duration-matching-for-liability-immunization — yield duration and convexity

Duration matching is the core discipline of liability-driven investing. When assets and liabilities have the same duration, parallel interest rate shifts affect both sides equally—leaving funded status unchanged. Research confirms that a 100% hedging ratio implies complete immunization, removing interest rate risk from the equation (Leibowitz & Weinberger, 1982). The point is: duration matching isn't about predicting rates; it's about making rate predictions irrelevant to your outcome.

What Immunization Actually Protects (And What It Doesn't)

Immunization ensures that a portfolio's terminal value meets a future liability regardless of how rates move between now and then. The mechanism works through offsetting effects: when rates rise, coupon reinvestment earns more but bond prices fall; when rates fall, reinvestment earns less but prices rise.

The magic number is duration. When your portfolio's duration exactly matches your liability's duration, these opposing forces cancel out for small parallel shifts in the yield curve.

The durable lesson: Immunization protects against interest rate risk. It does not protect against credit defaults, non-parallel curve shifts, or dramatic rate moves where convexity effects dominate.

The Immunization Formula (Why Duration Must Match)

For a single liability due in T years, you need assets with Macaulay duration equal to T. The present value of assets must also equal or exceed the present value of the liability.

Two conditions for immunization:

Duration Match: Asset Macaulay duration = Liability duration (time to payment)
Present Value Match: PV of assets >= PV of liability

If you have $7.11 million today earning 5% and need to pay $10 million in 7 years, your portfolio duration must equal 7 years. Any bonds you select—whether 5-year or 10-year maturities—must combine to a weighted average duration of exactly 7 years.

Worked Example: Constructing an Immunized Portfolio

You manage a pension plan with a single liability: $10 million due in exactly 7 years. Current yields are 5% across the curve. Your investment horizon requires immunization against rate changes.

Step 1: Calculate Present Value of Liability

PV = $10,000,000 / (1.05)^7 = $7,106,813 (approximately $7.11 million)

You need $7.11 million in assets today.

Step 2: Determine Required Duration

Since the liability is a single payment in 7 years, the liability's Macaulay duration is simply 7 years. Your asset portfolio must have a weighted average Macaulay duration of 7 years.

Step 3: Select Bond Positions

You have access to two bonds:

Bond A: 5-year maturity, 5% coupon, duration = 4.5 years
Bond B: 10-year maturity, 5% coupon, duration = 7.8 years

Weight calculation to achieve 7-year duration:

w(A) x 4.5 + w(B) x 7.8 = 7.0, where w(A) + w(B) = 1

Solving: w(A) = 24.2% and w(B) = 75.8%

Allocate $1.72 million to Bond A and $5.39 million to Bond B.

Step 4: Verify Portfolio Duration

Portfolio Duration = (0.242 x 4.5) + (0.758 x 7.8) = 1.09 + 5.91 = 7.0 years

The portfolio is now immunized against parallel yield curve shifts.

Why this matters: If yields immediately jump from 5% to 6%, both your liability's present value and your assets decline by approximately 7%. The funded status remains stable.

The Convexity Problem (Why Duration Matching Isn't Enough)

Duration matching works perfectly for infinitesimally small rate changes. For larger moves—say +100 bps or more—convexity differences between assets and liabilities create tracking error.

The mechanics: Duration captures the first-order price sensitivity. Convexity captures curvature. Two portfolios with identical duration but different convexity will diverge under large rate moves.

A common mistake: matching duration without considering convexity creates exposure to large rate moves. In the 2022 hiking cycle, when 10-year yields rose 236 bps in a single year, convexity mismatches caused immunized portfolios to drift off target (Federal Reserve, 2022).

The fix: Match both duration and convexity, or use key-rate duration matching to hedge curve risk at specific maturity points. The CFA Institute notes that key-rate duration-matching is superior to simple duration-convexity matching for liability-driven strategies (CFA Institute, 2026).

Rebalancing: When Duration Drifts Too Far

Duration isn't static. As time passes and rates change, your portfolio's duration shifts—even if you don't trade. This creates duration drift, requiring periodic rebalancing.

