3. Assume a liability that consists of making two payments: $5,000 in 5 years and $5,000 in 10 years. Suppose that you can only use 3-year and 15-year zero coupon bonds with the face value of $100. Assume the yield curve is flat, i.e. all bond maturities have the same YTM. The current YTM (BEY) is 8 %.

a) Calcuate the modified duration and present value of your liability.

b) How many shares for each bond you need to hold in your portfolio to fully immunize your obligations?

c) Suppose yield changes the next day after you chose the immunization portfolio and stays there until liability is due (assume reinvesting proceeds from the 3-year zero coupon bond at the market rate). Complete the following table. Write down all your steps.

Present Values (Year 0 - next day)

Yield Liability Immunization Portfolio

4%

6%

8%

10%

12%

Final Values (Year 2)

Yield Liability Immunization Portfolio

4%

6%

8%

10%

12%

Final Values (Year 10)

Yield Liability Immunization Portfolio

4%

6%

8%

10%

12%

d) Suppose yield changes right before your first liability is due (that is at the end of year 5). Assume reinvesting proceeds from the 3-year bond at the market rate. Complete the following table. Write down all your steps.

Final Values (Year 5)

Yield Liability Immunization Portfolio

4%

6%

8%

10%

12%

e) Suppose yield changes right before your second liability is due (that is at the end of year 10). Assume reinvesting proceeds from the 3-year bond at the market rate. Complete the following table. Write down all your steps.

Final Values (Year 10)

Yield Liability Immunization Portfolio

4%

6%

8%

10%

12%

f) What is the discrepancy in durations between your lability and immunization portfolio at the beginning of year 9?

g) Suppose at the beginning of year 4, 2-year zero coupon bond is available on the market. To rebalance your immunization portfolio, how many shares of this 2-year zero coupon you need to purchase? Assume the current YTM (BEY) is still 8 %.