Introduction

In the previous chapters, we explored the fundamental concepts of bond duration and convexity. While duration provides an estimate of the percentage change in a bond’s price for a given change in interest rates, it has limitations, particularly for larger interest rate changes. Convexity, on the other hand, addresses these limitations by providing a more accurate measure of bond price sensitivity. In this chapter, we delve deeper into the calculation of bond price changes using both duration and convexity, illustrating how these measures work together to provide a more comprehensive understanding of how bond prices respond to interest rate fluctuations.

Bond Price Change Calculation

Bond price change calculation involves using both duration and convexity to estimate the impact of changes in interest rates on a bond’s price. The formula for calculating the approximate percentage change in bond price due to a change in yield is as follows:

$$\Delta P \approx -D \times \Delta y + \frac{1}{2} \times Convexity \times (\Delta y)^2 $$

Where:

• $\Delta P$ = Percentage change in bond price
• $D$ = Macaulay duration of the bond $\Delta y=$ Change in yield (expressed as a decimal)
• $C$ = Convexity of the bond

The first term, $-D \times \Delta y$, represents the change in bond price due to changes in yield using duration. The second term $\frac{1}{2} \times Convexity \times (\Delta y)^2$, represents the change in bond price due to changes in yield using convexity.

Example: Consider a bond with a Macaulay duration of 5 years and a convexity of 40. If the yield increases by 0.02 (2%), calculate the estimated percentage change in bond price. Given:

• Macaulay duration $(D)$ = 5 years
• Convexity $(C)$ = 40
• Change in yield $(\Delta y)=$ 0.02 (2%)

Using the formula for percentage price change in bond price, we get:

$$\Delta P \approx -D \times \Delta y + \frac{1}{2} \times Convexity \times (\Delta y)^2 $$

Putting given values, $$\Delta P \approx -5 \times \Delta 0.02 + \frac{1}{2} \times 40 \times (\Delta 0.02)^2 $$ $$\Delta P \approx -0.1 + 0.002 $$ $$\Delta P \approx -0.098 $$

The estimated percentage change in bond price is approximately -0.098% or -9.8 basis points.