What is bond convexity?

Q: Duration vs Convexity: visualised

Price–Yield Relationship Price Yield Current price/yield Convexity gain ↑ Less loss than predicted — Duration approximation — Actual bond price

Q: Positive vs negative convexity

Most standard government and corporate bonds have positive convexity — you always benefit from the curvature in both directions. However, some bonds have negative convexity :

In brief: Bond convexity measures the curvature in the relationship between bond prices and interest rates. Duration tells you the first approximation of price change; convexity corrects for the fact that this relationship is curved, not straight. Positive convexity is desirable — it means bonds gain more when rates fall than they lose when rates rise by the same amount.

Duration is an excellent tool for measuring interest rate risk, but it has a flaw: it assumes the price-yield relationship is linear. In reality, it's curved — the price of a bond falls less than duration predicts when rates rise, and rises more than duration predicts when rates fall. Convexity captures this curvature and is one of the most important advanced concepts in bond portfolio management.

Duration vs Convexity: visualised

The formula (for context)

The complete price change incorporating both duration and convexity is:

  ΔP/P ≈ −Modified Duration × Δy + ½ × Convexity × (Δy)²

The second term — the convexity correction — becomes significant for large yield moves. For a 1% rate move, duration dominates. For a 3% or 5% move, convexity matters substantially.

Positive vs negative convexity

Most standard government and corporate bonds have positive convexity — you always benefit from the curvature in both directions. However, some bonds have negative convexity:

Callable bonds: The issuer can redeem them early when rates fall — just when you'd want to hold them most. This caps the price upside, creating negative convexity.
Mortgage-backed securities: Homeowners prepay mortgages when rates fall (they refinance), which effectively calls the bond. MBS have notoriously complex convexity profiles.

Negative convexity is generally undesirable — you lose more when rates rise than you gain when rates fall by the same amount. Investors demand a yield premium to hold negatively convex bonds.

~100–150Typical convexity of a 30-year government bond — much higher than a 5-year bond's convexity of ~20–25

Why convexity is valuable

Positive convexity is free optionality — your bond performs better than duration predicts in both rate scenarios. In volatile rate environments, high-convexity bonds outperform low-convexity bonds of the same duration. Long-dated zero-coupon bonds have the highest convexity of standard bonds — making them powerful instruments when rate volatility is high. Hedge funds and pension funds actively manage convexity alongside duration in their fixed income portfolios.

What this means for you

Convexity explains why long-dated government bonds often outperform their duration would suggest in sharp rate rallies. If you hold bond funds, the stated duration tells you the first-order risk; convexity tells you how the fund benefits asymmetrically from large rate moves. In periods of extreme rate volatility — like 2022-2023 — the convexity profile of a bond portfolio can significantly affect actual versus expected performance.