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## June 19, 2005

### Notes: URAP Project 2: Fall 2005: Analyzing Marty Weitzman, "A Unified Bayesian Theory of Equity 'Puzzles'"

Time to start setting out potential projects for undergraduate research assistants for the forthcoming fall...

Here's another possibility: one that requires somebody with enough statistics to not be scared of moment-generating functions, enough math to not be scared of stochastic Taylor expansions, and enough programming skills to run a number of Monte Carlo simulations:

Over the past century in the United States, the equity premium has been on average 5% per year. Over the past century in the United States, the standard deviation of a diversified portfolio of equities has--accounting for apparent mean reversion--compounded at a rate corresponding to about a 10% standard deviation per year.

This means that if you buy-and-hold stocks for sixteen years, you have a "t statistic" of 2: under a normal distribution, your portfolio will outperform the alternative risk-free portfolio 97.5% of the time. If you buy-and-hold stocks for thirty-six years, you have a "t-statistic" of 3: 99.5% of the time your portfolio will outperform the risk-free portfolio.

This is the asset market version of the equity-premium puzzle. Why don't agents--at least agents with secure other forms of wealth that they can pledge--borrow at nearly the risk-free rate and invest in stocks? Why don't they do this on a large enough scale to drive the risk-free rate up and the return on equities (and the equity premium) down, and so avoid the paradox of a market where it looks like there is a thirty-six-year portfolio strategy that earns you a sixfold profit on your original long position (and a sixty-fold profit if you were able to borrow 90% of your original investment)?

Now comes Marty Weitzman (2005) with a very impressive and well-worked out paper on the importance of taking into account our ignorance of the structure of the economy. His is the observation that investors do not know but have to estimate the parameters of the economy's structure, and that under Constant Relative Risk Aversion utility this structural uncertainty adds fat tails to the subjective return distribution and could easily account for the equity premium and for related puzzles.

^{1}

- Is this a reformulation of Rietz (1988)--and thus subject to the same critiques?
- Is this a reformulation of Geweke (2001)--and thus primarily a statement about the limited usefulness of CRRA utility (with its asymptote at Wealth = 0) in addressing economic questions?
- Or is this a statement about the way the world works--a successful assertion that our ignorance about the true structure of the economy is such that rational investors with reasonable preferences are right to be at least somewhat shy of equities?
Rietz's (1988) answer to the equity premium puzzle was this: a long, fat lower tail to the return distribution. A small probability of very bad things happening to stock returns could support both (a) a relatively small sample variance of returns, and (b) rational aversion to large-scale stock ownership large enough to produce the observed equity premium. The question that Rietz was unable to answer was: "What exactly are these very bad things?" Remember that the equity premium is a premium relative to the return on relatively short-period U.S. Treasury securities. Any macroeconomic factor to drive the equity premium must therefore be a factor that leaves the real value and real return on short-period U.S. Treasury securities unaffected. But almost all true macroeconomic disasters that could halve or do worse to the real value of equities are likely to produce at the very least rapid and substantial inflation, if not confiscatory taxes on or outright repudiation of government bonds.

Geweke's (2001) observation was that the CRRA utility function is an extraordinarily fragile tool when confronted with alternative distributions than the Gaussian Normal. We use CRRA utility because it buys us extraordinary analytical simplicity at the price of accepting:

lim(x->0) U(x)=-∞, U'(x)=+∞

but we are not thereby committed to riding the taxi of this vertical asymptote at Wealth=0 to its final destination. The CRRA implication that investors limit the size of their leveraged equity positions because bankruptcy is seen as infinitely painful does not appear to correspond with our world.

The third possibility is that Weitzman (2005) really is a profoundly powerful statement about the world: that structural uncertainty, even conditional on the requirement that whatever bad news comes does not materially affect the real rate of return on relatively short-term U.S. Treasury securities, combined with plausible preferences and risk aversion would lead us to expect a considerable equity premium.

It is pretty clear to me that Weitzman (2005) is saying considerably more than Rietz (1988)--that it is either the second or the third. It's clear to me that it's both, but I'm not sure of the weights. I'm not at all sure whether it's primarily the second (in which it is a very useful illustration of the fragility of CRRA-based models, and thus in most part, as Daniel Davies puts it, "a fact about applied math"), or primarily the third (in which case it is the solution to a puzzle that has stood effectively unanswered for a generation).

^{1}Note: Weitzman's paper does not seem to me to deal withleverageproperly. The r_{e}in the Consumption CAPM isn't the return on equities, but a return on a portfolio that is a claim on output as a whole--say, 60% labor income, 10% real estate, 20% bonds, 20% stocks... an equity premium of 2-3% per year instead of 5-6% per year...How fat do the tails have to get to generate a large equity premium for preferences that do not regard bankruptcy as infinitely painful? I need somebody to run some simulations and calculate a bunch of Taylor moment expansions for different distributions

John Geweke (2001), "A Note on Some Limitations of CRRA Utility,"

Economics Letters71, 341-345.Rajnish Mehra and Edward Prescott (2003), "The Equity Premium in Retrospect," chapter 14 in Constantinides, Harris, and Stulz, eds.,

Handbook of the Economics of Finance(Amsterdam: Elsevier B.V.).Rajnish Mehra and Edward Prescott (1985), "The Equity Premium: A Puzzle,"

Journal of Monetary Economics.Tom Rietz (1988), "The Equity risk premium: a solution."

Journal of Monetary Economics21: 117-132.Martin Weitzman (2005), "A Unified Bayesian Theory of Equity 'Puzzles'" (Cambridge: Harvard).

Posted by DeLong at June 19, 2005 10:19 PM