Three triggers for rebalancing:

Time passage: Duration naturally shortens as bonds age (the 10-year becomes a 9-year)
Rate changes: Duration itself is rate-sensitive (higher rates = lower duration)
Cash flows: Coupon receipts change portfolio composition

Rebalancing rule of thumb: When portfolio duration drifts more than 0.25 years from target, rebalance to restore the match.

Practical consideration: Transaction costs matter. Some managers use futures overlays to adjust duration cheaply without selling bonds. A pension fund with $25 million in assets and 6.1-year duration targeting 4.0 years would sell approximately 255 Treasury futures contracts to close the gap (CME Group, 2020).

Detection Signals: You're Likely Exposed If...

Your immunized portfolio hasn't been rebalanced in over 6 months
You matched duration but ignored convexity (vulnerable to +100 bps moves)
Your portfolio uses only bullet bonds while the benchmark uses barbells (curve risk)
You assumed parallel shifts when the 2-10 spread has moved 50+ bps
Your funded status swung despite "immunization" during the 2022 rate shock

Barbell vs. Bullet: Same Duration, Different Risk

Two portfolios can have identical duration but radically different curve exposure:

Bullet strategy: Concentrate holdings around the liability horizon (e.g., all 7-year bonds for a 7-year liability). Lower convexity. Less sensitive to curve reshaping.

Barbell strategy: Mix short and long bonds (e.g., 50% 2-year, 50% 12-year). Higher convexity. Outperforms in parallel shifts but vulnerable to curve steepening.

Consider the 2022-2023 experience: a barbell with 50% 2-year and 50% 10-year bonds had duration of approximately 5 years but suffered during the curve flattening when the 2y-10y spread inverted to -100 bps (Federal Reserve, 2022).

The test: A → B → C chain: Curve steepens → Long end falls more → Barbell underperforms bullet

Checklist: Building an Immunized Portfolio

Essential (Start Here)

Calculate liability duration (for single payment, it equals time to payment)
Ensure asset present value exceeds liability present value
Weight bond positions to achieve target duration (within 0.1 years)
Document rebalancing triggers (time-based and threshold-based)

High-Impact Refinements

Match convexity as well as duration for large-move protection
Use key-rate durations if liability stream has multiple maturities
Consider futures overlays for cost-efficient duration adjustments

When Immunization Fails: Real-World Stress Tests

The 2022 rate cycle tested immunization discipline. Portfolios that:

Matched duration but ignored convexity saw 2-3% funded status swings
Used key-rate matching maintained tighter tracking
Relied on model assumptions about parallel shifts underperformed

The Bloomberg US Aggregate Index duration was 6.0 years versus its long-term average of 4.97 years entering 2022 (Hartford Funds, 2025). Many LDI portfolios benchmarked to the Agg carried more rate risk than historical norms suggested.

The point is: Immunization works when you respect its assumptions. Assuming parallel shifts in a world of curve twists introduces basis risk.

Common Mistakes and How to Avoid Them

Mistake 1: Ignoring duration drift

Your 7-year portfolio becomes a 6.5-year portfolio just from time passage. Without rebalancing every 6-12 months, immunization erodes.

Mistake 2: Assuming YTM equals holding period return

If rates change, reinvestment returns differ from YTM projections. Duration matching accounts for this—but only for parallel shifts.

Mistake 3: Matching modified duration instead of Macaulay duration

For immunization targeting a specific horizon, Macaulay duration is the correct measure. Modified duration measures price sensitivity—related but not identical.

Your Next Step

Identify a fixed liability—a balloon payment, tuition due date, or retirement target. Calculate its present value at current rates and design a bond portfolio with matching duration. Track both over 6 months to see how rate changes affect each side equally.

Related: Modified Duration and Price Sensitivity | Convexity Concept and Calculation | Using Futures and Swaps to Adjust Duration

Sources: CFA Institute (2026). Liability-Driven and Index-Based Strategies. | CME Group (2020). Key Rate Duration Adjustment Case Study. | Hartford Funds (2025). Duration of the Bloomberg US Aggregate Bond Index. | Federal Reserve Board (2025). Nominal Yield Curve Methodology.

Duration Matching for Liability Immunization