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IAI Actuarial Core Principles · Business Economics · Consumer demand and behaviour

A risk-averse person with U(W) = ln(W) owns wealth of Rs 1,00,000 facing a 50% chance of losing Rs 36,000 from a fire. Full insurance costs a premium equal to the expected loss, Rs 18,000. Which statement is correct?

She buys insurance. Full cover leaves her with certain wealth of Rs 82,000, the same as expected uninsured wealth. Because ln(W) is concave, expected utility of the risky outcome is lower than the utility of the certain amount, so risk aversion makes insurance attractive.

  1. AShe is indifferent, because the premium equals the expected loss and expected utility is unchanged
  2. BShe declines insurance, because the premium exceeds the utility cost of the risk
  3. CShe buys insurance, because certain wealth of Rs 82,000 gives higher utility than the uninsured gamble, whose expected utility is lowerCorrect
  4. DShe buys insurance only if the premium is below Rs 10,000
  5. She declines because ln(W) is linear in wealth

Explanation

Insured wealth is 1,00,000-18,000=82,000 with certainty. Uninsured wealth is 64,000 or 1,00,000 with equal chance, so expected wealth is also 82,000. Because ln is concave, by Jensen's inequality E[ln W] < ln(82,000). The insured outcome gives higher utility, so a risk-averse person buys. Indifference applies only to risk-neutral people.

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