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IAI Actuarial Core Principles · Economic Modelling · Rational choice theory and utility

Which statement about an investor with a strictly concave utility function who is offered a fair gamble (expected gain zero) is correct?

She will reject the fair gamble. For a strictly concave utility function Jensen's inequality gives expected utility below the utility of the expected outcome, which equals current wealth. Higher variance widens this shortfall, so she would need a positive risk premium to accept it.

  1. AShe will accept it because expected wealth is unchanged
  2. BShe will reject it because expected utility falls below the utility of current wealthCorrect
  3. CShe will accept it only if her utility is also quadratic
  4. DShe is indifferent to it because the expected value is zero
  5. She will accept it if the variance of the outcomes is high enough

Explanation

For strictly concave U, Jensen's inequality gives E[U(W)] < U(E[W]) = U(current wealth) for a fair gamble with non-zero variance. She therefore rejects it. Higher variance makes the gap larger, not smaller, so accepting does not become attractive.

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