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IAI Actuarial Core Principles · Business Management · Decision-making process, attitude to risk and competition

An actuary advises an insurer's board that, on the expected utility hypothesis, a decision-maker with utility function U(w) = ln(w) will rank uncertain outcomes in a particular way. Which statement about this decision-maker is correct?

Prospects are ranked by expected utility, E[ln(w)]. The function is increasing, since U' is positive, and concave, since U'' is negative, so the decision-maker is risk averse. Risk aversion comes from the second derivative, not from the first.

  1. AShe ranks prospects by the expected value of U(w), and is risk averse because U''(w) is negativeCorrect
  2. BShe ranks prospects by the expected monetary value, and is risk neutral
  3. CShe ranks prospects by the variance of outcomes only, and is risk seeking
  4. DShe is risk averse because U'(w) is negative
  5. She is risk seeking because U(w) is increasing

Explanation

Under expected utility, choices are ranked by E[U(w)]. For ln(w), U'(w) = 1/w is positive (more wealth preferred) and U''(w) = -1/w^2 is negative, giving concavity and risk aversion. A negative first derivative would mean less wealth is preferred, which is not the case here.

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