Bond Convexity Calculator

What Is Bond Duration? Duration measures how sensitive a bond’s price is to changes in interest rates. Macaulay Duration is the weighted average time (in years) to receive a bond’s cash flows. Modified Duration = Macaulay Duration / (1 + YTM/n), where n = payments per year. A bond with modified duration of 5 will lose approximately 5% of its value if interest rates rise 1%.

The Limitation of Duration Duration gives a linear approximation of price change — it works well for small rate moves. For larger rate moves, duration alone underestimates price gains and overestimates price losses. This is because the true price-yield relationship is curved (convex), not straight. Convexity captures this curvature and makes the estimate more accurate.

What Is Convexity? Convexity is the second derivative of a bond’s price with respect to yield. Higher convexity = greater price gain when rates fall, smaller price loss when rates rise. All else equal, bonds with higher convexity are more valuable. Zero-coupon bonds have the highest duration but moderate convexity. Callable bonds can have negative convexity (price rise is capped when yields fall).

The Full Price Change Formula ΔP/P ≈ −D* × Δy + ½ × C × (Δy)² Where D* = modified duration, C = convexity, Δy = change in yield. For a 1% rate rise on a bond with duration 7 and convexity 60: ΔP/P ≈ −7 × 0.01 + 0.5 × 60 × (0.01)² = −0.07 + 0.003 = −6.7% Without convexity adjustment: −7.0%. The convexity correction adds 0.3%.

Factors That Affect Convexity Longer maturity → higher convexity (more time for cash flow timing to vary). Lower coupon rate → higher convexity (cash flows concentrated at maturity). Lower yield → higher convexity (small denominator magnifies the effect). Embedded options (call/put) → modify convexity: callable bonds have lower/negative convexity.

Practical Use in Portfolio Management Portfolio managers use convexity to compare bonds when duration is similar. When immunizing a bond portfolio against rate changes, both duration matching and convexity matching improve the hedge. Convexity is especially important when managing long-duration bonds or in volatile rate environments. The convexity of a 30-year zero-coupon Treasury bond is approximately 900 — enormously sensitive to rate moves.