Duration is the mandate's first constraint and it says nothing about where on the curve the duration sits. Three structures at an identical modified duration of 5.00, built from the same US par bonds, make the difference visible.
How do the three structures compare at equal duration?
| Structure | Composition | Convexity | Dispersion | Yield |
|---|---|---|---|---|
| Bullet | 29.0238% five-year, 70.9762% six-year | 29.375705 | 2.015735 | 4.376781% |
| Ladder | 19.7202% each at 2, 4, 6, 8 and 10 years, 1.3989% overnight cash | 35.482490 | 8.207526 | 4.361453% |
| Barbell | 48.6661% two-year, 51.3339% ten-year | 41.044449 | 14.330864 | 4.342603% |
The ordering is the whole lesson and it never reverses: at equal duration the bullet has the least convexity and the most yield, the barbell has the most convexity and the least yield, and the ladder sits between them on both. The barbell buys 11.668745 points of convexity for 3.42 basis points of yield. Whether that is a good trade depends entirely on how large the coming move is and what shape it takes.
Convexity alone is worth 0.5 x 11.668745 x (delta y)^2, so at a 25 basis point parallel move the barbell's convexity edge is 0.003646% and at 200 basis points it is 0.233375%. Against a yield give-up of 0.034177% a year, the barbell needs a parallel move of 76.5373 basis points before convexity pays for the carry it costs. That is the honest version of "barbells are good in volatile markets".
Advertisement
How do they behave when the curve changes shape?
Parallel moves are also not the interesting case, because at equal duration the three structures behave almost identically under one. A 25 basis point parallel selloff costs the bullet 1.240870%, the ladder 1.238990% and the barbell 1.237279%, a spread of under four tenths of a basis point. The structures separate when the curve changes shape, and the reason is visible in where each one's duration sits.
| Structure | 2y | 4y | 5y | 6y | 8y | 10y |
|---|---|---|---|---|---|---|
| Bullet | 1.292697 | 3.707303 | ||||
| Ladder | 0.375219 | 0.718579 | 1.030047 | 1.311096 | 1.565060 | |
| Barbell | 0.925974 | 4.074026 |
Each figure is a holding's contribution to the portfolio's 5.00 years of duration, and for par bonds struck at these maturities it is a serviceable stand-in for the key-rate profile the duration work builds properly. Four fifths of the barbell's duration sits at ten years. Three quarters of the bullet's sits at six. Under a steepening in which the two-year rallies 40 basis points and the ten-year rallies 5, the bullet gains 1.189760%, the ladder 0.869943% and the barbell 0.576381%, because the barbell's duration is parked where the rally is smallest. Under a flattening in which the two-year sells off 30 basis points and the ten-year 5, the ranking inverts and the barbell loses least, at -0.480001% against the bullet's -0.910517%.
A structure choice is therefore a curve-shape view wearing the clothes of a risk-management decision, and pretending otherwise is how managers end up with a slope bet they never authorised.