Relating Bond Convexity Under Discrete and Continuous Compounding

Summary

The document introduces duration as a measure of a bond price’s sensitivity to interest rates and notes that its formula depends on the rate convention. For a discretely compounded rate, it identifies modified duration as the relevant measure and gives its relationship to duration under continuous compounding. It then asks whether an analogous conversion exists for convexity, defined in the text using the second derivative of bond price with respect to the rate.

The text supplies the question and definitions but no answer, derivation, or worked bond example. It therefore raises a useful modeling issue rather than establishing a conversion formula: derivatives with respect to rates expressed under different compounding conventions need to account for the transformation between those rate variables. No assumptions about cash flows, yield definitions, or the exact convexity convention are further specified, so the requested relationship cannot be determined from the document alone.

Key ideas

• Bond duration measures price sensitivity to interest rate changes.
• The document distinguishes duration formulas for continuous and discrete compounding.
• Modified duration is presented as the discrete-rate adjustment to duration.
• The text asks for a corresponding convexity relationship but provides no derivation or answer.

Tags

# Bond Convexity: Relationship between discrete and continuous interest rate


# 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: $$C=\frac{\frac{d^2P}{P}}{dr^2}$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.