Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 29J / a

Put $b=(1+r)w_0$, $a=\mu -(1+r)p$ and $d=m-b$. The terminal mean and variance are $b+{\theta }^{\mathsf{T}}a$ and ${\theta }^{\mathsf{T}}V\theta$. A nonsingular covariance matrix is positive definite. If $a\ne 0$, define $D=a^{\mathsf{T}}{V}^{-1}a>0$. The Lagrange multiplier equations for the fixed-mean constraint give $2V\theta =\lambda a$, and thusFor a direct global proof, every feasible portfolio is ${\theta }^{\ast }+h$ with $a^{\mathsf{T}}h=0$, so its variance is $d^2/D+h^{\mathsf{T}}Vh$, minimized uniquely at $h=0$. If $a=0$, only $m=b$ is feasible and its minimum variance is zero, attained by the all-bank portfolio; no other required mean is achievable.