Q$4$ In this problem the term structure of interest rates is flat at $5\%.$ The following bonds and

liabilities are given:

Bond A: A zero$-$coupon bond with a face value of $$100$ and a time to maturity of $3$ years.

Bond $B$ : A zero$-$coupon bond with a face value of $$100$ and a time to maturity of $6$ years.

Bond $C$ : A zero$-$coupon bond with a face value of $$100$ and a time to maturity of $10$ years.

Liability $x$ : A one$-$time liability of $$100$ maturing in $4$ years.

Liability $Y$ : A one$-$time liability of $$100$ maturing in $8$ years.

a$.$ Suppose you have liability $x$ and want to immunize it using bonds A and $B.$ How would you

invest in each bond?

b$.$ Suppose you have liability $x$ and want to immunize it using bonds $B$ and $C.$ How would you

invest in each bond?

c$.$ Suppose you have both liabilities $x$ and $Y$ and want to immunize your position using bonds

$B$ and $C.$ How would you invest in each bond?