If $T\mapsto P_t^T$ is nonincreasing, then
$$
P_t^{t+1}\leq P_t^t=1,
$$
so $1+r_t=(P_t^{t+1})^{-1}\geq1$ and $r_t\geq0$.

Conversely, if every spot rate is nonnegative, then $B_{T+1}=B_T(1+r_T)\geq B_T$. Under a <risk-neutral measure>,
$$
P_t^{T+1}
=B_t\mathbb E^Q[B_{T+1}^{-1}\mid\mathcal F_t]
\leq B_t\mathbb E^Q[B_T^{-1}\mid\mathcal F_t]
=P_t^T.
$$
Hence
$$
\boxed{T\mapsto P_t^T\text{ is nonincreasing}
\iff r_t\geq0\text{ for every }t.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 5 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-211/5/c/solution