Convexity in Bonds: Definition, Meaning, and Examples

What Is Convexity?

Convexity is apparent in the relationship between bond prices and bond yields. Convexity is 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 measures 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.

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.

As interest rates fall, bond prices rise. Conversely, rising market interest rates lead to falling bond prices. The bond yield is the earnings or returns an investor can expect to make by buying and holding that particular security. The bond price depends on several characteristics, including the market interest rate, and can change regularly.

If market rates rise, new bond issues must also have higher rates to satisfy investor demand for lending money. 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 with 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. If rates rise by 1%, a bond or bond fund with a 5-year average duration would likely lose approximately 5% of its value. Conversely, when this figure is low, the debt instrument will show less movement to the change in interest rates.

The higher a bond's duration, the larger the change in its price when interest rates change and the greater its interest rate risk. 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 the time during which the bondholder will receive coupon payments until the principal is paid.

If market rates rise by 1%, a one-year maturity bond price should decline by an equal 1%. 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.

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. Where duration assumes that interest rates and bond prices have a linear relationship, convexity 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. For a fixed-income portfolio, 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.

Example of Convexity

A bond issuer, XYZ Corporation, has two bonds 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%.

If the interest rate increase by 2%, 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 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. 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, and the price would fall. As interest rates rise, the opposite is true.

If a bond's duration rises and yields fall, the bond is said to have positive convexity. As yields fall, bond prices rise by a greater rate or duration than if yields rise. Positive convexity leads to increases in bond prices. If a bond has positive convexity, it would typically experience 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. 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. A portfolio of bonds with high yields would have low convexity and subsequently less risk of existing yields becoming unattractive as interest rates rise.

The Bottom Line

Convexity is a measure of the curvature of its duration or the relationship between bond prices and yields. It describes how the duration of a bond changes in response to changes in interest rates. Convexity can impact the value of investments. Several factors impact the convexity of a bond, including the bond's coupon rate, maturity, and credit quality. Bond investors can use convexity to their advantage by managing their bond portfolios to take advantage of changes in interest rates.

Correction—Jan. 23, 2024: The article was amended to clarify that the higher the convexity, the more the bond price will increase when rates fall, and the less the bond price will drop when rates rise.