Sorry to disappoint: I don’t mean inappropriate in the Jeff Ely sense, but in the overmathematical sense. This post is intended for people who have had an intro econ PhD sequence, at least as far as expected utility theory.
Nabil was showing me a question he asks the MBAs about their preference between two gambles with the same mean and variance. There is no right answer; the idea is just to show them that distinct distributions can have the same mean and variance. With some normalization, the gambles boil down to: A: (1, .5;-1, .5) vs. B:(2, .125; 0, .75; -2, .125). You could think of the units as thousands or ten-thousands to make it more interesting. When Nabil showed me the problem, I said, only half-joking, “I don’t know, I’d have to think about it; I’ve never decided whether I’m kurtosis-averse.” Indeed (as my discussion will confirm), neither gamble second-order stochastically dominates the other, i.e. being risk-averse (having a concave utility function) doesn’t tell you which to choose. I decided to see what would be preferred for various CARA or CRRA utilities, and discovered the following:
Theorem: Let A and B be two bounded gambles which are symmetric about the same mean. If each moment of B is at least as big as each moment of A, A is weakly preferred to B by every CARA or CRRA utility function. If at least one inequality is strict, the preference is strict.
Proof: Note that symmetry means all odd moments are 0. Expanding any CARA or CRRA utility as a power series centered at the mean, we find that all even coefficients are negative. These series converge absolutely and uniformly in the range of definition, so linearity of expectation applies. Q.E.D.
That is, people with such utility functions are kurtosis-averse (and 6th-moment-averse, and 8th-moment-averse…) So any CARA or CRRA person prefers A to B above. Apparently most MBAs also pick A; my sense is that a small probability of a large risk tends to loom large in one’s mind. I admit to a similar psychological bias; I would force myself to overcome it if there were a good reason, but if I can support the decision with any CARA or CRRA function, that sounds all right to me.
So what kind of function chooses B? By my claim above there is a concave function that does, and indeed:
Claim: Let A and B be gambles with the same mean. Normalize this mean to 0, and suppose . Then any concave, piecewise linear function with unique kink at 0 will prefer B to A.
Proof: Simple calculation is left to the reader.
Such piecewise linear functions are often used in a simplified version of prospect theory. So, that part of prospect theory tends to select B, but the overweighting of small probabilities tends to select A. Florian Herold has a paper about disentangling these aspects of prospect theory. I’m getting too tired to think about how it applies; perhaps he would like to comment.
Bottom line: I’ve always thought that CRRA sounded pretty reasonable. Today I learned that if I want to stick with this, I’m kurtosis-averse, and also…what should we call the 6th moment? Sextosis-averse? (nod to Jeff.)
P.S.: An interesting not-so-technical question about reference points: On the actual homework, the mean was 5 and not 0. I know when I looked at this, my psychological reference point instantly became 5, and my feelings about gains and losses went accordingly. Would this also be true of MBAs? Or would they not be so quick to recognize symmetry and “translate” their expectations?