For a fixed-rate, option-free bond, the bond price \(P\) is a function of its yield to maturity \(y\):
\[ P = P(y) \]Duration measures the first-order sensitivity of bond price to changes in yield.
Convexity measures the second-order sensitivity of bond price to changes in yield. More precisely, it measures the curvature of the bond price-yield relationship.
The key distinction is:
\[ \text{Duration} \rightarrow \text{first derivative} \] \[ \text{Convexity} \rightarrow \text{second derivative} \]For an ordinary option-free bond, the price-yield relationship is generally convex:
\[ \frac{d^2P}{dy^2} > 0 \]Consider a bond with cash flows \(CF_t\) at times \(t=1,\ldots,n\), assuming annual compounding.
\[ P(y) = \sum_{t=1}^{n} \frac{CF_t}{(1+y)^t} \]For a standard coupon bond:
\[ CF_t = \left\{ \begin{array}{ll} C & \text{if } t < n,\\ C+F & \text{if } t = n. \end{array} \right. \]where:
Differentiate the bond price with respect to yield:
\[ \frac{dP}{dy} = -\sum_{t=1}^{n} \frac{tCF_t}{(1+y)^{t+1}} \]Divide by \(P\):
\[ -\frac{1}{P}\frac{dP}{dy} = \frac{ \displaystyle \sum_{t=1}^{n} \frac{tCF_t}{(1+y)^{t+1}} }{ P } \]The Macaulay duration is:
\[ D_M = \frac{ \displaystyle \sum_{t=1}^{n} t\frac{CF_t}{(1+y)^t} }{ P } \]For annual compounding, the modified duration is:
\[ D_{\mathrm{mod}} = \frac{D_M}{1+y} \]Therefore:
\[ \frac{dP}{P} = -D_{\mathrm{mod}}\,dy \]For a finite yield change, this becomes the familiar first-order approximation:
\[ \frac{\Delta P}{P} \approx -D_{\mathrm{mod}}\Delta y \]This is a linear approximation to the actual price change.
Differentiate the price equation a second time:
\[ \frac{d^2P}{dy^2} = \sum_{t=1}^{n} \frac{t(t+1)CF_t}{(1+y)^{t+2}} \]Define convexity as:
\[ Cvx = \frac{1}{P} \frac{d^2P}{dy^2} \]Therefore, for annual compounding:
\[ Cvx = \frac{1}{P} \sum_{t=1}^{n} \frac{t(t+1)CF_t}{(1+y)^{t+2}} \]For a bond with positive cash flows, every term in the summation is positive. Hence:
\[ \frac{d^2P}{dy^2}>0 \]and the price-yield relationship is convex.
The most direct way to understand convexity is through a second-order Taylor expansion of \(P(y+\Delta y)\) around \(y\):
\[ P(y+\Delta y) \approx P(y) + \frac{dP}{dy}\Delta y + \frac{1}{2} \frac{d^2P}{dy^2} (\Delta y)^2 \]Divide by \(P\):
\[ \frac{\Delta P}{P} \approx \frac{1}{P} \frac{dP}{dy}\Delta y + \frac{1}{2} \frac{1}{P} \frac{d^2P}{dy^2} (\Delta y)^2 \]Substituting modified duration and convexity:
This is really the gist of the idea, that convexity is the second order perturbation to the duration. This will become more evident in the numerical example below. And where is it different from the duration term is that it is always positive, so it increase price of the bond whether the yield go up or down.
This is the duration-convexity approximation.
Suppose:
\[ \Delta y>0 \]The duration term is:
\[ -D_{\mathrm{mod}}\Delta y<0 \]while the convexity term is:
\[ \frac{1}{2}Cvx(\Delta y)^2>0 \]Thus, convexity offsets part of the price decline predicted by duration alone.
Suppose:
\[ \Delta y<0 \]The duration term becomes positive:
\[ -D_{\mathrm{mod}}\Delta y>0 \]but the convexity term remains positive because:
\[ (\Delta y)^2>0 \]Therefore, for a conventional option-free bond, convexity increases the price gain from a decline in yield relative to the duration-only approximation.
The mathematical reason is simply:
\[ (\Delta y)^2>0 \]regardless of whether \(\Delta y\) is positive or negative.
Suppose:
\[ P=100 \] \[ D_{\mathrm{mod}}=5 \] \[ Cvx=40 \]and the yield increases by one percentage point:
\[ \Delta y=0.01 \]Therefore:
\[ \Delta P\approx-5 \]The duration-convexity approximation is therefore:
\[ \frac{\Delta P}{P} \approx -0.05+0.002 = -0.048 \]Hence:
\[ \Delta P\approx-4.8 \]rather than the duration-only estimate of \(-5\).
Then:
\[ \frac{\Delta P}{P} \approx -5(-0.01) + \frac{1}{2}(40)(0.01)^2 \] \[ = 0.05+0.002 = 0.052 \]Therefore:
\[ \Delta P\approx+5.2 \]Duration approximates the price-yield curve using its tangent:
\[ \Delta P \approx P'(y)\Delta y \]Convexity measures how quickly the slope itself changes:
\[ P''(y)>0 \]For a conventional bond:
Thus the price-yield curve bends upward.
A duration-only model treats the curve as a straight line. Convexity corrects for that curvature.
Recall the convexity formula:
\[ Cvx = \frac{1}{P} \sum_{t=1}^{n} \frac{t(t+1)CF_t}{(1+y)^{t+2}} \]Convexity weights a cash flow at time \(t\) by:
\[ t(t+1) \]whereas duration has a weighting proportional to:
\[ t \]Therefore, convexity places relatively more weight on distant cash flows than duration does.
Holding other relevant characteristics comparable, longer-maturity bonds generally have greater convexity.
| Measure | Mathematical Object | Meaning |
|---|---|---|
| Price | \(P(y)\) | Bond value as a function of yield |
| Duration | \(-P'(y)/P\) | First-order yield sensitivity |
| Convexity | \(P''(y)/P\) | Second-order yield sensitivity |
The complete second-order approximation is:
\[ \frac{\Delta P}{P} \approx -D_{\mathrm{mod}}\Delta y + \frac{1}{2}Cvx(\Delta y)^2 \]The mathematical distinction is:
\[ \text{Duration}=\text{slope of the price-yield relationship} \] \[ \text{Convexity}=\text{rate of change of that slope} \]Equivalently:
\[ D_{\mathrm{mod}} = -\frac{1}{P}\frac{dP}{dy} \] \[ Cvx = \frac{1}{P}\frac{d^2P}{dy^2} \]