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What is 'Convexity'

Convexity is a measure of the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes. Convexity is used as a risk-management tool, which helps measure and manage the amount of market risk to which a portfolio of bonds is exposed.

Graph of bond price and yield; convexity.


As interest rates increase, bond yields increase, and consequently, bond prices decrease. Conversely, as interest rates fall, bond yields fall and bond prices rise. In the example figure shown above, Bond A has a higher convexity than Bond B, which indicates that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall.

Convexity and Risk

Convexity is a better measure of interest rate risk, in relation to duration, because the concept of duration assumes that interest rates and bond prices have a linear relationship. Duration can be a good measure of how bond prices may be affected due to small and sudden fluctuations in interest rates. However, the relationship between bond prices and yields is typically more sloped, or convex. Therefore, convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates.

As convexity increases, the systemic risk to which the portfolio is exposed increases. As convexity decreases, the exposure to market interest rates decreases and the bond portfolio can be considered hedged. In general, the higher the coupon rate, the lower the convexity (or market risk) of a bond. This is because market rates would have to increase greatly to surpass the coupon on the bond, meaning there is less risk to the investor.

Negative and Positive Convexity

If a bond's duration increases as yields increase, the bond is said to have negative convexity. In other words, the shape of the bond is said to be concave. Therefore, if a bond has negative convexity, its price would increase in value as interest rates rise, and the opposite is true. Some examples of bonds that exhibit negative convexity are bonds with a traditional call provision, preferred bonds and most mortgage-backed securities (MBS).

If a bond's duration rises and yields fall, the bond is said to have positive convexity. If a bond has positive convexity, it would typically experience larger price increases if yields fall, in relation to price decreases when yields increase. The typical types of bonds with positive convexity are bonds with make-whole call provisions and non-callable bonds. Under normal market conditions, the higher the coupon rate, the lower a bond's degree of convexity. Consequently, zero-coupon bonds have the highest degree of convexity because they do not offer any coupon payments.

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