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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?

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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.

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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.

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