A bullet bond has duration 7 and convexity 49. A barbell of 2 year and 15 year bonds with convexities 4 and 225 is built to the same duration of 7. What extra return does the barbell earn on a 100 basis point parallel shift, and what is given up for it?

A bullet bond has duration 7 and convexity 49. A barbell of 2 year and 15 year bonds with convexities 4 and 225 is built to the same duration of 7. What extra return does the barbell earn on a 100 basis point parallel shift, and what is given up for it?

Approach: Solve for the weights that match duration, take the weighted convexity, and price the difference through the second order term. Then ask which curve moves the trade is exposed to.

0.20%. Matching duration requires w*2 + (1 - w)*15 = 7, so w = 8/13 = 61.5% in the 2 year and 38.5% in the 15 year, and the portfolio convexity is 0.615*4 + 0.385*225 = 89.0 against 49 for the bullet. On a 100 basis point parallel shift in either direction the extra return is 0.5*(89 - 49)*0.01^2 = 0.0020, or 20 basis points. Two things are given up. The barbell yields less, and the annual yield give up on a curve of normal shape is usually close to the same 20 basis points, so a year without a large shift is a loss. The barbell is also short the belly, so a curve move where the 7 year point richens by 15 basis points against the wings costs roughly 7*0.0015 = 1.05%, five times the convexity gain. The advantage exists for parallel moves only.

Follow-up: At what size of parallel shift does the convexity gain cover a 25 basis point annual yield give up?

Key concepts: convexity, barbell, bullet, yield give up.