The Math Equation that Destroyed the World

Last week, Wired came out with "Recipe for Disaster: The Formula That Killed Wall Street." The cover story, by Felix Salmon, does a killer job of explaining why so many banks made so many screwy decisions. 

For five years, David X. Li's formula, known as a Gaussian copula function, looked like an unambiguously positive breakthrough, a piece of financial technology that allowed hugely complex risks to be modeled with more ease and accuracy than ever before. With his brilliant spark of mathematical legerdemain, Li made it possible for traders to sell vast quantities [$100's of billions] of new securities, expanding financial markets to unimaginable levels.

His method was adopted by everybody from bond investors and Wall Street banks to ratings agencies and regulators. And it became so deeply entrenched—and was making people so much money—that warnings about its limitations were largely ignored.

The crux of the problem:

A bond, of course, is just an IOU, a promise to pay back money with interest by certain dates. If a company—say, IBM—borrows money by issuing a bond, investors will look very closely over its accounts to make sure it has the wherewithal to repay them. The higher the perceived risk—and there's always some risk—the higher the interest rate the bond must carry.

Bond investors are very comfortable with the concept of probability. They invest in pools of hundreds or even thousands of mortgages. The potential sums involved are staggering: Americans now owe more than $11 trillion on their homes. 

There's no easy way to assign a single probability to the chance of default. Wall Street solved this problem through a process called tranching, which divides a pool and allows for the creation of safe bonds with a risk-free triple-A credit rating. Investors in the first tranche, or slice, are first in line to be paid off. Those next in line might get only a double-A credit rating on their tranche of bonds but will be able to charge a higher interest rate for bearing the slightly higher chance of default. And so on.

"…correlation is charlatanism" 
Photo: AP photo/Richard Drew
The reason that ratings agencies and investors felt so safe with the triple-A tranches was that they believed there was no way hundreds of homeowners would all default on their loans at the same time. One person might lose his job, another might fall ill. But those are individual calamities that don't affect the mortgage pool much as a whole: Everybody else is still making their payments on time.

One word: Doh!

Friction is the only conception which, in a general way, corresponds to that which distinguishes real war from war on paper. The military machine, the army and all belonging to it, is in fact simple; and appears, on this account, easy to manage. But let us reflect that no part of it is in one piece, that it is composed entirely of individuals, each of which keeps up its own friction in all directions. 

Theoretically all sounds very well; the commander of a battalion is responsible for the execution of the order given; and as the battalion by its discipline is glued together into one piece, and the chief must be a man of acknowledged zeal, the beam turns on an iron pin with little friction. But it is not so in reality, and all that is exaggerated and false in such a conception manifests itself at once in war. The battalion always remains composed of a number of men, of whom, if chance so wills, the most insignificant is able to occasion delay, and even irregularity. The danger which war brings with it, the bodily exertions which it requires, augment this evil so much, that they may be regarded as the greatest causes of it.

Li wrote a model that used price rather than real-world default data as a shortcut (making an implicit assumption that financial markets in general, and CDS markets in particular, can price default risk correctly).

Li didn't just radically dumb down the difficulty of working out correlations; he decided not to even bother trying to map and calculate all the nearly infinite relationships between the various loans that made up a pool. What happens when the number of pool members increases or when you mix negative correlations with positive ones? Never mind all that, he said. The only thing that matters is the final correlation number—one clean, simple, all-sufficient figure that sums up everything.

Li's approach made no allowance for unpredictability: It assumed that correlation was a constant rather than something mercurial.