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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%.
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.
Posted by Alex Tabarrok on May 13, 2010 at 07:34 AM in Economics | Permalink
Excellent explanataion!
Posted by: Sune at May 13, 2010 7:43:09 AM
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.
So is Congress like the Good Witch of the North?
And is this "Good Witch Congress" trying to ban "the practice of magic by any other witch"?
It all makes sense now!
Posted by: anon at May 13, 2010 7:43:30 AM
"the probability of default of the CDO jumps from p=.0005 to p=24.7"
There is a typo at that second probability.
Posted by: Marcos at May 13, 2010 7:54:10 AM
But that is, if I may say so, bleedin' obvious. And so I am confident that it was understood by the blokes that designed these toys. From which, I conclude....what? Hang 'em?
Posted by: dearieme at May 13, 2010 7:57:23 AM
Good catch, Marcos!
Posted by: Alex Tabarrok at May 13, 2010 8:03:38 AM
This math assumes that security defaults are independent, which is an assumption nobody ever made. It was not until David Li's seminal work on using the Gaussian copula to model correlations that anyone had any confidence in valuing these securities.
Unfortunately, tranches, especially safe, senior tranches, are very sensitive to the correlation assumption - even worse than to misspecifications of the probability of default. So when you are used to looking at historical correlations of 20%-30%, and you make a safe tranche even to correlations of 50%, when every underlying blows up and correlations are >90%, the tranche blows up. For a securitization of a securitization (e.g. CDO of ABS), this is even more so.
The example above only works because the attachment point is too low - if you use a more realistic 15%-25% subordination you will still get very small probabilities of default for senior tranches even with higher probabilities of default. But high attachment point tranches will blow up easily if you incorporate correlation.
If you want a better calculator for valuing structured finance securities, you can go to my site at http://www.battleagainstentropy.com/cdo/ (which still uses the single factor Gaussian which is hardly state of the art, but a useful illustrative model)
Posted by: njnnja at May 13, 2010 8:10:30 AM
Good catch, Marcos!
Posted by: replica iwc at May 13, 2010 8:46:45 AM
njnnja, Cool site, but in IE the background is black so it's nearly unusable (savy users will realize that they can select everything but that's pretty unweildy).
Posted by: nelsonal at May 13, 2010 8:48:47 AM
It is the essence of risk management to recognize the probabilities used for calculations are nearly unknowable. This is why Fischer Black is a crank first and a genius second. He knew, but chose to ignore that small changes in probability and correlation can mean vastly different outcomes. He used the idea that everything goes to equilibrium to drive real world dynamics and human error out of finance - a crime of the highest order.
A huge majority of our financial world and economics world doesn't recognize this idea and refuse to understand it when they do recognize it. We've optimized an economic system for speed, but it is not robust.
Posted by: Mr. E at May 13, 2010 9:05:38 AM
@ njnnja
Of course you're correct that the correlations rule the day, especially with respect to CDO of other structured products. The Coval, Jurek and Stafford paper linked above makes precisely this point. It would have been a bit tougher for Alex to demonstrate this in a few cells in Excel. I think the point is that the example is quite accessible and demonstrates how sensitive the structured product risk is to parameters that can't be estimated with any high degree of certainty.
I like the calculator by the way. thanks for the link.
Posted by: AaronG at May 13, 2010 9:08:46 AM
I think garbage in, garbage out is a better description than 'black magic'.
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