Calculate the duration of the following two bonds:
a. A $1,000 face value bond, with a coupon of 7% (paid annually), a yield to maturity of 8%, and a maturity date 15 years from today.
b. A $1,000 face value bond, with a coupon of 4% (paid annually), a yield to maturity of 3%, and a maturity date 12 years from today.
c. Explain why the 12 year bond has the longer duration, despite having the shorter time to maturity.