Pension Immunization with a Callable Bond
A pension plan must pay 6 million in 5 years and 10 million in 12 years. It currently holds 13.5 million in cash. The actuary uses a flat 4.5% annual effective yield curve. Two bonds are available. Bond X is a 6-year bond with a 3% annual coupon, priced to yield 4.5%. Bond Y is a 12-year zero-coupon bond, priced to yield 4.5%; it is callable at par after year 6. The investment committee proposes 60% in X and 40% in Y, arguing that the portfolio's average maturity is close to the liabilities. The plan is closed to new entrants, and the sponsor wants Redington-style immunization with a surplus cushion. The committee asks whether the proposal is adequate and how the call feature affects the result.
1. Compute the present value and duration of the liabilities at 4.5%. Determine whether the 60/40 allocation matches the liability duration.
Read the worked answer
At 4.5%, the liability present value is
$$PV = \frac{6}{1.045^5}+\frac{10}{1.045^{12}} \approx 4.815+5.897=10.711 \text{ million}.$$
The Macaulay duration is
$$D_L=\frac{5(4.815)+12(5.897)}{10.711}\approx 8.85.$$
For matching, compare modified durations: $D_L^*=8.85/1.045\approx 8.47$.
Bond X has Macaulay duration about 5.56 and modified duration about 5.32. Bond Y is a zero, so its modified duration is $12/1.045\approx 11.48$. The proposed 60/40 portfolio has modified duration
$$0.60(5.32)+0.40(11.48)\approx 7.79.$$
That is below the liability modified duration of 8.47, so the proposal is short-duration, not duration-matched. A duration match would require roughly 49% in X and 51% in Y before considering the call feature.
2. Assess the call feature on Bond Y. In what interest-rate scenario does it hurt immunization, and what should the committee do?
Read the worked answer
A callable bond has negative convexity. If rates fall, the issuer is likely to call Bond Y at par after year 6, truncating its effective duration. The asset then stops participating in price gains just when the liability present value rises most. If rates rise, the bond is not called and its long duration remains, so the asset falls more. In both directions the call feature worsens the asset-liability match relative to a non-callable zero.
The committee should not treat Bond Y as a 12-year zero for immunization. It should use effective duration and option-adjusted spread, or simply avoid the callable bond for the matched portfolio. If the callable bond is retained for yield, the allocation should be adjusted upward in effective duration and the surplus should be stress-tested for a call at year 6. A non-callable zero or a non-callable bond with suitable duration is cleaner for Redington matching.
3. Recommend a revised allocation and two risk controls.
Read the worked answer
On a market-value basis, move toward about 49% Bond X and 51% Bond Y to match the liability modified duration, but first replace or haircut the callable Bond Y for effective duration. If no non-callable long bond is available, hold a larger surplus and shorten the liability hedge using the call-adjusted duration. The plan has 13.5 million against a liability PV of 10.711 million, so there is a surplus cushion, but the cushion should not be used to ignore duration mismatch.
Two controls: first, set rebalancing triggers when asset or liability modified duration diverges by more than a small tolerance, such as 0.25 years. Second, run key-rate and call-scenario tests, including a 200 bp rate fall with Bond Y called at year 6, and require the surplus to remain above a policy floor. Report both market-value and cash-flow projections to the committee.