Convexity

Duration treats the link between a bond's price and its yield as a straight line, but the real relationship is curved. Convexity measures that curvature. For an ordinary bond with fixed payments, convexity is positive: as yields fall, the price rises by more than duration predicts, and as yields rise, the price falls by less. The effect is small for modest moves but grows with the size of the change in yield. It is larger for long-dated bonds and for bonds with low coupons, whose cash flows are spread further into the future.

Positive convexity is helpful to bondholders, and investors generally pay for it through slightly lower yields. Some bonds have negative convexity, however. A callable bond, which the issuer can repay early, and mortgage-backed securities, where homeowners refinance when rates fall, both see their price gains capped when yields drop, because the high-paying debt is likely to be repaid. For most private investors, convexity is a second-order detail: duration explains most of a bond's rate sensitivity, and convexity explains why that estimate is off for large moves.