Bond convexity is a measure of how the price of a bond changes when the interest rate changes. It shows how much the bond price will increase or decrease for a given change in the interest rate.
For example, suppose you buy a bond for $100 that pays 5% interest per year. If the interest rate in the market goes up to 6%, your bond will be less attractive to other investors, because they can buy new bonds that pay 6%. Therefore, the price of your bond will go down, say to $95. This means that your bond has a negative relationship with the interest rate: when the interest rate goes up, the bond price goes down, and vice versa.
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. For example, a payment that is due in one year will be less affected by the interest rate than a payment that is due in 10 years, because the interest rate has more time to compound in the latter case.
This means that the bond price will change more when the interest rate is low than when it is high. For example, if the interest rate goes from 5% to 4%, your bond price will increase more than if it goes from 5% to 6%. This is because a lower interest rate makes the future payments more valuable, and a higher interest rate makes them less valuable.
Bond convexity measures how much the bond price changes for a given change in the interest rate, and how this change varies depending on the level of the interest rate. A bond with high convexity will have a large price change for a small interest rate change, and a bond with low convexity will have a small price change for a large interest rate change.
Bond convexity is important for investors and portfolio managers, because it helps them assess the risk and return of different bonds and bond portfolios. 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.
Basic Theory
1. Definition of Convexity:
Convexity is the second derivative of the bond’s price with respect to the yield. In simpler terms, it measures the curvature of the price-yield relationship. A higher convexity implies a greater price change in response to interest rate fluctuations.
2. Importance of Convexity:
While duration estimates the linear price-yield relationship, convexity accounts for the non-linear effects. This is crucial in scenarios where interest rates change, as convexity helps refine the estimate of bond price changes.
Procedures for Calculating Bond Convexity in Excel
1. Excel Formula for Convexity:
The formula for convexity is given by:
Convexity = 
2. Steps to Calculate Convexity in Excel:
- Determine the bond’s yield, modified duration, and number of compounding periods per year.
- Plug these values into the convexity formula.
- Utilize Excel functions to streamline the calculations.
Real-World Scenario
Let’s consider a 5-year bond with a face value of $1,000, a coupon rate of 5%, and a yield of 4%. The bond pays interest semi-annually.
Calculation in Excel:
Assuming the bond’s modified duration is 4.2 years, the convexity calculation in Excel would look like this:
=1 / (1 + (0.04 / 2))^2 * (4.2 / 2) * 1000
Result:
The calculated convexity for the given scenario is approximately 18.43.
Other Approaches:
1. Macaulay Duration:
Another way to estimate convexity is by using the Macaulay duration:
Convexity = 
2. Excel Add-ins:
There are Excel add-ins and financial functions that can automate convexity calculations, making it more convenient for analysts and investors.
