What Is Convexity?
Duration approximates a bond's price change for a small move in yields, but the true price-yield relationship is a curve rather than a straight line. Convexity is the second-order term that corrects for this curvature. For an ordinary bond the curve bows in the investor's favor: a one-point drop in yields lifts the price by more than a one-point rise lowers it.
Convexity increases with maturity and is highest for long-dated bonds with low coupons; a 30-year zero-coupon bond is far more convex than a 5-year note. Bonds with embedded options can display negative convexity, meaning price gains stall as yields fall, which is characteristic of callable bonds and mortgage-backed securities.
How to Use It in Bond Math
The standard second-order approximation is: percentage price change ≈ (−Duration × Δy) + (0.5 × Convexity × Δy²). Suppose a bond has a duration of 7 and a convexity of 60. For a 1% rise in yields, duration alone predicts a 7% loss, while the convexity term adds back 0.5 × 60 × 0.01² = 0.3%, for a net loss of about 6.7%. For a 1% fall, the same term lifts the gain to roughly 7.3%.
Because the convexity adjustment scales with the square of the yield move, it barely matters for tiny fluctuations but becomes significant in large rate shocks. Traders often quote the effect per 100 basis points and compare bonds by how much extra performance the curvature adds in volatile markets, effectively treating convexity as embedded optionality on rates.
Why It Matters
Investors pay for convexity: between two bonds with identical duration and yield, the more convex one outperforms whenever rates move meaningfully in either direction, so it commands a slightly higher price and lower yield in equilibrium. Barbell portfolios, which pair short and long maturities, carry more convexity than a bullet portfolio of intermediate bonds with the same overall duration.
Negative convexity drives real market events. When rates rise, mortgage-backed securities extend in duration as homeowners stop refinancing, forcing hedgers to sell Treasuries and amplifying the move, a dynamic known as convexity hedging. For fixed income interviews, sketching the price-yield curve and explaining why callables are negatively convex demonstrates command of bond math beyond duration.
