A Model Explanation

It's easy and tempting to seek a scapegoat for the current financial crisis we face. I've variously seen Evil Bush, the Current Occupant (hey, Keillor's not using it anymore), George Soros, Christopher Dodd and Barney Frank, Franklin Raines, Henry Paulson, Karl Rove and perhaps the Mormon Tabernacle Choir. Not once had I thought to blame an obscure Chinese mathematician. But this guy might be the key to understanding what has happened.

The gentleman's name is David X. Li and the mathematical formula he developed is the subject of a fascinating piece that appeared in a recent issue of Wired Magazine. The longish piece from financial writer and blogger Felix Salmon is well worth your time. You can read it for yourself, and you really should, but I wanted to bring up a key point or two about it.

As many of you know, in a previous life I was a program analyst for Bank of America, which is now on the verge of financial collapse alongside Citi and many other financial institutions. I was an odd choice for an analyst, since I'm an English major and have no formal training in finance. Still, I got the gig and crunched numbers for B of A's corporate relocation program from 2003-2006, in the midst of the real estate bubble. I had the ability to coax the correct numbers out of the gigantic B of A mainframe computers and plug them into the financial reports that our team used to run our business. I always enjoyed the challenge of getting the numbers and presenting the information. We were proud of our business and we rode along with the boom, making many mortgages for rising executives. I worked with many highly talented, perceptive people during my tenure there, which ended when B of A closed its Minnesota office and we decided not to relocate.

Looking back on that experience, a few things were pretty clear. While B of A wasn't nearly as reckless as some of its competitors in providing bad loans, there was a lot of wishful thinking involved in what we did. Our program numbers were always accurate, because they had to be. The problem was always in forecasting. The higher-ups demanded that we deliver significant incremental growth every year and we built a lot of our business models in ways that really wouldn't have stood up to scrutiny. The best way to advance in that environment was to deliver a model that would give you the answer you wanted. You could call this optimism, or you could call it pulling numbers out of your ass.

This brings us back to the matter of Mr. Li and his financial model. One of the more intractable challenges for any financial analyst is understanding the actual amount of risk involved in a transaction, especially in a scenario with a lot of moving parts. Felix Salmon explains:

How could one formula pack such a devastating punch? The answer lies in the bond market, the multitrillion-dollar system that allows pension funds, insurance companies, and hedge funds to lend trillions of dollars to companies, countries, and home buyers.

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. If there's a 1 percent chance of default but they get an extra two percentage points in interest, they're ahead of the game overall—like a casino, which is happy to lose big sums every so often in return for profits most of the time.

Bond investors also 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. But mortgage pools are messier than most bonds. There's no guaranteed interest rate, since the amount of money homeowners collectively pay back every month is a function of how many have refinanced and how many have defaulted. There's certainly no fixed maturity date: Money shows up in irregular chunks as people pay down their mortgages at unpredictable times—for instance, when they decide to sell their house. And most problematic, there's no easy way to assign a single probability to the chance of default.

In other words, mortgage backed securities really have a lot of moving parts, probably too many to wrap one's arms around. Unless you don't. And that is where Mr. Li comes in.

In 2000, while working at JPMorgan Chase, Li published a paper in The Journal of Fixed Income titled "On Default Correlation: A Copula Function Approach." (In statistics, a copula is used to couple the behavior of two or more variables.) Using some relatively simple math—by Wall Street standards, anyway—Li came up with an ingenious way to model default correlation without even looking at historical default data. Instead, he used market data about the prices of instruments known as credit default swaps.

If you're an investor, you have a choice these days: You can either lend directly to borrowers or sell investors credit default swaps, insurance against those same borrowers defaulting. Either way, you get a regular income stream—interest payments or insurance payments—and either way, if the borrower defaults, you lose a lot of money. The returns on both strategies are nearly identical, but because an unlimited number of credit default swaps can be sold against each borrower, the supply of swaps isn't constrained the way the supply of bonds is, so the CDS market managed to grow extremely rapidly. Though credit default swaps were relatively new when Li's paper came out, they soon became a bigger and more liquid market than the bonds on which they were based.

When the price of a credit default swap goes up, that indicates that default risk has risen. Li's breakthrough was that instead of waiting to assemble enough historical data about actual defaults, which are rare in the real world, he used historical prices from the CDS market. . . 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).

It was a brilliant simplification of an intractable problem. And 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.

Emphasis mine. Boiled down to its essence, what this meant was pretty simple -- don't worry about things you can't measure. Look at something else that will give you the answer you want.

The problem was, Li never meant for his model to be the basis on which the entire financial world operated. But that is what happened. And we have to be honest about this -- a hell of a lot of people benefited. Bond traders got rich, people who couldn't afford houses suddenly were creditworthy, loan officers originated loans, closers closed them, analysts tallied them up and built projections that weren't sustainable.

So why did this happen? Let Salmon explain.

The damage was foreseeable and, in fact, foreseen. In 1998, before Li had even invented his copula function, Paul Wilmott wrote that "the correlations between financial quantities are notoriously unstable." Wilmott, a quantitative-finance consultant and lecturer, argued that no theory should be built on such unpredictable parameters. And he wasn't alone. During the boom years, everybody could reel off reasons why the Gaussian copula function wasn't perfect. Li's approach made no allowance for unpredictability: It assumed that correlation was a constant rather than something mercurial. Investment banks would regularly phone Stanford's Duffie and ask him to come in and talk to them about exactly what Li's copula was.
Every time, he would warn them that it was not suitable for use in risk management or valuation.

In hindsight, ignoring those warnings looks foolhardy. But at the time, it was easy. Banks dismissed them, partly because the managers empowered to apply the brakes didn't understand the arguments between various arms of the quant universe. Besides, they were making too much money to stop.

Again, emphasis mine. There's more, a lot more, in the article. There's something entirely normal in the human need to find an explanation that is comprehensive. It is the reason that people who are adrift join mass movements, the reason that you still find college students dressed in black who babble on about Foucault. And there is always greed and a bit of larceny in the human heart. Having a model that explains everything is awfully seductive. And it's almost always wrong. And as our friend Doug Williams pointed out not that long ago, we still have a lot of other models out there.