Risk and insurance

Compare expected utility with expected wealth and derive the meaning of a certainty equivalent.

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1. What you will learn

You will compare expected utility with expected wealth and derive the meaning of a certainty equivalent, showing the calculation and stating the assumptions that make the conclusion valid.

2. Starting point

Use the supplied definitions and units. Separate an accounting identity, a behavioral assumption, and a normative criterion before drawing conclusions.

3. Terms to use precisely

4. A risky prospect specifies states, probabilities and consequences

A lottery in this lesson is a mathematical description of uncertain wealth, not necessarily a gambling product. It lists mutually exclusive states, the probability of each state and the wealth received in that state. Probabilities must be nonnegative and sum to one. Wealth levels must use the same units and must lie in the domain of the utility function. Without those ingredients, an expected-utility calculation is not fully specified.

Suppose wealth is thirty-six with probability one half and one hundred with probability one half. Expected wealth is the probability-weighted average, sixty-eight. That average need not be one of the wealth outcomes actually realized. It summarizes the distribution's mean rather than predicting a guaranteed payoff of sixty-eight. Repeated independent copies can behave differently from one individual's single exposure, so the mean alone does not describe all economically relevant risk.

An expected-utility model evaluates each wealth outcome through a utility function u and then averages those utilities using the probabilities. For u(w)=square root of w, the two utility levels are six and ten. Expected utility is eight. Applying utility to expected wealth instead gives the square root of sixty-eight, which is larger than eight. The two operations generally do not commute.

This distinction is the core computational rule: transform each state first, then weight and add. Averaging wealth before applying a nonlinear utility function answers a different question. A reliable solution records both the expected-wealth calculation and the expected-utility calculation so that units and order of operations remain visible.

Term	What it means
Expected utility	Probability-weighted average of utility evaluated separately in each state.
Certainty equivalent	Certain wealth producing the same utility as the risky prospect.
Risk premium	Expected wealth minus certainty equivalent for the specified prospect and preferences.
Actuarially fair premium	Expected covered claims in the simplified no-loading cost benchmark.
Correlated losses	Loss outcomes that move together, limiting diversification relative to independent risks.

Another way: Curvature connects the model to risk attitudes

A concave increasing utility function represents risk aversion within expected-utility theory. A person with such a function weakly prefers certain wealth equal to a lottery's expected wealth to the lottery itself. With strict concavity and genuinely different possible wealth outcomes, the preference is strict. This is a statement about preferences over the specified prospects, not a psychological judgment about courage or a universal description of every person's choices.

The square-root function is increasing and concave for positive wealth. Its marginal utility declines as wealth increases: adding one unit matters more at a low level than at a high level in this representation. The reduction in utility from a downward wealth change can therefore outweigh the gain from an equally sized upward change around a common starting point. That curvature produces the gap between utility of the mean and mean utility.

A linear utility function gives risk neutrality: only expected wealth matters for ranking these monetary lotteries. A convex increasing utility function can represent risk preference over the specified domain. The same distribution can therefore have different certainty equivalents under different preferences. Probabilities and outcomes alone do not determine an individual's willingness to bear risk.

Expected-utility representations have more structure than ordinal utility over deterministic bundles. Positive affine transformations, multiplying utility by a positive constant and adding a constant, preserve expected-utility rankings. An arbitrary increasing nonlinear transformation need not preserve rankings of lotteries after expectations are taken. This is why the earlier rule about ordinal deterministic utility cannot be transferred unchanged to risky prospects.

Another way: A certainty equivalent is a wealth amount

The certainty equivalent is the certain wealth that yields the same utility as the risky prospect. If u is increasing, solve u(CE)=expected utility. For the square-root representation, this means CE equals expected utility squared. In the wealth-thirty-six-or-one-hundred example, expected utility is eight, so the certainty equivalent is sixty-four wealth units.

The risk premium is expected wealth minus the certainty equivalent. Here it is sixty-eight minus sixty-four, or four wealth units. This measures how much of the mean wealth the person would give up to remove this particular risk under the supplied utility model. It is not an insurance company's quoted premium and is not the expected loss from a starting wealth level.

For a risk-averse person the certainty equivalent is no greater than expected wealth, so the risk premium is nonnegative. In a degenerate lottery with the same wealth in every state, they are equal and the risk premium is zero even with a concave utility function. This provides a useful boundary check: a calculation should not attach a positive premium to uncertainty that does not change wealth outcomes.

Utility units and wealth units must remain separate. Expected utility eight is not eight credits of wealth in the example. Squaring it converts through the stated inverse utility function. Adding expected utility directly to an insurance premium would combine incompatible units. The numerical value of expected utility also changes under an allowed affine rescaling, while the implied certainty equivalent remains the same.

Another way: Insurance exchanges contingent wealth for a more stable allocation

Suppose initial wealth is one hundred and a loss of sixty-four occurs with probability one half. Without insurance, final wealth is thirty-six in the loss state and one hundred otherwise. Full insurance at a fixed premium t gives final wealth one hundred minus t in both states, assuming the insurer pays the entire specified loss and the contract is honored.

