What conditions must be satisfied to immunize a portfolio against multiple liabilities using duration and convexity?

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AcadiFi · Accepted Answer

Multi-liability immunization extends the single-liability framework to protect against interest rate risk when the liability stream has multiple payment dates. Additional conditions beyond simple duration matching are required. **Single vs. Multi-Liability Immunization** For a **single liability**, immunization requires: 1. Duration of assets = Duration of liability. 2. Convexity of assets >= Convexity of liability. 3. Portfolio value >= Present value of liability. For **multiple liabilities**, additional conditions apply to handle the **dispersion** of payments across time. **Multi-Liability Immunization Conditions:** 1. **Present value match:** PV(assets) >= PV(liabilities). 2. **Dollar duration match:** Dollar duration of assets = Dollar duration of liabilities. (This ensures parallel shifts are hedged.) 3. **Dispersion condition:** The **range of asset durations** must **bracket** the range of liability durations. That is, the shortest-duration asset must have duration <= shortest liability duration, and the longest-duration asset must have duration >= longest liability duration. 4. **Convexity condition:** Portfolio convexity must exceed liability stream convexity, but **not excessively** — very high convexity introduces exposure to non-parallel shifts. **Worked Example — Meridian Life Insurance:** Meridian has three liability payments: | Year | Liability | PV (at 5%) | Duration Contribution | |---|---|---|---| | 3 | $10,000,000 | $8,638,000 | 0.88 years | | 7 | $15,000,000 | $10,660,000 | 2.54 years | | 12 | $20,000,000 | $11,140,000 | 4.55 years | | **Total** | | **$30,438,000** | **D_L = 7.97 years** | Meridian constructs an immunizing portfolio using three bonds: | Bond | Duration | Weight | Weighted Duration | |---|---|---|---| | Ashworth 2-Year | 1.9 years | 15% | 0.285 | | Carrington 8-Year | 7.6 years | 55% | 4.180 | | Whitfield 15-Year | 13.8 years | 30% | 4.140 | | **Portfolio** | | **100%** | **D_A = 8.605** | Wait — the asset duration (8.605) does not match liability duration (7.97). Meridian must adjust weights: | Bond | Revised Weight | Weighted Duration | |---|---|---| | Ashworth 2-Year | 18% | 0.342 | | Carrington 8-Year | 55% | 4.180 | | Whitfield 15-Year | 27% | 3.726 | | **Portfolio** | **100%** | **D_A = 8.248** | Further iteration converges to D_A = 7.97 years. The key constraint is that asset durations (1.9, 7.6, 13.8) **bracket** liability durations (3, 7, 12). \`\`\`mermaid flowchart TD A["Multi-Liability
Immunization"] --> B["Condition 1:
PV Assets ≥ PV Liabilities"] A --> C["Condition 2:
Dollar Duration Match"] A --> D["Condition 3:
Asset Durations Bracket
Liability Durations"] A --> E["Condition 4:
Convexity ≥ Liability Convexity
(but not excessive)"] D --> F["Min Asset Duration ≤
Min Liability Duration"] D --> G["Max Asset Duration ≥
Max Liability Duration"] B --> H["Immunized Against
Parallel Shifts"] D --> I["Protected Against
Some Non-Parallel Shifts"] \`\`\` **Why Excessive Convexity Is Dangerous:** While convexity generally helps (gains more when rates fall, loses less when rates rise), in a multi-liability context, a portfolio with very high convexity but matching duration may be **concentrated in barbell positions** (very short and very long bonds). This portfolio is exposed to **yield curve twist risk** — if the curve flattens or steepens, the barbell responds differently than the intermediate liabilities. **Exam Points:** 1. The **bracketing condition** is unique to multi-liability immunization — it does not apply to single-liability cases. 2. **Periodic rebalancing** is required as durations drift over time and after rate changes. 3. Multi-liability immunization protects primarily against **parallel shifts**. Protection against twists requires additional constraints or key rate duration matching. 4. The minimum number of bonds needed equals the number of distinct liabilities (in theory), but practitioners use more for diversification. Explore immunization strategies in our CFA Level II Portfolio Management course.

What conditions must be satisfied to immunize a portfolio against multiple liabilities using duration and convexity?

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