Correlations

This morning, I read a fascinating article on the Financial Times website, here, on the general subject of copulas. (I would have guessed it's the little house you sometimes set atop somebody's roof, but that's a cupula, not a copula. )

The ever-helpful Wikipedia defines a copula as ".... a general way of formulating a multivariate distribution in such a way that various general types of dependence can be represented." They give the example of statistically improbable death occurrence -- he died shortly after her; I think he died of a broken heart -- and move casually through neurobiology, quants, LTCM, and related matters to the concept of 'default correlations'. Quoting from the article:

How correlated are the default probabilities on bonds issued by our Irish dairy farm and those issued by a software company in Malaysia? Not at all, you might think: the businesses not only provide totally different products and services, they’re also geographically remote from each other. Suppose, though, that both companies have been lent money by the same troubled bank that is now calling in its loans.In fact, this is exactly what sank LTCM. How correlated are Russian government bonds and those in Mexico? Not at all, according to LTCM’s model, which, it should be noted, crunched data going back a hundred years. And yet it turned out for the hedge fund that both markets were dominated by the same few investors. The 1998 financial crisis in Russia, when Boris Yeltsin’s government defaulted on its bonds, caused panic selling in Mexico as investors rushed to de-risk their portfolios. Li realised that his insight was groundbreaking. Speaking to The Wall Street Journal seven years later, he said: "Suddenly I thought that the problem I was trying to solve [as an actuary] was exactly the problem these guys were trying to solve. Default [on a loan] is like the death of a company." And if he could apply the broken hearts maths to broken companies, he’d have a way of mathematically modelling the effect that one company’s default would have on the chance of default for others.

(That's one of the points where I had to stop a couple of times to say wait, what?)

The article goes on to describe how this led to banks and others thinking about risks not so much as 'a cost of doing business, to be minimized as much as possible' and instead as a form of asset -- you could generate a value for it, and then trade it. I'll trade you five million mortgages with a X probability of default for six thousand corporate bonds with a Y probability of default. But, how to relate those risks to each other, to be sure that they weren't affected by the same external events? Again, quoting from the article:

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.

And then came the Gaussian cupola as applied to the valuation of securities, which assumed that risks were distributed along a range, and not correlated. If this were a horror movie, this is where the dark, brooding background music would start. Because, as it turned out, they were.

I still don't entirely understand the article, but what I understood, I liked.