Immunisation Strategy
Immunisation Strategy
Bond immunisation aligns the duration of your bond portfolio with the timing and magnitude of your liabilities, protecting your returns from interest-rate shocks by offsetting price losses with reinvestment gains.
Key takeaways
- Immunisation is a mathematical strategy: if your bond portfolio's duration equals the duration of your liability, interest-rate moves create offsetting gains and losses, leaving your total return unchanged.
- A liability has a duration just like a bond: the time-weighted average of when you must pay obligations.
- Immunisation requires periodic rebalancing to maintain duration matching, especially as interest rates change.
- Works for single obligations (e.g., a one-time 500,000 dollar payment in 10 years) or multiple obligations (e.g., a stream of annual expenses).
- More complex than a bullet or ladder but more flexible: you are not locked into specific maturity dates.
Understanding liability duration
A liability is a future obligation. Just as a bond's duration measures its time-weighted cashflow, a liability's duration measures when you need to pay.
Example: One-time obligation You must pay 500,000 dollars exactly 10 years from now (to redeem a debt, fund a project, etc.). This liability has a duration of 10 years.
To immunise, you need a bond portfolio whose duration is also 10 years. If rates rise after you buy the bonds, the portfolio's value falls (bad), but you can reinvest maturing coupons at higher rates (good). If designed correctly, the two offset.
Example: Multiple obligations (pension) You must pay:
- 50,000 dollars in 5 years.
- 75,000 dollars in 10 years.
- 100,000 dollars in 15 years.
What is the duration of this liability stream?
Duration = (5 × 50k + 10 × 75k + 15 × 100k) / (50k + 75k + 100k) Duration = (250k + 750k + 1,500k) / 225k Duration = 2,500k / 225k ≈ 11.1 years
The liability's duration is 11.1 years. To immunise, your bond portfolio's duration should also be 11.1 years.
The immunisation principle
When interest rates rise:
- The market value of your bonds falls (duration risk).
- But coupons and maturing principal can be reinvested at higher rates (reinvestment gain).
When designed correctly, these two effects offset.
Numerical example:
You have a 500,000 dollar liability due in 10 years. Current 10-year bond yield: 4%.
Strategy 1: Naive approach (buy 10-year bonds, hope nothing changes)
- Invest 500,000 dollars in 10-year bonds at 4% yield.
- In 10 years, you have principal (500,000) + coupons (500,000 × 0.04 × 10 = 200,000).
- Total: 700,000 dollars.
But suppose rates rise to 5% after year 1:
- Your 4% bond is now worth less (you own a below-market coupon bond).
- When you rebalance or sell, you realize a loss.
- New 10-year bonds yield 5%, so your coupon is now locked below market.
This is duration mismatch: you are exposed to rate changes.
Strategy 2: Immunisation (match duration)
- Calculate your liability duration: 10 years.
- Build a portfolio with 10-year duration (e.g., a ladder of 2y, 4y, 6y, 8y, 10y bonds, where the weighted-average duration is 6 years—wait, that does not match).
Let me recalculate. A 5-rung ladder with equal weight: Duration = (2 + 4 + 6 + 8 + 10) / 5 = 6 years.
This is a 6-year duration portfolio, not 10 years. To get 10 years, you would need bonds tilted toward longer maturities.
Better approach for immunisation:
- Buy a mix of bonds such that the average duration is exactly 10 years.
- Example: 50% in 5-year bonds (duration 5), 50% in 15-year bonds (duration 15). Average duration: (0.5 × 5) + (0.5 × 15) = 10 years. Perfect.
Now suppose rates rise 1% (from 4% to 5%):
- The 5-year bond loses ~5% of value (5% × 1-year effective maturity).
- The 15-year bond loses ~15% of value (15% × 1-year duration).
- Weighted loss: (0.5 × 5%) + (0.5 × 15%) = 10%.
- But you can reinvest coupons at the new, higher 5% rate. Over the remaining 9 years, this reinvestment gain approximately offsets the 10% price loss.
- Result: You still have enough capital to meet your 10-year obligation.
This is the magic of immunisation: duration matching protects your return.
Multiple obligations: a pension example
Suppose you are a pension fund administrator and must pay:
- 1,000,000 dollars in 3 years.
- 1,200,000 dollars in 7 years.
- 1,500,000 dollars in 12 years.
Step 1: Calculate liability duration
Duration = (3 × 1m + 7 × 1.2m + 12 × 1.5m) / (1m + 1.2m + 1.5m) Duration = (3m + 8.4m + 18m) / 3.7m Duration = 29.4m / 3.7m ≈ 7.95 years
Your liabilities have a weighted-average duration of 7.95 years.
Step 2: Build a bond portfolio with 7.95-year duration
You need to invest 3.7 million dollars (the total of all liabilities) in bonds such that the portfolio's duration is 7.95 years.
One approach: Buy a mix of bonds such that the weighted-average duration is 7.95.
Example:
- 1.2 million in 3-year bonds (duration 3).
- 1.2 million in 8-year bonds (duration 8).
- 1.3 million in 12-year bonds (duration 12).
Weighted-average duration: (1.2m × 3 + 1.2m × 8 + 1.3m × 12) / 3.7m = (3.6m + 9.6m + 15.6m) / 3.7m = 28.8m / 3.7m ≈ 7.78 years. Close enough.
Step 3: Rebalance periodically
As rates change and time passes, the portfolio's duration will drift. Semi-annually or quarterly, rebalance to maintain 7.95-year duration.
When the 3-year bonds mature in year 3, redeploy proceeds into longer bonds (e.g., 15-year) to keep the portfolio duration on target.
