An interesting paper just out, suggests that modern financial instruments are computationally intractable: specifically finding that inverting the synthesis of CDOs is equivalent to finding whether there is a dense subgraph of the CDO graph.
Thus figuring whether CDOs are manipulated to dump bad collateral selectively on a subset of buyers is probably NP hard!
Ouch.
Good discussion at Freedom to Tinker
If correct, then the current structure of Collateralized Debt Obligations, and their higher derivatives - CDOs, CDO2s etc - is intrinsically not just asymmetric between seller and buyer, but finding out whether the blending of mixed collateral into a specific set of CDOs was random and fair, or deliberately skewed to dump "lemons" on a subset of buiers, is, in general, impossible to reliably find, for the buyer.
And this is even assuming the construction of the collaterlized debt is fair to begin with and the seller is only working with the asymmetric information later on which random part of the collateral is lemons.
The paper demonstrates that finding whether CDOs are fair is equivalent to the dense subgraph problem, which is believed to be NP-complete.
So with finite computational resources, CDO buyers, "even Goldman Sachs", can not determine a fair price, or detect whether they are being selectively screwed with and having lemons dumped on them in CDO purchases.
The authors suggest computational models for constructing "fair CDOs" using different structures for slicing collateralize debt, but it may be easier to prove P=NP than to reform the financial industry...
h/t kos
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Oh my. If Marvin the Martian were in the financial industry, he would call this an "Earth-shattering kaboom".
An update at the Kos post notes that the problem of future CDOs/CDSs is essentially solved: their fair value is effectively zero. But money tied up in existing CDOs/CDSs, which amounts to several times world GDP, is basically stuck there.
But it just goes to prove the Warren Buffett mechanism that you should not invest in things you do not understand.
If Buffett practices what he preaches he has some explaining to do. I seem to recall that his companies were peddling plenty of derivatives. And he made a fair bit of money investing in one of the ratings agencies just before they raked in the $$s promoting CDOs and their ilk. Moody's I think.
Unfortunately much of wall street "research" including things that we think we know is an echo chamber that is one dubious set of numbers away from being junk, as Enron etc. proved. Buffett's general advice about spreading the risk and relying on the overall trend of the market is most sound though.
Reading the paper, I'm a little underwhelmed. Computer scientists go nuts over the precise complexity class, all well and good for their purposes. But that just tells you about the behavior at large N. At small N, the problem may be very feasible. I never see anywhere in the paper an estimate of what N might be for these CDO's.
For instance, although factorization is computationally hard, I can factor the number 15 in my head. On good days I can even do it pretty fast :)
Good question, Andrew. I don't know the answer, so let me try it as a Fermi problem.
There are on the order of 100 million housing units (including multifamily rental buildings) in the US. I'll guesstimate that 70% of those are houses or condos. I'll also guesstimate the average time someone lives in a house as 7 years (this is consistent with post-WWII moving habits), which implies ~10 million houses sold per year. While not all of them require mortgages (there are investors who do all-cash deals), this is offset by people who refinance or take out a second mortgage or HELOC, so call it 10 million new mortgages per year. Taking 200k as a typical loan balance (it will be higher in places like California and lower in places like Detroit), that's $2T in mortgages. Let's assume half of that is kept in house by smaller community banks and half is done by larger banks who plan to bundle them into CDOs. The amounts per CDO issue need to be big enough to be worth the bank's while, and this is the biggest guess on my part: it could be 1000 issues of $1b each or 10k issues of $100m each (I doubt it will be much smaller than $100m). Those frequencies of issue, if anything, look a little high to me, but in any case we are looking at 5000 mortgages in the big pool and 500 even in the smallest pools. And this is just a plain vanilla CDO; there are also CDO^2 and CDO^3 products out there.
I don't doubt that you can factor a 4-bit number pretty fast. So can I. But a 500-bit number, let alone a 5000-bit number, is a much harder problem, especially when (as is true of the numbers that arise in encryption algorithms) the number is the product of two primes.
Part of the problem is that information is lost in the derivative generation process, is it not? Which is resulting in both 'good' and 'bad' outcomes, like the actual holder of a note becoming obfuscated, causing some foreclosures to be rejected by the courts. Does that also mean the lucky owner gets the title outright and can stop making payments altogether?
My understanding is that information is hidden not lost in the procees. It is somewhat analogous to encryption, in that choosing how to slice CDOs is simple from the seller side, but reconstructing how the slicing was done and whether the stated algorithm for doing so was actually used is hard.
As a rule, until and unless there is a judicial rule (and I ANAL), the mortgage payee must keep paying, and that is handled by servicing firms, distinct from the holder of the mortgage (which may be spread between many creditors after CDO^n).
@Foland the fiscussion at "Freedom to Tinker" covers some of the practical computational aspect - the authors show up to discuss the issue.
Subgraphs smell to me like a combinatoric problem, so the numbers get large fast. Looks like Erdos solved this, but his notation ain't exactly transparent.
Hm, looks interesting - I'm taking the #subgraphs > 2n/4k for n vertext graphs and k some well defined small number.
So, easily > 2100 subgraphs to compute for modest size CDOs, if I have parsed this right - and if not, someone is sure to correct me...
Gray Gaffer, you are presumably referring to the handful of cases where, in a contested foreclosure, the foreclosing entity fails to produce the note and/or a valid chain of endorsements giving them the right to foreclose. That's a simple matter of sloppy paperwork, as the late Tanta explained so long ago on Calculated Risk. While the sloppy paperwork is often correlated with the creation of a CDO, it is a separate issue from the computational intractability of CDO creation.