Thursday, February 26, 2009

The Quant who Killed your IRA: David X. Li and the Guassian Copula Function: The Risk Management Model that Imploded

When the economy started to collapse, one of the first things I wondered was how it could happen so fast, in so many sectors. I was baffled that so many bright math & finance minds did not seem to have the tools to communicate the depths of the problem in a way that decision-makers could understand.

At that point in time, pre- Bernie Madoff, I still had some faith in our system, but still, I wondered how so many math "geniuses" could be asleep at the wheel, all at the same time! I came across an article by Saul Hansell in the New York Times in September 2008 that explained things just enough to open my eyes to the reality of the situation: "How Wall Street Lied to Its Computers: So where were the quants?"

Well, that was five months ago.The economy continues to plummet, gasping for breath from time-to-time. The article below does a good job of explaining some of the more confusing sets of events:

Recipe for Disaster: The Formula That Killed Wall Street Felix Salmon, Wired Magazine, 02/23/09
"Here's what killed your 401(k) David X. Li's Gaussian copula function as first published in 2000. Investors exploited it as a quick—and fatally flawed—way to assess risk. A shorter version appears on this month's cover of Wired."


Specifically, this is a joint default probability—the likelihood that any two members of the pool (A and B) will both default. It's what investors are looking for, and the rest of the formula provides the answer.

Survival times

The amount of time between now and when A and B can be expected to default. Li took the idea from a concept in actuarial science that charts what happens to someone's life expectancy when their spouse dies.


A dangerously precise concept, since it leaves no room for error. Clean equations help both quants and their managers forget that the real world contains a surprising amount of uncertainty, fuzziness, and precariousness.


This couples (hence the Latinate term copula) the individual probabilities associated with A and B to come up with a single number. Errors here massively increase the risk of the whole equation blowing up.

Distribution functions

The probabilities of how long A and B are likely to survive. Since these are not certainties, they can be dangerous: Small miscalculations may leave you facing much more risk than the formula indicates.


The all-powerful correlation parameter, which reduces correlation to a single constant—something that should be highly improbable, if not impossible. This is the magic number that made Li's copula function irresistible.

More from the Wired article:

"Li's copula function was used to price hundreds of billions of dollars' worth of CDOs filled with mortgages. And because the copula function used CDS prices to calculate correlation, it was forced to confine itself to looking at the period of time when those credit default swaps had been in existence: less than a decade, a period when house prices soared. Naturally, default correlations were very low in those years. But when the mortgage boom ended abruptly and home values started falling across the country, correlations soared."


"Responding to FAQ: What is Tranching?" Mark Chu-Carroll, Bad Math, Good Math blog, 9/19/2007

"In many cases, the investors have no clue how many levels of re-bundling are going on to create their top-tranch low-risk bond. And you've got all sorts of people who thought they were buying conservative investments who are now stuck with their money invested in bundles of low-tranch shit loans."

Conde Nast's Interactive Graphic: What's a CDO? 5/5/07

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