Fixed Income & Portfolio Analytics
Barbell vs Bullet Bond Convexity Advantage Formula
Match modified duration via w_short = (D_long - D_bullet)/(D_long - D_short) and calculate the Barbell Convexity Advantage and parallel-shift outperformance 0.5 * Delta_C * (Delta_y)^2.
Why a duration-matched Barbell ALWAYS has higher convexity than a Bullet bond!
Fixed-income portfolio managers choosing between a single intermediate Bullet bond (modified duration D_b, convexity C_b) and a Barbell combining a short note (D_s, C_s) with a long bond (D_l, C_l, D_s < D_b < D_l) lock in identical first-order yield sensitivity by setting Short Weight w_s = (D_l - D_b) / (D_l - D_s) and Long Weight 1 - w_s! Because bond convexity scales roughly with duration SQUARED (T*(T + 1)), Jensen’s inequality guarantees Barbell Convexity C_barbell = w_s * C_s + (1 - w_s) * C_l > C_b — earning an instant second-order return bonus 0.5 * (C_barbell - C_b) * (Delta_y)^2 on ANY large parallel rate rally OR selloff (traded off against curve-flattening twist exposure)!
Check by hand with 2-yr Short Note (D_s = 2 in B2, C_s = 6 in B3), 30-yr Long Bond (D_l = 18 in B4, C_l = 406 in B5), 10-yr Bullet (D_b = 8 in B6, C_b = 75 in B7), and Parallel Yield Shock Delta_y = 0.02 (±200 bps in B8): Short Weight is w_s = (18 - 8) / (18 - 2) = 0.6250 (62.50% Short + 37.50% Long -> Duration = 0.625*2 + 0.375*18 = 8.00!), Barbell Convexity jumps to 0.625*6 + 0.375*406 = 156.00 (+81.00 above Bullet 75.00!), and ±200 bps parallel outperformance equals 0.5 * 81 * (0.02^2) = +0.0162 (+1.62% = +162 bps)!
| Barbell vs Bullet Metric (D_target = 8.00) | Formula | Verified Output |
|---|---|---|
| Duration-Matching Short Weight (w_s) | (D_l - D_b) / (D_l - D_s) | 0.6250 (62.50% 2Y / 37.50% 30Y) |
| Barbell Portfolio Convexity | w_s * C_s + (1 - w_s) * C_l | 156.0000 (vs Bullet 75.0000) |
| Net Convexity Pick-Up (Delta_C) | C_barbell - C_bullet | +81.0000 |
| Parallel ±200 bps Return Bonus | 0.5 * Delta_C * (Delta_y)^2 | +0.0162 (+162 bps Outperformance) |
Duration-Matched Barbell Portfolio Convexity Formula
Validate strict bracketing D_short <= D_bullet <= D_long with D_short < D_long and positive convexities.
=LET(d_s, B2, c_s, B3, d_l, B4, c_l, B5, d_b, B6, c_b, B7, IF(OR(d_s <= 0, d_l <= d_s, d_b < d_s, d_b > d_l, c_s <= 0, c_l <= 0, c_b <= 0), "Require D_short <= D_bullet <= D_long and Positive Inputs", LET(w_s, (d_l - d_b) / (d_l - d_s), ROUND(w_s * c_s + (1 - w_s) * c_l, 4))))Second-Order Parallel Yield-Shock Outperformance Formula
Test both Delta_y = +0.02 (+200 bps selloff) and Delta_y = -0.02 (-200 bps rally): because (Delta_y)^2 is symmetric, both shocks produce +0.0162!
=IF(ISTEXT(E3), E3, ROUND(0.5 * (E3 - B7) * (B8^2), 4))Try the example
Use the original values or change them to test your understanding.