How does immunization work for fixed-income portfolios and what are the conditions for it to succeed?

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AcadiFi · Accepted Answer

Immunization is a strategy that structures a fixed-income portfolio so that its value will be sufficient to meet a future liability regardless of interest rate changes. It works by balancing **price risk** (rates rise, bond prices fall) against **reinvestment risk** (rates rise, coupons reinvested at higher rates). ```mermaid graph TD A[Immunization Strategy] --> B{Type of Liability?} B -->|Single liability| C[Classical Immunization] B -->|Multiple liabilities| D[Cash Flow Matching
or Multi-Period Immunization] C --> E[Conditions for Success] E --> E1[1. Portfolio Duration =
Liability Duration] E --> E2[2. PV of Portfolio >=
PV of Liability] E --> E3[3. Minimize Portfolio
Convexity subject to
duration match] E --> E4[4. Rebalance as
duration drifts] C --> F{Interest Rate Shift?} F -->|Parallel shift up| G[Price Loss offset by
Reinvestment Gain] F -->|Parallel shift down| H[Price Gain offset by
Reinvestment Loss] F -->|Non-parallel shift| I[Immunization at Risk!
Convexity mismatch matters] D --> J[Duration Match +
Distribution of Durations
must bracket liabilities] style A fill:#c9a84c,color:#000 style I fill:#7f1d1d,color:#fff ``` **Worked Example — Immunizing a Pension Obligation:** Redfield Pension Fund has a single liability of $25M due in 8 years. Current interest rates are 5% across the curve. **Step 1: Calculate PV of liability.** PV = $25,000,000 / (1.05)^8 = **$16,916,781** Redfield must invest at least $16,916,781 today. **Step 2: Build a portfolio with Macaulay duration = 8.0 years.** Consider two bonds: | Bond | Coupon | Maturity | Mac Duration | Yield | |---|---|---|---|---| | Bond A | 4.0% | 5 years | 4.63 years | 5.0% | | Bond B | 5.5% | 12 years | 9.42 years | 5.0% | To get a portfolio duration of 8.0: w_A x 4.63 + (1 - w_A) x 9.42 = 8.0 4.63 w_A + 9.42 - 9.42 w_A = 8.0 -4.79 w_A = -1.42 w_A = 0.2964 (29.64%) w_B = 0.7036 (70.36%) Invest $5,013,832 in Bond A and $11,902,949 in Bond B. **Step 3: Verify convexity condition.** For immunization to work against non-parallel shifts, minimize convexity — a barbell portfolio (very short + very long) has excess convexity that creates risk under curve twists. The 5yr + 12yr combination is moderately barbelled, which is acceptable but a bullet portfolio centered near 8 years would be ideal. **What Can Go Wrong:** 1. **Non-parallel yield curve shifts:** If short rates rise while long rates fall (curve flattening), a barbell portfolio gains more than expected on the long end but loses reinvestment income on the short end — the gains/losses may not offset perfectly. 2. **Duration drift:** As time passes, bond durations change. After 1 year, the portfolio duration is no longer 8.0 (it might be 7.2). You must rebalance — sell some bonds and buy others to restore the duration match. 3. **Large rate movements:** Duration is a linear approximation. For large moves, convexity effects become significant, potentially creating surplus or deficit. **Classical Immunization vs. Cash Flow Matching:** | Feature | Immunization | Cash Flow Matching | |---|---|---| | Concept | Duration match | Each liability matched by a specific bond's cash flow | | Rebalancing | Required | Not required | | Cost | Lower (more flexibility) | Higher (exact matching is constraining) | | Risk | Non-parallel shifts | Reinvestment risk on intermediate flows | **Exam Tip:** The constructed response often asks you to identify why an immunization failed. Common answers: portfolio was not rebalanced, convexity was too high (barbell), or the yield curve shift was non-parallel. Explore more liability-driven strategies in our CFA Level III Fixed Income question bank.

How does immunization work for fixed-income portfolios and what are the conditions for it to succeed?

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