Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 1 / d / Solution

For a numéraire portfolio $\eta$, the normal random variable $\eta \cdot P_1$ is strictly positive almost surely. A nondegenerate normal variable has support on all of $\mathbb{R}$, so it must be degenerate: ${\eta }^{T}V\eta =0$, equivalently $V\eta =0$. Thus $\eta \cdot P_1=\eta \cdot \mu >0$ deterministically. The scaled portfoliohas terminal value $B\cdot P_1=1$ almost surely and therefore replicates a risk-free bond.