Monday, March 23, 2009

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.


  1. Krugman was only employing a classic binomial model used to value options. Actually the numbers were given only to illustrate his case, i.e. to show laymen how the systwm works out

  2. Maybe you should do the arithmitic fully out? Because of the binary structure of the payout to the investor, one only has to consider two payout points, that of loss an profits. How the payouts are distributed besides this is irrelevant, since one can substitute it with an expected value and a probability of win/loss, exactly as Krugman has. If you care to elaborate on the different probability structure that you propose, you will see that it comes up with exatcly the same result as Krugman

  3. I tend to agree with a lot of what Krugman writes, but it seems this person has a point, and I don't follow the argument in the April 4th post.

    As a hypothetical example, let's say the variance around the expected value was 0. Then the asset would have 100% chance of paying out $100, meaning I'd always get back $15. But I put up $19.50, so I always lose $4.50.

    Maybe there's some property of bell curves Krugman is relying on to reach his conclusion, but to me, it seems the variance around the expected value should make a difference.

    At any rate, when I read Krugman's argument, I didn't take it to mean the Geithner plan actually resulted in the 30% subsidy, and more I take my brother literally when he says he will 'kill me in tennis'. It was just a toy example to illustrate the overpayment would take place, right?

  4. I never saw these comments originally so I'm responding VERY late, but I encourage anonymous April 2 and anonymous April 4 to also see my post here, which is a bit more detailed, if you haven't already:

    Justin, I mostly agree with you -- the variance matters tremendously (and you give a good example except I think the investor loses all 19.5 if profits and losses are shared equally yes? Bid is 130, variance is 0, investor automatically takes straight losses down to recourse loan that starts at 110.5 yes?). However, you say that you didn't take Krugman literally that there would be an overpayment of 30%; it was just a "toy example." Krugman's own defense is similar to this, in that he says academic economists use simplified models all the time.

    I would say "fine," except for two things: 1. the very premise of his model is wrong to begin with, and however simple the model is, that premise must be correct. 2. his "toy example" gives a significantly large overpayment -- 30% -- that is considerably off, I think. I don't think the overpayment would be anywhere near that much.

    Why that matters (before I return to point #1): if the public thinks there's a huge overpayment (as opposed to perhaps a modest or small overpayment), they will be hostile to this plan, the banks will probably resist as well if for no other reason than because the public will vilify them for getting perceived big overpayments when they know they really aren't (see my nearby entry about 5 reasons the big banks are terrified of Geithner's plan -- which, incidentally, in line with my predictions, has sort of quietly faded out of sight, you may have noticed, as we move into the fall of 2009).

    Now, point #1: the premise of the model is all wrong. Remember, this is a model of overpayment. All that matters in a model of overpayment is what THE BIDDER THINKS (he's the one overpaying right?). The bidder expects the value to be $100, Krugman says. Okay. Now what will the bidder's thought process be? Something like this: "Hmm, it could be worth $100, but then again, it could be a little more or a little less, and there's a small chance it could be worth a lot more, or a lot less, and there's an even smaller chance the eventual value could lie at one of two extremes, $150 or $50." How many bidders do you think say to themselves, "I'm going to assume its final value will be either $50 or $150"? (And by the way: it doesn't even matter if you posit that the final value IS $150 or $50, and the seller, with a Godlike omniscience, somehow knows this. It doesn't matter because the bidder DOESN'T KNOW THIS and will bid simply based on an expectation that if its expected value is $100, chances are good its final value will come in close to that figure, but could be off some, and could even be way off -- say even $50 high or low). Joe Bidder will probably construct a model of the possible outcomes for the final value, and will get some kind of bell shape-y distribution of points.

    For more on this, see my blog entry I linked above. In short, I just think Paul (and I love his blog, so no venom here) made a bad assumption: that when you have an asset of uncertain value, a bidder will assume its eventual value will lie at one of two extremes and not anywhere in the middle and make his bid accordingly. And this doesn't really make sense, even at its simple heart.