There are lots of mistakes that can be made when calculating odds, whether the likelihood of double-yoked eggs or mortgage default, but some mistakes are much more costly than others, writes TIM HARFORD
IN THE NORTH of England, early in 2010, a lady named Fiona Exon bought a carton of six eggs and discovered that all six of them had double yolks. Newspapers reported this as beating odds of 1,000,000,000,000,000,000 to one. The mathematics seemed straightforward.
According to the egg-watchers to whom the newspapers spoke (the British Egg Information Service – who knew?), the chance of any given egg having two yolks is one in a thousand. The chance of any two given eggs both having double yolks would therefore appear to be, from multiplying the two probabilities together, one in a million. Three in a row would be a one in a billion chance; four would be a trillion, five a quadrillion, and six double-yolked eggs in a row would be a one-in-a-quintillion chance.
If that calculation is right, then if each and every person in the world bought six eggs each morning, we’d expect to see a carton of double-yolk eggs being sold somewhere on Earth roughly every four centuries.
The trouble is, after Mrs Exon’s apparently astonishing discovery was reported in the media, lots more people popped up to announce that the same thing had happened to them. It seemed that double-yolks are rare, but not that rare.
What had gone wrong with the calculation? Perhaps double-yolk eggs are more common than the British Egg Information Service thinks. But let’s be charitable to the British Egg Information Service and assume that it provides accurate information about British eggs. (It would be a minor tragedy if it did not.) The problem is that the newspapers made an insidious assumption: that double-yolk eggs do not come in clusters. Allowing for the fact that they might, we get a very different equation, with the chance of seeing a carton full of eggs trillions of times more likely.
All this talk of eggs might seem a curiosity. But the banking crisis came about because banks and other financial institutions were taking very big bets on the chance of events that, like Fiona Exon’s discovery, seemed too unlikely to contemplate. They were wrong.
The financial egg cartons were called mortgage-backed securities, and the rotten eggs that filled them were the now-infamous subprime loans. Banks bought these risky – or “subprime” – mortgages from “originators”, companies who initially made the loans.
Then the banks repackaged them into financial products, which provided the rights to a stream of mortgage repayments and the risks that mortgage would not be repaid. The repackaging was often repeated many times. (The first package was called a residential mortgage-backed security, or RMBS. The repackaged versions were called collateralised debt obligations, or CDOs, then CDOs-squared and CDOs-cubed.)
These RMBSs, CDOs and the rest were tremendously complex products, but their aim was simple enough: to create financial assets that promised a safe and predictable income. This predictable income could then be traded between banks, pension funds and other financial institutions. If the underlying risks had been misunderstood, however, the value of these financial assets would collapse, threatening the companies that owned them with bankruptcy.
So how did these strange mortgage-backed products work? Let’s switch the analogy from double-yolk eggs to rotten ones. Imagine that a bunch of mortgages is simply a basket full of eggs, some of which are rotten – say 5 per cent. The bank repackages the eggs into cartons of six, keeping one egg and selling the rest.
The price of those other eggs reflects the risk of who gets the rotten eggs. It turns out that, assuming the rotten eggs don’t cluster, the chance that at least one egg will turn out to be rotten is 27 per cent. If the first egg is rotten, the bank is first in line and takes the hit. But it’s a risk they’re happy to take, because they’ve sold the rest of the carton at a profit.
The buyer of the second egg – the “junior investor” – is next in line if there’s another rotten egg. It turns out that the chance of a second bad egg comes out at just over 3 per cent. The junior investor gets a decent discount on the price of the egg for taking this fairly small but still meaningful risk.
Senior investors buy the remaining eggs, and as the chance of any further bad eggs is very low – only about 0.2 per cent – they pay top dollar. They’re expecting a safe and predictable income; in the very unlikely event that there’s a third rotten egg, they’ll get a nasty surprise.
The repackaging makes some sense, because some investors prefer to pay for a guarantee of safety while others would prefer to get a cheap egg and take a risk. And the continued repackaging into CDOs-squared, CDOs-cubed and so on had the same logic behind it.
