 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-40/2/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

S_{I} = α S

and

S_{R} = (1 - α) S

. Write

v = Var (S) = λ E [X^{2}]

. Then

Var (S_{I}) + Var (S_{R}) = v (α^{2} + (1 - α)^{2}) = \frac{v}{2} + 2 v (α - \frac{1}{2})^{2} .

α^{*} = \frac{1}{2}, min {Var (S_{I}) + Var (S_{R})} = \frac{λ E [ X ^{2} ]}{2} .

For positive claims and

λ > 0

this minimizer is unique. If the aggregate has zero variance, every retained fraction gives the same zero objective.

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