What's wrong with Paul Krugman's arithmetic?

I like Paul Krugman a lot. I like his skepticism a lot. Further, he has a magic touch for simplifying complex and arcane topics in economics. But in this blog entry, I think he goes a simplification too far.

He is attacking the “subsidy effect” of the Geithner plan. The Treasury Secretary unveiled a proposal Monday that would have the government partner with private investors to buy distressed banking assets. The FDIC would finance 85 percent of the purchases with non-recourse loans. That, Krugman argues with the following example, invites what looks like a great deal of overpaying:
Let me offer a numerical example. Suppose that there’s an asset with an uncertain value: there’s an equal chance that it will be worth either 150 or 50. So the expected value is 100.
But suppose that I can buy this asset with a nonrecourse loan equal to 85 percent of the purchase price. How much would I be willing to pay for the asset?
The answer is, slightly over 130. Why? All I have to put up is 15 percent of the price — 19.5, if the asset costs 130. That’s the most I can lose. On the other hand, if the asset turns out to be worth 150, I gain 20. So it’s a good deal for me.
Notice that the government equity stake doesn’t matter — the calculation is the same whether private investors put up all or only part of the equity. It’s the loan that provides the subsidy.
And in this example it’s a large subsidy — 30 percent.
The trouble with his example: he's using a casino-type model that doesn't translate well to the real world. In his attempt to simplify, he assumes an equal chance of the asset ending up worth either 150 or 50. In other words, you win if you land on black but you lose if you come up red -- like a roulette wheel.

What's more likely with an asset of uncertain value is a bit more complex. You crunch the numbers (and indeed, this is exactly what these private investors preparing bids will do) for a bunch of scenarios. Let's say you conclude the asset will wind up with a value of between 50 (worst scenario) or 150 (best scenario).

Now, if you run different scenarios, this doesn't lead to a binary data set of outcomes. In fact, “50” and “150” are the least likely values, lying at the far end of the distribution curves. Assuming a fairly normal bell curve, if you looked at 1,000 scenarios, most resulting values would cluster in the middle, then taper off toward the ends.

So you're most likely to wind up with an asset having a value between 90 and 110. A smaller batch of data points will occur between 80-90 and 110-120. A smaller batch still will populate the next bands (70-80 and 120-130). Let’s say, for the sake of argument, that 90 percent of the probable values lie between 70 and 130 (if anything, this may be conservative).

Returning to Krugman's example, he states that the private investor, who stands to lose at most only 15 percent of the purchase price, would be willing to pay slightly over 130 for the asset. That's a whopping overpayment of 30 percent.

He's right if you accept the casino model. But for a more realistic model (as laid out above), it's not true at all. In that model, the investor who bids 130 stands a 90 percent chance of losing part, or all, of his money. Actually, it's even grimmer than that because most results are clustered around the 90-110 mark.

It may seem like I'm picking a nit, but it's worth putting the arithmetic in the proper perspective. Krugman is right that there will be overpaying, but I don't think it will be nearly as bad as he envisions (a few percent?). (Of course I'm assuming that loss- and profit-sharing are fairly split between the public and private entities.)

What seems like a greater potential threat, and the one the government needs to keep a sharp eye on, is the possibility of investors gaming the system. I'm not sure exactly how it would work, or if anyone on Wall Street still has the chutzpah to attempt it, considering how vilified the Street has become.

But the possibility is certainly there because private investors will be buying highly leveraged investments with dumb money partners (yup, that’s us, the U.S. taxpayer) who will take a huge chunk of the downside risk.

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