 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/30k/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Put

q = μ - (1 + r) S_{0}, h = m - (1 + r) x .

For a risky holding vector

θ

and the remaining wealth in the bank,

X_{1} = (1 + r) x + θ^{T} (S_{1} - (1 + r) S_{0}),

E X_{1} = (1 + r) x + θ^{T} q, Var (X_{1}) = θ^{T} V θ .

When

V

is nonsingular it is positive definite. Cauchy--Schwarz in the

V

(θ^{T} q)^{2} \leq (θ^{T} V θ) (q^{T} V^{- 1} q),

with equality exactly when

θ = λ V^{- 1} q

. Under

θ^{T} q = h

θ^{*} = \frac{h}{q ^{T} V ^{- 1} q} V^{- 1} q, min Var (X_{1}) = \frac{h ^{2}}{q ^{T} V ^{- 1} q} .

For the constraint

E X_{1} \geq m

, if

h \leq 0

the zero risky portfolio is feasible and the minimum variance is zero. If

h > 0

, the inequality binds and the preceding optimizer and minimum apply.

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