Bond Convexity Calculator

Understanding Bond Convexity and Duration

Bond convexity measures how sensitive a fixed-income security's price is to changes in interest rates, specifically accounting for the non-linear curvature of the price-yield relationship. While modified duration provides a linear approximation of price changes for small rate shifts, convexity measures the rate of change of duration itself (the second derivative of bond price with respect to market yield).

Why Bond Convexity Matters

For standard fixed-rate bonds with positive convexity, as market yields decrease, bond prices increase at an increasing rate. Conversely, when yields increase, bond prices fall at a decreasing rate. This asymmetric price movement works in favor of bondholders:

• Greater Price Appreciation: When interest rates drop, the actual price gain is higher than duration alone predicts.
• Lower Price Loss: When interest rates rise, the actual price drop is smaller than duration alone predicts.

Bond Convexity Formula

The exact formula for annual bond convexity ($C_{vx}$) is:

$$C_{vx} = \frac{1}{P (1 + y)^2} \sum_{t=1}^{N} \frac{t(t+1) C}{(1+y)^t} + \frac{N(N+1) M}{(1+y)^N}$$

where $P$ is the current bond price, $y$ is the yield to maturity per period, $C$ is the periodic coupon payment, $M$ is the par value, and $N$ is the total number of compounding periods.

Price Change Estimation Formula

Combining modified duration ($D_{mod}$) and convexity ($C_{vx}$), the percentage price change ($\Delta P / P$) for a change in yield ($\Delta y$) is calculated as:

$$\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \frac{1}{2} C_{vx} \cdot (\Delta y)^2$$