Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 2 / a

Assume the covariance matrix $\Sigma$ is positive definite and the excess-mean vector $d=z-R1$ is nonzero. Put $A=d^t{\Sigma }^{-1}d>0$. Substituting $w=\lambda {\Sigma }^{-1}d$ in the mean constraint gives $\lambda A=\mu -R$. ThereforeThese weights are indeed the minimum-variance solution, not just a stationary point. Any other feasible vector is $w+v$ with $d^{t}v=0$. Its variance is $w^t\Sigma w+2\lambda d^{t}v+v^t\Sigma v=w^t\Sigma w+v^t\Sigma v$, which is strictly larger unless $v=0$. If $d=0$, only mean $R$ is feasible, and the minimum-risk portfolio invests entirely in the risk-free asset; the multiplier formula with zero denominator is not applicable.