Why regulation CANNOT work

There is an interesting paper discussed here on derivatives and computational complexity - the paper makes the case that, while it may theoretically be possible to value a CDO, in practice it is likely to require intractable amounts of computation. They reduce the pricing problem to the Densest Subgraph problem, which is believed to be NP Complete (read that as "impossibly slow" if you're not interested in the details), and also show that some deliberate fraud in the CDO construction would be undetectable. Essentially it would be the same difficulty as factoring a large number - which is so hard it's used as the basis for cryptosystems.

The paper is an interesting read, and suggests that neither counterparties nor regulators nor ratings agencies could hope to know the true situation, even after the fact. That obviously implies a lot more risk than people were expecting. Hindsight is 20/20, eh? Future regulations will hit the same problem, though - there are some theoretically calculable things that you just can't work out fast enough, and we're not talking "a few days" here, we're talking "billions of years". The FSA can't do the risk calculations here, and nor can anyone else.

Ken's comment down at the bottom of that blog post makes a much more important point: it's possible to construct undecidable derivatives. I'd have gone one step further, had a paper C which pays out if B doesn't - making it superficially similar to A - and have the holdings consist only of C. With that chain, the solution isn't just unknown, it's unknowable: C will pay out if - and only if - C doesn't pay out.

This is fundamental computation theory, closely related to the Halting Problem and some mathematical results (Russell's Paradox, Gödel's Incompleteness Theorem). If the FSA had infinite computing resources, they still couldn't solve it. No matter how smart the people involved are, or how fancy the techniques, or how shiny the machine room, it's impossible. Truly, mathematically, impossible.

It follows from this that Mervyn King is completely right: the solution is to firewall the retail banks from the risks in the market. You can't tax, or risk-weight things which cannot be calculated. If two "casino" banks want to tie themselves in knots over undecidable derivatives, fine - their lawyers can make money negotiating a solution, and if one or both collapses, so be it. But the banks we rely on have to be insulated - and this can be done.

Undecidability only exists in "sufficiently powerful" computation models. Arbitrary derivatives are clearly sufficiently powerful. Long positions are not, even when the companies and funds own shares in each other. The solution, and it's a nice simple one, is to work out what instruments a retail bank can safely trade in, and limit it to only those things. You could even let the bank invest some small amount of its assets in a dodgy-as-you-like hedge fund - with the proviso that it can expect to lose 100% (but no more) of its investment and plan the risk accordingly.

Northern Rock may still have collapsed, of course - it made very poor lending decisions - but its assets would have been snapped up as a going concern. All people would have noticed is a change in the letterhead on their statements. Bear Sterns and Lehman Brothers couldn't have affected the high-street banks, as they wouldn't have been allowed to gamble on them - and the other investment banks (Goldman Sachs, et al) and their shareholders would simply have had to eat their losses. Rather than the taxpayer.

- KoW