Insurance trades a small sure loss (the premium) for protection against a large uncertain one. Risk Aversion and Certainty Equivalents tells us why that can be a good trade even when the premium is more than the expected loss.
Key idea
A premium is actuarially fair when it equals the expected payout: probability times loss. At a fair price, a risk-averse person buys full coverage. The most she would pay for full coverage is $W - CE$, where $CE$ is the certainty equivalent of going uninsured; the amount above the expected loss she would accept to pay is her risk premium.
Why it works
Buy $K$ dollars of coverage at $\gamma$ per dollar. Wealth is $W - \gamma K$ with no loss and $W - L + (1 - \gamma)K$ with the loss. Maximize $(1 - \pi)u(W - \gamma K) + \pi\,u(W - L + (1 - \gamma)K)$: $$\pi(1 - \gamma)\,u'(W_{\text{loss}}) = (1 - \pi)\gamma\,u'(W_{\text{no loss}}).$$ At the fair price $\gamma = \pi$ this says $u'(W_{\text{loss}}) = u'(W_{\text{no loss}})$. Because $u'$ falls as wealth rises, the two wealth levels must be equal: full coverage, $K = L$. When $\gamma > \pi$ the buyer keeps part of the risk.
Worked example: Fair premium and the most she would pay
Wealth 400 (thousand dollars); a loss of 256 with probability 1/4; $u = \sqrt{W}$.
- Fair premium: $\tfrac{1}{4}(256) = 64$.
- Uninsured: $E[u] = \tfrac{3}{4}\sqrt{400} + \tfrac{1}{4}\sqrt{144} = 15 + 3 = 18$, so $CE = 324$.
- Full coverage at premium $P$ leaves $400 - P$ for sure. She buys if $400 - P \ge 324$: the most she would pay is 76.
- Risk premium: $76 - 64 = 12$.
Answer: Fair premium 64; she would pay up to 76.
Worked example: When the price is above fair
Wealth 100 (thousand), loss 50 with probability 0.1, $u = \ln W$, coverage costs 0.15 per dollar. How much coverage does she buy?
- The first-order condition with log utility, $$\dfrac{0.1(0.85)}{50 + 0.85K} = \dfrac{0.9(0.15)}{100 - 0.15K}$$ solves to $K = \dfrac{\pi(1 - \gamma)W - (1 - \pi)\gamma(W - L)}{\gamma(1 - \gamma)}$.
- $$K = \dfrac{0.1(0.85)(100) - 0.9(0.15)(50)}{0.15(0.85)} = \dfrac{8.5 - 6.75}{0.1275} \approx 13.7$$
- Far less than full cover: at an unfair price she insures only part of the loss.
Answer: About 13.7 thousand dollars of coverage.
Common mistake
Thinking insurance is a bad buy whenever the premium is above the expected loss. Real insurers must charge more than the expected payout to cover their costs, yet buying can still raise a risk-averse person's expected utility: in the first example she gains from any premium up to 76, well above the expected loss of 64. Check: compare the certain wealth after the premium with the certainty equivalent of staying uninsured.
Insurance also changes behavior and attracts some buyers more than others; Moral Hazard and Incentives and Adverse Selection and Lemons take up those problems.