---
description: "A regression-weighted butterfly uses an empirically estimated β to account for the higher volatility of short-term rates relative to long-term rates, improving yield-curve-neutrality beyond the fifty-fifty butterfly."
tags: [fixed-income, butterfly, duration-neutral, yield-curve, curvature, regression]
---

# Regression-Weighted Butterfly

**Section**: 5.8 | **Asset Class**: Fixed Income | **Type**: Yield Curve / Curvature

## Overview
Empirically, short-term interest rates are significantly more volatile than long-term rates. The regression-weighted butterfly accounts for this by weighting the short wing's dollar duration by a factor β > 1, estimated from historical data via regression. This produces better curve-neutrality than the fifty-fifty butterfly in practice.

## Construction / Mechanics
Using positions P_1, P_2, P_3 with modified durations D_1, D_2, D_3 (T_1 < T_2 < T_3):

**Dollar-duration neutrality** (parallel shift immunity):
```
P_1·D_1 + P_3·D_3 = P_2·D_2                                         (407)
```

**Regression-weighted curve-neutrality**:
```
P_1·D_1 = β · P_3·D_3                                               (408)
```

where β > 1 is the regression coefficient from regressing the spread change between the body (T_2) and the short wing (T_1) on the spread change between the body and the long wing (T_3), using historical data.

## Payoff / Return Profile
- Immune to both parallel shifts (407) and, approximately, to yield curve slope changes in proportion β.
- Profits from yield curve curvature moves: gains when the body yields rise relative to the wings.
- More robust curve-neutrality than the fifty-fifty butterfly in practice due to the empirically calibrated β.

## Key Parameters / Signals
- β: regression coefficient (typically β > 1, calibrated from historical spread data)
- P_1, P_3: determined by solving (407) and (408) given P_2
- T_1, T_2, T_3: the three maturity points on the yield curve

## Variations

### 5.8.1 Maturity-Weighted Butterfly
Instead of estimating β from historical regressions, it is set analytically from the three bond maturities:

```
β = (T_2 - T_1) / (T_3 - T_2)                                       (409)
```

This is proportional to the ratio of the short-wing maturity distance to the long-wing maturity distance from the body. It is a simpler, model-based alternative that does not require historical calibration.

## Notes
- β is empirically greater than 1 because short-term rates fluctuate more than long-term rates; the short wing therefore needs less dollar duration to hedge the same spread move.
- The regression β should be re-estimated periodically as the volatility relationship between short and long rates can change over time.
- The maturity-weighted variant (5.8.1) provides a model-based β that requires no estimation but may not capture the true empirical volatility asymmetry.
- All butterfly strategies share the exposure to transaction costs, financing costs, and bid-ask spreads that can erode theoretical curve-neutrality profits.