Convexity Adjustment

A bond's price and yield generally move in opposite directions. The relationship curves rather than forming one straight line, so a duration-only prediction becomes less accurate as the yield move grows.

Modified duration measures local sensitivity: a duration of five suggests about a 5% price decline for a one-percentage-point yield increase, before considering curvature, and that is an approximation, not a promise about the traded price. Convexity measures how that sensitivity itself changes as yield changes.

An adjustment based on half of convexity times the squared yield change adds a second-order term to the duration estimate. The Massachusetts Institute of Technology's fixed-income lecture presents duration and convexity together for a second-order bond-price approximation, and notes that simple duration and convexity analysis is most natural for parallel shifts in a flat term structure.

For positive convexity, the squared yield change makes the correction positive for a rate move in either direction, while the first-order duration term changes sign with the yield move. Consider a bond with modified duration 5 and convexity 40, using yields as decimals.

A one-percentage-point rise gives a duration effect of negative 5% and a convexity term of positive 0.2%, suggesting about negative 4.8% before other effects. A one-percentage-point fall gives positive 5% from duration and positive 0.2% from convexity, suggesting a 5.2% gain.

The asymmetry is a central reason fixed-income managers track convexity. Callable bonds may show negative convexity in some rate ranges because falling yields make a call or refinancing more likely, so the positive-convexity intuition from an option-free bond fails and the model must include the embedded option.

A large shock can exceed the range where a two-term approximation is reliable. The estimate also assumes a particular yield movement, so changes in credit spread, liquidity, prepayment behaviour and the curve's shape require separate analysis.

The units must also be consistent: moving from 5% to 6% means a decimal yield change of 0.01, not 1.00, and squaring the wrong input creates an enormous error. There is another use of the phrase convexity adjustment in interest-rate futures and forward comparisons, reflecting differences in payoff timing and rate sensitivity rather than the same numerical correction to one bond's duration estimate.

A manager should name the instrument and state which adjustment is being calculated before carrying a result into a hedge or pricing memo. For a risk report, show the duration-only estimate, convexity term and combined estimate separately, then compare with repricing from the full cash-flow model, since the approximation explains sensitivity and is not a substitute for a market quote.