# Understanding Convexity in Bonds: Definition, Meaning, and Examples

## What is Convexity in Bonds?

Convexity in bonds is a measure of the curvature of the relationship between bond prices and interest rates. It is an important concept in fixed income investing as it helps investors understand how the price of a bond will change in response to changes in interest rates.

Convexity is derived from the concept of duration, which measures the sensitivity of a bond’s price to changes in interest rates. While duration provides a linear approximation of price changes, convexity takes into account the non-linear relationship between bond prices and interest rates.

When interest rates decrease, bond prices generally increase, and when interest rates increase, bond prices generally decrease. However, the relationship between bond prices and interest rates is not a straight line. Convexity measures the extent to which the relationship curves, or bends, and helps investors understand the magnitude of price changes for a given change in interest rates.

A positive convexity indicates that the relationship between bond prices and interest rates is convex, meaning that the price of the bond will increase more than it would decrease for a given change in interest rates. This is beneficial for bondholders as it provides a potential for higher returns if interest rates decrease.

On the other hand, a negative convexity indicates that the relationship between bond prices and interest rates is concave, meaning that the price of the bond will decrease more than it would increase for a given change in interest rates. This is detrimental for bondholders as it exposes them to greater downside risk if interest rates increase.

In summary, convexity in bonds is a measure of the curvature of the relationship between bond prices and interest rates. It helps investors understand the non-linear nature of this relationship and the potential for price changes in response to changes in interest rates. Positive convexity can provide higher returns in a decreasing interest rate environment, while negative convexity exposes bondholders to greater downside risk in an increasing interest rate environment.

## Examples of Convexity in Bonds

Convexity is an important concept in the world of bonds. It measures the sensitivity of a bond’s price to changes in interest rates. A bond with higher convexity will experience smaller price changes when interest rates fluctuate, while a bond with lower convexity will experience larger price changes.

Here are a few examples that illustrate the concept of convexity in bonds:

### Example 1: Zero-Coupon Bond

A zero-coupon bond is a type of bond that does not pay periodic interest payments. Instead, it is sold at a discount to its face value and pays the full face value at maturity. Zero-coupon bonds have high convexity because their prices are more sensitive to changes in interest rates. For example, if interest rates decrease, the price of a zero-coupon bond will increase more than that of a coupon-paying bond with the same maturity.

### Example 2: Callable Bond

A callable bond is a bond that can be redeemed by the issuer before its maturity date. Callable bonds have negative convexity because they have an embedded call option. When interest rates decrease, the issuer may decide to call the bond and refinance it at a lower interest rate, resulting in a loss for the bondholder. This loss offsets the potential gain from lower interest rates, leading to negative convexity.

### Example 3: Floating-Rate Bond

A floating-rate bond is a bond with an interest rate that adjusts periodically based on a reference rate, such as LIBOR or the prime rate. Floating-rate bonds have lower convexity compared to fixed-rate bonds because their interest rates reset periodically. As a result, the price of a floating-rate bond will be less affected by changes in interest rates compared to a fixed-rate bond with the same maturity.

Example Convexity
Zero-Coupon Bond High Convexity
Callable Bond Negative Convexity
Floating-Rate Bond Lower Convexity