Duration is a linear measure, meaning it assumes that for a certain percentage of change in interest rates, an equal percentage change in price will occur. As interest rates change, however, a bond’s price is unlikely to change linearly, and instead would change over some curved, or “convex” function of interest rates.

How Convexity Works

The graph that shows the relationship between bond price and yield will be convex. A bond’s convexity measures the curvature of its price/yield relationship. The degree of convexity shows how much a bond’s yield changes in response to a change in price.

When used together, duration and convexity offer a better approximation of the percentage of price change resulting from a particular change in a bond’s yield than using duration alone. In addition to improving this estimate, convexity can also be used to compare bonds with the same duration.

If two bonds offer the same duration and yield but one exhibits greater convexity, changes in interest rates will affect each bond in a different manner. A bond with greater convexity is less affected by interest rates than a bond with less convexity. In addition, bonds with greater convexity will have a higher price than bonds with lower convexity, regardless of what’s happening with interest rates.

Some convexity points to remember:

• The graph of the price-yield relationship for plain vanilla bonds shows positive convexity. The price-yield curve increases as yield decreases, and vice versa. And as market yields decrease, the duration increases, and vice versa.
• The higher the coupon rate, the lower a bond’s convexity. Zero-coupon bonds have the highest convexity.
• Callable bonds will show negative convexity at certain price-yield combinations. Negative convexity means that as market yields decrease, duration decreases as well.

Advanced Bond Concepts: Formula Cheat Sheet
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