# Convexity in Bonds: Definition, Meaning, and Examples

## What Is Convexity?

Convexity is a measure of the curvature, or the degree of the curve, in the relationship between bond prices and bond yields.

Convexity is thus a measure of the curvature in the relationship between bond prices and interest rates. It reflects the rate at which the duration of a bond changes as interest rates change. Duration is a measure of a bond's sensitivity to changes in interest rates. It represents the expected percentage change in the price of a bond for a 1% change in interest rates.

### Key Takeaways

• Convexity is a risk-management tool, used to measure and manage a portfolio's exposure to market risk.
• Convexity is a measure of the curvature in the relationship between bond prices and bond yields.
• Convexity demonstrates how the duration of a bond changes as the interest rate changes.
• If a bond's duration increases as yields increase, the bond is said to have negative convexity.
• If a bond's duration rises and yields fall, the bond is said to have positive convexity.
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## Understanding Convexity

Convexity demonstrates how the duration of a bond changes as the interest rate changes. Portfolio managers will use convexity as a risk-management tool, to measure and manage the portfolio's exposure to interest rate risk.

In the example figure shown below, 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.

Before explaining convexity, it's important to know how bond prices and market interest rates relate to one another. As interest rates fall, bond prices rise. Conversely, rising market interest rates lead to falling bond prices. This opposite reaction is because as rates rise, the bond may fall behind in the payout they offer a potential investor in comparison to other securities.

The bond yield is the earnings or returns an investor can expect to make by buying and holding that particular security. The price of the bond depends on several characteristics including the market interest rate and can change regularly.

For example, if market rates rise, or are expected to rise, new bond issues must also have higher rates to satisfy investor demand for lending the issuer their money. However, the price of bonds returning less than that rate will fall as there would be very little demand for them as bondholders will look to sell their existing bonds and opt for bonds, most likely newer issues, paying higher yields. Eventually, the price of these bonds with the lower coupon rates will drop to a level where the rate of return is equal to the prevailing market interest rates.

## Bond Duration

Bond duration measures the change in a bond's price when interest rates fluctuate. If the duration of a bond is high, it means the bond's price will move to a greater degree in the opposite direction of interest rates. Conversely, when this figure is low the debt instrument will show less movement to the change in interest rates. Essentially, the higher a bond's duration, the larger the change in its price when interest rates change. In other words, the greater its interest rate risk. So, if an investor believes that interest rates are going to rise, they should consider bonds with a lower duration.

Bond duration should not be confused with its term to maturity. Though they both decline as the maturity date approaches, the latter is simply a measure of time during which the bondholder will receive coupon payments until the principal must be paid.

Typically, if market rates rise by 1%, a one-year maturity bond price should decline by an equal 1%. However, for bonds with long-dated maturities, the reaction increases. As a general rule of thumb, if rates rise by 1%, bond prices fall by 1% for each year of maturity. For example, if rates rise by 1%, the two-year bond price would fall 2%, the three-year bond price by 3%, and the 10-year price by 10%.

Duration, on the other hand, measures the bond's sensitivity to the change in interest rates. For example, if rates were to rise 1%, a bond or bond fund with a 5-year average duration would likely lose approximately 5% of its value.

## Convexity and Risk

Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change. Convexity is a better measure of interest rate risk, concerning bond duration. Where duration assumes that interest rates and bond prices have a linear relationship, convexity allows for other factors and produces a slope.

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. The term systemic risk became common during the financial crisis of 2008 as the failure of one financial institution threatened others. However, this risk can apply to all businesses, industries, and the economy as a whole.

The risk to a fixed-income portfolio means that as interest rates rise, the existing fixed-rate instruments are not as attractive. As convexity decreases, the exposure to market interest rates decreases and the bond portfolio can be considered hedged. Typically, the higher the coupon rate or yield, the lower the convexity—or market risk—of a bond. This lessening of risk is because market rates would have to increase greatly to surpass the coupon on the bond, meaning there is less interest rate risk to the investor. However, other risks, like default risk, etc., might still exist.

## Example of Convexity

Imagine a bond issuer, XYZ Corporation, with two bonds currently on the market: Bond A and Bond B. Both bonds have a face value of \$100,000 and a coupon rate of 5%. Bond A, however, matures in 5 years, while Bond B matures in 10 years.

Using the concept of duration, we can calculate that Bond A has a duration of 4 years while Bond B has a duration of 5.5 years. This means that for every 1% change in interest rates, Bond A's price will change by 4% while Bond B's price will change by 5.5%.

