Matching a Fixed Liability with Two Bonds
A UK defined-benefit pension scheme has a single £10 million lump-sum payment due on 31 December 2029, exactly five years from today. The trustees want to fund it with a buy-and-hold bond portfolio and are choosing between two sterling investment-grade bonds. Bond A: 4.0% annual coupon, matures in five years, priced at par to yield 4.0%; modified duration 4.45. Bond B: 2.5% annual coupon, matures in eight years, priced at 88.50 to yield 4.25%; modified duration 6.90. The actuary discounts the liability at 4.0%. Coupons can be reinvested at the prevailing market yield. The scheme is currently 96% funded on this basis. A trustee asks whether Bond B is better because it offers the higher yield and more duration. No credit or liquidity differences are material.
1. Which bond is the better match for the five-year liability, and why? Show the cash-flow and duration reasoning, not just a preference.
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Bond A is the better match. The liability is a single cash flow in exactly five years, so the portfolio should have a duration close to the five-year horizon and cash flows that arrive by that date. Bond A matures at the horizon: coupons in years 1–4 are reinvested, and the principal is repaid at year 5. With yields unchanged at 4%, £100 nominal grows to approximately 4 × 5.4163 + 100 = £121.67, so the terminal value is largely locked in by the purchase yield.
Bond B has a modified duration of 6.90, well beyond the five-year horizon. At the end of year 5 the scheme would still hold a three-year bond and would have to sell it at an unknown yield. A rise in yields would reduce the sale price, while a fall would increase it; the higher running yield does not eliminate that horizon risk. Approximate terminal value at unchanged yields is about £109 per £100 nominal, or £123 per £100 invested, which is not clearly superior after allowing for the extra price risk. Duration matching, not headline yield, is the deciding factor.
2. Suppose yields rise by 100 basis points immediately after purchase and remain there. Using the duration and reinvestment information, explain how the two bonds' five-year terminal values are likely to compare.
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For Bond A, the immediate price fall is offset by holding to par and by reinvesting coupons at higher yields. If coupons are reinvested at 5%, the year-5 value is approximately 4 × 5.5256 + 100 = £122.10 per £100 nominal. The starting price was par, so the terminal value is slightly higher than under unchanged yields. This is the reinvestment gain from a rate rise for a bond whose horizon matches its duration.
For Bond B, the scheme still has to sell a three-year bond at the end of year 5. At a 5.25% yield, the sale price of the remaining 2.5% bond is roughly £92.59 per £100 nominal. Reinvested coupons add about £13.84, giving approximately £106.43 per £100 nominal. Because Bond B was bought at 88.50, that is about £120.26 per £100 invested. So Bond A's terminal value is higher despite the rise in yields: Bond B's duration gap of almost two years means the price loss at the sale date outweighs the benefit of higher coupon reinvestment. The comparison supports the horizon-matching choice in the first task.
3. Recommend an implementation and state one condition that would make Bond B more appropriate.
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Implement with Bond A, or a ladder of bonds whose cash flows arrive by the liability date, and set a reinvestment policy for the coupons. Monitor the funding ratio and the liability discount rate; if the scheme's liability profile changes, revisit the match. Do not choose Bond B solely for its higher yield.
Bond B would become more appropriate if the liability were extended, for example to eight years, or if the scheme could tolerate a surplus/deficit around the five-year date and wanted longer-duration exposure to hedge a longer liability. A second condition would be a change in the shape of the yield curve that made the eight-year bond's horizon return genuinely superior after allowing for sale-price risk. Even then, the decision should be based on liability duration and cash-flow timing, not on the highest running yield.