Suppose you purchase a six$-$year, $8$ percent coupon bond $($paid annually$)$ that is priced to yield $9$ percent. The face value of the bond is $$1,000.$

a$.$ Show that the duration of this bond is equal to five years

b$.$ Show that if interest rates rise to $10$ percent within the next year and your investment horizon is five years from today, you will still earn a $9$ percent yield on your investment.

c$.$ Show that a $9$ percent yield also will be earned if interest rates fall next year to $8$ percent