Now, let's say that interest rates suddenly increase by 2%. This means that the price of Bond A should decrease by 8% while the price of Bond B will decrease by 11%. However, using the concept of convexity, we can predict that the price change for Bond B will actually be less than expected based on its duration alone. This is because Bond B has a longer maturity, which means it has a higher convexity. The higher convexity of Bond B acts as a buffer against changes in interest rates, resulting in a relatively smaller price change than expected based on its duration alone.

## 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 bond price will decline by a greater rate with a rise in yields than if yields had fallen. Therefore, if a bond has negative convexity, its duration would increase—the price would fall. As interest rates rise, and the opposite is true.

If a bond's duration rises and yields fall, the bond is said to have positive convexity. In other words, as yields fall, bond prices rise by a greater rate—or duration—than if yields rose. Positive convexity leads to greater increases in bond prices. If a bond has positive convexity, it would typically experience larger price increases as yields fall, compared to price decreases when yields increase.

Under normal market conditions, the higher the coupon rate or yield, the lower a bond's degree of convexity. In other words, there's less risk to the investor when the bond has a high coupon or yield since market rates would have to increase significantly to surpass the bond's yield. So, a portfolio of bonds with high yields would have low convexity and subsequently, less risk of their existing yields becoming less attractive as interest rates rise.

Consequently, zero-coupon bonds have the highest degree of convexity because they do not offer any coupon payments. For investors looking to measure the convexity of a bond portfolio, it's best to speak to a financial advisor due to the complex nature and the number of variables involved in the calculation.

Most mortgage-backed securities (MBS) will have negative convexity because their yield is typically higher than traditional bonds. As a result, it would take a significant rise in yields to make an existing holder of an MBS have a lower yield, or less attractive, than the current market.

## What Is 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 bond price will decline by a greater rate with a rise in yields than if yields had fallen. Therefore, if a bond has negative convexity, its duration would increase as the price decreased and vice versa.

If a bond's duration rises and yields fall, the bond is said to have positive convexity. In other words, as yields fall, bond prices rise by a greater rate—or duration—than if yields rose. Positive convexity leads to greater increases in bond prices. If a bond has positive convexity, it would typically experience larger price increases as yields fall, compared to price decreases when yields increase.

## Why Do Interest Rates and Bond Prices Move in Opposite Directions?

As interest rates fall, bond prices rise and vice versa. For example, if market rates rise, or are expected to rise, new bond issues must also have higher rates to satisfy investor demand for lending the issuer their money. However, the price of bonds returning less than that rate will fall as there would be very little demand for them as bondholders will look to sell their existing bonds and opt for bonds, most likely newer issues, paying higher yields. Eventually, the price of these bonds with the lower coupon rates will drop to a level where the rate of return is equal to the prevailing market interest rates.

## What Is Bond Duration?

Bond duration measures the change in a bond's price when interest rates fluctuate. If the duration is high, it means the bond's price will move in the opposite direction to a greater degree than the change in interest rates. Conversely, when this figure is low the debt instrument will show less movement to the change in interest rates.

Essentially, the higher a bond's duration, the larger the change in its price when interest rates change. In other words, the greater its interest rate risk. So, if an investor believes that a sizable change in interest rates could have a negative effect on their bond portfolio, they should consider bonds with a lower duration.

## The Bottom Line

Convexity is a measure of the curvature of its duration, or the relationship between bond prices and yields. It is used to describe the way in which the duration of a bond changes in response to changes in interest rates. When a bond's price is more sensitive to changes in interest rates, it is said to have higher convexity. Convexity is important for bond investors because it can impact the value of their investments. For example, when interest rates rise, the prices of most bonds tend to fall, and the magnitude of the price decline is typically greater for bonds with higher convexity. Conversely, when interest rates fall, the prices of most bonds tend to rise, and the magnitude of the price increase is typically greater for bonds with higher convexity.

There are several factors that can impact the convexity of a bond, including the bond's coupon rate, maturity, and credit quality. Higher coupon bonds, for example, tend to have higher convexity than lower coupon bonds because they are more sensitive to changes in interest rates. Similarly, longer-term bonds tend to have higher convexity than shorter-term bonds because they are exposed to interest rate risk for a longer period of time.

Bond investors can use convexity to their advantage by managing their bond portfolios to take advantage of changes in interest rates. For example, an investor who anticipates rising interest rates might choose to hold a portfolio of bonds with low convexity, while an investor who anticipates falling interest rates might choose to hold a portfolio of bonds with high convexity.