Duration Matching
Duration Matching
Duration matching is the practice of choosing bonds such that the weighted-average duration of your bond portfolio equals the weighted-average duration of your liabilities, creating natural protection against interest-rate risk.
Key takeaways
- A liability's duration is the time-weighted average of when you must pay obligations, calculated similarly to bond duration.
- When asset duration equals liability duration, interest-rate moves create offsetting price and reinvestment effects, leaving your total return stable.
- Duration matching is more flexible than cashflow matching (you can buy any bond maturity) and less demanding than full immunisation (you rebalance less often).
- Requires periodic rebalancing, especially after rates change or time passes, to maintain the duration alignment.
- Widely used by pension funds, insurance companies, and sophisticated individual investors for liability-driven investing.
What is liability duration?
Just as a bond has duration (the weighted time to receive cashflows), a liability has duration.
Example: Single payment You owe 500,000 dollars in exactly 10 years. Liability duration = 10 years.
Example: Multiple payments You owe:
- 100,000 in 5 years.
- 150,000 in 10 years.
- 200,000 in 15 years.
Liability duration = (5 × 100k + 10 × 150k + 15 × 200k) / (100k + 150k + 200k) = (500k + 1,500k + 3,000k) / 450k = 5,000k / 450k ≈ 11.1 years
Why duration matching works
When you match asset duration to liability duration, interest-rate movements create two offsetting effects:
Effect 1: Price change on existing bonds If rates rise, bond prices fall. A portfolio with 11.1 years duration loses roughly 11.1% of its value if rates rise 1%.
Effect 2: Reinvestment rate improvement If rates rise, new coupons and maturing principal can be reinvested at higher rates. Over the next 11.1 years, the reinvestment gain approximately offsets the price loss.
Result: Your total return (price change + reinvestment gain) is stable regardless of rate changes.
Numerical example:
You have a 500,000 dollar liability due in 10 years. Current rates: 4%.
Portfolio A: Buy only 10-year bonds (duration 10, maturity mismatch)
- Initial investment: 500,000 at 4% yield.
- Rates rise to 5% after year 1.
- Price loss: Roughly 10% (duration 10 × 1% rate rise) = 50,000 dollar loss.
- New bonds yield 5%, so coupons reinvest at 5% (better).
- Over remaining 9 years, reinvestment gain ≈ 35,000–40,000 dollars.
- Net: Price loss partially offset by reinvestment gain.
Portfolio B: Buy a duration-matched blend (duration 10, exact match)
- 50% in 5-year bonds (duration 5).
- 50% in 15-year bonds (duration 15).
- Weighted duration: 10 years. Perfect match.
- Rates rise to 5% after year 1.
- 5-year bonds lose ~5%, 15-year bonds lose ~15%.
- Weighted loss: 10%. Same as Portfolio A.
- But the blend is more efficient: the short bonds' lower loss offsets the long bonds' higher loss.
- Reinvestment gain from higher rates applies to the short bonds' coupons immediately (reinvested at 5% right away).
- Net: The offset is more complete because duration is 10, not just by chance, but by design.
Duration matching makes the offset intentional and precise.
Calculating portfolio duration
Once you select bonds, calculate the portfolio's duration as a weighted average.
Example portfolio:
| Bond | Maturity | Amount | Duration | Weight | Weighted Duration |
|---|---|---|---|---|---|
| Treasury | 3 years | 50,000 | 2.9 | 0.25 | 0.73 |
| Corporate | 8 years | 100,000 | 7.5 | 0.50 | 3.75 |
| Long bond | 15 years | 50,000 | 13.8 | 0.25 | 3.45 |
| Total | 200,000 | 7.93 |
Portfolio duration ≈ 7.93 years.
If your liability duration is also 7.93 years, you have a duration match.
Rebalancing to maintain duration match
As time passes and rates change, the portfolio's duration drifts. You must rebalance to maintain the match.
Example: Year 1
Original portfolio duration: 7.93 years. Liability duration: 7.93 years (liability is now 6.93 years away, so its duration changes).
After 1 year:
- The 3-year Treasury is now a 2-year bond (duration ~1.9).
- The 8-year Corporate is now a 7-year bond (duration ~6.5).
- The 15-year Long bond is now a 14-year bond (duration ~13).
New portfolio duration: (50k × 1.9 + 100k × 6.5 + 50k × 13) / 200k = (95 + 650 + 650) / 200 = 6.975 years.
New liability duration: The liability is now 6.93 years away instead of 7.93 years, so its duration is roughly 6.93 years.
Portfolio duration (6.98) is still very close to liability duration (6.93). No immediate action needed.
After 5 years:
Original liability duration: 7.93 years. Liability duration now: 2.93 years (only 2.93 years until the obligation is due).
The portfolio has aged. The 3-year bond has matured. You still hold the 8-year corporate (now a 3-year bond) and the 15-year long bond (now a 10-year bond).
Estimated portfolio duration: (0 + 100k × 3 + 50k × 10) / 200k = 500 / 200 = 2.5 years.
Liability duration: 2.93 years.
Close enough. Still matched.
But if you had reinvested the matured 3-year Treasury into a new short bond (1 year), the portfolio duration would have dropped to roughly 2 years, creating a mismatch.
To maintain the match, you could:
- Reinvest the 3-year maturity into a longer bond (e.g., 5-year) to keep portfolio duration around 2.93 years.
This is the discipline of duration matching: after each maturity or rate change, rebalance the portfolio to realign asset duration with liability duration.
