# Showing the Math on Structured Finance

Like many Americans, my quantitative skills aren’t all they could be. That said, I’ve found the mathophobia of elite American culture to be very frustrating throughout the financial crisis because it’s hard to actually understand what’s happening unless you show us some numbers. Instead journalists have tended to rely on some fairly vague language about “complex securities” and “bundles of mortgages” that leaves things more confusing than they need to be.

This right here from Alex Tabarrok, by contrast, is a model of clarity. The example is apparently pulled from Robert Pozen’s *Too Big to Save? How to Fix the U.S. Financial System* and he in turn poached it from Coval, Jurek and Stafford:

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%.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.

Importantly, this is not a scam. The math really checks out. The whole reason finance works is that pooling and redividing risks really can improve the situation. You’re not producing nothing, you’re producing risk-mitigation. In this case, you took a bunch of mortgages with a 5% chance of defaulting and creating, among other things, a CDO with just a 0.05% chance of defaulting. If demand for super-safe loan-backed securities exceeds the actual quantity of such securities, this kind of activity can be very profitable but can also be useful in meeting that consumer demand and make it easier for relatively bad credit risks to get loans.

But you never really know exactly what the default risk on a loan is. So what if you make a mistake? Disaster:

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!

That’s terrible, obviously. What’s more, once people start paying attention to the fact that some of the probabilities have been specified wrong, you can see why there’s going to be a panic. Now suddenly not only are all those AAA-rated securities suspect, but one really has no idea how suspect they are. Even a much smaller mistake than the .05 to .06 one could have major consequences.