The actuarially fair premium in the simplified model equals expected claims, one half times sixty-four, or thirty-two. Full insurance at that premium yields certain wealth sixty-eight, equal to uninsured expected wealth. Under the strictly concave square-root utility function, utility of certain sixty-eight exceeds the uninsured expected utility eight. Removing risk raises utility without changing expected wealth in this frictionless comparison.

The largest premium the person would weakly pay for full insurance solves one hundred minus t equal to the uninsured certainty equivalent sixty-four. Thus the maximum premium is thirty-six. It exceeds expected claims by four, exactly the uninsured risk premium. Distinguish these three quantities: fair insurance premium thirty-two, maximum acceptable full-insurance premium thirty-six, and risk premium four.

This willingness-to-pay calculation assumes that the insurance changes only the person's contingent wealth through the stated indemnity and fixed premium. Exclusions, default risk, changing preventive behavior, administrative costs and private information can alter the comparison. The preceding information lesson explains why actuarially fair coverage for a proposed population may not be available to every voluntary participant under asymmetric information.

Another way: Pooling depends on dependence as well as group size

Insurance pools combine many uncertain claims. If individual losses are sufficiently independent, the average claim per policy can become more predictable as the number of contracts grows. That reduces some aggregate uncertainty about average costs. It does not make every claim vanish or guarantee that the insurer never experiences an unusually large total payout.

Common shocks can make losses move together. A flood affecting many insured properties at once is different from many independent equipment breakdowns. Adding more policies exposed to the same event may leave substantial systematic risk. An analyst must consider the joint distribution, not merely count contracts, before asserting that diversification has removed the uncertainty.

The expected value of the sum of claims equals the sum of their expected values even without independence. Independence matters for the variability of the sum and the strength of diversification, not for that linear expectation identity. Keeping these two facts separate prevents a common mistake: requiring independence just to add expected claims or assuming that additivity proves a pool is safe.

Reserves, reinsurance, capital requirements and contract design can address some risks, but each has costs and institutional conditions. The introductory calculation of a fair premium is an expected-cost benchmark. It is not a complete solvency model and does not imply that every premium above expected claims is pure profit or evidence of exploitation.

Another way: Risk measures and evidence require a stated question

Two lotteries can have the same expected wealth and different certainty equivalents. Their variance can also fail to determine their ranking for every utility function because higher moments and the distribution's shape may matter. If one distribution is a mean-preserving spread of another, a risk-averse expected-utility decision maker weakly prefers the less spread distribution, but a general comparison should not be reduced to a slogan about standard deviations alone.

Probabilities in these exercises are given. In an empirical decision they may be estimated imprecisely, disputed, or dependent on future behavior. Sensitivity analysis can show how the result changes under alternative plausible probabilities or utility assumptions. It cannot convert an unsupported probability into a known fact. Report which uncertainty is represented by the lottery and which uncertainty remains about the model itself.

Finally, the utility function here evaluates the individual's own wealth outcomes. A public policy affecting many people requires additional choices about aggregation, rights, distribution and external effects. Adding everyone's utility numbers as though the scales were automatically comparable would introduce an assumption not provided by the individual expected-utility model. The calculation offers a disciplined conditional comparison, not a complete social decision rule.

5. A fictional archive's equipment risk

A fictional archive uses an accounting model for an equipment reserve of one hundred units. A breakdown would cost sixty-four units and occurs with probability one half. The decision maker's stipulated utility is the square root of remaining wealth. The model therefore has wealth thirty-six in the breakdown state and one hundred otherwise; it does not claim these probabilities describe any real archive.

Expected wealth is sixty-eight, while expected utility is half of six plus half of ten, or eight. The certainty equivalent is sixty-four and the risk premium four. An analyst keeps these labels on the worksheet to prevent eight utility units from being mistaken for eight accounting credits.

Full insurance at a premium of thirty-two would leave certain wealth sixty-eight and is actuarially fair under the exercise's no-cost, no-default assumptions. The maximum premium the decision maker would weakly accept is thirty-six, leaving certain wealth equal to the certainty equivalent. A quote of thirty-eight would fall outside that modeled willingness to pay even though it also removes the loss uncertainty.

Before applying the result elsewhere, the analyst would need to investigate exclusions, the insurer's reliability, other correlated losses and whether coverage changes maintenance incentives. A large pool of archives exposed to the same flood cannot be treated as independent merely because it contains many members. The worked result therefore demonstrates how risk preference and contingent wealth enter a comparison while making the missing empirical and institutional assumptions explicit.

6. Check the tempting inference

Do not take the square root after averaging wealth when asked for expected utility. The certainty equivalent has wealth units; expected utility does not. Distinguish the risk premium from expected claims and from a provider's quoted insurance premium.

7. Transform states before averaging

8. Unequal probabilities change both means

9. Distinguish three different premiums

10. Complete a new risky prospect

11. Guided practice

Wealth is 4 with probability 0.5 and 100 otherwise. Utility is sqrt(w). Produce expected wealth, expected utility, certainty equivalent and risk premium in that order.

12. Guided practice

Wealth is 4 or 36 with equal probability; utility is sqrt(w). Complete expected utility, certainty equivalent and risk premium.

13. Guided practice

Wealth is 25 or 81 with equal probability and u(w)=sqrt(w). Calculate expected wealth, expected utility, certainty equivalent and risk premium.

	Your result
Expected wealth
Expected utility
Certainty equivalent
Risk premium