Tim Olson
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51 lines
2.8 KiB
Markdown
51 lines
2.8 KiB
Markdown
---
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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."
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tags: [fixed-income, duration, convexity, barbell, yield-curve]
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---
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# Barbells
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**Section**: 5.3 | **Asset Class**: Fixed Income | **Type**: Duration / Convexity
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## Overview
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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.
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## Construction / Mechanics
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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):
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**Duration** (equals a bullet at T_*):
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```
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D = (w̃_1·T_1 + w̃_2·T_2) / (w̃_1 + w̃_2) (390)
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T_* = D_* = D (391)
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```
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**Convexity** (exceeds the equivalent bullet):
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```
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C = (w̃_1·T_1² + w̃_2·T_2²) / (w̃_1 + w̃_2) (392)
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C_* = T_*² (393)
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```
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The convexity advantage:
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```
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C - C_* = (w̃_1·w̃_2 / (w̃_1 + w̃_2)²) · (T_2 - T_1)² > 0 (394)
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```
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## Payoff / Return Profile
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- Higher convexity than an equivalent bullet means the barbell outperforms when yields move significantly in either direction (parallel shifts).
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- The long-maturity bonds benefit from high yields; the short-maturity bonds provide protection if rates rise (proceeds reinvested at higher rates).
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- Flattening of the yield curve (short-term rates rise relative to long-term) has a positive impact; steepening has a negative impact.
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## Key Parameters / Signals
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- T_1 (short maturity), T_2 (long maturity): the two maturities defining the barbell
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- w_1, w_2: dollar allocations to short and long maturities
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- Target duration D: matched to the equivalent bullet at T_*
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- Convexity advantage C - C_*: larger the spread T_2 - T_1, the greater the convexity benefit
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## Variations
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- Combine with duration matching to an intermediate bullet for controlled rate exposure.
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## Notes
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- Higher convexity comes at the expense of lower overall yield (yield curve typically slopes upward, so the mid-point bullet earns more carry).
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- Duration scales approximately linearly with maturity; convexity scales quadratically — this is why the barbell's convexity exceeds the equivalent bullet.
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- The barbell is more complex to manage than a bullet due to two distinct maturity exposures.
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