Expected Utility and Risk Aversion¶

Definition¶

The expected utility model compares uncertain wealth $W$ through $\mathbb{E}[u(W)]$ . An increasing concave utility function $u$ represents risk aversion: for integrable wealth with well-defined utilities, $\mathbb{E}[u(W)]\le u(\mathbb{E}[W])$ $u (E [W])$ . A linear increasing utility represents risk neutrality.

Interpretation¶

A risk-averse policyholder can prefer a fixed payment to an uncertain loss with the same mean. Utility is applied to final wealth, so larger losses usually reduce utility.

Concave utility curve with wealth outcomes w_1 and w_2: the chord midpoint E[u(W)] above the mean E[W] lies below the curve's value u(E[W]), as Jensen's inequality states.

The figure, drawn schematically, takes wealth $W$ equal to $w_1$ or $w_2$ with probability $\tfrac{1}{2}$ each. The expected utility $\mathbb{E}[u(W)]$ is the midpoint of the dashed chord, while $u(\mathbb{E}[W])$ is the height of the curve at the mean; concavity keeps the curve above the chord, so $\mathbb{E}[u(W)]\le u(\mathbb{E}[W])$ $u (E [W])$ . In the Example $w_1=80$ $80$ , $w_2=100$ and $\mathbb{E}[W]=90$ under logarithmic utility: $[\log100+\log80]/2<\log90$ $lo g 90$ , so the certain loss of 10 is preferred.

Choose a utility and move the wealth $w$ , the loss and its probability: the chord value $\mathbb E[u(W)]$ stays below $u(\mathbb E[W])$ , and the horizontal gap to the certainty equivalent is the risk premium. The default logarithmic case gives $[\log100+\log80]/2<\log90$ $lo g 90$ .

Convexity and concavity describes the downward curvature of a risk-averse utility. Applying Jensen's inequality to this concave function gives $\mathbb{E}[u(W)]\le u(\mathbb{E}[W])$ $u (E [W])$ , the comparison between uncertain wealth and its certain mean.

Key Results¶

Replacing $u$ by $au+b$ with $a>0$ preserves preferences.
Concavity gives the Jensen comparison; strict preference requires suitable strict concavity and nondegenerate wealth.
Fractional power utility requires positive wealth. Quadratic utility is increasing only on a restricted domain.
The expected value of a payoff and expected utility are different criteria.

Examples¶

At wealth 100, a loss of 0 or 20 with equal probabilities gives expected logarithmic utility $[\log100+\log80]/2$ $lo g 80] /2$ . A certain loss of 10 gives $\log90$ , which is larger.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 7

Aliases: utility of wealth, risk-averse decision maker, expected utility model, expected utility theory, risk aversion, expected utility hypothesis, risk-averse Keywords: risk neutrality, concave utility, E[u(W)] ≤ u(E[W]), Jensen's inequality for utility, logarithmic utility, certainty equivalent

Last updated: 17 October 2026 21:53