---
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."
tags: [fixed-income, butterfly, duration-neutral, yield-curve, curvature]
---

# Dollar-Duration-Neutral Butterfly

**Section**: 5.6 | **Asset Class**: Fixed Income | **Type**: Yield Curve / Curvature

## Overview
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.

## Construction / Mechanics
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.

**Zero-cost** (dollar neutrality): the long barbell finances the short bullet position:
```
P_1 + P_3 = P_2                                                      (404)
```

**Dollar-duration neutrality** (parallel shift immunity):
```
P_1·D_1 + P_3·D_3 = P_2·D_2                                         (405)
```

These two equations determine P_1 and P_3 given P_2.

## Payoff / Return Profile
- 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.
- Immune to small parallel shifts in the yield curve (both level and dollar-duration matched).
- Exposed to changes in the slope and curvature of the yield curve.

## Key Parameters / Signals
- T_1 (short wing), T_2 (body), T_3 (long wing): the three maturities; T_1 < T_2 < T_3
- D_1, D_2, D_3: modified durations of the three bonds
- P_2: the reference position size (determines P_1 and P_3 via the two constraints)

## Variations
- See also: fifty-fifty butterfly (5.7) and regression-weighted butterfly (5.8), which relax the zero-cost condition.

## Notes
- Dollar-duration neutrality (Eq. 405) protects against parallel shifts only; non-parallel changes in slope or curvature can still generate losses or gains.
- The zero-cost constraint (Eq. 404) means no initial capital is required, making it attractive as an overlay strategy.
- In practice, bid-ask spreads, financing costs, and liquidity differences across maturities affect profitability.