Duration and Convexity in Practice

A worked example with 7.2 duration and 65 convexity shows why the duration-only estimate overstates a yield-driven bond loss.

Modified duration estimates how much a bond's price moves for a small change in yield, and convexity corrects that estimate for the fact that the price-yield relationship curves rather than moves in a straight line. Used together on an actual position, the two numbers show that a duration-only estimate overstates the loss on a yield increase and understates the gain on a yield decrease.

Modified duration converts a change in yield into an approximate percentage change in price through a simple multiplication: the price change is approximately negative modified duration times the change in yield. A bond with a modified duration of 7.2 facing a yield increase of 50 basis points, or 0.50 percent, would see an estimated price decline of 7.2 times 0.50 percent, which comes to a loss of 3.60 percent.

That estimate is a straight line drawn tangent to the actual price-yield curve at the bond's current yield. It is accurate for very small moves, but the true relationship between a bond's price and its yield is not linear, it curves, and the further the yield moves from its starting point the more the straight line estimate departs from the true price.

Convexity measures the curvature of that price-yield relationship, and the correction it contributes to the estimate is one half times convexity times the change in yield, squared. For a bond with a convexity of 65 and the same 50 basis point move, the correction is one half times 65 times 0.50 percent squared, which works out to approximately 0.08 percent.

Adding the two terms together, a 3.60 percent loss from duration alone is offset by a 0.08 percent gain from convexity, leaving a corrected estimate of roughly 3.52 percent. The duration-only number overstated the loss, because it ignored the fact that the bond's price falls by slightly less than a straight line would predict as yields rise, and would rise by slightly more than a straight line predicts as yields fall.