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Home / Financial Tools / Duration-Convexity Bond Immunization Optimizer

Graduate-Level Modeling · Fixed Income Portfolio Management

Duration-Convexity Bond Immunization Optimizer

Match a bond portfolio's duration to a future liability, and you're protected against a small rate move. Build in extra convexity on top, and you're protected against a large one too, in either direction.

How To Use This Model

Reading This Tool

Enter your liability's duration and convexity, and the duration/convexity of two candidate bonds.

The tool solves the exact weights that match portfolio duration to the liability, then shows whether the resulting convexity gap protects or exposes you across a range of yield shifts.

Liability & Bond Inputs

Bond 1 duration must be below the liability duration and Bond 2 duration above it (a barbell), otherwise no valid long-only weight combination can match the target duration exactly.

Immunizing Portfolio Weights

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Weight In Bond 1

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Weight In Bond 2

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Portfolio Convexity Vs. Liability Convexity

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Asset Vs. Liability Value Change Across A Yield Shift

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Why The Barbell Beats A Bullet Here

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What The Surplus Line Actually Means

At zero yield shift, assets exactly equal liabilities by construction, duration was matched precisely for that. Away from zero, if the surplus line stays positive in both directions, the portfolio is protected against parallel shifts either way; if it goes negative in either direction, immunization has failed on that side.

What This Doesn\u2019t Protect Against

This entire analysis assumes a parallel shift, every maturity's yield moving by the same amount. A non-parallel shift, the curve twisting or steepening, can still create a mismatch even with duration and convexity both matched, a limitation worth stating explicitly to any risk committee relying on this.

Asset-Liability Management

The Core Formulas

Duration matching: w1D1 + w2D2 = DL, w1+w2=1
w1 = (DL−D2)/(D1−D2)
%ΔPrice ≈ −D·Δy + ½C·(Δy)²

Once duration is matched, the second-order convexity term determines protection against larger yield moves. A portfolio convexity exceeding the liability's convexity outperforms the liability in both directions of a parallel shift, this is the classical Redington immunization result.

When To Actually Use This Model

  • Pension fund and insurance liability-driven investing, matching a bond portfolio to a known future obligation.
  • Teaching classical immunization theory in a fixed income or asset-liability management course.
  • Structuring a barbell versus bullet bond portfolio decision with an explicit convexity objective.
  • Explaining to a risk committee why a barbell structure was chosen over a simpler single-bond duration match.

Key Assumptions & Limitations

  • Assumes only parallel yield curve shifts; twist and steepening risk require key-rate duration analysis instead.
  • Duration and convexity matching must be actively rebalanced over time as both drift with the passage of time and yield changes.
  • Two-bond solutions are illustrative; real immunized portfolios often use several bonds or a bond ladder for practical liquidity reasons.
  • Ignores credit risk, liquidity risk, and reinvestment risk on coupon cash flows.

Foundational Reference

Redington, F. M. (1952). Review of the Principles of Life-Office Valuations. Journal of the Institute of Actuaries, 78(3), 286-340.

Want single-bond duration and convexity math too?

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