$6.$ You are considering an investment in two different bonds. One bond matures in eight years and has a face value of $$1,000.$ The bond pays an annual coupon of $3\mathrm{.}5\%$ and has an $8\%$ yield to maturity. The other bond is a $7-$year zero coupon bond with a face value of $$1,000$ and has a yield to maturity of $8\%.$

a$.$ What is the price of each bond?

b$.$ What is the duration of each bond?

c$.$ If the yield to maturity of each bond were to immediately increase to $12\%,$ what would be the percentage change $($including the correct sign$)$ in the price of each bond $($from the price found in part a$)?$

d$.$ If the yield to maturity of each bond were to immediately decrease to $4\%,$ what would be the percentage change $($including the correct sign$)$ in the price of each bond $($from the price found in part a$)?$

e$.$ Assume that you plan on holding the coupon bond for eight years and reinvesting all the coupons when they are received at the going interest rate $($which is the yield to maturity$).$ Assume that after the zero matures you invest in a $1-$year security that earns the going interest rate.

i$.$ Set up a table where you show what happens to the value of each investment $($zero and coupon bond$)$ over time if the yield to maturity remains at $8\%.$ Specifically, show what the cumulative value of each investment $($including $4$ the value of the reinvested coupons for the coupon bond$)$ would be at the end of each of the next eight years. For example, at the end of Year $1,$ you would calculate the value of the coupon bond $($with one year less remaining until maturity$)$ and you would receive the interest coupon payment. However, no interest would be earned on that interest coupon payment in Year $1.($However$,$ that coupon payment will remain in your account and earn interest in the years that follow.$)$ The sum of these $2$ amounts would be the cumulative value for the coupon bond at the end of Year $1.$

ii$.$ Set up a table where you show what happens to the value of each investment $($zero and coupon bond$)$ over time if the yield to maturity immediately and permanently increases to $12\%.$ Specifically, show what the cumulative value of each investment $($including the value of the reinvested coupons for the coupon bond$)$ would be at the end of each of the next eight years.

iii. Set up a table where you show what happens to the value of each investment $($zero and coupon bond$)$ over time if the yield to maturity immediately and permanently decreases to $4\%.$ Specifically, show what the cumulative value of each investment $($including the value of the reinvested coupons for the coupon bond$)$ would be at the end of each of the next eight years. iv$.$ Plot on two graphs $($one for the coupon bond and one for the zero$)$ the cumulative value of the investment over the next eight years under each of the three scenarios outlined above. v$.$ Roughly speaking, after how many years would the value of the investment be the same regardless of what happens to interest rates? What can explain this?