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Bond Convexity: The Relationship Between Price and Yield

Last week we discussed bond duration. This week, we're talking convexity.

If you didn't read the duration piece, please go back and do so. I'm going to rely heavily on it in this piece.

Last week, we talked about how you can use duration to estimate how much the price of a bond will change if its yield changes. In the example we used, a bond with a duration of 6.4 years will go up about 6.4% in price if its yield drops by 1% (100 basis points), and down about 6.4% in price if its yield rises by 100 basis points.

If duration describes the approximate relationship between price and yield, convexity describes the actual relationship between price and yield.

Take a look at this graph. For now, just concentrate on the red line, which describes the relationship between price and yield for a standard, noncallable bond. (We'll get to callable bonds in a little bit.)

As you can see, as the bond's yield falls, its price increases, and vice versa. But what's important here is the shape of the line. It's curved, or convex -- not straight. That's because as the yield falls, the price increases at an ever-faster rate. As the yield rises, the price falls at an ever-slower rate.

That's due to something else we talked about last week: As a bond's yield falls, its duration lengthens, indicating increased price sensitivity. So as the yield falls further, the price will rise at a faster rate. Likewise, as a bond's yield rises, its duration shortens, making it less sensitive to additional increases in yield.

That's the aspect of the relationship between price and yield that the duration generalization above (a 6.4 year duration means a 6.4% price change if the yield changes by 100 basis points) doesn't capture. Imagine in the graph above a third line describing the relationship between price and yield predicted by the duration generalization. It wouldn't be curved, like the convexity line, which is derived from duration. It would be straight. It would touch the red line at the point where price equals 100 and yield equals 8%. But as it extended to the left, it wouldn't rise as much as the red line, and as it extended to the right, it would decline more than the red line. Hence, for a noncallable bond, duration underpredicts the amount by which the price will rise as the yield falls, and overpredicts the amount by which the price will fall as yields rise. Moreover, the longer the duration of a bond, the greater the margin of error.
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