How do I immunize a portfolio against multiple future liabilities with different dates?

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AcadiFi · Accepted Answer

Multi-liability immunization extends the single-liability case with three conditions that must all hold. First, portfolio PV equals liability PV. Second, portfolio duration equals liability duration. Third, portfolio convexity (distribution of cash flows) spans a wider range than liability cash flows; specifically, portfolio cash flows should bracket each liability payment. This convexity condition is the key extension beyond single-liability immunization. Intuition. For parallel yield curve shifts, matching duration and convexity ensures the portfolio reinvestment risk and price risk offset. For non-parallel shifts (twists, bends), the distribution of cash flows matters: a portfolio with barbell structure (short + long bonds) bracketing a single-bullet liability is vulnerable to yield curve steepening. Multi-liability immunization with bracketing cash flows protects against these. Implementation. Consider Whitehollow Life with liabilities at years 3, 8, 15, 22. PV $2.4B, duration 11.8, cash-flow-weighted dispersion measure M² = 68. Construct a portfolio with PV $2.4B, duration 11.8, and M² ≥ 68, with bonds providing cash flows close to each liability date. Practical: use a linear program minimizing PV cost subject to duration and convexity constraints, bond positions ≥ 0. Allow corporates for yield pickup but constrain credit (AA+ minimum at 70%+ weight). Rebalance when duration drifts beyond 0.3 years or rates move 50 bps. Key rate duration (KRD) refinement. Match KRDs at 2, 5, 10, 20, 30-year points within 5% tolerance. This protects against non-parallel shifts better than a single duration. Most insurance ALM teams use KRD or principal-component-based immunization today rather than Macaulay duration alone.

How do I immunize a portfolio against multiple future liabilities with different dates?

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