Tim Olson
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54 lines
3.0 KiB
Markdown
54 lines
3.0 KiB
Markdown
---
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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."
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tags: [fixed-income, butterfly, duration-neutral, yield-curve, curvature, regression]
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---
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# Regression-Weighted Butterfly
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**Section**: 5.8 | **Asset Class**: Fixed Income | **Type**: Yield Curve / Curvature
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## Overview
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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.
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## Construction / Mechanics
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Using positions P_1, P_2, P_3 with modified durations D_1, D_2, D_3 (T_1 < T_2 < T_3):
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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 (407)
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```
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**Regression-weighted curve-neutrality**:
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```
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P_1·D_1 = β · P_3·D_3 (408)
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```
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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.
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## Payoff / Return Profile
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- Immune to both parallel shifts (407) and, approximately, to yield curve slope changes in proportion β.
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- Profits from yield curve curvature moves: gains when the body yields rise relative to the wings.
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- More robust curve-neutrality than the fifty-fifty butterfly in practice due to the empirically calibrated β.
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## Key Parameters / Signals
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- β: regression coefficient (typically β > 1, calibrated from historical spread data)
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- P_1, P_3: determined by solving (407) and (408) given P_2
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- T_1, T_2, T_3: the three maturity points on the yield curve
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## Variations
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### 5.8.1 Maturity-Weighted Butterfly
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Instead of estimating β from historical regressions, it is set analytically from the three bond maturities:
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```
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β = (T_2 - T_1) / (T_3 - T_2) (409)
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```
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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.
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## Notes
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- 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.
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- The regression β should be re-estimated periodically as the volatility relationship between short and long rates can change over time.
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- 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.
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- All butterfly strategies share the exposure to transaction costs, financing costs, and bid-ask spreads that can erode theoretical curve-neutrality profits.
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