The paper shows the example of a high-volume seller who builds 1000 CDOs from 1000 assert-classes of home mortages. Suppose the seller knows that a few of those asset classes are "lemons" that won't pay off. The seller is supposed to randomly distribute the asset classes into the CDOs; this minimizes the risk for the buyer, because there's only a small chance that any one CDO has more than a few lemons. But the seller can "tamper" with the CDOs by putting most of the lemons in just a few of the CDOs. This has an enormous effect on the senior tranches of those tampered CDOs.
In principle, an alert buyer can detect tampering even if he doesn't know which asset classes are the lemons: he simply examines all 1000 CDOs and looks for a suspicious overrepresentation of some of the asset classes in some of the CDOs. What Arora et al. show is that is an NP-complete problem ("densest subgraph"). This problem is believed to be computationally intractable; thus, even the most alert buyer can't have enough computational power to do the analysis.
Arora et al. show it's even worse than that: even after the buyer has lost a lot of money (because enough mortgages defaulted to devalue his "senior tranche"), he can't prove that that tampering occurred: he can't prove that the distribution of lemons wasn't random. This makes it hard to get recourse in court; it also makes it hard to regulate CDOs.
Intractability of Financial Derivatives
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Sounds about like something the bankers would like...
Now, admittedly this isn't my speciality, but would it stand that there were ways to make a mint by having the seller pass on to the prospective buyers info on which where lemons and which weren't?
Long-term plan, mind you, since it'd take a good decade or so for the market to recover, but...you'd own the only good liferafts after the ship sank, right?
It is irrelevant. The assumption that they are making is that somehow the defaults of the assets are independent of each other. They aren't they are linked. ie. When mortgages do well, they all do well. When mortgages go bad, lots go bad. If your mortgages are in Detroit, you have lots of problems.
The other problem is that they are assuming in some way that it is only the seller or bundler who has access to the information. The buyer has access too.
It's a typical problem with academics who stray outside their sphere of expertise and end up making assumptions that are wrong, and then coming to a conclusion that isn't valid.
Caveat emptor.
But fuck me, Goldmans and JPM are announcing humungo profits - and didn't the former engineer the crisis?
Sounds like the sort of thing a bunch of 'rocket scientists' with limited feelings of responsibility and empathy might cook up to please the 'smartest guys in the room'.
The fix is in. The fix has been in since long before you or I were born.
"Goldmans and JPM are announcing humungo profits - and didn't the former engineer the crisis?"
I wouldnt say they engineered the crisis, but they definatlty profited from irresponsibility and short term thinking. They are posting huge profits because the American tax payers were forced to bail them out with hundreds of billions of our tax money. Its hard not to make a profit if someone else covers you in the case of a loss.
It should be noted that this is a purely theoretical result. What it shows is that the amount of computing time which it would take a buyer to figure out the exact value of a CDO grows substantially faster than the amount of computing time it would take a seller to create biased CDOs which benefit them. There are two immediate issues with applying this to the real world. The first is that they don't actually calculate the time needed for existing CDOs. Obviously some CDOs involve a large number of assets, but just because the time is exponential in the number of assets does not establish that it is so large that the buyer cannot compute it. In practice, there are quite a number of NP complete (and even exponential) problems which are entirely tractable for the concrete instances of them which frequently occur. To establish whether or not this is the case, the authors would need to gather some real industry numbers and then run some simulations to determine the actual wall time involved in solving instances of the problem.
The second issue is that they don't address whether or not the solution can be approximated in polynomial time. Even if the buyer cannot establish the exact price within polynomial time, that does not imply that the buyer cannot approximate the price within polynomial time. It may be the case that the buyer can find a sufficiently good approximation of the solution to give sufficient information to make good enough purchasing decisions.
They should also consider addressing the question of whether or not the problem is in co-NP. If it is, then the seller could offer a short certificate which could be used to verify that they had not created a biased CDO. The validity of such a certificate could be verified in polynomial time. Mind you, I'm not saying that the problem is in co-NP, just that they should check whether or not it is.
