A tool often used by investors when making decisions about bonds is convexity. Bond convexity shows the relationship between the price of a bond and its yields due to changes in interest rates. It is a tool often used along and confused with bond duration. While bond duration assumes the relationship between a bond’s price and its yield is directly proportional, convexity is different.

### What is bond convexity?

The word ‘convex’ in English means having an outline or surface curved like the exterior of a circle or sphere. Therefore, bond convexity refers to the curvature of the relationship between the price of a bond and its yield. The steeper or more convex the price-yield curve of a bond is, the more investors will prefer it. It is because more convexity means that the price of the bond is less susceptible to changes due to changes in interest rates.

According to bond duration, the relationship between bond yield and price is a straight line. Therefore, bond duration assumes that any changes in the interest rate of a bond will directly affect its price. However, in practice, the relationship isn’t direct. Practically, when the interest rates fluctuate, the price of a bond does not alter by the same amount. For example, a big change in interest rates may cause small price changes in the bond. It is known as bond convexity.

### How to calculate bond convexity?

The formula to calculate the convexity of a bond is as follows.

**C = d ^{2} (P (r)) / P x d x r^{2}**

In the above formula, ‘C’ represents the convexity of the bond. ‘d’ denotes the duration of the bond. Similarly, ‘P’ signifies the price of the bond. Finally, ‘r’ represents the interest rate.

Generally, the lower the interest is, the higher the bond convexity will be. For example, a bond with a coupon rate of 4% is more prone to changes in interest rates than bonds with a higher rate. Bonds with a higher convexity are also susceptible to price changes due to changes in interest rates. On the other hand, for bonds that have a lower convexity, changes in interest rates won’t affect their prices as much.

### What is negative bond convexity?

Sometimes, bond convexity can also be negative. For example, callable bonds will always illustrate a negative convexity. When a bond shows positive convexity, it means the relationship between the yield and the duration of the bond is a positive one. Simply put, positive bond convexity represents a positive correlation between the yield and duration of a bond. For positive bond convexity, the yield curve moves upward.

In contrast, negative bond convexity represents a negative correlation between the yield and duration of a bond. Therefore, the yield curve moves downwards rather than upwards.

### Why is bond convexity important?

Bond convexity plays a critical role for investors when making decisions regarding their bond portfolios. It helps them determine the impact that the changes in the yield of a bond will have on its duration. Investors use bond convexity as a risk management tool against interest rate risk. Furthermore, bond convexity can be useful in times of high market volatility.

### Conclusion

Bond convexity is a tool closely relate to the bond duration and shows the relationship 1between the price of a bond and its yield. According to bond convexity, the relationship between the price and the yield of a bond isn’t a straight one but rather a curved one, which means they are not directly proportional.

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