Question

Problem 4. You work for a pension fund that has an obligation that must be paid in 10 years. Currently this obligation, which has a present value of $200 million, is exactly funded (i.e., the market value of the fund's assets equals the present value of its obligation; net equity is exactly zero). Te duration of the fund's assets is currently 8. The yield-to-maturity on zero coupon bonds of all maturities is currently 6%. (a) Using the duration information, what would be the approximate change in the equity of the fund if interest rates decrease to 5% at all maturities? (b) Your boss tells you that she would like you to shift all of the fund's assets into a combination of 5-year and 20-year zero-coupon bonds. How would you structure your investment to immunize the fund against interest rate risk? That is, how much should you invest in the 5-year zero, and how much in the 20-year zero?

Answer 1

a). Change in value of fund's assets will be current value*duration*change in interest rate/(1+interest rate)

= 200*8*1%/(1+6%) = 15.09 million. This will be the equity value.

b). Present value of the fund's obligation is 200 million.

Let the weight of 5-year zero be w. Then, weight of 20-year zero is (1-w). Remember that for a zero, the duration equals its maturity.

Duration of obligation = sum of weighted durations of investments

8 = w*5 + (1-w)*20

8 = 5w + 20 - 20w

w = 12/15 = 0.80 (amount invested in 5-year zero)

(1-w) = 1-0.80 = 0.20 (amount invested in 20-year zero)

The PV of the obligation has to match the amounts invested in both bonds, so

amount invested in 5-year zero is 0.80*200 = 160 million

amount invested in 20-year zero is 0.20*200 = 40 million