Move beyond certainty by modeling lotteries, risk, and ambiguity. Expected utility theory converts probabilistic outcomes into comparable certainty equivalents, powering risk management and asset pricing in modern finance.
Demonstrates the inadequacy of maximizing expected monetary value, motivating utility-based evaluation of lotteries.
Introduces diminishing marginal utility of wealth via logarithmic utility, creating finite expected utility values.
Illustrates alternative risk-averse functional forms aligned with observed decision behavior.
The St. Petersburg lottery has infinite expected payoff but finite expected utility under logarithmic preferences:
Concavity of U ensures diminishing marginal utility, rationalizing investors' finite willingness to pay.
Decision-makers can compare any two lotteries.
Consistent rankings across lotteries prevent preference cycles.
Intermediate lotteries exist between preferred and less-preferred options.
Preference between lotteries is unaffected by mixing with a third lottery proportionally.
Higher probability of preferred outcomes enhances lottery attractiveness.
Simple Lottery: A finite set of outcomes with associated probabilities. Example: (\$500, 0.3; \$200, 0.7).
Compound Lottery: Lotteries whose outcomes are themselves lotteries; the reduction of compound lotteries axiom ensures simplification to simple lotteries.
Certainty Equivalent: The guaranteed amount delivering the same utility as a risky lottery. Solved by equating U(CE) = E[U(X)].
Expected utility justifies pricing contingent claims using state prices, foundational to complete market models and derivative valuation.
Stochastic discount factors emerge from intertemporal expected utility, linking consumption growth to asset returns.
Expected utility maximization yields optimal asset allocations, extending deterministic models to risky environments with multiple states.
Daniel Bernoulli proposes logarithmic utility, solving the St. Petersburg paradox.
Von Neumann and Morgenstern publish expected utility axioms in 'Theory of Games and Economic Behavior'.
Arrow and Debreu embed expected utility in general equilibrium models.
Markowitz popularizes mean-variance analysis as an expected utility approximation under quadratic utility or normal returns.
Empirical evidence reveals deviations from expected utility. The Allais paradox exposes violations of independence, while the Ellsberg paradox highlights ambiguity aversion. These findings inspire alternative frameworks such as prospect theory, rank-dependent utilities, and ambiguity-sensitive models.
Bayesian updating offers a normative benchmark for belief revision—useful when contrasting with ambiguity-averse behavior.
Research ExtensionsExpected utility underpins insurance pricing, risk limits, and derivative hedging strategies. Investors compute certainty equivalents and risk premiums to determine whether an uncertain project dominates a safe alternative, aligning decisions with firm risk appetite.
Measure Risk PreferencesIndependence ensures investor preferences are linear in probabilities, enabling us to price securities by weighting state payoffs with probabilities and marginal utilities—core to expected utility and risk-neutral valuation.
The Allais paradox violates independence by showing real-world preference reversals, while Ellsberg demonstrates ambiguity aversion. Both motivate alternative models like prospect theory and ambiguity-sensitive utilities.
When returns are jointly normally distributed or investors have quadratic utility, mean-variance optimization mirrors expected utility maximization, providing computational simplicity in practice.