There is plenty of blame to go around from the current global financial crisis. But, I would like to point out that it is not “all the quants’ fault.” We are all now, unfortunately, sitting in the middle of a high quality (and extremely expensive) lesson in financial mathematics. I would hate for some of the truly important points to be lost to paying too much attention to some of the shiny details.

One fascinating article ( Recipe for Disaster: The Formula That Killed Wall Street by Felix Salmon, Wired, February 2009) has so popularized assigning blame to one formula (and one mathematician) that posting the image of a formerly obscure statistical formula (called a “copula”) is now considered good for a laugh.

However, the original mathematical paper being castigated (“On Default Correlation: A Copula Function Approach” by David X Li, Risk Metrics (2000)) is in fact good work. What is wrong is not the formula but the over-reliance on the formula. If we place all the blame on “copulas” we will be too ready to repeat the current disaster with some other “better” model.

We need to think more like Michael Lewis and use specific failures as miniature laboratories to learn larger lessons. A great example is his write-up of the Iceland financial collapse ( Wall Street on the Tundra by Michael Lewis, Vanity Fair, April 2009 ) which, if you read carefully, contains a general indictment of speculative greed and getting rich by pushing around bits of paper (instead pursing activities that create actual value).

So back to the copulas: what is to be learned (now at great expense) there? I would like to work through some of the important points of Dr. Li’s paper and explain some of the painful points in our current lesson. In my opinion none of the flaws are mathematical (or in the paper)- the flaws are all deep defects in logic and reason (and found in the later behavior of traders).

The main purpose of Dr. Li’s paper was to figure out how to price a newer and more complicated financial instrument (the credit default swap) in terms of older underlying instruments (mortgages). In addition to developing the necessary mathematics the paper contains several clever ideas based on the logic of reasonable markets. As the markets became very large and very unreasonable the logic no longer applied. That is what went wrong.

Credit default swaps (in their simplest form) essentially started as insurance policies against mortgages defaulting. Unfortunately, credit default swaps were unregulated financial instruments instead of regulated insurance policies. Credit default swaps degenerated into “bets” (or derivative securities) when they were separated from the underlying mortgages. You could, in essence, buy or sell insurance on your neighbor defaulting on their mortgage.

The legitimate use of credit default swaps would be to set up a market for insurance and re-insurance. If you borrowed money to buy a house a bank might buy a credit default swap to partially insure against you defaulting on the loan. However, the market went somewhat insane. Since everything was measured in dollars and probabilities (instead of specific contracts and records) a bank that had a million dollars of exposure from lending you money (to buy your mansion) would end up buying insurance on your neighbor’s mansion (which they did not finance) from somebody with no really ability to pay-off in the event of default. From a pure balance-sheet point of view the numbers “made sense” a bank with a million dollars of exposure brought the appropriate amount of insurance. From a business point of view: they purchased insurance on the wrong property and purchased it from somebody they should not be doing business with.

So credit default swaps eventually made no sense as insurance policies. How did they fare as financial instruments? Even if credit default swaps made no sense for the institutions originating (creating) them there was a market trading them. So, ignoring what they were: if you could buy them when they were cheap and sell them when they were expensive you could make money. This is where Dr. Li’s paper comes in: he figured out how to estimate the underlying theoretical value of a credit default swap. With this knowledge you would know when the market price for a credit default swap was cheap (the trading price would be below the theoretical price) or expensive (the trading price would be above the theoretical price). Traders could make more money.

And this is where things went very very wrong. With more profit there were more traders. With more traders there was a larger market to accept credit default swaps. Since there were no rules anybody could originate (create) them. In particular there was no rule that said there could not be more credit default swaps than underlying mortgages. And this is where the insanity of the market no longer matched the reasonable logic of Dr. Li’s original paper.

The idea of assigning a theoretical value to items using information from another market depends critically on two financial concepts. The first one is well known and is called “price taker.” The second one is more obscure and I will call it “information taker.” Due to extreme scale both reasonable assumptions became false.

A “price taker” is a trader in the market that is small enough that the trader does not radically change prices. This is the opposite of “price maker” who is a trader who’s activity is so great that they essentially drive prices. The assumption was that the credit default market would be a “price taker” with respect to other markets. The theory was that you could disassemble a credit default swap into some mortgages, some interest bearing annuities and some other pieces. You could then get the prices for all of these components from other markets and know if the credit default swap was cheap or expensive relative to the current price of its constituent parts. This works for a single credit default swap. But what happens if you needed to take apart a larger number of them at once? That might require acquiring more mortgages than actually exist. Attempting to acquire or dump the components would have a huge price-making impact on all of the other markets. The idea that the credit default swap should price at the current price of its components falls apart, the very attempt to dissemble them would re-price the other markets. Even worse: the markets could “lock up” and stop trading (if for example you dumped so many mortgages into the markets that nobody wanted to buy any at any price).

What I call an “information taker” is a newer idea. One of the clever steps in Dr. Li’s paper that the some of the unknown quantities needed by the theoretical model for credit default swaps could be estimated from the market pricing of mortgages. For example: an estimate of future mortgage default rates is one component needed to correctly price credit default swaps. One way to estimate future mortgage default rates is to learn a lot about actual mortgage holders, learn a lot about macro economics and try to predict future default rates in a number of plausible future scenarios. This is expensive and it is by no means certain (since you really can not predict the future). Another way to estimate future mortgage default rates is to examine the “credit spread” or difference in market pricing of mortgages as compared to less risky securities. If these other markets are working correctly (or “in equilibrium”) you can infer the future default rates from the pricing. This idea works, until too many people use it. If everybody else in the market is performing expensive research on future default rates then: the pricing of mortgages (relative to other less risky assets) will necessarily give you the information needed to solve for your model’s unknowns. However, once everybody is an “information taker” (using market pricing to try to estimate unmeasured fundamentals) the market is just one big “echo chamber” with no actual data being injected. You can no longer correctly estimate parameters from the market because there are no informed players to steal from. Even worse if those markets go out of equilibrium, lock up or stop trading you don’t even hear echoes- you become completely deaf.

These simple flaws in reasoning (in addition to bubble-driven greed) are behind the current global disaster. We need to protect ourselves from all of these fundamental causes (which will occur again and again), not vilify some formerly obscure financial mathematics (which will never appear in the same skin twice).

Categories: Expository Writing Opinion

### jmount

Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.