Newsworthy Math

Credit default swaps are a type of insurance. I buy something that has a small risk of costing me a lot of money, it might get stolen or broken, if it’s a bond or a mortgage it might default. Rather than risk losing my whole $100, I will buy insurance for $5. You promise to pay me if one of these terrible things happens.

Credit

Companies and governments, just like individuals, have a credit rating. The rating is some number on a scale from good to bad. For individuals in the US the scale is in the hundreds, for companies and governments the ratings are on a scale that sounds more like school grades, AAA, AA, A, BBB, BB, etc, with, perhaps, depending on the rating agency (we’ll get to them later), some gradations in between and some differences in nomenclature.

Default

Whatever the scale is, the rating is supposed to be a measure of how good a risk you are. Specifically, what’s the probability is of you defaulting on your debt. Sovereign debt – money borrowed by governments – generally have a very good rating. Until recently the probability of the US government defaulting on its debt was considered to be 0% (it’s still considered very close to that, but some nerves have been frayed and unaskable questions asked). Other governments have been considered a much higher risk. Argentina used to be (reasonably enough) considered a high risk borrower, Greece and Iceland are currently considered as having a decidedly non zero chance of default.

Credit ratings matter because the higher I perceive the risk of you defaulting, the more I’m going to charge you to borrow money from me.

Swap

Suppose we all agree that there’s a 1 in 10 chance of you defaulting: you might lose your job, go bankrupt, go out of business, or be unable to raise enough in taxes to service your debt. (We’re not all going to agree. You’re going to say the odds are 1 in 50, a nervous investor might consider you a 1 in 5 risk. This is where the rating agencies come in. We will get back to them later.)

But suppose we agree. And suppose that I want to lend you money but I don’t want to take a 1 in 10 chance of not getting it back? Well, why not insure it? Insurance on a 1 in 10 risk should a tenth of the amount insured. The “Expected Value” of a 10% chance of making $100 is 10% of $100. That’s really all you need to know. The rest is arithmetic. (How much is a 3% chance of losing $2,500,000 worth, etc. It’s all just proportion. Being able to use proportion is by far the most important mathematical skill beyond simple arithmetic.)

If I insure 10 $10 loans each with a 1 in 10 chance of defaulting, then, on average, one of them can be expected to default. I’ll have to pay $10 to the person who bought the insurance, but I’ve already taken in $1 from each of lenders, so I’ve broken even. Of course I’m trying to make a profit here, so I’ll charge you more than $1. A credit default swap is insurance against a lender defaulting. (More subtly, I can buy insurance to turn a 1 in 10 risk to a 1 in 20, or to turn a A rating into a AAA rating. Once you get the idea of expected value, that a 1 in 10 chance of winning $10 is worth $1, the mathematic is no more complicated.

What Could Possibly Go Wrong?

First off, to human beings, with our finite life spans and our asymmetric financial needs, a 1 in 10 chance of winning $10 is not necessarily worth $1, but that’s another topic. The real question is the reliability of the ratings. How do I know that this company has a 0.1% chance of going bankrupt, and the other company has a 4% chance of doing the same?

We could, in theory, just let the market figure it out. Let some people charge too much for insurance while other underestimate their risks, surprisingly (to me) this might actually work and could be a valuable and legitimate application of CDSs.

Markets can assess risk within reasonable degrees of accuracy and given some fairly reasonable assumptions about what’s likely to happen. GE will probably not go bust because it has been around a long time, and survived some pretty bad economic crises, it seems to adapt well to changing circumstances, all good things. On the other hand, mighty giants have fallen before, so there is some risk. This would be a valid role for Credit Default Swaps – let people bid on the price of guaranteeing a particular debt, and use the market price as a measure of its danger of default. This system is still exposed to serious systemic risk: the danger that the entire thing blows up at once. If I promise to pay if you default, then what happens if I go broke too? If we’re both in the same industry,or country in some cases, then there is likely to be extreme correlation between both our probabilities of defaulting. It’s hard to see a way around this, but if we were looking for a system that’s far worse that, the one we have now answers nicely.

The Ratings Agencies

What we do now is call in the experts.

There is such a thing as expertise in judging risk. The life insurance business has it, But’s that business’s success is built on certain well confirmed assumption. If uncle Earnest gets killed by a giant meteorite, or a war breaks out and he’s hit by a V2 rocket, his insurance company is going to point to the “Acts of God” exemption, and wave goodbye. Events that are so improbable and so cataclysmic that it is impossible to measure their likelihood cannot be insured against.

The rating agencies claim to have similar expertise, but they don’t have an explicit “Acts of God” clause, and they have an inherent incentive to underestimate risk.

Rating agencies are paid by banks to estimate the risk of things the banks are selling defaulting. That’s right. I’m going to pay you to say how good the second hand car I’m selling is. I’m going to hire the movie critics to say whether my movie is 5 star good or only 3 star good. I could go on, but you see the problem. If car assessors or movie critics were paid by car salesmen and studios then getting people to pay any heed to these ratings would take a pretty serious branding campaign (at least as good as De Beers’ successful persuasion of the public that the diamonds they are buying are rare and valuable).

So the rating agencies are incentivized to give good ratings, because otherwise the banks won’t hire them. An inevitable outcome of this, given that the rating is a proxy for the probability of the debt defaulting, is that these probabilities are systematically underestimated. This bias added another wing to the glorious structure that became the great economic meltdown.

The Debt Guarantors

The risk of a given tranche of a CDO depends entirely on the risk of the debt used as its collateral (and on the size and price of the tranches above it). If that risk is underestimated, or if the correlation between the debts used as collateral is higher than assumed (the probability of a car company’s debt defaulting is one thing, the probability of it defaulting if another two equally large car companies have already done so is significantly higher), if either of these things is true then the rating of the tranche is going to be far, far higher than it should be.

AIG bought it. They really believed that the probability of the AAA tranche of a CDO defaulting was the same as that of the AAA debt issued by some well established, well tested corporation. So they insured it. They took the probabilities at face value. Actually, what may be even worse, is that they looked at them too, and agreed with the rating agencies valuations. After all, they were getting paid to sell insurance. If they said it was too risky they’d miss out on their commission. That was where their incentives lay.

So they insured it on the assumption that there was a given, low, probability of them ever having to pay off. They thought they were getting money for old rope, collecting everyone’s overpriced lottery money and pocketing it. Sure, there was a chance that some low grade debt would default and they’d have to pay out (like someone winning some minor lottery prize), but the chances of any of that AAA stuff defaulting could be ignored. They thought they were selling insurance that there was no chance they’d have to pay out on. So they did it as much as possible.

It all contributed to a beautifully vicious circle. Overvalued debt was used to create CDOs the tranches of which were similarly overvalued, then their valuations was raised again by buying insurance that seriously underestimated the tranches’ riskiness, then synthetic CDOs were created with these overrated tranches as collateral. Lather, rinse and repeat. Every time round the circle everyone gets paid.

At every stage the incentives of those in the game were to keep the game going. In the short term they were getting paid, in the long term, well, they would still have got paid.

The problem is not that people were greedy, selfish and short sighted. There is never going to be a shortage of these things. The problem is that it paid to be greedy, selfish and short sighted.