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A Long-Dead Mathematician and Some Very Lively Problems

April 23, 2013

The dour-looking fellow you see before you is Thomas Bayes, a mid-eighteenth century English mathematician, dead since 1761, the ramifications of whose work continue to unfold, and in directions of interest to CAIA members.


In Bayes’ day, the study of probability was mostly about issues of this form: if we know a certain underlying fact, what is the probability that we will make certain observations? If we know there are 100 blue balls and 40 red balls in the urn, and the urn has been thoroughly shaken, what is the probability that we will draw two blue balls in succession from the urn?

Bayes’ famous posthumous manuscript was about “inverse probability” in that it inverted this problem. If we know only the observed data, what can we infer about the underlying matters of fact? If we have just drawn two blue balls from the urn, what can we infer about the mixture of balls inside?

That ms itself didn’t lay down what is nowadays known as Bayes’ theorem, but it took some big steps in that direction, steps further developed at the start of the nineteenth century by the astronomer Pierre-Simon Laplace into the present formulation.

Excuse a simple bit of notation:

That’s one common formulation of the theorem, and among Bayesians, this stands for the relationship between a hypothesis (H) and the existing data (D).

From this formulation, Bayesians have also developed the idea of a Bayesian game: that is, a game in which a player has to try to figure out the characteristics of other players despite incomplete information on the basis of various signals. People are more complicated than urns filled with balls, but otherwise this is a natural development of the underlying idea. Think of a poker player trying to decide whether the other guy’s facial tick means that he has lousy cards, or just that he has a facial tic.

Uncertainty and the Oil Fields

The reason for the review is that an Australian scholar has recently published a paper offering “A Bayesian Understanding of Information Uncertainty and the Cost of Capital.” The gist of it is that traders face information uncertainty, that is, the risk of a misleading signal about the value of an asset.

“The Bayesian position,” says D.J. Johnstone of the University of Sydney Business School, “is that even a highly informative signal … can bring an increase in uncertainty, and hence an increase in the cost of capital.” This is at least somewhat counter-intuitive. Surely the highly informative signals (also known as “greater transparency”) will lessen uncertainty and risk, thus reducing the cost of capital.

Ah, Johnstone says, perhaps not. Consider a world in which there are two possible geological formations involved in the search for oil: A and B. There may be oil under either plot. Geologists tell us that plot A belongs to a type of geological formation with a 0.5 frequency of oil. B-type plots, on the other hand, have a 0.95 frequency of oil. It isn’t always obvious which is which, and oil companies like to figure out which is which before making the final decisive test to determine whether there is oil there.

Suppose also that the prior probability of oil under a random site, before we even know if the site is A or B, is 0.635.

Now, on Day 1, an oil company owns a piece of land that has not yet been tested for oil, or even tested to determine whether it is A or B. The market will presumably assess the value of this land accordingly. Prospective buyers will consider it as having a 0.635 likelihood of bearing oil.

On Day 2, the land is tested and found to be of Type A.

Thereafter, the market will lower the value of that land, because its likelihood of bearing oil has fallen to 0.5. There is greater uncertainty post-test than there was pre-test.

Final Thoughts

So, although common intuition indicates we would invest more in an asset that is more transparent than we would in one that is opaque, other things being equal, in this case clearly a prospective buyer would pay less for the land after its character has become more transparent to him.

And didn’t “other things” remain equal? No oil that might have been there before has leaked out. The drilling costs haven’t increased. All that has changed is the new signal.

Intriguing as that is. Johnstone’s real point is about accounting. Different accounting methods are often hailed because they will bring greater transparency and thus presumably decrease the cost of capital. The Bayesian argument indicates that is the wrong reason to support improvements in financial recording. Sometimes, improved information will reduce the value of a company.

That is not an argument against the improvement. It does require that we understand, though, that solid accounting “should be understood not in terms of its effect on the cost of capital per se, but as aiding investors to assess the probability distributions of future cash flows more accurately….”