ECON7870 Chap.8 Expected Utility, Risk and Insurance
Expected Utility, Risk and Insurance
Apply utility inside each state
A lottery is a distribution of outcomes with probabilities. Expected utility evaluates utility at state-dependent wealth and then averages, while expected value averages money.
Concavity gives diminishing marginal utility and makes a fair gamble unattractive to a risk-averse decision maker.
Return to money with certainty equivalence
The certainty equivalent is the sure wealth matching expected utility. Its distance below expected wealth is the risk premium.
Full insurance is attractive up to the premium that leaves certain wealth equal to the certainty equivalent.
Build a state table first
List each probability, final wealth and utility before adding. Insurance premiums are paid in every state, while indemnities appear only after the covered loss. An investment stake affects both good and bad returns.
Writing final wealth explicitly prevents a certain payment from disappearing in one branch and keeps probabilities attached to the correct outcome.
Use ordering as a checksum
For concave utility, the certainty equivalent should not exceed expected wealth. Its gap is positive risk premium. Under fair full insurance, expected wealth is preserved and dispersion is removed, so a risk-averse consumer prefers coverage.
If the computed order fails, inspect whether utility was applied after averaging wealth instead of state by state.
Distinguish a loaded premium
A premium above expected claim cost reduces certain wealth. The consumer accepts it only while insured wealth remains above the certainty equivalent of remaining uninsured.
Partial coverage can become optimal when the loading grows because it balances retained risk against the certain payment. Risk aversion alone does not imply purchasing every available policy.
Set reservation prices from both sides
Expected loss gives the actuarially fair premium, while initial wealth minus the certainty equivalent gives maximum willingness to pay for full coverage.
The difference is the value of removing risk. A quoted premium below expected loss is favourable even before risk preferences matter; one above maximum willingness to pay is rejected even by this risk-averse consumer. Between those bounds, calculate expected utility carefully. State wealth must subtract the premium in every branch and add the indemnity only in the loss branch.
This accounting keeps insurance value separate from a mistaken change in the probabilities themselves.
What this chapter covers
- 01
Lotteries and states
- 02
Expected value
- 03
Expected utility
- 04
Concave utility
- 05
Certainty equivalents
- 06
Risk premia
- 07
Fair and loaded insurance
Worked example · free
Find an insurance reservation premium
- 3Write final wealth and utility in each state.
- 3Compute expected utility and invert it.
- 2Convert the certainty equivalent into a premium.
Key terms
- Lottery
- A list of possible outcomes paired with probabilities that sum to one.
- Expected Utility
- The probability-weighted average of utility evaluated at wealth in each state.
- Risk Aversion
- A preference represented by concave utility that rejects fair gambles around current wealth.
- Certainty Equivalent
- The sure amount of wealth valued exactly like a specified risky prospect.
- Fair Insurance
- Coverage priced at the insurer's expected claim cost without an additional loading.
Expected Utility, Risk and Insurance FAQ
Why not choose the option with highest expected money?
Expected money ignores how the decision maker values wealth across states. Concave utility makes a loss reduce utility more than an equal gain raises it, so a fair gamble can lower expected utility even when expected wealth is unchanged.
How is a certainty equivalent calculated?
First compute expected utility by applying utility to wealth in every state and weighting by probability. Then apply the inverse utility function to that average. The result is a sure monetary amount, which can be compared directly with expected wealth or an insurance premium.
Will a risk-averse person reject every risky investment?
No. Risk aversion implies rejection of fair gambles under the standard assumptions, but a risky asset with a sufficiently favourable expected return can raise expected utility. The choice also depends on initial wealth, utility curvature, exposure and the distribution of returns.
Exam move
Set up a state table with probability, final wealth and utility. Verify probabilities sum to one before adding. After finding the certainty equivalent, check its ordering relative to expected wealth. Use that ordering to catch mistakes before interpreting insurance or investment decisions.
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