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gateway/knowledge/trading/strategies/fixed-income/bond-immunization.md

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