Question

Suppose that you are responsible for managing a $100 million bond portfolio at a large pension fund. Because the pension fund has obligations to retirees in 2029, you want your portfolio to have a duration of 10 years. Suppose that you can only use 2-year and 12-year Zeros to immunize your portfolio. Assume that new 2-year Zeros and 12-year Zeros are issued only in 2019, 2021, etc. (i.e., odds years). Assume that interest rates are the same for all these securities.

a. (10 pts) In 2019, how much of your $100 million do you need to invest in each of the two bonds to immunize your portfolio from interest rate risk?

b. (10 pts) Suppose that one year has passed (i.e., it’s 2020) and your portfolio now has a target duration of 9 years. What are the portfolio weights on the 2-year and 12-year Zeros (issued in 2019) required to immunize your portfolio? (Hint: consider what happens to the maturity of the Zeros in your portfolio as time passes.) If the yield curve has been flat and constant, do you need to rebalance your portfolio in 2020?

Answer 1

A zero coupon bond or ZCB does not pay dividend. Hence, the duration of a ZCB is approx. equal to its maturity. By this logic, the duration of 2 year ZCB is ~2 years and the duration of 12 year ZCB is ~12 years and
Consider a bond portfolio with 20% in 2 year ZCB and 80% in 12 year ZCB.
Portfolio duration is the weighted average of the bonds in the portfolio.
Hence, portfolio duration = 20%*2 + 80%*12= 10
Hence, in 2019 the portfolio should have $20Mn in 2 year ZCB and $80Mn in 12 year ZCB. This portfolio has an asset duration of ~10 years. The liability duration is given as 10 years. Since the asset duration and liability duration match, the portfolio is immunized and there is very minimal interest rate risk.

If the yield curve has been flat and stable, there has been no change in the asset-liability mismatch. The net duration of the portfolio will be zero. As time passes, the duration of coupon paying bonds change. However, for ZCBs, the duration is almost always equal to time to maturity. Hence, the duration of 2 year and 12 year ZCB has now changed to 1 year and 11 years respectively. The new portfolio asset duration is 20%*1 + 80%*11= 9. This matches exactly with the liability duration of 9 years. Hence, we do not need to rebalance the portfolio in 2020. This is the advantage of using ZCBs for immunizing the portfolio. Note that if we had used coupon paying bonds for immunization, we would have had to rebalance the asset duration as the duration of coupon paying bonds changes with time to maturity