For this exercise it is expected that you will use Excel for all of your calculations.

Term Structure of Interest Rates and Forward Rates

$($a$)$ Recall from lecture that $b_t$ is the price today of a zero$-$coupon bond that pays $$1$ at time $t,$ for a given value of $r_t,$ the t$-$period spot rate. Using the values for $b_t$ in Table $1$ on Page $2,$ together with Equation $(1),$ find the values for $r_t,$ for $t=1,2,3.$

$b_t=\frac{1}{(1+r_t{)}^t}$

Rearranged:

$(1+r_t{)}^t=\frac{1}{b_t}$

$r_t=\sqrt[t]{\frac{1}{b_t}}-1$

$r_t=(\frac{1}{b_t}{)}^{\frac{1}{t}}-1$

$($b$)$ Define $f_{jk}$ to be the $($hedgeable$)$ forward rate from time $j$ to time $k.$ Use the calculated spot rates, $r_t,$ from Part $(1$a$),$ together with Equation $(2),$ to calculate the three forward rates: $f_{12},f_{23},$ and $f_{13}.$

$(1+r_j{)}^j(1+f_{jk}{)}^{k-j}=(1+r_k{)}^k$

where:

$r_t-=$ the $t-$period spot rate

$f_{jk}-=$ the forward rate from Period $jto$ Period $k,$ where $k>j$

$1$

Table $1$: Zero$-$Coupon Bond Prices per $$1$ Face

$\backslash$table$[[t,b_t$