A $30-$year$-$maturity bond making annual coupon payments with a coupon rate of $8\mathrm{.}2\%$ has duration of $12\mathrm{.}11$ years and convexity of $211\mathrm{.}01.$ The bond currently sells at a yield to maturity of $8\%.$

Required:

a$.$ Find the price of the bond if its yield to maturity falls to $7\%.$

b$.$ What price would be predicted by the duration rule?

c$.$ What price would be predicted by the duration$-$with$-$convexity rule?

d$-1.$ What is the percent error for each rule?

d$-2.$ What do you conclude about the accuracy of the two rules?

e$-1.$ Find the price of the bond if its yield to maturity increases to $9\%.$

e$-2.$ What price would be predicted by the duration rule?

e$-3.$ What price would be predicted by the duration$-$with$-$convexity rule?

e$-4.$ What is the percent error for each rule?

e$-5.$ Are your conclusions about the accuracy of the two rules consistent with parts $($a$)($d$)?$