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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/29k/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 29K / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Using the representation

X = μ + σ Z

and differentiation under the integral sign,

\frac{\partial f}{\partial μ} = E [U^{'} (X)] > 0

because

U

is strictly increasing. Similarly, part (a) gives

\frac{\partial f}{\partial σ} = E [Z U^{'} (X)] = σ E [U^{''} (X)] \leq 0

because a twice differentiable concave function has

U^{''} \leq 0

. Therefore

f_{μ} > 0, f_{σ} \leq 0 .

In mean-variance optimization, an expected-utility investor consequently prefers a larger mean at fixed standard deviation and a smaller variance at fixed mean. This is the usual risk-averse ordering of Gaussian payoffs.

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