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

A risk-averse individual with a concave utility function of wealth faces a random loss. Under which condition will she be willing to buy full insurance at a premium equal to the expected loss?

She will always buy, because for a concave utility function the utility of certain wealth equal to the mean is at least the expected utility of the risky wealth (Jensen's inequality). Full insurance at a fair premium therefore raises or maintains her expected utility.

  1. ANever, because a premium equal to the expected loss gives lower expected utility than remaining uninsured
  2. BAlways, because by Jensen's inequality the utility of certain wealth is at least the expected utility of uncertain wealth with the same meanCorrect
  3. COnly if her utility function is linear in wealth
  4. DOnly if the loss is certain to occur
  5. Only if the insurer is also risk averse

Explanation

With concave U, E[U(W)] <= U(E[W]). Full insurance at the expected loss gives certain wealth E[W], so utility U(E[W]) is at least the uninsured expected utility. Option 0 reverses the inequality, and a linear utility would only make her indifferent.

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