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 $$1,000$ and it promises a coupon rate of $8\mathrm{.}6\%$ 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 $6\mathrm{.}5\%($annually$)$

ii$.$ The Yield to Maturity $($YTM$)$ is $10\mathrm{.}0\%($annually$)$

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

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

$\backslash$table$[[$Bond$,$Face Value,Time to Maturity,Market Price$],[1,$$$1,000,$One year,$$945\mathrm{.}00$