By considering convexity, investors can better predict the potential impact of interest rate changes on their bond investments. It represents the total return an investor would earn if they held the bond to maturity and reinvested all coupon payments at the same rate. Let us take the example of a bond that pays an annual coupon of 6% and will mature in 4 years with a par value of $1,000.

It follows from the above equation that the bond price P falls with increase in the market interest rate r and vice versa. Duration provides a linear approximation of a bond’s price change in response to interest rate changes. Bonds with higher duration and convexity tend to experience more significant price changes in response to interest rate shifts. A technique called gap management is a widely used risk management tool, where banks attempt to limit the “gap” between asset and liability durations. Gap management heavily relies on adjustable-rate mortgages (ARMs), as key components in reducing the duration of bank-asset portfolios. Unlike conventional mortgages, ARMs don’t decline in value when market rates increase, because the rates they pay are tied to the current interest rate.

- Therefore, convexity can be used to enhance the performance of a bond portfolio, by selecting bonds that have the desired level of convexity for the expected interest rate environment.
- It provides a more accurate assessment of a bond’s price sensitivity to changes in interest rates than duration alone.
- This has made the cash flows of these bonds somewhat unpredictable, hence more susceptible to interest rate movements.
- For instance, for a callable bond, the bond will not rise above the call price when interest rates decline because the issuer can call the bond back for the call price, and will probably do so if rates drop.

In the image below, the curved line represents the change in prices, given a change in yields. The straight line, tangent to the curve, represents the estimated change in price, via the duration statistic. The shaded area reveals the difference between the duration estimate and the actual price movement. As indicated, the larger the change https://personal-accounting.org/ in interest rates, the larger the error in estimating the price change of the bond. Bond convexity is a measure of how the shape of the bond price curve changes when the interest rate changes. It is related to the concept of duration, which is the average time it takes for a bondholder to get back the money invested in the bond.

A bond with higher convexity will have a higher price sensitivity to interest rate changes, which means it will have higher returns when the interest rate falls, and lower losses when the interest rate rises. A bond with lower convexity will have a lower price sensitivity to interest rate changes, which means it will have lower returns when the interest rate falls, and higher losses when the interest rate rises. Therefore, convexity can be used to enhance the performance of a bond portfolio, by selecting bonds that have the desired level of convexity for the expected interest rate environment. It can easily be seen that modified duration changes as the yield changes because it is obvious that the slope of the line changes with different yields.

It shows how much the bond price will increase or decrease for a given change in the interest rate. If a bond is callable or has prepayment risk, such as with mortgage bonds, then the maturity of the bond, and therefore its duration, may be less than what the stated maturity would suggest. 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%.

## Relevance and Use of Convexity Formula

Therefore, convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates. Coupon-paying bonds typically have lower convexity than zero-coupon bonds, as their periodic coupon payments reduce their overall price sensitivity to interest rate changes. Bonds with negative convexity have price decreases that are larger than the price convexity formula excel increases when interest rates change by equal amounts. The yield to maturity (YTM) is the interest rate that equates the present value of a bond’s future cash flows to its current price. Convexity measures the curvature of the relationship between bond prices and interest rates. It represents the rate at which the bond’s duration changes in response to changes in interest rates.

Bond convexity is one of the most commonly used metrics to assess the non-linear effect of interest rate changes. It is a crucial metric for analyzing the interest rate risk of your bond investments. Duration is an effective analytic tool for the portfolio management of fixed-income securities because it provides an average maturity for the portfolio, which, in turn, provides a measure of interest rate sensitivity to the portfolio. If the duration is high, the bond’s price will move in the opposite direction to a greater degree than the change in interest rates. Convexity demonstrates how the duration of a bond changes as the interest rate changes.

## Effective Duration for Option-Embedded Bonds

Bond managers will often want to know how much the market value of a bond portfolio will change when interest rates change by 1 basis point. Interest rates vary continually from high to low to high in an endless cycle, so buying long-duration bonds when yields are low increases the likelihood that bond prices will be lower if the bonds are sold before maturity. This is sometimes called duration risk, although it is more commonly known as interest rate risk. Duration risk would be especially large in buying bonds with negative interest rates.

## Calculation in Excel:

However, it’s important to note that bond convexity is an approximation and may not perfectly predict the actual bond price movements. Several factors can affect bond prices, such as market liquidity, credit risk, and changes in interest rates. Additionally, bond convexity assumes a constant yield curve and small changes in yield, which may not hold true in all market conditions. Therefore, while bond convexity provides valuable insights, it should be used as a tool alongside other fundamental and technical analyses to assess the potential impact on bond prices.

The convexity of coupon-paying bonds varies depending on factors such as the coupon rate, time to maturity, and yield to maturity. Calculate the present value of each cash flow (coupon payments and principal repayment) using the bond’s yield to maturity. The magnitude of the price change in response to interest rate fluctuations depends on the bond’s duration and convexity. Duration and convexity are two tools used to manage the risk exposure of fixed-income investments.

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. Bond duration is the change in a bond’s price relative to a change in interest rates. A higher duration means a bond’s price will move to a greater degree in the opposite direction that interest rates move. As you can see, the graph is curved which shows that the rate of change in price is different at different points on the graph. Lower convexity suggests that the bond’s price will experience larger fluctuations in response to interest rate shifts, increasing the potential for losses.

## Calculating the convexity

Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change. Where duration assumes that interest rates and bond prices have a linear relationship, convexity produces a slope. Convexity is a measure of the curvature of the relationship between bond prices and interest rates. It provides a more accurate assessment of a bond’s price sensitivity to changes in interest rates than duration alone. Conversely, when interest rates fall, the present value of a bond’s future cash flows increases, leading to a higher bond price.

## How Do I Calculate Convexity in Excel?

When graphed, this relationship is non-linear and forms a long-sloping U-shaped curve. A bond with a high degree of convexity will experience relatively dramatic fluctuations when interest rates move. These options can cause bond prices to be more sensitive to interest rate changes in one direction than the other, leading to asymmetric price responses to rate fluctuations.

By selecting bonds with different convexity characteristics, investors can create a bond portfolio with a more balanced exposure to interest rate fluctuations, reducing the potential impact of rate changes on their investments. Callable and putable bonds often exhibit negative convexity due to the embedded options that allow the issuer or bondholder to alter the bond’s cash flows. Convexity-adjusted duration combines duration and convexity to accurately measure a bond’s price sensitivity to interest rate changes. Callable bonds and mortgage-backed bonds typically exhibit negative convexity due to their embedded options, which allow the issuer or borrower to alter the bond’s cash flows. Bonds with positive convexity experience price increases that are larger than the price decreases when interest rates change by equal amounts.

A bond with high convexity will have a higher return when the interest rate falls, but also a lower loss when the interest rate rises, compared to a bond with low convexity. Therefore, a bond with high convexity is more desirable than a bond with low convexity, all else being equal. However, this relationship is not linear, meaning that the bond price does not change by the same amount for every change in the interest rate. This is because the bond price depends on the present value of all the future payments that the bond will make, and these payments are affected differently by the interest rate.