 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-1/30k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Because

S_{1}

is Gaussian, the terminal wealth

W_{1}

U (x) = - e^{- γ x},

the moment-generating function of a normal variable gives

E [U (W_{1})] = - exp {- γ E W_{1} + \frac{γ ^{2}}{2} Var (W_{1})} .

Maximizing this expected utility is equivalent to maximizing

θ^{T} b - \frac{γ}{2} θ^{T} V θ,

after dropping a constant. The unique first-order condition is

b - γV θ = 0,

and strict concavity gives

θ = \frac{1}{γ} V^{- 1} b = \frac{1}{γ} θ_{m} .

It agrees with part (a) exactly when

γ = \frac{1}{λ} = \frac{( μ - ( 1 + r ) S _{0} ) ^{T} V ^{- 1} ( μ - ( 1 + r ) S _{0} )}{w _{1} - ( 1 + r ) w _{0}} .

For the usual risk-averse convention

γ > 0

, the asserted correspondence assumes

w_{1} > (1 + r) w_{0}

; under that natural target-return condition the choice is unique.

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