Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 30K / a / Solution

PutFor a risky holding vector $\theta$ and the remaining wealth in the bank,soWhen $V$ is nonsingular it is positive definite. Cauchy--Schwarz in the
$V$ inner product giveswith equality exactly when $\theta =\lambda V^{-1}q$. Under
${\theta }^{T}q=h$,

For the constraint $\mathbb{E}{X}_1\ge m$, if $h\le 0$ 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.