Tyler Cowen, Mark Thoma, Brad DeLong and *Washington Monthly*'s Kevin Drum were among those who picked up my post last week, Has Barro solved the equity premium puzzle? Here's a summary of what they had to say. Tyler Cowen is typically succinct:

If you want my view of the equity premium paradox, we have to askwhether Bryan Caplan is stupid. Choice is context-dependent, and we should not be too worried if no single utility function can account for all of our investment decisions.

Brad DeLong* thinks that "Barro has missed something important":

Finance economists talk of the equity premium as a suspiciously low price of stocks relative to the prices of risk-free real bonds. But that's not really correct. In the real world, the equity premium takes the form of a suspiciously low price of stocks relative to *short-term nominal government bonds.* To say that stock prices relative to nominal bond prices can be explained by the risk of disaster - having a Great Depression or becoming the battleground in a World War - requires that the real value of nominal government bonds not be affected by such a disaster. Yet one standard response of governments to the budget crisis brought on by disaster is inflation.

In order to make Barro's theory work, you need a (1) significant probability of (2) economic disaster that nevertheless (3) does not lead to significant inflation and (4) does not lead to a formal government default. Besides the Great Depression itself, it's hard for me to think of a disaster in the past that fits those four requirements. Typically, inflation means that economic disasters that reduce stock values also reduce real bond values as well: the 1% disaster that removes half your capital stock and cuts real GDP in half also needs to leave the price level and the government's commitment to repaying its bonds unaffected.

Kevin Drum* says Barro's answer "is that investors are fundamentally irrational":

[as] they are overestimating the probability of unlikely but catastrophic losses, and this fear makes the demand for risk-free investments larger than it rationally ought to be and thus drives down their price. In other words, the mystery is not so much that returns from stocks are higher than they should be, but that returns from bonds are so low.

...Needless to say, I have nothing to add to this, although it's an explanation I find appealing because of my fondness for Prospect Theory, which is based on the fact that people are not so much risk averse as loss averse. It turns out that most people feel much more strongly about the probability of a loss than they do about the probability of an equivalent gain, and it seems like this is partly what's going on here.

...This debate is far from over, but Barro's contribution is to propose an analytic framework that can be tested. I don't think it can be easily tested, since it relies heavily on perceptions, which are not straightforward things to measure, but at least it's something.

New Canadian blogger Stephen Gordon notes that, as with Martin Weiztman's recent paper, "the main result rests on a returns density with fat tails":

I'm sceptical of these approaches. In the standard expected-utility framework (Arrow, 1971; Lucas, 1978), utility isbounded- as far as I know, no-one has yet extended expected utility analysis to the case where utility is unbounded. The way I see it, the best way to interpret the unbounded utility functions and the densities defined over the entire real line that we use in applied work is that they are convenient approximations to what the theory requires.

But if the approximation is to be a good one, then we need to use distributions such that the model's expected utility over the entire real line is a good approximation for the true expectation over [u_{0}, u_{1}]. In other words, the contributions of the tails in the calculation of expected utility should be 'small.' Since utility is non-decreasing in consumption, the only way that we can be certain that the tails' contribution will be small is if the density for utility goes to zero fairly quickly outside [u_{0}, u_{1}]. In this context, it would seem to me that thin-tailed distributions would be preferred to those with fat tails...

Winterspeak provides an account of a buddy who heard Barro present the paper at a University of Chicago seminar:

My take away is that Barro's main point is right - we should consider the possibility of low-probability events. But I still don't think it explains the equity premium. He is still working on his paper - he just discovered Reitz's paper 6 months ago - and right now he uses a very simplistic framework.

One parameter in the framework is the level of risk-aversion in a CRRA framework. Most economists believe a risk-aversion coefficient of about 5 is reasonable, and he finds that with a coefficient of 5 you can explain a lot with a 1% chance per year of an "end of the world" type event for financial markets.

...Well I think that is crazy. I think you might, at most give up $200 in the 60k year to get $100 in the [$30k] bad year--that is a risk-aversion of 1, not 5. So while Barro/Reitz model is good for most economists, it still doesn't explain things to me. Because I think the only reasonable coefficient of risk-aversion based on intuition is about 1, but that predicts virtually no equity premium.

So in conclusion, I think Barro's insight is important and much better than what most economists argue is driving the equity premium. But the equity premium is still a puzzle to me.

*Incidentally, both Kevin and Brad's posts are also worth reading for the extensive comments posted there.

Just a heads-uy - Zimran at Winterspeak has a first-hand account from a PhD candidate who attended Barro's lecture there on the topic.

Posted by: Ironman | Tuesday, October 04, 2005 at 02:17 AM

Has anyone checked out Mandelbrot's opinion, founder of fractal geometry? In his book, "The (Mis)Behavior of Market" (2004), he argues (quite convincingly) that the use of Gaussian distribution (normal distribution) in CAPM is not justified. Historical prices rarely if ever converge to a mean; therefore, orthodox calculations of risk and return have been inaccurate. Using Guassian estimations, a catastrophic event would occur once every hundred years (ie drastic price declines fall several deviations from the mean) but, in fact, empirical data shows that the frequency of such "improbable" losses is far greater than predicted. Goeztmann confirms Mandelbrot's assesment via cross country analysis showing that, of 39 countries (mostly developed), only one or arguably two have not experienced a permanent break in the market. "The U.S is the exception, not the rule." Mandelbrot suggests a more accurate probablity distribution.... using fractals. (I will do my best to describe this momentarily)

Posted by: zoe | Saturday, December 03, 2005 at 11:39 PM

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