$1.$ Atlantis has been planning to develop a new warning system. The installation of the system costs more than what their budget allows so the mayor decides to issue a $20$year bond to finance the project. The bonds have a face value of $\backslash (\backslash$$ $1,000\backslash )$ and it promises a coupon rate of $\backslash (8\mathrm{.}6\backslash \%\backslash )$ which will be paid quarterly to the bond holders.

a$.$ Calculate the price you have to pay to purchase the bond if

i$.$ The Yield to Maturity $($YTM$)$ is $\backslash (7\mathrm{.}5\backslash \%\backslash )($annually$)$

ii$.$ The Yield to Maturity $($YTM$)$ is $\backslash (12\mathrm{.}0\backslash \%\backslash )($annually$)$

b$.$ Let's assume you would like to buy $\backslash (\backslash$mathbf$\{50\}\backslash )$ bonds issued by Atlantis with a $20-$year tenure. If the YTM is $\backslash (8\mathrm{.}6\backslash \%\backslash )$ and the coupon rate is $\backslash (8\mathrm{.}0\backslash \%\backslash ),$ calculate how much you have to pay when you purchase a bond which makes annual coupon payments rather than quarterly.

$2.$ Consider the following three zero$-$coupon $($discount$)$ bonds:

a$.$ Calculate the one$-,$ two$-,$ and three$-$year spot rates.

b$.$ Calculate the forward rate over the second year and the forward rate over the third year.