In defense of the Gaussian copula

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QUANT models and their architects are so misunderstood, often by people working in finance. It pains me, though I am biased. I spent the better part of a decade devoted to studying elegant (and sometimes clumsy) mathematical expressions that describe messy and complex relationships. I still see beauty in these models, but am aware (much more so lately) of their weaknesses. Understanding these models means being mindful of their limitations.

Models are not perfect, but that doesnt mean theyre not useful. Bob Merton (full disclosure: Ive worked with him and consider him a mentor) recently explained a weakness of macro models. You can have a complex, non-linear relationship, but macro models, in the name of tractability, often make simplifying assumptions that involve describing the problem in linear terms. This may make sense locally, which means it works under a variety of static assumptions. But if circumstances suddenly change and you are no longer on the linear part of the curve, your model does not do a good job of explaining the world. The model still has value, but using it correctly requires an awareness that it does not hold globally.

Bankers, most finance writers, and even many economists failed to appreciate this. Now the trend is to condemn these models as completely useless.

The Gaussian copula is not an economic model, but it has been similarly misused and is similarly demonised. In broad terms, the Gaussian copula is a formula to map the approximate correlation between two variables. In the financial world it was used to express the relationship between two assets in a simple form. This was foolish. Even the relationship between debt and equity changes with the market conditions. Often it has a negative correlation, but other times it can be positive.

That does not mean it was useless. The Gaussian copula provided a convienent way to describe a relationship that held under particular conditions. But it was fed data that reflected a period when housing prices were not correlated to the extent that they turned out to be when the housing bubble popped. You can have the most complicated and complete model in the world to explain asset correlation, but if you calibrate it assuming housing prices wont fall on a national level, the model cannot hedge you against that happening.

In last weekends FT. Sam Jones did a good job explaining how the misunderstood formula came to be so widely used.

By 2001, correlation was a big deal. A new fervour was gripping Wall Street one almost as revolutionary as that which had struck when the Black-Scholes model brought about the explosion in stock options and derivatives in the early 1980s. This was structured finance, the culmination of two decades of quants on Wall Street. The basic idea was simple: that banks no longer had to hold on to risks. Instead they could value them, using complex maths and modelling, then package and trade them like any other, ordinary security. Mortgages were the prime example. Rather than make a mortgage loan and gradually collect interest over its lifespan, banks began to bundle the loans together and sell them into specially created off-balance-sheet shell companies. These companies in turn issued bonds to raise cash. And by using the modelling and maths being cranked out by quants, banks were able to tailor the structure of mortgage portfolios to ensure that bonds of varying risks could be issued to investors. The problem, however, was correlation. The one thing any off-balance-sheet securitisation could not properly capture was the interrelatedness of all the hundreds of thousands of different mortgage loans they owned. As a consequence, structured finance had remained a niche and highly bespoke practice throughout the 1990s. On August 10 2004, however, the rating agency Moodys incorporated Lis Gaussian copula default function formula into its rating methodology for collateralised debt obligations, the structured finance instruments that subsequently proved the nemesis of so many banks. Previously, Moodys had insisted that CDOs meet a diversity score that is, that each should contain different types of assets, such as commercial mortgages, student loans and credit card debts, as well as the popular subprime debt. This was standard investing good practice, where the best way to guard against risk is to avoid putting all your eggs in one basket. But Lis formula meant Moodys now had a model that enabled it to gauge the interrelatedness of risks and that traditional good practice could be thrown out of the window, since risk could be measured with mathematical certainty. No need to spread your eggs across baskets if you knew the exact odds of your one basket being dropped. A week after Moodys, the worlds other large rating agency, Standard & Poors, changed its methodology, too.

The Gaussian copula was a statistical convenience which merely provided an approximation of a complex relationship. It cannot be compared to Black-Scholes, which revolutionised finance. Black-Scholes created a new market by providing traders with a whole new way to think about and price risk.

The big and fatal change that occured after 2001 was not so much securitisation, but that banks began to hold mortgage assets on their balance sheets. If they did this as a result of the Gaussian copula, it screams incompetence. The same can be said for rating agencies who should have known better.

Like any elegant model, the Gaussian copula has some value. If you appreciate the local conditions, it provides a convenient way to describe a complex relationship. But like many useful innovations, when it is exploited by the wrong people it can cause great harm.

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