Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 2 / a

Indifference means equality of the risky expected utility and the utility of the sure reduced wealth. A second-order Taylor expansion, using $\mathbb{E}\varphi =0$, givesFor the sure alternative,Equating these and retaining the leading terms produces the small multiplicative risk premiumFor increasing concave utility functions this is positive. The approximation assumes genuinely small shocks with controlled Taylor remainders and positive final wealth; small variance alone would not justify it for rare large shocks.