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

Under expected utility theory, an individual with utility function U(w) is said to be risk averse if, for any non-degenerate gamble on wealth, which of the following holds?

A risk-averse person has a strictly concave utility function, so by Jensen's inequality the expected utility of a gamble is less than the utility of its expected wealth. They therefore prefer a certain sum equal to the gamble's mean to the gamble itself.

  1. AE[U(W)] > U(E[W]), which is equivalent to U being convex
  2. BE[U(W)] < U(E[W]), which is equivalent to U being strictly concaveCorrect
  3. CE[U(W)] = U(E[W]), which is equivalent to U being linear
  4. DE[U(W)] < U(E[W]), which is equivalent to U'(w) being increasing in w
  5. E[U(W)] > U(E[W]), which is equivalent to U''(w) < 0

Explanation

Jensen's inequality: for strictly concave U (U''<0), E[U(W)] < U(E[W]). Hence the person prefers the certain expected wealth to the gamble. Convex utility gives the reverse (risk seeking) and linear utility gives risk neutrality.

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