A $3-$year maturity bond has a coupon rate of $7\%$ and a yield to maturity $($YTM$)$ of $8\%.$ The bond pays annual coupons and has a face value of $$1,000.$

a$)$ Calculate the duration of the bond. Please show all of your work in a table format.

b$)$ If the yield to maturity $($YTM$)$ immediately rises to $11\%,$ compute the predicted $($dollar$)$ change in the bond price using the duration approximation rule.

c$)$ For large changes in yields $($or interest rates$),$ the predicted change in the bond price $($obtained via the duration approximation rule$)$ is likely to be different from the actual change in the bond price. Why? Explain briefly.