There is no profit into putting the fix on certain deals (unless you are keeping the good ones for yourself, which I haven't seen in practice, typically they sell everything of quality and keep the unsellable residual.) You can't earn additional profits on the good ones because it is a secret, right? And as soon as one of the bad ones has problems, all "equivalent" deals are going to get punished in the market, and viewed as lemons themselves.
Investment derivatives of all kinds -- other than very straightforward index funds -- have always been a good way to lose your shirt unless you spend Entirely Too Much Time managing them. I'm told that for a pro, a _small_ percentage of well-defined derivatives in a larger portfolio can be a way of hedging your bets... but that for anyone else, this is a case where you're definitely playing against the pros rather than with them, and "If you don't see the sucker at the poker table, he's sitting in your chair."
No offence, but the article is complete and utter BS, to use the technical term.
"Trading in derivatives brought down Lehman Brothers" - err, no. What brought down Lehman's (speaking as an ex-employee) was a) they weren't GS, and the treasury was looking to prove their moral hazard theory, b) they were too leveraged, c) their funding was too short term. Most of the derivatives desks at Lehmans were very profitable.
Bear in mind that Lehman's was largely a seller of CDOs, so the article doesn't make sense.
Anyway, in a CDO the issuer usually keeps the senior or equity tranche, rather than selling it on. Which means that they eat the first X% of the losses that happen before they pass losses on to the next tranche. So the result, while I'm sure it's theoretically valid (notwithstanding the other comments that the while P=NP issue only arises on large N etc...), it's not actually relevant to the issue.
@technogeek: Err, no. Index funds are not derivatives, and derivatives are a great way of making money, or insuring yourself against losses. Of course your definition of $(Too Much Time) may differ from mine ;)
So, we're continuing with the Economic Weapons of Mass Destruction, despite all that has already happened?
kraut doesn't seem to understand CDOs. Senior tranches are the safest investment, not the riskiest. They only lose principal if both the junior and mezzanine tranches are completely wiped out. It's kind of sad that even Lehman employees don't understand these products, given their pivotal role in the current state of affairs.
I am one of the authors of the cited paper. Note that the commenter commenting on the complexity aspect is quite mistaken (the paper uses avg case complexity, not P vs NP) and the problems of approximation, certification etc. are also conjectured to be intractable.
Also, current rating algorithms at the ratings agencies use simple monte-carlo simulation. That definitely will not detect dense subgraphs. The commenter who thinks he can do any version of densest subgraph efficiently should definitely try to publish this algorithm because the experts would be very interested.
I also want to address the issue of simple model vs complicated model. As pointed out in the paper introduction, the simpler model with independent asset yields considered in the paper can be *embedded* (ie is a subcase of) inside more complicated models. That makes the result *stronger* not weaker. In general you want to prove negative results in as simple a model as possible, and a positive result (in this case, that would be a pricing algorithm) in as complicated model as possible.
The details about CDOs in the paper are correct. The commenter Kraut who thinks that issuer keeps the senior tranche is clearly mistaken, as anonymous has noted.
Heh, guess why the whole house of cards collapsed? I think you put your finger on it...
The philosophical statement about hardness for a simple model implying hardness for a more complicated model does not seem to be true for many natural systems.
One example is Protein Folding. The problem is hard for simple lattice models. Once the model gets more complicated (more complicated lattices, more degrees of freedom), the problem is not known to be NP-hard and becomes much easier to approximate.
Why shouldn't it be the case that certain correlations between mortgage prices allow for better algorithms for detecting bad derivatices? Granted it may be the case that the industry does not use such algorithms, but I do not believe the current work rules out the possibility that in certain prevalent conditions, the problem can be solved.
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What capitalists often forget is that the "Invisible Hand" guiding the market can quickly become a punishing fist wrapped around a lump of iron when things go wrong...
Recently, a significant number of economists have been expressing the opinion that "The reason the 'invisible hand' is invisible appears to be that it isn't there."
When real life and theory don't agree, it's time to refine the theory.
It's possible to make a rational market-driven argument (although I still think that economic theory is mostly voodoo), but people talk about the invisible hand of the market like they talk about the toe-bone of Saint Euphemia; an unverifiable source of miracles.