Convexity
What is Convexity?
Convexity measures the curvature in the relationship between a bond's price and yield and improves duration-based estimates for larger yield changes.
Why convexity matters
Duration gives a linear approximation, but most option-free bond price curves are bowed. Positive convexity means the price gain from a yield decline is larger than the price loss from an equal yield rise, relative to the duration estimate. Convexity becomes more important for longer durations and larger rate moves.
How it is used
A second-order price estimate adds one half multiplied by convexity and the squared yield change to the modified-duration estimate. Conventions and scaling differ among systems, so raw convexity numbers require formula documentation. Effective convexity is estimated by repricing instruments whose cash flows can change when rates move.
Example
Two bonds have the same yield and modified duration, but one has higher positive convexity. For a sufficiently large rate move in either direction, the higher-convexity bond is expected to retain more value, all else equal. Investors generally pay for this favorable shape through a higher price or lower yield.
Positive and negative convexity
Option-free government bonds usually have positive convexity. Callable bonds and mortgage-backed securities can develop negative convexity when falling rates increase the likelihood of early repayment, limiting price gains and shortening cash flows. Convexity therefore reflects optionality and can change materially as the instrument moves between rate environments.
Limitations
Convexity remains a local model measure and does not capture every large or path-dependent move. Scaling conventions can make vendor values incomparable. The estimate generally isolates yield effects while holding credit, liquidity, volatility, and currency constant. For complex instruments, model assumptions about exercise and prepayment behavior can dominate the reported result.
Practical checklist
Confirm formula, units, curve, bump size, and whether the measure is analytical or effective. Use duration and convexity together, then compare their estimate with full revaluation under several curve scenarios. Identify instruments with negative convexity and test volatility changes. Evaluate whether additional convexity justifies its price, carry, liquidity, and model risk. Track how convexity changes across relevant yield and volatility ranges throughout the holding period and document the results.