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

An investor has quadratic utility U(w) = w - b w^2 with b > 0, defined where w < 1/(2b). Which statement about this utility function is correct?

Absolute risk aversion rises with wealth. For quadratic utility it equals 2b divided by (1 minus 2bw), which increases as wealth grows. This implies wealthier investors hold fewer risky assets, an unrealistic property of the quadratic form.

  1. AAbsolute risk aversion increases with wealth, which is regarded as an unrealistic featureCorrect
  2. BAbsolute risk aversion is constant and equals 2b
  3. CRelative risk aversion is constant and equals b
  4. DAbsolute risk aversion decreases with wealth, consistent with empirical behaviour
  5. The investor is risk seeking because the second derivative is -2b

Explanation

U' = 1 - 2bw and U'' = -2b. A(w) = 2b/(1-2bw), which rises as w increases toward 1/(2b). Increasing absolute risk aversion means richer investors hold less in risky assets, which is unrealistic. Negative U'' means risk averse, not seeking.

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