Expected Utility & Risk Aversion

Why decline a fair bet? Concave utility makes the average of utilities less than the utility of the average.

The idea

Expected utility is the standard model of decision-making under risk: a decision maker values not expected wealth but the expected usefulness of wealth. One observation shows the need for it. Offered double-or-nothing on their whole wealth on the toss of a fair coin, almost everyone declines, although the bet leaves expected wealth unchanged — whatever people maximize, it is not expected dollars.

Bernoulli's proposal: fix a utility function $u(W)$, the value of holding wealth $W$, and let the decision maximize the expected utility $\mathbb{E}[u(W)]$ rather than the expected wealth $\mathbb{E}[W]$. The property $u$ needs is concavity — each extra dollar is worth slightly less than the one before it — and $\sqrt{W}$ is the standard concave example.

Concavity explains the refusal, through Jensen's inequality in its concave form.

Theorem (Jensen's inequality, concave form).

If $u$ is concave, then for every random wealth $W$, $u(\mathbb{E}[W]) \ge \mathbb{E}[u(W)].$

So a sure amount is always at least as good as a gamble averaging to it: a fair bet loses in utility. On the graph of $u$, the gamble's value $\mathbb{E}[u(W)]$ lies on the chord between its two outcomes, below the arc at $u(\mathbb{E}[W])$; the gap between them is the utility a fair bet loses. This preference for the sure mean is risk aversion.

Two definitions convert the aversion back into money.

Definition (Certainty equivalent and risk premium).

The certainty equivalent $CE$ of a gamble $W$ is the sure amount valued exactly as highly as the gamble, so that $u(CE) = \mathbb{E}[u(W)]$. The risk premium is $\mathbb{E}[W] - CE$, the expected value the agent will give up to be rid of the uncertainty.

That premium is what an insurer collects.

Ways to work on it

• Walkthrough. Weigh a double-or-nothing bet in dollars and in utility, meet Jensen's inequality, and price the fear via the certainty equivalent.
• Practice. Compute expected utilities, certainty equivalents and risk premia, and make accept-or-decline calls.
• Hardest. A gamble that wins on expected wealth and loses on expected utility — resolve the tension.

Not sure where to start? Take the ten-question placement test.