# The price sensitivity of bonds is measured using convexity, duration..

#### jlahuerta

##### Member
Please, could you anyone explain me why the convexity increase at an increasing rate as duration increase. I think that the duration should be decrease at an increasing rate.

Best regards

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @jlahuerta Dollar convexity (as the 2nd derivative) can be defined as the rate of change of dollar duration (with respect to yield). Dollar duration is the slope of the tangent line; duration divides this dollar duration by price; but it can be helpful to visualize the slope of the tangent line (to the Price/Yield curve) to interpret some relationships.

A typical coupon-bearing bond has a duration that decreases as the yield increases; i.e., as yield increases, duration decreases. We can visualize this by shifting to the right on the typical P/Y curve. This decreasing duration (as yield increases) is visualized by a tangent that is getting flatter; it is decelerating. In this way, a typical coupon-bearing bonds is also exhibiting a decrease in convexity; i.e., higher yield --> lower duration --> lower convexity, as the rate of change of duration is decreasing. This can be loosely visualized by the lessening of the curvature as we shift to the right and the curve becomes flatter ...

But I personally find "convexity increase at an increasing rate" to be a confusing, and perhaps nonsensical, phrase ... I also think the phrase "duration should be decrease at an increasing rate" to be imprecise at best, nonsensical at worst, FWIW. It has a mathematical interpretation, but it's easy to get confused without actual numbers. The reason is that we tend to confuse dollar duration (the first derivative) with duration and dollar convexity (the 2nd derivative) with convexity. I hope that helps,

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