101 Concepts for the Level I Exam

Essential Concept 96: Short-term rates and the business cycle

The price of a default-free nominal coupon-paying bond can be expressed as:

$\mathrm{P}}^{\mathrm{i}}_{\mathrm{t}}\mathrm{=}\sum\limits_{s=1}^{N}{\frac{\mathrm{C}{\mathrm{F}}^{\mathrm{i}}_{\mathrm{t+s}}}{{\left(\mathrm{1+}{\mathrm{l}}_{\mathrm{t,s}}+{\mathrm{\theta }}_{\mathrm{t,s}}+{\mathrm{\pi }}_{\mathrm{t,s}}\right)}^{\mathrm{s}}}}\mathrm{\ \ \ }$

where:

$\theta$ = additional return required by investors to compensate for inflation

$\pi$ = risk premium for uncertainty in future inflation

With short-term nominal interest rates inflation uncertainty can be ignored and short-term T-bills can be priced as:

Short-term nominal interest rates are positively related to short-term real interest rates and short-term inflation expectations. The interest rates are higher in economies with higher inflation and higher, more volatile growth.

Central banks set short-term interest rates in response to the economy’s position in the business cycle.

They cut rates when economic activity and/or inflation is slow.
They increase rates when economic activity and/or inflation is high.

The Taylor rule is a rule for setting policy rates and determining whether the rate is at an appropriate level. The rule is given below:

$\mathrm{p}{\mathrm{r}}_{\mathrm{t}}\mathrm{=}{\mathrm{l}}_{\mathrm{t}}\mathrm{+}{\mathrm{i}}_{\mathrm{t}}\mathrm{+0.5}\left({\mathrm{i}}_{\mathrm{t}}\mathrm{-}{\mathrm{i}}^{\mathrm{*}}_{\mathrm{t}}\right)\mathrm{+0.5}\left({\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{Y}}^{\mathrm{*}}_{\mathrm{t}}\right)\mathrm{=}{\mathrm{l}}_{\mathrm{t}}\mathrm{+1.5}{\mathrm{i}}_{\mathrm{t}}\mathrm{-}\mathrm{0.5}{\mathrm{i}}^{\mathrm{*}}_{\mathrm{t}}\mathrm{+0.5}\left({\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{Y}}^{\mathrm{*}}_{\mathrm{t}}\right)$

where:

$\mathrm{p}{\mathrm{r}}_{\mathrm{t}}$ = the policy rate at time t

$\mathrm{l}}_{\mathrm{t}}$ = the level of real short-term interest rates that balance long-term savings and borrowing in the economy

$i_t$ = the rate of inflation

$i^*_t$ = the target rate of inflation

$\mathrm{Y}}_{\mathrm{t}}~\mathrm{and}~{\mathrm{Y}}^{\mathrm{*}}_{\mathrm{t}}$ = the logarithmic levels of actual and potential real GDP respectively.