Convexity
Quantitative Finance glossary · Quantitative Finance
Definition
Second-order sensitivity of bond price to yield: C = 1/P(partial^2 P)/(partial y^2), with the second-order price change Δ P / P ≈ -D,Δ y + 1/2C,(Δ y)^2. Always positive for option-free bonds (the price-yield curve is convex), and therefore always favourable to the long: gain on a rate fall exceeds loss on an equal rate rise. Callable bonds exhibit negative convexity near the call strike (price compression as yields fall, since the call becomes more likely to be exercised) — the structural reason MBS pass-throughs underperform in rate rallies.
Convexity in plain English
A bond's price and its interest rate (yield) move in opposite directions, but not in a perfectly straight line — the relationship curves, and convexity measures how much it curves. Why care? Because that curve is a gift to the bond's owner: when rates drop you gain a bit more than you'd lose if rates rose by the same amount, so the bend is always in your favor for ordinary bonds. It's like a seesaw that's slightly tilted to reward you on the upside — a small built-in bonus. The exception is bonds the issuer can pay off early ('callable' bonds, including many mortgage-backed securities): near the point where early payoff gets likely, the curve bends the wrong way ('negative convexity'), which is why those bonds disappoint when rates fall fast.
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