HOW THE FORMULA WORKSA description of the Gaussian copula function from an article in Wired magazine by Felix Salmon:
Consider something simple, like a kid in an elementary school: Let's call her Alice/> />. The probability that her parents will get divorced this year is about 5 per cent, the risk of her getting head lice is about 5 per cent, the chance of her seeing a teacher slip on a banana peel is about 5 per cent, and the likelihood of her winning the class spelling bee is about 5 per cent. If investors were trading securities based on the chances of those things happening only to Alice/> />, they would all trade at more or less the same price.
But something important happens when we (also look at) the girl she sits next to, Britney. If Britney's parents get divorced, what are the chances that Alice/> />'s parents will get divorced, too? Still about 5 per cent: The correlation there is close to zero. But if Britney gets head lice, the chance that Alice will get head lice is ... about 50 per cent – which means the correlation is probably up in the 0.5 range. If Britney sees a teacher slip on a banana peel, what is the chance that Alice/> />will see it, too? ... It could be as much as 95 per cent, which means the correlation is close to 1. And if Britney wins the spelling bee, the chance of Alice/> />winning it is zero, which means the correlation is negative: -1.
If investors were trading securities based on the chances of these things happening to both Alice and Britney, the prices would be all over the place, because the correlations vary so much.
Math whiz proposed applying this statistical formula to credit risk, and financial meltdown ensued
Mar 18, 2009 04:30 AM
Cathal Kelly
STAFF REPORTER
Note: This article has been edited to correct a previously published version.Former/> University/> of Waterloo/> />statistician David X. Li didn't burn down the American economy. He just supplied the matches.As economists and market watchers cast about for people to blame for the U.S./> />market meltdown, Li has surfaced as a scapegoat. Recently, Wired magazine ran an article on Li's work subtitled, "The Formula That Killed Wall Street."The formula in question is the so-called Gaussian copula function. On the most basic level, the formula allows statisticians to model the behaviour of several correlated risks at once.In a scholarly paper published in 2000, Li proposed the theorem be applied to credit risks, encompassing everything from bonds to mortgages. This particular copula was not new, but the financial application Li proposed for it was.
Disastrously, it was just simple enough for untrained financial analysts to use, but too complex for them to properly understand. It appeared to allow them to definitively determine risk, effectively eliminating it. The result was an orgy of misspending that sent the U.S./> />banking system over a cliff."To say David brought down the market is like blaming Einstein for Hiroshima/> ," says Prof. Harry Panjer, Li's mentor at the University/> of Waterloo/> />. "He wasn't in charge of the financial world. He just wrote an article."When David X. Li first arrived from China/> />in 1987, he was known as Xiang Lin Li. He already held a masters in economics from Tianjin/> 's Nankai/> University/> />. He was one of a group of the faculty there who won a scholarship to study business in Canada/> />through CIDA. In order to claim his prize at Quebec City/> 's Laval/> University/> />, Li was given four months to master French."We were all highly motivated," says Jie Dai, who was in the program with Li. "He was from a small town in the south of China/> />. A small family, very ordinary, not poor or rich. There wasn't anything distinguished about his personality."Li graduated with an MBA in 1991. Most of his Chinese classmates were bound for academia. Li saw a more worldly future. Says Dai: "I clearly remember him mention that if you are an actuarial guy, you can earn a lot more money."Li had recently married a colleague from Nankai when he decided to study at Waterloo/> />'s department of statistics and actuarial sciences. He was drawn by the work of Panjer, a world leader in the study of loss modelling, especially as it applies to the world of insurance."He had the ability to take ideas from different fields and synthesize them," Panjer says.
In Waterloo/> />, Li lived the hand-to-mouth life of a grad student. He anglicized his name. He and Panjer became close, and still correspond. Over six years, he earned his third masters and a PhD.After graduation in 1997, Li taught briefly. He worked for CIBC World Markets. But his ambition quickly drove him to New York/> />. He tore up the corporate ladder. By 2000, he was a partner in J.P. Morgan's RiskMetrics unit, trying to find ways to leverage a new generation of risk-based financial assets.His breakthrough was an article published that year entitled, "On Default Correlation: A Copula Function Approach."Many of the ideas contained within it were drawn from statistics research Li had observed firsthand at Waterloo/> />. His insight was to transfer the work to financial models.Li's model sidestepped the problem of trying to correlate all the variables that determine risk. Instead, it based its assumptions on the historical dips and swells of the market itself. In essence, Li used the past to map the future.
"It was a very simple mathematical answer almost anyone could use," Panjer says "And when you've got a hammer, everything suddenly looks like a nail. They jumped on it."Through the lens of Li's theorem, even the shakiest investments suddenly looked viable. The Gaussian cupola created the sort of financial alchemy that made high-risk mortgages and credit card debt look like triple-A rated gold.Money poured into CDSs (credit default swaps), a financial device that acts as an insurance policy against defaults. By the end of 2007, the total investment in credit default swaps had swelled to $62 trillion ( U.S./> />), a 6,700 per cent increase in only six years.Li didn't make money directly off the idea, but it made him famous.
Maybe he sensed the danger inherent in the system he'd help establish. By 2005, Li was among those warning about the limitations of his model. "The most dangerous part is when people believe everything coming out of (the model)," he told The Wall Street Journal.
What Li's theorem could not do was predict what might happen in extreme economic environments, what experts call "tail dependency." And one was arriving.The 2008 collapse of the U.S./> />housing bubble rendered Li's model useless. Defaults that the model had not predicted piled up, rippling through U.S./> />banks and wiping out trillions of dollars in investment.But Li's colleagues say he's not to blame. "We have a saying in statistics, `All models are wrong, but some are useful,'" says Panjer. "He supplied something, a tool kit, for financial analysts. They took one small part of it and used it in ways he had never intended."Li since moved to Beijing/> />, where he heads the risk management department for the China International Capital Corp., a major investment bank. He has not commented on the meltdown or his role in it.
