Module 3 — Choice under Risk and Insurance

Core question

How should a decision-maker compare uncertain outcomes, and when does transferring risk through insurance create value?

Learning outcomes

You will be able to:

• represent a lottery with expected utility and state its key assumptions;
• calculate certainty equivalents, risk premia, and Arrow–Pratt risk aversion;
• derive benchmark insurance and portfolio conditions;
• distinguish risk, ambiguity, adverse selection, and moral hazard;
• use climate-insurance evidence to identify where the benchmark is incomplete.

1. Outcomes, probabilities, and timing

A lottery L=(p₁,x₁;…;pₙ,xₙ) pays outcome x_s with probability p_s. Expected utility is:

The von Neumann–Morgenstern representation follows from completeness, transitivity, continuity, and independence over lotteries. Unlike ordinary ordinal utility, expected-utility representations are unique only up to a positive affine transformation a+bu, b>0; an arbitrary monotone transformation changes attitudes toward lotteries.

Expected utility separates:

• beliefs about states, p_s;
• consequences in each state, x_s;
• utility curvature over consequences, u.

Confusing these makes “risk aversion” absorb bad information, distorted beliefs, liquidity constraints, or market frictions.

2. Certainty equivalent and risk premium

The certainty equivalent CE solves:

The risk premium is:

If u''<0, Jensen's inequality gives u(E[X])>E[u(X)], so ρ>0 for a non-degenerate risk.

Worked lottery

Initial wealth is 100. With equal probability, wealth becomes 120 or 80. Let u(w)=√w:

Therefore:

The lottery's expected wealth is 100, yet the agent would exchange it for about 98.99 with certainty.

3. Measure local risk aversion

Absolute and relative risk aversion are:

For a small zero-mean risk ε with variance σ², a second-order approximation gives:

This is a local comparison. Two utilities can rank small risks similarly and large, skewed risks differently.

4. Insurance as state-contingent consumption

Let wealth be W, loss L occur with probability p, indemnity be I, and premium π(I). Consumption is:

The consumer chooses I to maximise:

With concave utility, actuarially fair linear pricing π=pI, no hidden action, and no other friction, full insurance equalises consumption across states. With a loading, wealth effects, background risk, or moral hazard, partial insurance and a deductible may be optimal.

Worked insurance decision

Let W=100, p=0.1, L=50, and u(w)=√w.

Without insurance:

so CE₀≈94.24. The most this consumer pays for full coverage is approximately:

The actuarially fair premium is pL=5, so full insurance is preferred. At a premium above about 5.76, it is not. This maximum depends on wealth, utility, loss size, and other risks—not merely the expected loss.

5. Portfolio choice

Invest share a in a risky asset with return R and the remainder at risk-free return r_f:

An interior optimum satisfies:

The expected excess return is weighted by marginal utility: losses in bad states matter more. Under quadratic/normal approximations, higher expected return raises the risky share while variance and risk aversion lower it. With fat tails, borrowing constraints, or multiple background risks, mean and variance are insufficient.

6. Risk is not ambiguity

• Risk: probabilities are treated as known.
• Ambiguity: probabilities or the model generating outcomes are uncertain.
• Parameter uncertainty: probabilities are estimated with error.
• Deep uncertainty: even relevant states or mechanisms are contested.

Expected utility can incorporate subjective probabilities, but it does not by itself explain Ellsberg-type ambiguity aversion or probability weighting. Module 9 treats behavioural alternatives; they are competing models with their own testable restrictions, not permission to explain any anomaly after the fact.

7. Current case: wildfire-risk classification

Boomhower, Fowlie, Gellman, and Plantinga combine parcel-level wildfire risk with insurer filings. Their 2024 working paper, revised in 2025, documents large differences in insurers' classification and pricing methods. Insurers using coarser risk measures can face adverse selection relative to firms with richer models, charge high prices in risky segments, or withdraw (NBER Working Paper 32625).

The case links three layers:

1. households value risk transfer through expected utility;
2. insurers differ in information and classification technology;
3. regulation constrains prices and participation.

It does not identify one optimal regulatory rule for every disaster market. The paper studies a particular market and uses proprietary risk data; climate trends, reinsurance, public backstops, and household adaptation also affect equilibrium.

Practice

1. Compute CE and ρ for a 50–50 lottery over 64 and 144 with u=√w.
2. Show that a risk-neutral consumer buys actuarially fair insurance but will not pay a positive loading in the benchmark.
3. Derive the insurance first-order condition for π(I)=qI.
4. Explain why a deductible can reduce moral hazard without eliminating insurance value.
5. Separate preference, belief, information, and regulation explanations for rising wildfire premiums.

Quick check

• Expected value and expected utility answer different questions.
• Utility curvature determines risk attitude only within the specified outcome domain.
• Certainty equivalents put utility comparisons back into outcome units.
• Full insurance is a benchmark requiring fair pricing and no hidden action.
• Real insurance markets combine risk preferences with information, contracts, and regulation.

Next: make individual plans jointly feasible in general equilibrium.