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 (assume independence). 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%.
Now let's try this trick again. Suppose we take 100 of these type-10 tranches and suppose we now pool and prioritize these into tranches creating 100 new securities. Now tranche 10 of what is in effect a CDO will have a probability of default of just 0.05 percent, i.e. p=.000543895 to be exact. We have now created some "super safe," securities which can be very profitable if there are a lot of investors demanding triple AAA.
To review we have assumed that the underlying mortgages each have a probability
of default of p=.05 and by pooling and prioritizing we have created a tranche with a probability of
default of just p=.0282 and a CDO with a probability of default of
p=.0005. In this way, structured finance was able to create many triple AAA securities from a pool of securities none of which were triple AAA. This point is widely understood. Now here is a much less well understood consequence.
Suppose that we misspecified the underlying probability of mortgage default and we later discover the true probability is not .05 but .06. In terms of our original mortgages the true default rate is 20 percent higher than we thought–not good but not deadly either. However, with this small error, the probability of default in the 10 tranche jumps from p=.0282 to p=.0775, a 175% increase. Moreover, the probability of default of the CDO jumps from p=.0005 to p=.247, a 45,000% increase!
The dark magic of structured finance conjured many low-risk securities
out of many risky securities. Like all dark magic, however, the
conjuring came at a price because if you didn't get the spell exactly
correct it was easy to create something much more risky and dangerous
than you were likely to have ever imagined.
Here is an excel file, StructuredFinanceMath, with the calculations.
Addendum: Adding in correlation among mortgage defaults makes the math more difficult but doesn't change the bottom line that I wanted to illustrate which is that small changes in the underling default risk (or correlation) are highly amplified in the tranches and CDOs.