Economics and similar, for the sleep-deprived
A subtle change has been made to the comments links, so they no longer pop up. Does this in any way help with the problem about comments not appearing on permalinked posts, readers?
Update: seemingly not
Update: Oh yeah!
Thursday, May 13, 2010
Correlations and finger exercises
Yglesias finds a working paper with a perfectly sensible little finger exercise demonstrating how CDO tranching creates safe-looking but highly fragile structures:
"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.
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!
This is actually very good. But the comments, dear God, the comments! About a dozen people (including, I chuckle to note, Robert Waldmann, who would certainly have got this right if he'd been concentrating) step in to lambast the authors for oversimplification. A sample:
It’s not just misestimating the probability of default as .05 rather than .06. It’s also assuming that the probability of a mortgage defaulting is independent of the probability of every other mortgage defaulting. If there are economic developments that affect many borrowers at once, then the probability of borrower A defaulting *and* borrower B defaulting is not simply the prob(A defaults) * prob(B defaults). This is indeed what happened in 2008.
Moreover, the very act of A defaulting can alter the probability of B defaulting. If A defaults, and lives near B, then the rate of foreclosures in B’s neighborhood increases, which decreases B’s property value, which could push B underwater, and lead to B defaulting, which leads to C, D, and E defaulting, etc.
You know all of this, but I wanted to point out that the independence assumption is perhaps even more fundamental than using a mistaken probability. You can at least see what happens when you change .05 to .06, but non-independence can be a lot more complicated and make your model wrong not just by changing a single parameter.
Thanks for the sermon vicar, but ...
There is no assumption of independence in the original finger example, is there? In the first case, there's an expectation of the default rate. In the second case, there's some noise around this expectation, and that noise is perfectly correlated, isn't it? It's a realisation that is 0.01 higher in every single pool. The finger example is specifically and definitely one in which the entire sting is delivered by the fact that variables which were assumed to be independent turn out to be perfectly correlated.
I think that the problem here is that people tend to think of "correlation" as something measured by a Pearson coefficient, and so if they don't see one explicitly mentioned (or a copula structure specified or something), they just progress straight to the sermon and miss the big picture. Something to remember.
(Update, five seconds after posting: Actually I think the problem is more about imprecise use of the word "probability" in the original example. The number .05 doesn't refer to "the probability of default", it refers to the expected default rate, that's the only way the example makes sense. Expected default rates can't be correlated (because they're not random variables - they're expectations of random variables) which is why there's no correlation structure specified. Then "the probability moves to 0.06" in the second paragraph has to mean that the experienced default rate is 0.06 - ie, that the random variables concerned are actually perfectly correlated. The mapping from the default rates to the probability of default of the tranches makes sense though, because different loans are allocated to different pools at random).
(Update, half an hour after that. I suppose that in calculating the expectations in the first paragraph, there's an implicit assumption of independence. But that assumption is immediately relaxed and shown to be wrong! The real driver of this toy model is the fact that when the true default rate is discovered to be .06, this is true in every single pool.)
this item posted by the management 5/13/2010 11:38:00 PM