Assume 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. Suppose you have liability X and want to immunize it using bonds B and C. How would you combine the two bonds to cover the liability?

Question

BlackTom AI · Accepted Answer

We’re given a flat yield curve at 5% and three zero-coupon bonds with different maturities (B at 6 years, C at 10 years) plus two liabilities (X at 4 years). To immunize the liability X using bonds B and C, we typically align the present value (PV) and the duration (immunization condition) of the asset mix with the liability. For zero-coupon bonds, the duration equals the maturity.

First, express the key relationships in terms of present values. Let PV_B be the present value of the position in bond B and PV_C be the present value of the position in bond C. The liability’s PV at 4 years is P_L = 100 / (1.05^4).

A basic immunization condition uses the duration equality: (PV_B * 6 + PV_C * 10) / (PV_B + PV_C) = 4. This is because we want the asset portfolio’s duration to equal the liability’s horizon.

From the durability equation, 6PV_B + 10PV_C = 4(PV_B + PV_C). Expanding gives 6PV_B + 10PV_C = 4PV_B + 4PV_C, which simplifies to 2PV_B + 6PV_C = 0, or PV_B = -3 PV_C. In other words, the present value in bond B must be three times the magnitude of the (negative) present value in bond C, meaning we would be long B and short C.

Now use the PV balance condition PV_B + PV_C = P_L to determine the actual magnitudes. Substituting PV_B = -3PV_C into the sum gives (-3PV_C) + PV_C = P_L, so -2PV_C = P_L, hence PV_C = -P_L/2 and PV_B = 3P_L/2. Since P_L is positive, this means we should hold 1.5 times the liability’s PV in bond B and short 0.5 times the liability’s PV in bond C.

Interpreting the result in terms of bond positions (as percentages of face value exposure): the appropriate strategy is to long 150% of bond B and short 50% of bond C. This aligns both the PV and the duration with the 4-year liability, satisfying the immunization condition.

Now consider the other options:
- Option b (Long 165% of B and short 65% of C) does not satisfy PV_B = 1.5 P_L and PV_C = -0.5 P_L; the ratios deviate from the required 3: -1 relationship implied by the duration condition, so the immunization would fail.
- Option c (Long 63% of B and long 37% of C) uses both bonds long; both PVs are positive, which cannot satisfy the negative PV_C required by the duration equation, so the duration target cannot be met.
- Option d (Long 211% of B and short 111% of C) represents magnitudes far from the derived 1.5 and -0.5 proportions and would overshoot both PV matching and duration alignment, failing immunization.

In summary, the only set of positions that satisfies both the PV balance with the liability and the duration (immunization) condition for a 4-year horizon with B (6-year) and C (10-year) zero-coupon bonds is to long 150% of bond B and short 50% of bond C.

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