Understanding Utility Theory

Uncertainty and risk have long confounded human reasoning: the syllogisms of basic logic assume certainty in all assertions, and allow no conclusions to be drawn until all factual states are decided.

And, in practice, people often demonstrate themselves to be remarkably poor at making decisions based on uncertain conditions, foregoing much needed insurances while over-paying for absurdly unlikely coverage, for example.

Within this confusion, the intuitively appealing notion of a "risk premium" has arisen. In general, the notion of a risk premium is that reasonable people will not accept risk without an expectation of, on average, profit above and beyond what is available without risk.

Utility theory is a method of quantifying the notion of a risk premium.

The main premise of utility theory is that we should concern ourselves not with the cash value ("wealth") of anything, but rather with the utility (which I guess is just a of saying "usefulness" with a word derived from Latin ...) of the wealth. And then there are just two parameters that the utility function of wealth are held to have to satisfy: the utility of more wealth is always greater than the utility of less wealth; and the rate of increase of utility relative to wealth decreases as the absolute level of wealth increases.

Mathematically, these two properties are stated as follows:

1. dU/dW > 0; and
2. d²U/dW² < 0.

And this is really all that standard utility theory tells us.

There are, however, a couple more things to know about utility theory.

The first is that, in standard financial theory today, utility theory is the beginning and end of how risk is understood. This can perhaps be seen most clearly through a 1963 article by Paul A. Samuelson, "Risk and Uncertainty: A Fallacy of Large Numbers", and through a 1995 article that is largely a reprise of the same argument: "On the Risk of Stocks in the Long Run" by Zvi Bodie (sorry, no link).

"Risk and Uncertainty" relies upon utility theory to demonstrate that if you will not accept, due to risk concerns, a single wager with a positive expected return, you also should not accept any multiple repetitions of the same wager, even if the possibility of any loss under the multiple repetitions is almost zero. "On the Risk of Stocks" uses Black-Scholes options pricing theory -- which is based on utility theory -- to dismiss the proposition that, if your time horizon is long enough, the higher expected return on stocks will make them a more suitable investment than investments with more certain returns, even though you would want the more certain investments for a shorter time horizon.

The second thing to know about utility theory is that it seems to have almost no predictive value whatever in determining how economic agents, like investors, actually behave. In traditional science, having no predictive value is the hallmark of a hypothesis (not a theory ...) that must be rejected.