Duration matching vs. cashflow matching
| Aspect | Duration Matching | Cashflow Matching |
|---|---|---|
| Matching criterion | Duration (weighted avg time to cashflow) | Maturity dates (exact payment dates) |
| Flexibility | High (any maturity mix works if duration is right) | Low (must match exact dates) |
| Rebalancing | Periodic (when duration drifts) | None (hold to maturity) |
| Complexity | Medium-high (requires duration calculations) | Medium (scheduling requirement) |
| Reinvestment risk | Low (duration offset handles it) | Zero (coupons reinvested on schedule) |
| Suitable for | General liabilities, pension funds | Specific, fixed obligations |
For a college fund with exact payment dates (2027: 25k, 2028: 25k, etc.), cashflow matching is cleaner.
For a pension with ongoing obligations, duration matching is more practical (you do not need exact maturity dates, just the right duration blend).
Interest-rate scenarios and duration
Duration-matched portfolios behave predictably under different rate scenarios.
Scenario 1: Rates rise (stress test)
- Asset duration: 10 years.
- Liability duration: 10 years.
- Rate shock: +1% (rates rise from 4% to 5%).
- Asset loss: ~10% (duration 10 × 1%).
- Liability present value change: ~10% (duration 10 × 1%). (A future liability's present value falls when rates rise, because discounting at 5% makes it more valuable to hold bonds today.)
- Result: Asset value and liability value both fall ~10%. The mismatch is preserved (both decline equally).
Wait, that is not quite right. Let me reconsider.
Actually, when rates rise:
- Bond values (your assets) fall ~10%.
- The present value of your liability falls too (because the 10-year obligation is discounted at 5% instead of 4%, making it less onerous).
- Both move in the same direction and by roughly the same magnitude, so the surplus (asset - liability) is stable.
This is the key insight: duration matching protects your surplus against rate shocks.
Scenario 2: Rates fall (favorable scenario)
- Rates fall from 4% to 3%.
- Asset value rises ~10% (duration 10 × 1% fall).
- Present value of liability rises ~10% (future obligation is discounted at 3%, making it more onerous).
- Surplus remains stable.
Rebalancing frequency
How often should you rebalance to maintain duration match?
Monthly rebalancing: Too frequent. Transaction costs outweigh benefits.
Quarterly rebalancing: Typical for pension funds. Manageable and cost-effective.
Semi-annual rebalancing: Suitable for individual investors. Balances accuracy with convenience.
Annual rebalancing: Acceptable for long-duration liabilities (20+ years). Drift is slow.
Rebalancing trigger: Instead of a fixed schedule, rebalance when duration drifts more than 0.5–1 years from the target. This avoids unnecessary trades while maintaining alignment.
Advanced duration concepts: convexity
Duration is a linear approximation. For large interest-rate moves (e.g., +3%), duration alone underestimates price changes. Convexity corrects this.
A bond with positive convexity (most bonds have this) benefits from large rate moves in either direction. A bond with negative convexity (some mortgage-backed securities) is harmed by large rate declines.
For duration matching, ignoring convexity is usually acceptable if:
- Your liability duration is not extreme (under 15 years).
- Interest-rate scenarios are mild (within ±1–2%).
- You rebalance regularly.
For institutional portfolios with large liabilities, convexity matching becomes important.
Practical example: a pension fund
A pension fund has 100 million dollars in obligations:
- 10 million per year for years 1–10.
- Liability duration: approximately 5.5 years.
The fund builds a portfolio:
- 40% in 3-year bonds (duration 2.8).
- 40% in 7-year bonds (duration 6.5).
- 20% in 12-year bonds (duration 11.0).
Weighted portfolio duration: (0.4 × 2.8) + (0.4 × 6.5) + (0.2 × 11.0) = 1.12 + 2.6 + 2.2 = 5.92 years.
Close to the 5.5-year liability duration. Reasonably matched.
As rates change and bonds age, the fund rebalances quarterly, adjusting the mix to maintain ~5.5-year duration.
Over the liability period, rates may rise or fall, but the duration-matched portfolio generates sufficient returns to fund the obligations.
When to use duration matching
Good fit:
- Multi-year, multi-payment liabilities (pensions, annuities).
- You need flexibility in bond selection (any maturity blend works as long as duration is right).
- You can rebalance at least semi-annually.
- Liabilities are reasonably stable (no major changes expected).
Poor fit:
- Liabilities are inflexible or highly specific (use cashflow matching instead).
- You want zero rebalancing (use cashflow matching or a bullet).
- Interest rates are extremely volatile or a major rate shock is expected (rebalancing cannot keep up).
Limitations and real-world considerations
1. Convexity mismatch: Even with duration matched, a large rate move can create issues if assets and liabilities have different convexities.
2. Yield curve risk: Duration assumes a parallel shift in the yield curve (all rates move equally). If the curve twists (short rates up, long rates down), the match breaks.
3. Credit risk: A bond default reduces your asset value but does not directly affect your liability. Duration matching does not hedge credit risk.
4. Rebalancing costs: Transaction costs, bid-ask spreads, and taxes (if in a taxable account) eat into returns. High turnover is inefficient.
5. Basis risk: The bonds you buy may not have exact duration matching the liability. There is always some basis risk (a gap between asset and liability durations).
Most practitioners accept these limitations and view duration matching as an approximation—good enough for most scenarios.
Next
Both duration matching and cashflow matching are passive liability-matching strategies. The next article shifts to active management: using the yield curve to take directional bets on interest rates via strategies like steepeners and flatteners.