David Li's Gaussian copula and the one number that broke

A borrowed idea

In 2000, David X. Li, a quantitative analyst who had worked at the Royal Bank of Canada and then Canadian Imperial Bank of Commerce (CIBC) before moving to J.P. Morgan's RiskMetrics unit in New York, published a short paper called "On Default Correlation: A Copula Function Approach" in the Journal of Fixed Income. He took the idea from actuarial science, where it had been used to model the correlated mortality of a married couple, sometimes called "broken heart syndrome": if one partner dies, the other's own risk of dying soon after rises. Li applied that same mathematical device, a copula function, to the correlated default times of thousands of separate mortgage borrowers, using credit default swap prices or bond yields to calibrate each borrower's own default probability.

The idea did not appear from nowhere. Oldrich Vasicek, a probability theorist then at KMV, had already built a version of this kind of model as early as 1987-1991 to price portfolios of correlated loans, though his work stayed inside the industry and was never published at the time. J.P. Morgan's CreditMetrics, launched in 1997, was the first widely adopted credit-risk model to use a multivariate Gaussian correlation structure. Li's 2000 paper added a clean, tractable, one-parameter version of the idea, one that could be dropped into a spreadsheet and run in seconds. Standard & Poor's introduced its own Monte Carlo Gaussian-copula tool, CDO Evaluator, in November 2001, reporting that 100,000 Monte Carlo scenarios took about two and a half minutes to run on a single PC of that era.

Correlation gets a market price

Between 2003 and 2005, investment banks, J.P. Morgan among them, created standardised, tradeable CDO index tranches, for example on a pool of 125 North American investment-grade corporate names. That gave correlation itself a market price for the first time: a bank could run the Gaussian copula backwards from traded tranche spreads to infer an implied correlation. A J.P. Morgan team introduced "base correlation" in 2004-05 to fix cases where that backward calculation gave two possible answers, or none. By this stage the model had stopped being a research curiosity and had become market infrastructure, used at scale to price and rate an enormous stock of mortgage-backed CDOs.

A warning that changed nothing

By September 2005, the model was well known enough that the Wall Street Journal profiled Li directly. He told the paper that "very few people understand the essence of the model." The admission did not slow its use. Banks and rating agencies continued to rely on the same one-parameter correlation input to price and rate securities across the mortgage market.

The assumption meets reality

Between 2006 and 2007, US national house prices, which had not fallen year-on-year since the 1930s, turned down. As prices fell across most regions at once, mortgage defaults that the copula models had priced as close to independent turned out to be highly correlated. Rating agencies spent 2007 to 2010 downgrading tens of thousands of structured-finance tranches. According to the US Financial Crisis Inquiry Commission's final report, 83% of the mortgage-backed securities that Moody's had rated triple-A in 2006 were downgraded to junk by 2010. On 23 February 2009, Wired published Felix Salmon's cover story "Recipe for Disaster: The Formula That Killed Wall Street," which argued that "Li's Gaussian copula formula will go down in history as instrumental in causing the unfathomable losses that brought the world financial system to its knees." The article won the American Statistical Association's Excellence in Statistical Reporting Award in 2010, and the phrase stuck. Li himself moved to China International Capital Corporation in Beijing in 2008 and did not give press interviews defending or explaining his model during the crisis.

This case is a model-risk case sitting behind Lecture 8's securitisation and tranching material, not a single trade or a single loss. A CDO tranche is a claim on a slice of the losses from a pool of mortgage-backed securities, or, for a "CDO of asset-backed securities," a pool built from other CDOs' mezzanine tranches. Holding a AAA senior tranche is, in effect, a short position in the assumption that defaults are independent. The Gaussian copula was the tool that priced that assumption. It compressed the pairwise default correlation among thousands of individual mortgages into a single parameter, usually written rho.

What broke was not the mathematics; it was the number fed into it. Because historical mortgage-performance and house-price data had shown almost no national decline, that data implied a low correlation, close to a world of independent defaults. The pool's actual behaviour, once house prices fell across the country at once, was closer to a world where one common factor drives nearly all defaults together. A senior tranche priced for the first world was badly exposed to the second. This failure mode differs from a hedge that broke on liquidity, or a basis trade that broke on funding. The mechanism here is calibration risk: an unobservable assumption, fed by data that had never contained the event the model was meant to price.

Re-securitisation made things worse rather than better for the same reason. Re-pooling a hundred mezzanine tranches into a new CDO does not diversify a correlated risk away, because the risk in each of those tranches was never independent in the first place. If the correlation assumption behind the first layer was understated, the same error compounds through every further layer built on top of it.