February 21, 2007
Economics Focus: Bayesian Asset Pricing
The less we assume we know, the smaller the asset pricing puzzles
When physicists test their theories against the real world, they are frequently correct to five decimal places after the comma. Economists tend not to be as lucky. Approximating within an order of magnitude is often all they can hope for when applying their theories to the data. However, even that has proven to be elusive in a quest to explain asset pricing phenomena with economic theory.
This inability to match theory with data has given rise to three related failures of neoclassical economics applied to financial markets: the equity premium, the risk-free rate, and the variability mismatch puzzles. Given historic trends, neoclassical theory for plausible risk parameters predicts that stocks ought to pay less than 0.1% more annual return than the safest assets. In reality, equities pay a premium over 6% above short-maturity U.S. treasury bonds – a discrepancy by a factor of more than 60 – much more than an order of magnitude. Reversing the logic, the risk-free rate for bonds is much too low compared to theoretical predictions. Lastly, the observed variability in the stock market is much higher than reasonable expectations about the economy itself. Yet theory says the two should be the same. Worse still is that attempts to solve one puzzle frequently exacerbate at least one of the others.
A defining characteristic of neoclassical economics is its adherence to the frequentist school of statistics. Frequentists believe that there is an underlying, true structure of asset prices. Their distribution is known; only the correct parameters are uncertain. With enough observations, we can approximate the correct mean and variance around it and act as though the parameters are known.
Bayesian statistics provides an alternative school of thought. Bayesians are not only uncertain about the correct parameters; they also do not know the true structure of the problem. Instead of setting out to accumulate massive amounts of data to approximate an unbiased sample, Bayesians rely on appropriate “priors,” pre-existing assumptions. Reverend Thomas Bayes first presented these ideas in eighteenth century England. Frequentists have managed to sideline them for most of their existence, but Bayesian statistics has experienced a renaissance of late, particularly in areas where data are sparse. Bayesians can draw inferences with as little as one data point.
Asset pricing is not an area that comes to mind in the context of paltry data. Hardly any field in economics can rely on as many observations. With decades of publicly available minute-by-minute pricing information from various stock markets around the world, equities are a fertile ground for frequentist statistical study. This abundance of data seems to have been the demise of theoretical economists trying to explain the asset pricing puzzles in the past. Many economists have suspected that Bayesian structural uncertainty may play a role in explaining these puzzles. Now Martin Weitzman, an economist at Harvard, has carried this idea much further. His paper* shows that all three asset pricing puzzles may be reversed when relying solely on Bayesian statistics. Some added mathematical wizardry brings us to a happy middle ground, where the puzzles disappear into the realm of statistical curiosities rather than massive failures of the theory.
Uncertainty is worse than risk
Mr Weitzman’s argument relies on the fundamental difference between risk and uncertainty. Risk refers to the frequentist worldview of unknown outcomes given known distributions. Uncertainty is linked to the Bayesian idea of unknown outcomes and unknown underlying structures. Poker players face risk. The distribution of a deck of cards is known. The risk of the game comes with not knowing the exact outcome of the next draw. Investors, according to Mr Weitzman, face risk and uncertainty. They do not know the exact outcome. More importantly, though, the underlying structure of the distribution is likewise unknown to some degree.
This distinction is not new to economics. It traces back to Frank Knight’s Ph.D. dissertation**. Mr Knight was a towering figure in early twentieth century economic thought. He went on to lead the Economics Department at the University of Chicago for two decades beginning in the 1920s. Most economists would recognize his distinction between risk and uncertainty. Few have known what to do with it.
Compared to the standard normal distribution often assumed by frequentists, a pure Bayesian analysis results in a “Student-t” distribution with significantly thicker tails. Cataclysmic crashes and miraculous price rallies now have a much larger chance of occurring than neoclassical theory allows. By putting a slightly higher probability on these extreme events, the asset pricing puzzles are even reversed. Pure Bayesian theory would predict a larger equity premium than is seen in reality. At the very least, it is no longer clear what puzzle needs to be explained: excessive equity premiums claimed by frequentists or overly modest premiums observed by Bayesians.
[Insert graph here: Student-t, “Bayesian-t learning”, and standard normal distributions shown in one graph, with decreasing thickness of tails.]
Mr Weitzman creates a hybrid dubbed “Bayesian-t learning” distribution with characteristics of both the frequentist and Bayesian approach. In fact, both are special cases of this more general theory. That is always a good way to find acceptance for one’s research. Instead of declaring past approaches wrong, simply sell your own idea as a generalization subsuming previous work. Most importantly, the task for economists now becomes a matter of model calibration to fit the data, rather than searching for the correct model in the first place.
As with every path-breaking piece of research, the question of why nobody has considered this previously remains. In this case, that question is the real puzzle. Economists needed only to take the ubiquitous warning for stock investments seriously: past performance does not guarantee future results. It appears that investors have stuck to that mantra all along.
* “Prior-Sensitive Expectations and Asset-Return Puzzles”. January 2007.
** “Risk, Uncertainty, and Profit”. 1921. Boston, MA: Houghton Mifflin.
Posted by Gernot Wagner on Wednesday, February 21, 2007. 
