Two years ago, Carbon Valley Bank bought $20$ units of a $20-$year treasury bond with a coupon rate

of $4\%,$ a face value of $$1,000$ and a YTM of $1\%.$ The bond pays semi$-$annual coupons.

$($a$)[3$ points$]$ How much did the bank pay for the $20$ bonds in total two years ago? Were the bonds

trading at premium, discount or par?

$($b$)[3$ points$]$ After the pandemic, the federal government has been raising interest rates to suppress

inflation. As a result, the YTM has increased to $5\%$ now. What is the bond price now, immediately

after the $4$th coupon payment?

$($c$)[4$ points$]$ Based on the result in $($b$),$ what is capital gain$/$loss in dollars and percentage respectively for Carbon Valley Bank over the two years?

$3$

$($d$)[1$ points$]$ As you know, treasury bonds are usually considered $$safe$$ and also the YTM of a

treasury bond is sometimes referred to as the risk$-$free rate. However, Carbon Valley Bank does

have a loss on paper according to the result in $($c$).$ Is there any contradiction here? Please explain.

$($e$)[5$ points$]$ Starting from today, assume the YTM of $5\%$ won$$t change for the future $18$ years

and the bank will reinvest the coupons at the $5\%($semiannual APR$)$ immediately after receiving

each of the $36$ coupon payments, what is the future value of all the coupons and the face value in

$18$ years, for the entire bond position?

$($f$)[3$ points$]$ What is the rate of return per year $($semi$-$annual APR$)$ earned by the bank in the

investment described in $($e$)$ over the $18$ years?