Tim Olson
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45 lines
2.5 KiB
Markdown
45 lines
2.5 KiB
Markdown
---
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description: "A dollar-duration-neutral butterfly combines a long barbell (short T_1 and long T_3 maturities) with a short bullet (intermediate T_2) at zero net cost, immunizing against parallel yield curve shifts to profit from yield curve curvature changes."
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tags: [fixed-income, butterfly, duration-neutral, yield-curve, curvature]
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---
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# Dollar-Duration-Neutral Butterfly
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**Section**: 5.6 | **Asset Class**: Fixed Income | **Type**: Yield Curve / Curvature
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## Overview
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The dollar-duration-neutral butterfly is a zero-cost combination of a long barbell (long T_1 and T_3 maturity bonds) and a short bullet (short the T_2 intermediate maturity bond), where T_1 < T_2 < T_3. Both zero cost (dollar neutrality) and dollar-duration neutrality conditions are imposed, immunizing the portfolio against parallel yield curve shifts. The strategy profits from changes in yield curve curvature.
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## Construction / Mechanics
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Let P_1, P_2, P_3 be the dollar amounts invested in the three bonds, and D_1, D_2, D_3 their modified durations.
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**Zero-cost** (dollar neutrality): the long barbell finances the short bullet position:
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```
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P_1 + P_3 = P_2 (404)
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```
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**Dollar-duration neutrality** (parallel shift immunity):
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```
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P_1·D_1 + P_3·D_3 = P_2·D_2 (405)
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```
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These two equations determine P_1 and P_3 given P_2.
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## Payoff / Return Profile
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- Profits when the yield curve becomes more curved (humped): the intermediate yield rises relative to the wings, or the wings fall relative to the body.
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- Immune to small parallel shifts in the yield curve (both level and dollar-duration matched).
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- Exposed to changes in the slope and curvature of the yield curve.
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## Key Parameters / Signals
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- T_1 (short wing), T_2 (body), T_3 (long wing): the three maturities; T_1 < T_2 < T_3
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- D_1, D_2, D_3: modified durations of the three bonds
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- P_2: the reference position size (determines P_1 and P_3 via the two constraints)
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## Variations
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- See also: fifty-fifty butterfly (5.7) and regression-weighted butterfly (5.8), which relax the zero-cost condition.
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## Notes
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- Dollar-duration neutrality (Eq. 405) protects against parallel shifts only; non-parallel changes in slope or curvature can still generate losses or gains.
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- The zero-cost constraint (Eq. 404) means no initial capital is required, making it attractive as an overlay strategy.
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- In practice, bid-ask spreads, financing costs, and liquidity differences across maturities affect profitability.
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