convexity
/ kon-VEK-si-tee /
Duration draws a straight line to estimate how a bond's price moves when rates change, but the true relationship between price and yield is not a straight line — it is a gentle curve that bows. Convexity is the measure of how much that curve bends. It is the correction that captures what the straight-line duration estimate misses, especially for larger movements in interest rates.
Mathematically, if duration is the slope of the price-yield curve, convexity is its curvature — the rate at which the slope itself changes. The price-yield curve for a normal bond is convex, meaning it bows upward: as yields fall the price rises faster than duration predicts, and as yields rise the price falls more slowly than duration predicts. Either way, convexity works in the holder's favour. Combining the two gives a better estimate: the percentage price change is approximately minus modified duration times the rate change, plus one-half times convexity times the rate change squared.
Convexity matters because two bonds with the same duration can behave differently when rates move a lot; the one with greater convexity gains more and loses less, so positive convexity is a desirable feature. In asset-liability management, matching duration alone immunizes against small rate changes, but matching convexity as well (the second Redington condition) protects against larger ones. The caveat to keep honest: some instruments, notably callable bonds and mortgage-backed securities, can have negative convexity over certain yield ranges, which flips the usual comfort into a hazard.
Two bonds both have modified duration 7. If rates jump 2 points, the one with higher convexity falls less — say 12 percent instead of 14 — because the curvature term, plus one-half convexity times 2 squared, cushions the drop.
Convexity is the second-order correction to duration; it cushions losses and amplifies gains for the holder.
Positive convexity helps the holder, but callable bonds and mortgage-backed securities can show negative convexity, where rising prices stall as the borrower's option to prepay kicks in — do not assume convexity is always benign.