Consider four bonds A$,$ B$,$ C and D$,$ trading at a $15\%$ yield, with the following respective

cash flows:

$1$ yr $2$ yr $3$ yr$0$

$$100$ $$100$ $$1100$

$1$ yr $2$ yr $3$ yr$0$

$$50$ $$50$ $$1050$

$1$ yr $2$ yr $3$ yr$0$

$$0$ $$0$ $$1000$

$1$ yr $2$ yr $3$ yr$0$

$$1000$ $$0$ $$0$

$($a$)$ Determine the price of each bond.

$($b$)$ Determine the duration of each bond $($not the modified duration$).$

$($c$)$ Which bond is most sensitive to its change in yield?

$($d$)$ Suppose you owe $$2000$ at the end of $2$ years. Concern about interest rate risk suggests

that a portfolio consisting of bonds and the obligation should be immunized.

Let VA$,$ VB $,$ VC $,$ VD be the total values, i$.$e$.$ face times price, of bonds purchased of types A$,$

B$,$ C and D respectively. What are the constraints to implement the immunization of this

obligation? $[$Hint: There are two equations. Do not solve.$]$

$($e$)$ In order to immunize the portfolio, we decide to use bond C and one other bond. Which

other bond should you choose? Find the amounts $($in total value$)$ of each of these to purchase.

$($f$)$ You decided in $($e$)$ to use bond C in the immunization. Would other choices or combi$-$

nations of bonds A$,$ B$,$ C and D lead to lower cost