Question
A traveling exhibition begins the month with £200 available to cover a potential theft loss. The loss is £0 with probability 0.5, £40 with probability 0.3, or £100 with probability 0.2. The owner must choose one of two policies, each charging a £20 premium upfront:
- Deductible: the insurer reimburses any loss above £30; the owner pays at most the first £30.
- Proportional: the insurer reimburses \(\frac{17}{32}\) of every loss.
The owner evaluates end-of-month wealth using \(u(w)=\sqrt w\) and prefers the policy with higher expected utility. The quoted loss distribution is the only uncertainty, and neither policy has exclusions or payout caps.
Both contracts have equal expected insurer payouts. Which should the owner buy? Calculate expected utility under each and explain why equal average compensation need not imply equal desirability.
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Hint 1
Make a three-row table of final wealth under each policy; premiums are paid even if nothing is stolen.
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Solution
Expected loss before insurance is \(0.3\times40+0.2\times100\) = £32. The deductible pays £0, £10, £70 in the three states, so its expected reimbursement is \(0.3\times10+0.2\times70\) = £17. The proportional policy pays \(\frac{17}{32}L\); its expected reimbursement is \(\frac{17}{32}\times32\) = £17. Equal payouts and equal £20 premiums imply the same expected final wealth, but not the same distribution.
After the premium, wealth before theft is £180. With the deductible the owner bears \(\min(L,30)\), giving final wealth \((180,150,150)\) at losses \((0,40,100)\). Under proportional coverage the owner bears \(\frac{15}{32}L\), giving \((180,161.25,133.125)\). Both expected wealth values are £165: for the deductible, \(0.5\times180+0.5\times150=165\); for proportional, \(0.5\times180+0.3\times161.25+0.2\times133.125=165\).
Expected square-root utilities are therefore
\[\begin{aligned}U_d&=0.5\sqrt{180}+0.3\sqrt{150}+0.2\sqrt{150}\approx12.8319283,\\U_p&=0.5\sqrt{180}+0.3\sqrt{161.25}+0.2\sqrt{133.125}\approx12.8253276\end{aligned}\].
The deductible is preferred by approximately 0.0066007 utility units. There is also a structural proof. Conditional on a positive loss (probability 0.5), deductible wealth is £150 for certain. Proportional wealth is £161.25 with conditional probability \(\frac{0.3}{0.5}=0.6\) and £133.125 with probability 0.4, with the same conditional mean £150. Strict concavity of \(\sqrt{w}\) makes the certain £150 strictly preferable to that spread. The deductible allocates more protection to the severe, low-wealth state. Equal insurer payouts alone miss this risk-distribution effect.
Assumptions
Loss outcomes are mutually exclusive and exhaust all possibilities.
Final wealth is initial wealth minus premium minus uncovered theft loss; sqrt utility is defined at every listed wealth level.
Final Answer
Choose the £30-deductible policy: \(\mathbb{E}[u(W_d)]\approx12.83193\) versus \(\mathbb{E}[u(W_p)]\approx12.82533\), a utility advantage of about 0.00660.