Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 30K / b / Solution

Put $D=b^T{V}^{-1}b$ and decompose any $\theta$ in the $V$-inner product asThenFor fixed $\lambda$, a nonzero $\eta$ leaves the mean unchanged and strictly raises variance, so strict monotonicity of $F$ excludes it from a maximizer. If $D>0$ and $\lambda <0$, the zero portfolio has a larger mean and smaller variance, so this is also impossible. Hence every maximizer isIf $b=0$, the unique variance-minimizing maximizer is ${\theta }^{\ast }=0$, which is the same conclusion with $\lambda =0$.