What Is Bond Convexity?

Convexity measures the curvature of a bond's price-yield relationship. Duration estimates how much a bond's price changes when yields move, but the true relationship is curved — price gains accelerate as yields fall and losses decelerate as yields rise. Convexity quantifies that curvature, acting as the second-order correction that makes price estimates accurate for larger rate moves.

How This Calculator Works

Enter the bond's face value, coupon rate, yield to maturity, and years to maturity. The calculator prices the bond by discounting each annual cash flow, then computes Macaulay duration, modified duration, and convexity. Convexity uses the standard formula: C = [1/(P(1+y)²)] × Σ t(t+1)·CFₜ/(1+y)ᵗ.

For the defaults — a 6% coupon bond with $1,000 face, 5% yield, and 10 years — the price is about $1,077.2 and convexity about 60.4. A zero-coupon bond with the same maturity would show the maximum convexity of n(n+1)/(1+y)² ≈ 99.8, illustrating how coupon payments reduce convexity.

Duration vs Convexity

Modified duration estimates the percent price change per 1% yield move: a modified duration of about 7.7 means a 1% rise in yield lowers price roughly 7.7%. That estimate is a straight line tangent to the price curve. Convexity bends the line to match reality — for a 2% yield drop, duration alone underestimates the price gain, and convexity supplies the missing amount. Positive convexity is always beneficial: gains in falling-rate scenarios exceed losses in rising-rate scenarios of equal size.

Using Convexity in Practice

Portfolio managers combine duration and convexity to immunize against rate changes and to compare bonds fairly. High-convexity bonds protect better against large rate moves, which is why investors pay a premium for them. The cash-flow chart in this calculator shows the coupon stream that drives both measures, making it easy to see how the timing of payments shapes the bond's risk profile.