$3.$ Suppose you are considering purchasing a coupon bond with a coupon rate of $5$ percent and a face value of $$10,000$ that matures in $5$ years.

$($a$)$ How much would you be willing to pay for this bond if the market interest rate is $10$ percent?$($B$)$ Suppose you have just purchased the bond, and the market interest rate suddenly rises to $12$ percent. What is the bond worth now?

$($c$)$ Suppose that one year has elapsed, you have received the first coupon payment, and the market interest rate is still $12$ percent. How much would another investor be willing to pay for your bond? If you sell your bond at this price, what would be your rate of return?