How Duration and Convexity Jointly Affect Bond Price Changes

Summary

The document resolves an apparent conflict between duration and convexity for bonds. Lower-coupon bonds tend to have both higher duration and higher convexity. Duration describes the first-order sensitivity of price to yield changes, while convexity captures the second-order curvature in that relationship; these effects act in opposite directions for a given yield move.

A price-change approximation combines a negative duration term with a positive convexity term proportional to the squared yield change. For small yield shifts, the squared term is much smaller, so duration generally dominates the price response. Higher convexity therefore moderates losses and adds to gains relative to a linear duration estimate, without negating the larger first-order sensitivity associated with higher duration. The explanation is a local approximation and does not quantify effects for large yield moves or compare specific bonds.

Key ideas

• Duration represents the first-order effect of yield changes on bond prices.
• Convexity represents a second-order adjustment based on the squared yield change.
• For small yield shifts, duration generally has the larger effect because it is first order.
• Higher convexity moderates price declines and increases price gains relative to a linear estimate.
• Higher duration and higher convexity can coexist without contradiction.

Tags

# Duration vs. Convexity Contradiction


# Duration vs. Convexity Contradiction












A lower coupon bond exhibits higher duration, which means higher price volatility with changing YTM.

A lower coupon bond also exhibits higher convexity. However, with higher convexity, bond prices rise more and fall less.

So, a low coupon bond has higher duration and higher convexity, yet has higher and lower price volatility at the same time?

Is someone able to help with this contradiction?

## Answer by Richi Wa (score 5, accepted)

https://quant.stackexchange.com/a/21688

The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$ \frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2, $$ where $D$ is the duration and $C$ is convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.

## Answer by gdlamp (score 0)

https://quant.stackexchange.com/a/37351

They are not really contradiction but rather forces that act in counter direction. Loosely speaking, duration would dominate convexity because duration is the first derivates and posts first-order effect while convexity posts second-order effect on the price.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.