(Short rate sensitivity) Gavin Jones sometimes has flashes of brilliance. He asked his instructor if duration would measure the sensitivity of price to a parallel shift in the short rate curve. (That is, $r_{k} \rightarrow r_{k}+\lambda$.) His instructor smiled and told him to work it out. He was unsuccessful at first because his formulas became very complicated. Finally he discovered a simple solution based on the running present value method. Specifically, letting $P_{k}$ be the present value as seen at time $k$ and $S_{k}=\mathrm{d} P_{k} /\left.\mathrm{d} \lambda\right|_{\lambda=0}$, the $S_{k}$ 's can be found recursively by an equation of the form $S_{k-1}=-a_{k} P_{k}+b_{k} S_{k}$, while the $P_{k}$ 's are found by the running method. Find $a_{k}$ and $b_{k}$.