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!

In short, the equation––like any other––was only as good as the assumptions it rested on. And as it turned out, this equation's assumptions were crap outside of a very narrow context. 

Moreover, there's a deeper flaw: a belief that markets––that the aggregate mass of investors big and small––are always right. In economics, this
is called the "Efficient Market Hypothesis." It means that the price of something sold in a free market naturally reflects all known information about that particularly thing being sold. The market, in short, can never price something incorrectly because the price necessarily reflects and takes into account the current state of all that is knowable about the item in question. (See tautology.)

Now, on its face, this concept is both absurd and retarded. It's like saying, "People always act in their best interests." Sure, with your head clouded by a few beers, the idea might make some sense. That is, until you realize that what the statement really means is that no matter what a person does––whether they jump off a bridge or eat fried chicken 24/7––it is in the best interests to do so because, well, they wouldn't do it if it wasn't.

Only a person shielded from actual human and organizational decision making could come up with and believe in such a naively stupid idea. Unsurprisingly, it was an academic who came up with this idea of the (necessarily) efficient market. And by imbuing the theory with academic legitimacy, it somehow gained traction in the broader society.

In a slightly different context, my man Clausewitz talked about the problem

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.

Friction, ahh, friction. It was this 17th century German general who not only realized but wrote down this fundamental element of not simply war fighting but of all human behavior: Action requires more than just knowledge. What people do is subject to more chance, to more uncertainty, to more influences than what they know they think or know they should do. To know is NOT execute. 

And let us not forget: trade––the buying and selling of goods and services––is a social endeavor, subject to all the whimsy, chance and bias of any other social interaction. That's why some people are better at it than others. That's why Berkshire Hathaway has consistently beaten the broader market. Back to Wired:

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.

Umm, yeah. See above.

And so it was that the global economy was poisoned by a single, simple, stupid little equation.

See you at the s
oup kitchen. 

Here's what killed your 401(k)   David X. Li's Gaussian copula function as first published in 2000. Investors exploited it as a quick—and fatally flawed—way to assess risk. A shorter version appears on this month's cover of Wired. 


Specifically, this is a joint default probability—the likelihood that any two members of the pool (A and B) will both default. It's what investors are looking for, and the rest of the formula provides the answer.

Survival times

The amount of time between now and when A and B can be expected to default. Li took the idea from a concept in actuarial science that charts what happens to someone's life expectancy when their spouse dies.


A dangerously precise concept, since it leaves no room for error. Clean equations help both quants and their managers forget that the real world contains a surprising amount of uncertainty, fuzziness, and precariousness.


This couples (hence the Latinate term copula) the individual probabilities associated with A and B to come up with a single number. Errors here massively increase the risk of the whole equation blowing up.

Distribution functions

The probabilities of how long A and B are likely to survive. Since these are not certainties, they can be dangerous: Small miscalculations may leave you facing much more risk than the formula indicates.


The all-powerful correlation parameter, which reduces correlation to a single constant—something that should be highly improbable, if not impossible. This is the magic number that made Li's copula function irresistible.