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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-35/2/a/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

Indifference means equality of the risky expected utility and the utility of the sure reduced wealth. A second-order Taylor expansion, using

E ϕ = 0

, gives

E U (W (1 + ϕ)) = U (W) + \frac{1}{2} W^{2} U^{''} (W) E ϕ^{2} + o (E ϕ^{2}) .

(1)

For the sure alternative,

U (W (1 - ψ)) = U (W) - W U^{'} (W) ψ + O (ψ^{2}) .

(2)

Equating these and retaining the leading terms produces the small multiplicative risk premium

ψ ≃ - \frac{1}{2} \frac{W U ^{''} ( W )}{U ^{'} ( W )} Var (ϕ) .

(3)

For 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.

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