Exercise $5.$ Consider a $($payer$)$ Interest Rate Swap that exchanges a $4\%$ annual fixed

rate for a floating rate every year. The floating rate is LIBOR, which at the time you are

examining this IRS is

Table $1$: LIBOR Rates $($continuously compounded$)$

Additionally, assume LIBOR is risk free, the Interest Rate Swap has a notional amount of

$$1MM$ and expires exactly in $5$ years, today's cashflows were already settled, and there is no

credit risk from the Interest Rate Swap.

$($a$)$ What is the value $($today$)$ of a floating rate bond indexed on LIBOR and maturing in

$5$ years?

$($b$)$ What is the value $($today$)$ of a $4\%$ fixed coupon bond with annual coupons and maturing

in $5$ years?

$($c$)$ What is the value of the $($receiver$)$ Interest Rate Swap you have? Use the results from

$($a$)$ and $($b$)$ to solve this question.

$($d$)$ Use the fact that an IRS can be expressed as multiple FRAs to compute the value of

the Interest Rate Swap from this question. Hints:

The value you get here should match what you got from $($c$)!$

The forward rates you get using the formulas from class give you the continuously

compounded $1$ yr forward rates

You can convert the $1$ yr forward rates to annual rates using the formula $R=$

$\{$:$e^{r1}-1)$

$($e$)$ Suppose the market rates just moved such that the $1$yr LIBOR increased by $1$bp$.$ That

is$,$ the new $1yr$ LIBOR is $1\mathrm{.}710\%.$ What is the new price of your IRS?

$($f$)$ Repeat the previous point $4$ times for an increase of $1bp$ for each rate, respectively.

$($g$)$ Plot a graph with the results from $($e$)$ and $($f$).$ The $$ axis should be the periods $-$

$1,$dots,$5$ years $-$ and the yy axis the dollar difference from a $1bp$ increase to each rate.

$($h$)$ What are your conclusions from $($g$)?$