VaR + Gaussian copula function: Magic wands for making risk vanish
Many have pointed fingers at the VaR (value at risk) formula for modeling financial risk as the cause of excessive Wall Street risk taking. An interesting Wired.com article wags its finger at another risk model driving excessive risk taking: David X. Li’s Gaussian copula function.
Both risk models have legitimate uses but were totally misused by Wall Street, either because bankers and ratings agencies failed to understand their limitations and the implicit assumptions underlying their applicability or because bankers and ratings agencies didn’t want to think about the formulas' limitations. Those formulas were based entirely on the (unreasonable) assumption that the future would look remarkably like the very recent past. For example, the assumption that housing prices would continue rising forever was baked into these models. Because bankers' pay was based on short-term “profit,” bankers had no incentive to model the truth because these formulas — by systematically underestimating risk, esp. systemic risk — enabled those banks to take on much larger risks and rake in billions in extra “profits” than they could have had they accounted more honestly for systemic risk that would drive down all asset prices.
You can read the details, but it boils down to this: Li’s formula was used to slap “AAA” ratings on aggregations of very risky mortgages by systematically ignoring the possibility that the housing market could ever turn down. Basically, the math claimed that pooling together many mortgages eliminated almost all risk. The result was an explosion of risky mortgages (which could now be slapped with “AAA” seals of approval and passed on to unsuspecting investors) and gambling on top of those risky mortgages in the form of tens of trillions of dollars of CDSes (Credit Default Swaps):
The CDS and CDO markets grew together, feeding on each other. At the end of 2001, there was $920 billion in credit default swaps outstanding. By the end of 2007, that number had skyrocketed to more than $62 trillion. The CDO market, which stood at $275 billion in 2000, grew to $4.7 trillion by 2006.
At the heart of it all was Li’s formula. When you talk to market participants, they use words like beautiful, simple, and, most commonly, tractable. It could be applied anywhere, for anything, and was quickly adopted not only by banks packaging new bonds but also by traders and hedge funds dreaming up complex trades between those bonds.
“The corporate CDO world relied almost exclusively on this copula-based correlation model,” says Darrell Duffie, a Stanford University finance professor who served on Moody’s Academic Advisory Research Committee…
The damage was foreseeable and, in fact, foreseen. In 1998, before Li had even invented his copula function, Paul Wilmott wrote that “the correlations between financial quantities are notoriously unstable.” Wilmott, a quantitative-finance consultant and lecturer, argued that no theory should be built on such unpredictable parameters. And he wasn’t alone. During the boom years, everybody could reel off reasons why the Gaussian copula function wasn’t perfect. Li’s approach made no allowance for unpredictability: It assumed that correlation was a constant rather than something mercurial. Investment banks would regularly phone Stanford’s Duffie and ask him to come in and talk to them about exactly what Li’s copula was. Every time, he would warn them that it was not suitable for use in risk management or valuation.
Posted by James on Wednesday, February 25, 2009