Find the Mistake

Robert Waldmann

Alex Tabarrok wrote

In Too Big To Save Robert Pozen gives a clever example, based on an excellent paper by Coval, Jurek and Stafford, which explains both the lure of structured finance and why the model exploded so quickly.

Suppose we have 100 mortgages that pay \$1 or \$0. The probability of default is 0.05. We pool the mortgages and then prioritize them into tranches such that tranche 1 pays out \$1 if no mortgage defaults and \$0 otherwise, tranche 2 pays out \$1 if 1 or fewer mortgages defaults, \$0 otherwise. Tranche 10 then pays out \$1 if 9 or fewer mortgages default and \$0 otherwise. Tranche 10 has a probability of defaulting of 2.82 percent. A fortiori tranches 11 and higher all have lower probabilities of defaulting. Thus, we have transformed 100 securities each with a default of 5% into 9 with probabilities of default greater than 5% and 91 with probabilities of default less than 5%.

The quoted claim is false. I explain why after the jump.

In his post, Tabarrok assumes that the correlation between payments on any two mortgages must be zero. The calculation is correct only if the correlation is zero. This crucial assumption was not stated. The claim is false as written.

One way in which structured finance almost destroyed the world economy was very close to this but not so very very extreme. CDO designers and raters assumed that they could estimate correlations between default on different bonds assuming that default occurred if a latent variable were below zero and that the latent variables were jointly normal. There was no justification for the assumption of joint normality. Then they assumed that the CDS market was efficient (schizzo-finance) so they could estimate the correlations of the latent variables.

This is less extreme than just assuming that correlations are zero without even stating the assumption. However, it was extreme enough to cause a disaster.

Amazingly, after the disaster based on casual assumptions about correlation, Alex Tabarrok makes a much more extreme assumption and doesn’t even state it.

Note there are no ellipses in my quote of Tabarrok. That is the post in full from the first word to the false claim of 2.82%.

I am sincerely shocked and appalled.

update: There is another problem with the example. Given independence of defaults across mortgages, the probability of default of the 10th tranche is dramatically different if the probability of a default of each mortgage is 6% not 5%. However, the probability of default of the 12 tranche is tiny in either case and the probability of default of the 8th tranche is large in either case.

The example illustrates the properties of 10, 5 and 6 not the effect of 5 vs 6.

A small difference in the probability of default for uncorrelated default risks matters a lot for a small interval of tranches. For probabilities other than 5 and 6 the change from very small to high probability would occur at a range different from 8th to 12th but only for a small range.

The market value of such small ranges was not large enough that serious misspricing of them (in the range from safe to worthless) could bring down the financial system.

Similarly the tranches of pools of 10th tranches are very different but tranches of pools of 12th tranches would be safe in either case and the 10th tranche of a pool of 8th tranches would be risky in either case. Again pools of 10th tranches of pools weren’t important.

You just can’t make huge unexpected losses out of a tiny change in the mean and variance of the flow of money from debtors unless people make huge side bets and there are huge unexpected gains too.

In contrast, underestimating the correlation of default across mortgages can lead to a major underestimate in the variance of the flow of money from debtors to the financial system. That error, althoug much less extreme than the unstated independence assumption in the example, was an important part of the crisis.

The example ignores the important issue and emphasizes a very unimportant issue.