Deriving Bond Convexity from Modified Duration
Summary
The document examines an attempted derivation connecting bond convexity with squared modified duration. It defines modified duration as the negative price sensitivity to yield, scaled by bond value, and convexity as the similarly scaled second derivative. Applying the product rule to the relationship between price sensitivity and modified duration should produce a term involving the square of the first derivative of bond value.
The accepted answer identifies a missing square on that first derivative in the quotient-rule step. Without it, the algebra incorrectly collapses to an impossible condition for duration. The correction repairs this specific derivation; the document does not complete the broader textbook formula involving Macaulay duration, dispersion, and periodic yield, or explain how its convention relates to continuously compounded yield. Readers should therefore treat it as a focused calculus correction, not a full proof of the stated bond convexity identity.
Key ideas
- Modified duration is the negative first derivative of bond value with respect to yield, divided by value.
- Convexity is the second derivative of bond value divided by value.
- Differentiating the duration expression requires the quotient rule.
- The derivative of bond value in the quotient-rule term must be squared.
- Omitting that square creates an erroneous conclusion about duration.
Tags
Full text
# Proving that Convexity approx. equals Duration squared but something goes wrong?
# Proving that Convexity approx. equals Duration squared but something goes wrong?
I am trying to derive a formula for bond convexity that I saw in a textbook which states that
$$\text{convexity} = \frac{\text{Macaulay duration}^2 + \text{Macaulay duration} + \text{dispersion}}{(1+\text{periodic IRR})^2}.$$
I am assuming that a bond portfolio's IRR is continuously compounded, IRR denoted as $y$, and that the bond portfolio value is a function of its IRR, denoted $V(y)$. Modified Duration and Convexity are defined as
$$\text{ModD}(y)= -\frac{1}{V(y)}\frac{\partial V(y)}{\partial y}$$ $$\text{C}(y)=\frac{1}{V(y)}\frac{\partial ^2 V(y)}{\partial y^2}.$$
I then start with the $C(y)V(y)$ with the goal of getting something that looks similar to the first expression (just with $\text{ModD}$ instead of $\text{Macaulay Duration}$). This is my work, however, it leads me to an obviously wrong result but I am not sure where is/are the mistake/s.
$$C(y)V(y)=\frac{\partial ^2 V(y)}{\partial y^2}$$ $$C(y)V(y)=\frac{\partial \left( -V(y) \text{ModD}(y) \right)}{\partial y}$$ $$C(y)V(y)=-\frac{\partial V(y)}{\partial y}\times \text{ModD}(y)-V(y)\times \frac{\partial (\text{ModD}(y))}{\partial y} $$ $$C(y)=-\frac{1}{V(y)}\frac{\partial V(y)}{\partial y}\times \text{ModD}(y)-\frac{\partial (\text{ModD}(y))}{\partial y}$$ $$C(y)=\left( \text{ModD}(y) \right)^2-\frac{\partial (- \frac{1}{V(y)}\frac{\partial V(y)}{\partial y} )}{\partial y}$$ $$C(y)=\left( \text{ModD}(y) \right)^2-\left( \frac{1}{\left( V(y)\right)^2} \times \frac{\partial V(y)}{\partial y } - \frac{1}{V(y)} \times \frac{\partial ^2 V(y)}{\partial y ^2}\right)$$ $$C(y)=\left( \text{ModD}(y) \right)^2- \frac{1}{ V(y)} \times \frac{1}{V(y)} \frac{\partial V(y)}{\partial y } + C(y)$$ $$C(y)=\left( \text{ModD}(y) \right)^2+ \frac{1}{ V(y)} \times \text{ModD}(y) + C(y)$$ $$\left( \text{ModD}(y) \right)^2+ \frac{1}{ V(y)} \times \text{ModD}(y) =0$$ $$\text{ModD}(y)\left( \text{ModD}(y) + \frac{1}{ V(y)} \right) =0$$
I end up with
$$\text{ModD}(y)=-\frac{1}{V(y)}$$
which is obviously wrong. Where am I going wrong?
## Answer by Attack68 (score 1, accepted)
https://quant.stackexchange.com/a/79591
One error is that:
$$C(y)=\left( \text{ModD}(y) \right)^2-\left( \frac{1}{\left( V(y)\right)^2} \times \frac{\partial V(y)}{\partial y } - \frac{1}{V(y)} \times \frac{\partial ^2 V(y)}{\partial y ^2}\right)$$
should be:
$$C(y)=\left( \text{ModD}(y) \right)^2-\left( \frac{1}{\left( V(y)\right)^2} \times \left ( \frac{\partial V(y)}{\partial y } \right )^2 - \frac{1}{V(y)} \times \frac{\partial ^2 V(y)}{\partial y ^2}\right)$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.