Introduction

In the world of fixed income investments, understanding the limitations of duration is crucial. While duration provides a useful approximation of a bond’s price sensitivity to changes in interest rates, it has its drawbacks, especially for bonds with non-linear cash flow patterns. This chapter explores how convexity comes to the rescue by enhancing our understanding of bond price changes, particularly when interest rates fluctuate. Convexity, as a complementary measure to duration, provides a more accurate estimation of a bond’s price movement and helps investors manage their interest rate risk more effectively.

Understanding Convexity

Convexity refers to the curvature of the relationship between a bond’s price and its yield or interest rate. Unlike duration, which assumes a linear relationship, convexity recognizes the non-linear nature of bond price changes. A bond’s convexity arises due to the uneven distribution of its cash flows over time. Bonds with longer maturities or lower coupon rates generally exhibit higher convexity because a larger proportion of their cash flows is received further into the future.

Convexity is calculated using the second derivative of the bond price-yield curve. Mathematically, it is represented as follows:

$$Convexity =\frac{1}{P} \times \frac{d^2 P}{d y^2}$$ Where:

• $P=$ Bond price
• $y=$ Yield to maturity

Price Change Estimates with Convexity

While duration provides an estimate of the percentage change in bond price for a given change in yield, it assumes a linear relationship. However, in reality, as yields change, the bond price-yield curve is curved. Convexity accounts for this curvature and provides a correction factor to the duration-based estimate. The formula to calculate the approximate percentage change in bond price incorporating convexity is given by:

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

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: Let’s consider a 10-year bond with a face value of USD 1,000, a coupon rate of 5%, and a yield to maturity of 4%. The bond’s duration is calculated as 7.58 years, and its convexity is 64.26. If the yield increases by 0.5%, the duration-based estimate predicts a price decrease of approximately 3.79%. However, incorporating convexity, the actual price decrease is around 3.71%, demonstrating how convexity refines the estimation.