Paradox plays a part in a modern fable of economics


Bear in mind the gambler’s advice: if you don’t know who is the patsy in the room, it’s you.

What is the link between a visit to Russia by an 18th century Swiss mathematician, collateralised debt obligations, and the philosopher’s stone?

The Swiss mathematician, Daniel Bernoulli, first propounded the St Petersburg Paradox. A version for today’s markets envisages a casino with a coin-tossing game at fair odds. You hope to win at the first toss with a £100 stake. But if you lose, you will stake £200 on the next throw. If you lose again, you raise your stake to £400, and so on. You keep doubling your bet until you win. Simple mathematics shows that when you do win, you recoup all your previous losses and make £100 profit. Since you are bound to win eventually, the scheme is a sure means of winning £100.

Still, I don’t recommend it. One problem is that you can only be certain that you can last out until you win if you are infinitely rich, and if you are infinitely rich, why would you risk so much to gain £100? As the philosopher Russell Hardin observed, the St Petersburg Paradox does not survive the injection of any dose of realism.

Does today’s structured debt market survive any dose of realism? Perhaps we shall learn in the next few weeks. Like the gamblers in Bernoulli’s fable, today’s sophisticated investors hope to make risky prospects into sure things. Their prospective winnings come from turning mortgages and corporate credits into triple A securities.

But does this philosopher’s stone they seek exist? The risk characteristics of a bet can never be eliminated, but can be changed by division and aggregation. In every B-rated bond, there is A- and C-rated paper waiting to get out. Discard the C and your B has become an A.

This scheme works, but at a cost: in an efficient market, the value of the risk reduction will be precisely offset by a reduction in return.

Perhaps someone will take the lowly rated paper from you at a good price. You might be lucky. But bear in mind the gambler’s advice: if you don’t know who is the patsy in the room, it’s you.

Another means of turning risk into certainty is “payment in kind”. Any deterioration in credit leads to the provision of sufficient additional security to maintain the initial value of the issued paper. This pledge guarantees the purchaser against volatility and if the pledge can be believed, every instrument that benefits from it is as safe as the US Treasury.

You don’t need the insight of Bernoulli to spot the flaw. But the insight of the vendors of these products is to bury that issue in complex mathematics of which Bernoulli had not imagined.

But the best way to get something for nothing in financial markets is to profit from mispricing: identify pairs of securities whose relative value differs from its historic levels. If history repeats itself, these arbitrage opportunities can be leveraged to both reduce risk and enhance returns. Of course, history might not repeat itself.

But history often does, as supplicants find when they search for the philosopher’s stone.

Yet medieval science did not know the atomic structure of gold, but only the superficial characteristics of the precious metal. Gold glistens and alchemists brought shiny objects to the gods.

They were sent away: it was not enough that gold was lustrous, true gold was malleable and heavy. The alchemists trudged back and forth, adding components to the mixture until their confection met these specifications.

Eventually, the blocks they produced were shiny, dense and soft. The gods agreed that since the metal had the characteristics of gold, it must indeed be gold.

The alchemists made generous donations to the temple. And so it came to pass that the amount of gold in the world far exceeded the quantity that had ever been mined.

Many centuries later, the computer replaced the philosopher’s stone, the gods were replaced by the rating agencies but financial supplicants were as importunate as alchemists. And so it came to pass that the volume of investment grade securities far exceeded the value of investment grade credits.

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