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gateway/knowledge/trading/strategies/fixed-income/regression-weighted-butterfly.md

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+---
+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.