IAI Actuarial Core Principles · Economic Modelling · Rational choice theory and utility
An investor has a strictly increasing utility of wealth function U(w). Which property of U(w) corresponds to the investor being risk-averse?
A risk-averse investor has a strictly concave utility function, so the second derivative is negative. Then E[U(W)] is less than U(E[W]) for any gamble, by Jensen's inequality. Non-satiation alone only requires a positive first derivative, and convexity would indicate risk-seeking behaviour.
- AU'(w) > 0 only
- BU''(w) < 0 (U is strictly concave)Correct
- CU''(w) > 0 (U is strictly convex)
- DU'''(w) > 0
- U(w) is linear in w
Explanation
Risk aversion means the investor prefers the expected value of a gamble for certain to the gamble itself. By Jensen's inequality this holds for all gambles exactly when U is strictly concave, i.e. U'' < 0. A positive first derivative only means more wealth is preferred, and convexity indicates risk-seeking.
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