# Bond Convexity: Relationship between discrete and continuous interest rate

The interest rate risk of a bond price $P$ is measured by its Duration: $$D=-\frac{\frac{dP}{P}}{dr}$$

However, the explicit formula for the Duration given a function $P$ is different if $r$ is expressed in continuous or discrete compounding.

If $r$ is calculated in discrete compounding, the modified Duration should be used: $$D^{Mod}=\frac{D}{1+r}$$

Is there also an expression for the relationship of Convexity under discrete vs. continuous compounding? The convexity of a bond is defined as the second derivative of the bond price: $$C=\frac{\frac{d^2P}{P}}{dr^2}$$

• Modified Convexity = $C^{Mod} = Convexity / (1+r)^2$ – Alex C Jan 18 '16 at 18:22
• @AlexC Can you show the proof for a bond with coupon $c$? – emcor Jan 18 '16 at 18:35