Bond Convexity

Bond Convexity

Convexity is the second-order measure of a bond's price sensitivity to yield changes. It captures the curvature in the price-yield relationship.

Full Price Change Approximation

$\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y + \frac{1}{2} \cdot C \cdot (\Delta y)^2$

The convexity term always adds value (positive for standard bonds), meaning bonds gain more when yields fall than they lose when yields rise.

Convexity Formula

$C = \frac{1}{P} \sum_{t=1}^{n} \frac{t(t+1) \cdot CF_t}{(1+y)^{t+2}}$

For a coupon bond:

$C = \frac{1}{P} \left[ \sum_{t=1}^{n} \frac{t(t+1) \cdot C}{(1+y)^{t+2}} + \frac{n(n+1) \cdot F}{(1+y)^{n+2}} \right]$

Properties

• Longer maturity → higher convexity
• Lower coupon → higher convexity
• Lower yield → higher convexity
• Convexity is always positive for standard bonds