## The Equation that Sank Wall Street?

Investment banks and hedge funds have a healthy appetite for hiring brainy "quants" - usually in the form of well-qualified mathematicians and physicists - to help them understand and model the markets using complex mathematics. It's called quantitiative finance, and it's being blamed in-part for the current GFC. How so? Well the root of most of the evil for the smouldering balance sheets of many banks are the infamous collateralized debt obligations (CDOs) - those funky instruments crafted by slicing and dicing large numbers of ordinary loans before selling them off to investors. They are the most common form of structured credit and have now been labelled by many as "toxic assets".

CDOs have been around since the late 80s but they only came to prominence in 2000 when new mathematical models were invented that, apparently, made it significantly easier to price CDOs. The quant responsible for the new model was **David X. Li**, a mathematician working at JPMorgan. His model was called the **Gaussian Copula Function** which, in essence, is a formula to determine the correlation between the default rates of different securities.

In other words, if the model was correct, it would tell you the likelihood that related CDOs will explode - important information in the pricing of CDOs and in risk management of CDO portfolios. Since these CDOs were in the hands of many larger corporations the model was also a predictor of the likelihood that a given set of corporations would default on their bond debt in quick succession. Naturally this measure of counter-party risk is something you want to know before buying such instruments from another bank. Because of the new simpler model the CDO market swelled in volume and every large bank in the world had exposure to these instruments. Many believed David Li was on his way to a Nobel prize, and worldwide acknowledgement for his contribution.

As profit margins on CDOs reduced and bankers were looking for more loans for securitisation they ventured into sub-prime housing loans and packaged lesser quality loans into CDOs. Then the market starting doing things the model hadn't expected. Time has shown that the true risk associated with default rates on these loans and the effect on the prices of the related CDOs was not at all explained by the Guassian Copula function. Since the majority of financial institutions had exposure to these CDOs there was heavy reliance on the Guassian Copula function in the pricing and risk management of these instruments. Couple that with a model built on assumptions that weren't extensively tested by people putting the model to use, and a significant rise in sub-prime home loan defaults due to dubious lending decisions and you get the financial meltdown that is the GFC.

Many experts believe one of the fundamental drivers of the GFC was the significant rise of sub-prime loan defaults. Actuaries and mathematicians are adept are deriving probabilities for stochastic events like a loan default but they generally seek greater accuracy by grouping a large number of data points together into a risk pool. When this pooling is done it's preferred that the data points being pooled are *independent *random events but this is not always possible. To see this, consider a simple example of home loan defaults: If Bob were to default on his 250k loan, you would think that it would have *no effect *on the default likelihood of another customer, Charlie, who also has a 250k loan and earns roughly the same amount of money as Bob. That is, the two random events/variables are *uncorrelated*. But what if Bob and Charlie both work at the same company and the reason Bob defaulted was because his employer shut down and he was suddenly out of a job? It would mean that Charlie would also be out of job and he *may *not be able to secure a new job and default just like Bob did. In other words, there is a correlation between the two random events because of the same-employer linkage. And there could be other correlations between groups of people in the risk pool based on other factors - like tolerance to interest rate rises, or home devaluations in a particular geographic region due to market conditions or acts of god. Statisticians have ways to deal with such **multicollinearity ** (correlations between seemingly independent random variable) but in reality, the common linkage between borrowers in a loan pool are extremely complex and difficult to analyse.

Mathematicians realised that they needed to account for these correlations but determining conditional probability distributions to model correlations requires lots of historical data and a large degree of computational analysis. Li's formula simplified this greatly thus making the analysis considerably more tractable. Perhaps not fully understanding it, many quants and trading desks adopted the model and assumed it to be accurate.

The quants are the people Warren Buffett was talking about when he said, "Beware of geeks bearing formulas" in his letter to shareholders this year. However, the quants aren't entirely to blame for the financial meltdown; there's plenty of guilt to be shared by regulators, top executives and the investors who bought the instruments the quants created!

To be fair, Li warned that the "most dangerous part is when people believe everything coming out of the model".

*02 Jun 2009*
*Damien Wintour*