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IAI Actuarial Core Principles · Economic Modelling · Mean-variance portfolio theory

An investor has a utility function U(w) with U'(w) > 0 and U''(w) < 0 for all wealth levels w. Which statement about this investor is correct?

The investor is risk averse. A positive first derivative shows more wealth is preferred, and a negative second derivative makes utility concave, so the certain expected value is preferred to a gamble with the same mean.

  1. AThe investor is risk averse and prefers a certain payoff to a gamble with the same expected valueCorrect
  2. BThe investor is risk neutral and is indifferent between a certain payoff and a gamble with the same mean
  3. CThe investor is risk seeking and prefers the gamble to its expected value
  4. DThe investor prefers less wealth to more wealth
  5. The investor's utility function must be quadratic

Explanation

Positive first derivative means more wealth is preferred (non-satiation). Negative second derivative means concavity, which by Jensen's inequality gives E[U(W)] < U(E[W]), so a certain amount is preferred to a fair gamble. Risk neutrality would need U''=0 and risk seeking U''>0.

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