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IAI Actuarial Core Principles · Economic Modelling · Role of insurance in reducing or removing risk

An individual with U(w) = ln w has wealth of ₹10,00,000 and faces a 10% chance of losing ₹5,00,000 and a 90% chance of no loss. Insurance is offered at a premium equal to the expected loss. Which statement is correct?

She prefers full insurance. The fair premium is ₹50,000, leaving certain wealth of ₹9,50,000 with log utility about 13.764, above the uninsured expected utility of about 13.746. Risk-averse individuals fully insure at actuarially fair premiums.

  1. AShe prefers to remain uninsured because the premium is above her certainty equivalent
  2. BShe is indifferent between insuring and not insuring
  3. CShe prefers full insurance because it gives wealth of ₹9,50,000 for certain, which has higher utility than the uninsured gambleCorrect
  4. DShe prefers partial insurance only, since full insurance is never optimal at a fair premium
  5. She cannot be risk averse because ln w is convex

Explanation

Expected loss = 0.1×5,00,000 = ₹50,000, so full cover leaves wealth ₹9,50,000 with ln = 13.764. Uninsured: 0.9 ln(10,00,000) + 0.1 ln(5,00,000) = 0.9(13.8155)+0.1(13.1224) = 13.746. Full insurance is higher. With a fair premium, a risk-averse person fully insures.

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