Tim Olson
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67 lines
3.3 KiB
Markdown
67 lines
3.3 KiB
Markdown
---
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description: "Bond immunization constructs a portfolio whose duration matches a future cash obligation's maturity, protecting the portfolio value against parallel yield curve shifts to meet a predetermined liability."
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tags: [fixed-income, duration, immunization, liability-matching, convexity]
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---
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# Bond Immunization
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**Section**: 5.5 | **Asset Class**: Fixed Income | **Type**: Duration / Liability Matching
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## Overview
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Bond immunization is used to ensure a portfolio can meet a predetermined future cash obligation F at time T_*. A portfolio is constructed so that its duration matches T_*, making its value insensitive to parallel shifts in the yield curve. It extends to matching convexity for additional protection with three bonds.
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## Construction / Mechanics
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**Total investment** P given a future obligation F at time T_*, constant yield Y, periodic compounding with period δ:
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```
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P = F / (1 + Yδ)^(T_*/δ) (396)
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```
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**Two-bond immunization** (matches duration only):
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With two bonds of maturities T_1, T_2 and modified durations D_1, D_2, dollar allocations P_1, P_2:
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```
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P_1 + P_2 = P (397)
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P_1·D_1 + P_2·D_2 = P·D (398)
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```
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where the target modified duration:
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```
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D = T_* / (1 + Yδ) (399)
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```
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**Three-bond immunization** (matches duration and convexity):
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With three bonds, durations D_1, D_2, D_3 and convexities C_1, C_2, C_3:
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```
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P_1 + P_2 + P_3 = P (400)
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P_1·D_1 + P_2·D_2 + P_3·D_3 = P·D (401)
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P_1·C_1 + P_2·C_2 + P_3·C_3 = P·C (402)
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```
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where the target convexity:
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```
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C = T_*(T_* + δ) / (1 + Yδ)² (403)
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```
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## Payoff / Return Profile
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- Immunized portfolio is protected against parallel yield curve shifts: the gain/loss from price changes offsets the loss/gain from reinvestment rate changes.
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- Matching convexity (three-bond) provides additional protection against larger rate moves.
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- The portfolio value converges to F at time T_* under parallel shifts.
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## Key Parameters / Signals
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- T_*: maturity of the future cash obligation (target duration)
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- F: size of the future obligation
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- Y: assumed constant yield (all bonds assumed same yield — a simplification)
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- D, C: target modified duration and convexity
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## Variations
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- **Zero-coupon immunization**: purchase a single zero-coupon bond with maturity T_* — the simplest solution, but may not be available.
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- **Two-bond**: matches duration only; sufficient for small parallel shifts.
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- **Three-bond**: matches both duration and convexity; handles larger shifts.
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- Extension to non-parallel yield curve changes requires additional sophistication.
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## Notes
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- The assumption that all bonds have the same yield is a simplification; in practice yields differ across maturities and issuers.
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- The portfolio must be periodically rebalanced as the yield curve changes, incurring transaction costs.
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- Immunization protects against parallel shifts only; slope and curvature changes can still cause losses.
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- Non-parallel shifts, credit spread changes, and transaction costs all introduce complexity in practice.
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