Introduction

When it comes to analyzing and managing bonds, duration and convexity are essential tools that provide insights into how bond prices react to changes in interest rates. However, as valuable as these metrics are, they have limitations that must be recognized to make well-informed decisions. In this chapter, we will explore the limitations of duration and introduce the concept of convexity as a means to address some of these limitations. Let’s delve into a deeper understanding of these concepts and their implications.

Limitations of Duration

Duration is a measure that estimates the sensitivity of a bond’s price to changes in interest rates. While it provides a useful approximation of price change, it comes with certain limitations. One of the primary limitations is its sensitivity to parallel shifts in the yield curve. Duration assumes that the entire yield curve shifts uniformly, which may not accurately capture real-world market dynamics.

Example: Consider two bonds with the same duration. If the yield curve experiences a twist where short-term rates increase while long-term rates decrease, the bond with longer maturity might actually experience a larger price change than the bond with shorter maturity, contrary to what duration would predict. This demonstrates how duration fails to account for non-parallel yield curve shifts.

Convexity

Convexity is introduced as a corrective measure to address some of the limitations of duration. It accounts for the curvature of the price-yield relationship and provides a more accurate estimate of price changes. Convexity quantifies the rate of change of duration itself and reflects how a bond’s price change accelerates or decelerates as interest rates change.

Convexity is particularly relevant when analyzing bonds with larger coupon payments or longer maturities. Bonds with higher coupon payments have more convexity because they exhibit less price volatility when interest rates change, as the coupon payments mitigate the impact.

Example: To better understand the limitations of duration and the role of convexity, let’s consider a practical example. Suppose we have two bonds with the same duration but different coupon rates. Bond A has a coupon rate of 2%, while Bond B has a coupon rate of 8%. If interest rates decrease, Bond A will experience a larger price increase compared to Bond B. This is because the larger coupon payments of Bond B reduce its sensitivity to interest rate changes, highlighting the influence of convexity.