Strategy #0000000031 · Liability-driven investing and risk management
Bond portfolio immunisation
Match asset and liability value and duration, then inspect convexity, curve risk and rebalancing.
Immunisation seeks to align the asset portfolio’s rate sensitivity with a liability so price and reinvestment effects offset under defined conditions. A two-bond example solves matching weights; multiple liabilities, non-parallel shifts, changing cash flows and credit risk require richer methods.
Duration matching reduces first-order differences for small parallel shifts; convexity and curve risks remain.
Three ways in
Understand the story. Then inspect the risk.
Start with a familiar situation, check the payoff, then test how the numbers change.
The idea · in two minutes
Rose funds a liability due in ten years, not just today’s yield
A fund owes a fixed payment in ten years. Jack wants the highest-yielding long bond. Rose asks whether asset value and rate sensitivity match the liability. Under a parallel rate rise, both asset and liability values fall; matching can make the first-order change in funded status close to zero.
They solve weights in two bonds so weighted modified duration equals the liability target. If the target lies outside the two durations, no long-only solution exists; forcing a negative weight introduces leverage and funding.
Rose checks convexity, key-rate duration, liability size and rebalance dates. Duration changes with time and yields, so immunisation needs maintenance; credit default and non-parallel shifts are not removed by one duration equation.
- Matched durationFor two assets, w₂=(Dᴸ−D₁)/(D₂−D₁) and w₁=1−w₂.
- Convexity residual under +100 bpFirst-order funded-status change is −(D_A−D_L)Δy; matching cancels it for a small parallel move.
- After assumed rebalance costConvexity leaves a second-order residual; cash-flow or curve changes require rebalancing.
Illustrative research examples only. They are not live signals or a recommendation.
The mechanics · payoff and exposure
w_1D_1+w_2D_2=D_L,\quad w_1+w_2=1,\qquad \Delta(FV_A-FV_L)\approx-(D_A-D_L)\Delta y+\frac12(C_A-C_L)(\Delta y)^2
D₁ and D₂ are bond modified durations; w₁ and w₂ are present-value weights; D_L is liability duration. C_A and C_L denote asset and liability convexity. Δy is a decimal yield change. The approximation assumes fixed cash flows and unchanged credit spreads.
Greeks: the directional intuition
- Duration drift
- Measures sensitivity to implied volatility.
- Convexity
- Measures sensitivity to implied volatility.
- Curve shape
- Measures sensitivity to implied volatility.
- Liability and credit
- Measures sensitivity to implied volatility.
w_1D_1+w_2D_2=D_L,\quad w_1+w_2=1,\qquad \Delta(FV_A-FV_L)\approx-(D_A-D_L)\Delta y+\frac12(C_A-C_L)(\Delta y)^2
Practical context · published observations
Bond portfolio immunisation: read the record with its caveats.
The chart uses the published figures or explicitly labelled study inputs below. A historical index or model replay is not the same as an investor’s realised account return.
此工具只做兩資產、一負債、平行利率衝擊的二階近似，不是實際負債現金流回測。它不含利息再投資、信用利差、非平行曲線、贖回權或負債變動。目標 duration 不在兩資產區間時會明示無長倉解。 Source ↗ · Methodology ↗
Change assumptions and inspect the scenario
Inputs illustrate the strategy mechanics; they are not live quotes, forecasts or investment advice.
Variants · same family, different risk levers
Related methods share a mechanism, not every assumption.
Each row identifies what changes in its signal, construction or risk.
| Related strategy | Relationship | What changes | New cost or risk |
|---|---|---|---|
| Duration-vector immunisation | Multiple rate factors | Matches sensitivities across several cash-flow or tenor factors. | Requires more instruments and may be infeasible. |
| Principal-component immunisation | Curve factors | Matches level, slope and curvature exposures. | Estimated factors drift and retain model risk. |
| Contingent immunisation | Safety-net trigger | Allows active management above a floor, then immunises at the trigger. | Errors in the floor or trigger timing are costly. |
| Key-rate duration | Key-tenor matching | Matches sensitivity to local rate moves at selected tenors. | Curve interpolation and hedges are imperfect. |
| Multiple-liability immunisation | Multiple payments | Constrains assets against several liabilities and cash-flow dispersion. | More restrictive horizons and complex rebalancing. |
This is a two-asset, one-liability second-order scenario under a parallel shock, not a liability cash-flow backtest. It omits reinvestment, credit spreads, non-parallel moves, embedded options and changing liabilities. It flags targets outside the two asset durations as infeasible for long-only weights.
Read beyond the summary
References
Primary sources and research used to describe the strategy and its limits.
- 01Redington (1952), Review of the Principles of Life-Office Valuations.research
- 02Chambers, Carleton & McEnally (1988), Immunizing Default-Free Bond Portfolios with a Duration Vector.research
- 03Barber & Copper (1996), Immunization Using Principal Component Analysis.research
- 04Leibowitz & Weinberger (1982), Contingent Immunization—Part I.research