12. The degree of convexity varies across bonds, mainly depending on their option characteristics and durations. Embedded short options decrease convexity. For bonds without embedded options, convexity increases roughly as a square of duration (see Exhibit 36–4). There also are convexity differences between bonds that have the same duration. A barbell position (with very dispersed cash-flows) exhibits more convexity than a duration-matched bullet bond. The reason is that a yield rise reduces the relative weight of the barbell’s longer cash-flows (because the present values decline more than those of the shorter cash-flows) and thereby shortens the barbell’s duration. The inverse relation between duration and yield level increases a barbell’s convexity, limiting its losses when yields rise and enhancing its gains when yields decline. Of all bonds with the same duration, a zero has the smallest convexity because its cash-flows are not dispersed, so its Macaulay duration does not vary with the yield level.