On 18 February 2010, the bonds were issued with a face value of $\$1,000$. The bonds mature in 15 April 2015 and have an annual coupon rate of $6\%$.
(a) Assuming today is 15 April 2013, how much is the bond worth if its yield to maturity (YTM) is $5\%$?
(b) A bank purchases this bond for the price in part (a) and holds it for 1 year, i.e. until 15 April 2014. The bond’s YTM remains at $5\%$ during the bank's holding period. After that the yield increases to 7% in the remaining year of the bond’s life. What is the bond price when the bank sells it to another institutional investor? What is the annual rate of return for the bank and why it is different from the $5\%$ YTM in part (a)?
I calculated the present value of the coupon payments as an annuity for two years and then added the present value of the final payment to get $\$1,018.59$ for part (a) but I'm not sure what to do when the YTM increases. Would the new price be $\$990.65$ and the rate of return = $8.63\%$?