Positive Correlation will Kill You

Big drops in the market, big investment houses going down, and big loans by the government to prop up AIG all indicate that the financial system is having severe issues. I've gotten enough questions about this to warrant an attempt at a coherent explanation. Here you go.

You want to buy a $100,000 house, but you only have $10,000 in cash. A mortgage company agrees to provide you with $90,000, and in return you'll pay them back over time. More importantly, you'll pay them back more than $90,000 (the interest). So far, so good.

Why do you pay interest? To compensate the mortgage company for a) the fact that they cannot invest the $90,000 someplace else, and b) the fact that you might default on the loan. Let's ignore a) for now, because b) is what got us into trouble. There is a 1/100 chance that you'll default on the loan (maybe you'll lose your job, or it will turn out your neighborhood is on a toxic dump and the house value goes to zero). To compensate the mortgage company, you agree to pay them back $90,909 with a 99/100 probability, and $0 with a 1/100 probability. Thus the mortgage company can expect to get 99/100*$90,909 = $90,000 back, exactly equal to what they loaned you.

But the mortgage company now has this risky loan - there is a 1% chance you'll default. They don't like risk, so they have two options. One, they could keep $90,000 in the vault, and if you default, they have the $90,000 to cover it. But that sucks because it's a waste of $90,000. The second option is to unload the risk to someone like Fannie or Freddie. The mortgage company sells the mortgage to Fannie for $90,000. The mortgage company now has no risk - they don't care if you default.

Fannie, though, now has a 99/100 chance of making $909 ($90,909 that you pay back minus the $90,000 they paid for the mortgage) and a 1/100 chance of having lost $90,000. Sounds great, but Fannie doesn't really like that risk of losing $90,000 either, so they decide to sell the risk onto another financial firm (let's call them Lehman Brothers). Fannie insures their risk by giving Lehman $909 in return for a promise by Lehman Brothers that they will give Fannie $90,909 if in fact you default on the mortgage. So now no matter what Fannie gets back $90,909 (but they paid $90,000 for the mortgage and $909 for insurance, so their net profit is zero).

So now we get to Lehman. They have a 1/100 chance of having to pay Fannie $90,909, but a 99/100 chance that they keep the $909. That's a 99% chance of making $909 for doing nothing. That sounds like good business. So Lehman keeps doing this. They offer to insure another mortgage (maybe from Fannie, or Freddie, or another firm) for $909, promising to pay off $90,909 in the 1/100 chance that the mortgage defaults. Let's say that Lehman buys up exactly 100 of these contracts.

Now Lehman has 100 contracts, each with a 1/100 chance of default. Here is the big assumption that Lehman makes: that those mortgages are all statistically independent. That is, if one mortgage defaults, that doesn't make the other ones more or less likely to default. If that is true, then Lehman can expect 100*1/100 = 1 contract to default. One contract default means Lehman has to pay off $90,909 to Fannie. Lehman made 100*$909=$90,909 (off due to rounding) from making these deals, so they have a zero net profit.

[Here's an aside: why do firms do this if they make zero net profits? At each stage, in addition to the contract, they are charging a fixed fee. Your mortgage company charged you a fee for originating the mortgage - that fee, not the interest, is the basis of their profits. Fannie charges a fee for every mortgage they buy. Lehman charges a fee for every contract they write. The fees make the profits and bonuses get big. The fees are why Lehman didn't stop buying contracts.]

For a couple of years, this keeps going on, and no-one defaults at all. Lehman is making $90,909 for doing not very much (in addition to the fee they charge) and is raking in the profits. But sometime around August 2007, someone defaults on their mortgage. This would be managable for Lehman. Except that the conditions that make ME default on my mortgage are positively correlated with the conditions that make YOU default on your mortgage. In other words, if the housing market pops, it pops for everyone at once, and all of the sudden I have 5 or 10 or 20 defaults all at the same time.

Now what? The mortgage company doesn't care, they sold your mortgage to Fannie. Fannie theoretically doesn't care - they bought insurance against this from Lehman. If 10 mortgages default, then Lehman will pay them $909,090 to cover their losses. Lehman, though, is now on the hook for $909,090. They are supposed to pay this to Fannie. Lehman doesn't have $909,090. They only have $90,909 available from insuring the mortgages. Now they could pull capital from other parts of their business (like commercial lending or investment banking or whatever) to cover this loss. This would lower Lehman's profitability, but they could make good on their contracts.

The problem is that enough mortgages default all at once that Lehman cannot cover its obligations even if they liquidate everything else they own. They go bankrupt. If Lehman goes bankrupt, then what happens to Fannie? Fannie is out $909,090. If Fannie is out $909,090, they cannot purchase more mortgages. If they can't purchase more mortgages, the originators will no longer offer mortgages, because they don't want the risk, they just want the fees.

So the government steps in to un-privatize Fannie, providing Fannie with more money so they can keep buying mortgages. They let Lehman go, but after Lehman's bankruptcy proceedings, they might be able to pay back Fannie say $800,000 of the $909,090 they owe them.

Why did this all happen? Because Lehman (and many others) thought (implicitly or explicitly) that all those mortgages were not correlated at all. That with a 1/100 chance of default, then there would be 1 default for every 100 mortgages. The problem is that mortgage defaults are actually positively correlated. This meant that when things were good, they were really good (no one defaults). When things went bad, they went really bad (lots of people default all at once).

Should Lehman (and others) have been smarter? Yes. But they wrote more and more contracts as the good years rolled by, assuming that they'd be able to cover the 1/100 losses that might occur. They didn't get greedy, they got dumb. Dumb enough to not see the positive correlations in the mortgage market. And this brought it all crashing down.

What are the ramifications for us, the little people? The big problem is that because of all this, financial firms are reluctant to lend out any more money. Lehman obviously cannot enter into new contracts, Fannie is reluctant to buy more mortgages because they can't buy insurance on them, and mortgage originators are not loaning money to new people. So the housing market is slowed down. In addition, places like Lehman have stopped lending to other businesses, because they need the money to cover their existing debts. So lots of businesses are not hiring new people, or buying new equipment, which means that the job market is bad.

When will it be over? Once all the financial firms can unwind who owes what, and figure out exactly how much they can expect to be paid from places like Lehman. Once they have certainty again, they will start to lend, and the economy will pick back up. I have no idea how long that could take.