But here comes the catch with this repackaging. If the banks had got their sums wrong – no prizes at this stage for guessing whether they had – then each level of repackaging dramatically amplified the effect of that mistake.
Remember, we assumed that the risk of a rotten egg picked at random from the basket of eggs was 5 per cent. This implies that, in the first carton of eggs, the chance of a second rotten egg is just 3 per cent. (Trust me, I’m an economist.)
The CDO, then, is made of eggs with a probability of being bad of 3 per cent. The chance that any one of these eggs is bad turns out to be about 18 per cent, and the chance that a second egg in the CDO is off is about 1.5 per cent. The CDO-squared is made of eggs with a probability of being rotten of 1.5 per cent, and the chance that the second egg in the CDO-squared is bad is less than one third of one per cent. That’s a very small risk.
What happens if the chance of a rotten egg isn’t 5 per cent at all, but really 10 per cent? Crunch the numbers and it turns out that the chance the second egg in the RMBS is rotten hasn’t fallen from 5 to 3 per cent, it’s risen from 10 to 11.5 per cent. The fact that it’s risen is more important than how much it’s risen – because now each successive repackaging will further increase the risk.
The second egg in the CDO now has a 15 per cent chance of being bad, rather than a 1.5 per cent chance. It’s 10 times as risky. The second egg in the CDO-cubed becomes almost 2,500 times riskier. The risk seemed laughable; suddenly, it’s no joke.
Much the same analysis applies if the eggs come in clusters, although the mathematics are harder. Why did the banks’ mathematicians assume that mortgage defaults wouldn’t cluster together as much as they did? Well, one borrower might go through a divorce, another lose her job, and a third fall ill. But one person’s divorce is nothing to do with a stranger’s health, so mortgage defaults caused by these problems should come in ones and twos, not clusters. As long as that assumption held true, the re-re-re-packaged loans were extremely safe, for much the same reason that Fiona Exon apparently faced a one-in-a-quintillion chance of picking up a carton with six double-yolk eggs in it.
But if the mortgage defaults come in clusters, disaster looms. And with hindsight, it is blindingly obvious why a cluster might appear – it would automatically emerge if there was a national slump in housing prices, as there was in the United States in 2007. And the very system of intricate CDO repackaging that was supposed to protect investors from the small chance of getting a bad egg instead hugely exaggerated the problem. Because the sums were wrong, each repackaging of the second egg actually concentrated the risk instead of reducing it. We ended up in a Looking-Glass world where investments that had been thought to be preposterously safe turned out to be preposterously dangerous.
The products and the calculations were far more baffling even than I’ve suggested, so all this is a simplification. But it is not an exaggeration. To give you a sense of just how badly wrong the sums of the financial mathematicians became in reality, ponder the words of David Viniar, the chief financial officer of Goldman Sachs. At the beginning of the credit crunch, he explained the sudden loss of 30 per cent of the value of a Goldman Sachs spin-off fund by saying, “We were seeing things that were 25-standard deviation moves, several days in a row.”
Viniar meant that Goldman Sachs had been unlucky. But just how unlucky, exactly? The financial economist Kevin Dowd has calculated that (given some reasonable assumptions) we’d expect to see three 25-standard deviation days in a row roughly once every 28,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000 years.
For reference, the universe is about 13,000,000,000 years old. Bad luck is not really an explanation. Somewhere, somehow, Goldman Sachs had got its sums wrong and then the inexorable complexity of the financial instruments it was dealing with magnified the error almost beyond comprehension. The financial crisis had many causes, but if I I had to point to a single idea that underpinned the disaster, it would be those endlessly repackaged cartons of rotten subprime eggs. It was an accident waiting to happen – indeed, bound to happen – as soon as the global economic equivalent of Fiona Exon came along.
Tim Harford is a Financial Timescolumnist and the author of Adapt: Why Success Always Starts with Failure
