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Bonds & fixed income

Convexity

Convexity measures how much a bond's price-yield relationship curves, correcting duration's straight-line estimate of price moves.

Published by Quintessentia Network Inc. · Updated 17 July 2026 · Sources & disclosures
Part of the Bonds, Rates & the Economy course · Lesson 5 of 12
Formula
Approx. % price change ≈ (−Modified Duration × Δy) + (0.5 × Convexity × Δy²), with Δy the yield change in decimal form

What it is

Bond prices and yields move in opposite directions: when yields rise, prices fall. Duration estimates that move as a straight line — roughly the percentage a bond's price changes per one-percentage-point move in yield. But the real relationship is a curve, not a line. Convexity measures how much that curve bends, and it is the correction term duration alone misses. For most ordinary bonds the curve bends upward (positive convexity), meaning duration overstates the loss when yields rise and understates the gain when yields fall.

Why it matters

Duration is a decent approximation for small yield moves and increasingly wrong for large ones. Convexity tells a beginner how wrong, and in which direction. Two bonds can share the same duration yet behave differently in a sharp rate move because their convexity differs — so duration alone does not fully describe a bond's interest-rate sensitivity. Convexity also flags an important asymmetry: bonds the issuer or borrower can repay early, such as callable bonds and mortgage-backed securities, can have negative convexity, where the price rises less on a rally than it falls on a sell-off. Knowing the figure exists explains why a bond's actual price move can differ from a duration-based estimate.

How it's calculated

Convexity is the second derivative of a bond's price with respect to yield, scaled by price. The closed-form version uses the cash flow schedule: each coupon and principal payment is discounted, weighted by the time until it arrives multiplied by that time plus one, summed, then divided by price and by (1 + yield) squared. Analysts more often approximate it numerically: reprice the bond at a slightly higher and a slightly lower yield, then measure how far those prices sit above duration's straight-line estimate. Bonds with a call or prepayment feature need effective (option-adjusted) convexity, found by repricing under shifted yield curves.

FAQ

Is more convexity always better for a bondholder?
Positive convexity is generally described as favourable to a bondholder: prices rise a little more than duration predicts when yields fall, and drop a little less when yields rise. But it is not free — all else equal, meaning similar duration and credit quality, a bond with more convexity tends to be priced richer, which shows up as a lower yield. Convexity describes price behaviour; it is not a judgement about whether a bond is worth owning.
What is negative convexity?
Some bonds have an embedded option that lets the issuer or borrower repay early — callable bonds and mortgage-backed securities are the common examples. When yields fall, early repayment becomes more likely, which caps how much the price can rise. The price-yield curve bends the other way over part of its range, so the bond gains less on a rally than it loses on a sell-off. That asymmetry is what negative convexity means.
Does convexity have to be calculated by hand?
Rarely. Convexity is a refinement of duration, and it matters most when yield moves are large or when comparing bonds with similar durations. Analytics tools and some bond fund fact sheets report it, though it is disclosed far less consistently than duration. For a beginner, understanding what it describes — the curve in the price-yield relationship — is more useful than computing it by hand.
Related terms
Duration (Bonds)Yield to MaturityYield Curve
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