Convexity

Duration assumes the relationship between price and yield is a straight line. It is not — it is a curve. Convexity measures that curvature and corrects duration's estimate, especially for large rate moves.

Why duration alone is incomplete

The true price-yield relationship bows toward the origin: it is convex. Because of this curve:

• When yields fall, prices rise more than duration predicts (a pleasant surprise).
• When yields rise, prices fall less than duration predicts (a cushion).

Duration draws the tangent line to the curve. For small yield changes the line and the curve nearly coincide, so duration alone is fine. For large changes, the line drifts away from the curve, and the gap is what convexity captures.

The improved estimate

Adding a convexity term refines the price-change formula:

Percent change is approximately (-Modified duration * dy) + (0.5 * Convexity * dy^2)

The convexity term is always positive (because of the dy^2), so it adds to price when yields fall and softens the loss when yields rise — convexity always works in the bondholder's favour.

Worked example

Suppose our bond has a modified duration of 2.70 and a convexity of 9.5, and yields jump by a large 2% (dy = 0.02):

Duration term:   -2.70 * 0.02            = -0.0540  = -5.40%
Convexity term:   0.5 * 9.5 * (0.02)^2   = +0.0019  = +0.19%
Total estimate:  -5.40% + 0.19%          = -5.21%

Duration alone predicted a 5.40% loss; adding convexity revises it to 5.21%. The 0.19% difference is the curvature benefit — small here, but it grows with the size of the move and with the bond's convexity.

The practical takeaway

Between two bonds of equal duration, the one with higher convexity is more desirable: it gains more when rates fall and loses less when they rise. Together, duration and convexity give a complete first- and second-order picture of how a bond responds to the interest-rate moves that drive every fixed-income market.