Duration assumes a bond’s price moves in a straight line with yields. It’s a great approximation for small moves — but the real relationship is curved, and that curve has a name: convexity.
The curve that duration misses
Plot a bond’s price against its yield and you don’t get a straight line — you get a curve that bends. Duration is just the slope of that curve at today’s yield. For a small rate move, the straight-line estimate is fine. For a large move, the curvature matters, and duration alone will mislead you.
Convexity measures how curved that line is — the correction to duration for bigger moves.
Why curvature helps you
Here’s the nice part. For a normal bond, the curve is shaped so that price rises (when yields fall) are slightly bigger than the price falls (when yields rise by the same amount). This is positive convexity, and it’s a small, free bonus for the bondholder: you gain a bit more on the way down in yields than you lose on the way up.
The longer and lower-coupon the bond, the more convexity it has — so on top of more duration, it also gets more of this cushioning curvature.
When convexity turns negative
Some bonds have negative convexity — most importantly callable bonds (covered later), where the issuer can repay early. That caps the upside when yields fall, bending the curve the wrong way and working against the holder. Negative convexity is a real risk to watch.
The takeaway
Convexity is the curvature in the price/yield relationship that duration (a straight-line estimate) ignores. For ordinary bonds it’s positive — a small bonus where gains slightly outpace equal losses — but some bonds, like callables, have negative convexity that hurts. (This is education, not investment advice.)