Interest-rate scenarios and rebalancing
Immunisation works only if you rebalance when rates change. Let me illustrate.
Year 0 setup:
- Portfolio: 50% 5y bonds, 50% 15y bonds.
- Liability: 500,000 in 10 years.
- Duration match: 10 years.
- Portfolio value: 500,000 dollars.
Year 1: Rates rise from 4% to 5%
- 5-year bond value: 500,000 × 0.5 × (1 - 5% × 5 years / 100) ≈ 237,500 (approximation).
- 15-year bond value: 500,000 × 0.5 × (1 - 5% × 15 years / 100) ≈ 212,500.
- Total portfolio: ~450,000 (down from 500,000).
Now, the durations have shifted:
- The 5-year bond is now a 4-year bond (duration 4).
- The 15-year bond is now a 14-year bond (duration 14).
- New portfolio duration: (0.5 × 4) + (0.5 × 14) = 9 years.
The liability is still 10 years away, so its duration is now 9 years. The portfolio and liability are still matched! This is good news: the immunisation held despite the rate shock.
But wait: the portfolio is now worth only 450,000, and you need 500,000 in 9 years. If rates stay at 5%, you receive 450,000 × (1.05)^9 ≈ 694,000 dollars, which covers your 500,000 obligation. You are still OK.
However, if you wait another year (year 2) and rates remain at 5%:
- Your portfolio is now a 4-year and 13-year blend (average duration 8.5 years).
- The liability is 8 years away (duration 8 years).
- Duration mismatch is widening.
So rebalancing is necessary: every 6–12 months, rebalance to maintain the duration match.
Rebalancing to maintain immunisation
When the portfolio's duration drifts from the liability's duration, rebalance.
Example rebalancing:
- Current portfolio: 40% 4y bonds, 60% 14y bonds (duration 10).
- Current liability duration: 9 years (liability is now 9 years away).
- Drift: Duration is 1 year too long.
Action: Shift 10% from long bonds to short bonds.
- New: 50% 4y bonds, 50% 14y bonds (duration 9 years).
This simple rebalancing—shifting 5% of assets from long to short—brings the portfolio back into duration matching.
Cash-flow matching vs. immunisation
Both are liability-matching strategies, but they differ:
| Aspect | Cashflow Matching | Immunisation |
|---|---|---|
| Concept | Maturity dates align with liability dates | Duration aligns with liability duration |
| Example | Bullet maturing in 10 years for a 10-year liability | Blend of 5y and 15y bonds (avg duration 10y) for a 10-year liability |
| Flexibility | Rigid: must match exact dates | Flexible: can use any maturity mix with matching duration |
| Rebalancing | Minimal (bonds held to maturity) | Frequent (maintain duration matching) |
| Complexity | Low | High |
| Rate risk | None if dates match | None if duration matches (theoretically) |
Cashflow matching is simpler but less flexible. Immunisation is more complex but offers flexibility.
When immunisation works
Immunisation is strongest when:
- Interest rates move (rising or falling). The duration offset protects you.
- Your liability is a single, large payment (e.g., a balloon payment 10 years from now).
- You can rebalance at least quarterly.
- The yield curve does not distort dramatically (a parallel shift in rates works perfectly; curve bending is messier).
Immunisation is weaker when:
- The liability is a stream of payments (like a pension). Multiple obligations mean duration is an average, not a precise match.
- You cannot rebalance frequently.
- The yield curve inverts or twists (short rates up, long rates down, or vice versa).
Practical limitations
In reality, perfect immunisation is difficult:
- Rebalancing costs: Trading bonds incurs bid-ask spreads (0.25–1%) and potential capital gains taxes.
- Curve risk: If short rates rise but long rates fall (a yield curve twist), your duration match may not protect you.
- Convexity: Duration is a linear approximation; for large rate moves, duration alone does not fully predict price changes. You need convexity adjustments.
- Default risk: A bond default breaks immunisation (your asset value drops, duration-matching is meaningless).
A practical example: college funding via immunisation
You have a 10-year-old child. College starts in 8 years, costing 200,000 dollars in years 8–10 (50k per year, rising slightly).
Liability duration: (8 × 50k + 9 × 50k + 10 × 50k) / 150k ≈ 9 years.
Build a portfolio with 9-year duration:
- 60% in 5-year bonds (duration 5).
- 40% in 13-year bonds (duration 13).
- Weighted duration: (0.6 × 5) + (0.4 × 13) = 3 + 5.2 = 8.2 years. (Not exactly 9, but close; adjust allocations as needed.)
Invest 200,000 dollars in this split. Rebalance annually to maintain ~9-year duration.
When the child turns 18, your portfolio has grown (via coupons) and evolved into a mix maturing around 8–10 years. Liquidate the portfolio and fund college.
This is more flexible than a bullet (you could redeploy if college is later or cheaper) but more complex than buy-and-hold.
Advanced immunisation: multiperiod
For complex liabilities (pensions with decades of payments), immunisation becomes more sophisticated. Pension funds use:
- Multiperiod immunisation: Match not just average duration but also duration at each payment date.
- Key-rate duration: Immunise against specific points on the yield curve.
- Stochastic immunisation: Use Monte Carlo simulations to test duration-matching against many interest-rate paths.
These are beyond the scope for individual investors; they are professional-level techniques.
Next
Immunisation matches duration to liabilities abstractly. The next strategy—cash-flow matching—takes a simpler, more literal approach: align actual maturity dates with actual payment dates, eliminating duration complexity.