Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 29J / ii

With $x=0$, the budget constraint is $S_0^T\theta =w_0$. On this hyperplane the riskless term is constant, so maximize ${\theta }^T\mu -(\gamma /2){\theta }^{T}V\theta$. The Lagrange multiplier equation gives $\theta =M(\mu -\eta S_0)$. If $S_0\ne 0$, let $d=S_0^{T}M{S}_0>0$ and impose the budget constraint:Positive definiteness makes the objective strictly concave on the hyperplane, proving this stationary point is the global constrained optimum. If $S_0=0$, the cash-free constraint is feasible only when $w_0=0$; in that degenerate case the optimum is $M\mu$. The displayed quotient, including the one in the question, presupposes $S_0\ne 0$.