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gateway/knowledge/trading/strategies/fixed-income/barbells.md

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+---
+description: "A barbell portfolio holds bonds at two extreme maturities (short and long) to achieve a target duration while gaining higher convexity than an equivalent bullet, providing better protection against parallel yield curve shifts."
+tags: [fixed-income, duration, convexity, barbell, yield-curve]
+---
+
+# Barbells
+
+**Section**: 5.3 | **Asset Class**: Fixed Income | **Type**: Duration / Convexity
+
+## Overview
+A barbell concentrates holdings at two maturities: a short maturity T_1 and a long maturity T_2. It is a combination of two bullet strategies. For a given modified duration (matching a bullet at intermediate maturity T_*), the barbell achieves higher convexity, providing better protection against parallel yield shifts at the cost of lower overall yield.
+
+## Construction / Mechanics
+For a simple barbell of w_1 dollars in zero-coupon bonds with maturity T_1 and w_2 dollars with maturity T_2 (continuous compounding, constant yield Y), with price-adjusted weights w̃_1 = w_1·exp(-T_1·Y) and w̃_2 = w_2·exp(-T_2·Y):
+
+**Duration** (equals a bullet at T_*):
+```
+D = (w̃_1·T_1 + w̃_2·T_2) / (w̃_1 + w̃_2)                          (390)
+T_* = D_* = D                                                        (391)
+```
+
+**Convexity** (exceeds the equivalent bullet):
+```
+C = (w̃_1·T_1² + w̃_2·T_2²) / (w̃_1 + w̃_2)                        (392)
+C_* = T_*²                                                           (393)
+```
+
+The convexity advantage:
+```
+C - C_* = (w̃_1·w̃_2 / (w̃_1 + w̃_2)²) · (T_2 - T_1)² > 0          (394)
+```
+
+## Payoff / Return Profile
+- Higher convexity than an equivalent bullet means the barbell outperforms when yields move significantly in either direction (parallel shifts).
+- The long-maturity bonds benefit from high yields; the short-maturity bonds provide protection if rates rise (proceeds reinvested at higher rates).
+- Flattening of the yield curve (short-term rates rise relative to long-term) has a positive impact; steepening has a negative impact.
+
+## Key Parameters / Signals
+- T_1 (short maturity), T_2 (long maturity): the two maturities defining the barbell
+- w_1, w_2: dollar allocations to short and long maturities
+- Target duration D: matched to the equivalent bullet at T_*
+- Convexity advantage C - C_*: larger the spread T_2 - T_1, the greater the convexity benefit
+
+## Variations
+- Combine with duration matching to an intermediate bullet for controlled rate exposure.
+
+## Notes
+- Higher convexity comes at the expense of lower overall yield (yield curve typically slopes upward, so the mid-point bullet earns more carry).
+- Duration scales approximately linearly with maturity; convexity scales quadratically — this is why the barbell's convexity exceeds the equivalent bullet.
+- The barbell is more complex to manage than a bullet due to two distinct maturity exposures.