Introduction

In this chapter, we delve into the concept of convexity in fixed income securities, a critical measure used alongside duration to assess the sensitivity of bond prices to changes in interest rates. Convexity is a valuable tool for bond investors, providing insights beyond what duration alone can offer. We will explore the definition, calculation, and interpretation of convexity, shedding light on how it influences bond prices in response to fluctuations in yields.

Convexity

Convexity measures the curvature of the relationship between bond prices and yields. Unlike duration, which provides a linear estimate of price changes, convexity acknowledges the non-linear nature of bond price-yield relationships. It accounts for the fact that as yields change, the percentage price change in the bond is not constant. Instead, it varies with the level of interest rates.

Convexity for a fixed income security is calculated as follows:

$$\text{Convexity} =\sum_{t=1}^n \frac{C}{(1+\mathrm{YTM})^t} \cdot \frac{t(t+1)}{(1+\mathrm{YTM})^t}$$

Where:

• $C$ = Cash flow (coupon or principal)
• $YTM$ = Yield to maturity
• $t$ = Time to receipt of cash flow

Convexity is calculated as the sum of the present values of each cash flow, adjusted by the square of the time to receipt of that cash flow.

Interpreting Convexity

• Positive Convexity: When convexity is positive, it indicates that the bond’s price increases at an increasing rate as yields fall and decreases at a decreasing rate as yields rise. This means that bondholders benefit from falling interest rates more than they lose from rising rates.

• Negative Convexity: Negative convexity implies that the bond’s price increases at a decreasing rate as yields fall and decreases at an increasing rate as yields rise. Bonds with negative convexity are less responsive to declining rates and more sensitive to rising rates.

Importance of Convexity

Convexity has practical implications for bond investors:

• It helps refine the estimation of bond price changes under different yield scenarios.
• Convexity can be used to improve bond portfolio construction, especially in managing interest rate risk.
• Investors can assess the risk and return trade-offs associated with bonds of varying convexity profiles.