsday proportions.