How does duration matching immunize a portfolio against a single liability?

Question

AcadiFi · Accepted Answer

Immunization is a strategy that ensures a bond portfolio will have sufficient value to meet a future liability regardless of interest rate changes. For a single liability, the key condition is that the portfolio's Macaulay duration equals the liability's time horizon.

**Why Duration Matching Works:**

When rates change, two opposing effects occur:
1. **Price effect:** If rates rise, bond prices fall (bad if you need to sell)
2. **Reinvestment effect:** If rates rise, coupon reinvestment earns more (good for future value)

At the duration point, these two effects exactly offset each other. Duration is the 'balancing point' where price risk and reinvestment risk cancel out.

```mermaid
graph LR
    subgraph "Rates Rise"
    A["Bond Prices Fall
(Negative)"] --> C["Net Effect ≈ 0
at Duration Point"]
    B["Reinvestment Income
Rises (Positive)"] --> C
    end
    subgraph "Rates Fall"
    D["Bond Prices Rise
(Positive)"] --> F["Net Effect ≈ 0
at Duration Point"]
    E["Reinvestment Income
Falls (Negative)"] --> F
    end
    style C fill:#10b981,color:#fff
    style F fill:#10b981,color:#fff
```

**Example — Bellingham Insurance (fictional):**

Liability: Pay $5 million in exactly 7 years. Current rate: 5%.

**PV of liability:** $5,000,000 / (1.05)^7 = **$3,553,408**

**Immunization portfolio:** Purchase bonds with Macaulay duration = 7 years, invest $3,553,408 today.

**Scenario 1: Rates immediately jump to 7%**
- Bond prices fall (portfolio value drops immediately)
- But coupons are reinvested at 7% (higher income accumulates)
- At year 7: portfolio value ≈ $5,000,000 (the effects offset)

**Scenario 2: Rates immediately drop to 3%**
- Bond prices rise (portfolio value increases immediately)
- But coupons are reinvested at 3% (lower income accumulates)
- At year 7: portfolio value ≈ $5,000,000 (again, the effects offset)

**Conditions for Successful Immunization:**
1. Portfolio Macaulay duration = Liability horizon
2. PV of assets >= PV of liability
3. Portfolio convexity >= Liability convexity (minimally, but not excessively)
4. Yield curve shift must be parallel (a key limitation)
5. Must rebalance as time passes and rates change

**Exam Tip:** The CFA exam tests the immunization conditions and the trade-off between price risk and reinvestment risk. Remember to use Macaulay duration (not modified) for the matching condition.

Practice immunization problems in our CFA Level I question bank.

How does duration matching immunize a portfolio against a single liability?

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