---
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.