It is sometimes useful to introduce variations of the spot rates that are different from an additive variation. Let $\mathrm{s}^{0}=\left(s_{1}^{0}, s_{2}^{0}, s_{3}^{0}, \ldots, s_{n}^{0}\right)$ be an initial spotrate sequence (based on $m$ periods per year). Let $s(\lambda)=\left(s_{1}, s_{2}, \ldots, s_{n}\right)$ be spot rates parameterized by $\lambda$, where

\[1+s_{k} / m=e^{\lambda / m}\left(1+s_{k}^{0} / m\right)\]

for $k=1,2, \ldots, n$. Suppose a bond price $P(\lambda)$, is determined by these spot rates. Show that

\[-\frac{1}{P} \frac{\mathrm{d} P}{\mathrm{~d} \lambda}=D\]

is a pure duration; that is, find $D$ and describe it in words.