Reading the Duration-Convexity Approximation
A bond’s price generally falls when its yield rises and rises when its yield falls. Duration describes the leading, first-order effect of a small yield change. Convexity adds a second-order adjustment that reflects the curvature of the price-yield relationship. The duration-convexity formula combines these effects to estimate a bond’s percentage price change.
This tutorial uses an annual effective yield \(i\), with yield changes expressed as decimals. For example, a rise from 5.00% to 5.50% is \(\Delta i=0.005\), not \(0.5\) or \(0.50\). The bond is assumed to have fixed, positive cash flows and no embedded options that change its cash flows as yields move. The approximation is intended for a yield change small enough that a second-order estimate is informative.
The first term, \(-D_{\mathrm{mod}}\Delta i\), is the duration effect. Its sign is opposite the sign of the yield change: an increase in yield gives a negative estimated price change, while a decrease gives a positive one. The second term, \(\frac{1}{2}C(\Delta i)^2\), is the convexity adjustment. Because the yield change is squared, this term has the same sign for upward and downward shocks. For a conventional fixed-cash-flow bond with positive cash flows, \(C\) is positive, so the adjustment is positive.
Why the Two Terms Have These Roles
The formula is the second-order Taylor approximation of the price function around the initial yield. The linear term uses the slope of the price-yield curve at the starting yield; the quadratic term uses its curvature. Dividing the Taylor expansion by the initial price gives the percentage-change form.
For fixed cash flows \(CF_t\) paid at times \(t=1,\ldots,n\), the price under annual effective yield \(i\) is \(P(i)=\sum_{t=1}^{n}CF_t(1+i)^{-t}\). Differentiating each discounted cash flow gives
If each cash flow is positive and \(i>-1\), then \(P'(i)<0\) and \(P''(i)>0\). Thus modified duration is positive, and normalized convexity is positive. This explains both the negative sign attached to the duration effect and the positive convexity adjustment. If cash flows are not fixed or include negative amounts, these sign conclusions need not hold without further analysis.
Macaulay duration and modified duration are related, but they are not interchangeable in this formula. Under annual effective yield, \(D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+i)\). Use modified duration with a yield change measured in annual decimal units. A different compounding convention requires the matching duration and convexity convention; do not combine figures defined under different yield conventions.
Worked Interpretations
Worked Example: An Increase in Yield
A conventional bond has fixed positive cash flows, an initial annual effective yield of 6%, modified duration \(D_{\mathrm{mod}}=4.8\), and normalized convexity \(C=32\). Estimate the percentage price change if the yield rises by 1.00 percentage point. Use the duration-convexity approximation.
Method. Convert the yield movement to a decimal: \(\Delta i=0.0100\). The initial yield establishes the point at which duration and convexity are measured; the yield movement, not the initial yield, goes into the approximation.
The duration effect is
The convexity adjustment is
Adding the two contributions gives
The estimated percentage price change is a decrease of 4.64%. The positive convexity adjustment offsets part of the negative duration estimate; it does not reverse the direction here. Check: the shock is positive, so the duration contribution is negative; the squared shock makes the convexity contribution positive.
Common mistake avoided: using only the duration term would give \(-4.80\%\), omitting the stated convexity adjustment. The requested two-term estimate is \(-4.64\%\).
Worked Example: Equal Shocks in Opposite Directions
A fixed-cash-flow bond has an initial annual effective yield of 5%, modified duration \(D_{\mathrm{mod}}=6\), and normalized convexity \(C=50\). Estimate the percentage price change for a 2.00-percentage-point yield rise and for a 2.00-percentage-point yield fall. Compare the estimates.
For either movement, the magnitude is \(0.0200\), and the convexity adjustment is the same:
For the yield rise, \(\Delta i=+0.0200\), so the duration contribution is \(-6(0.0200)=-0.1200=-12.00\%\). Adding the convexity adjustment gives
For the yield fall, \(\Delta i=-0.0200\), so the duration contribution is \(-6(-0.0200)=+0.1200=+12.00\%\). The convexity adjustment remains positive:
The estimates are a decrease of 11.00% for the rise and an increase of 13.00% for the fall. The duration contributions are equal in magnitude and opposite in sign, while the convexity contributions are equal and positive. Check: the midpoint of the two estimated changes is \((-11.00\%+13.00\%)/2=1.00\%\), equal to the common convexity adjustment.
Common mistake avoided: changing the sign of the convexity term when yield falls. The square \((\Delta i)^2\) is positive in both cases.
Worked Example: Comparing Two Bonds with Equal Duration
Two hypothetical conventional bonds each have an initial annual effective yield of 4% and modified duration \(D_{\mathrm{mod}}=5.5\). Bond A has normalized convexity \(C=25\); Bond B has \(C=65\). Estimate each percentage price change for a 2.00-percentage-point yield rise, then interpret the difference.
The yield change is \(\Delta i=+0.0200\). Since both bonds have the same duration, each has the same duration contribution:
For Bond A, the convexity adjustment is
Its estimated percentage price change is \(-11.00\%+0.50\%=-10.50\%\). For Bond B, the adjustment is
Its estimated percentage price change is \(-11.00\%+1.30\%=-9.70\%\). Under this approximation, Bond B’s estimated decline is 0.80 percentage points smaller. The difference comes from convexity, not duration: with the same yield shock, the difference between the adjustments is \(\frac{1}{2}(65-25)(0.0200)^2=0.0080=0.80\%\). Check: both estimates remain negative, and the bond with higher positive convexity has the less negative estimate for this upward shock.
Common mistake avoided: treating equal duration as implying equal estimated percentage price changes. Equal duration gives equal first-order contributions, but different convexities give different second-order adjustments.
Interpreting the Approximation Carefully
The formula is useful because it separates two distinct effects. Duration indicates the direction and approximate first-order scale of a price response. Convexity adjusts that estimate in a direction determined by the sign of convexity, not by the direction of the yield shock. For a conventional bond with positive convexity, the adjustment raises the estimate relative to the duration-only result for both rising and falling yields.
The size of the convexity term depends on the square of the yield change. If the yield change is multiplied by two, the duration contribution doubles in magnitude, while the convexity contribution becomes four times as large. This is why convexity can have a more visible influence as the shock grows, even though duration is the leading effect for small movements. The formula remains an approximation: it does not assert that the estimated change equals the bond’s exact repriced change.
A useful way to organize a calculation is to keep the contributions separate until the final addition:
Confirm that the duration, convexity, and yield change all use the same convention. Here the yield change is an annual decimal change.
Evaluate \(-D_{\mathrm{mod}}\Delta i\). Its sign should oppose the yield movement when modified duration is positive.
Evaluate \(\frac{1}{2}C(\Delta i)^2\). For positive convexity, this contribution is positive for either yield direction.
Report the approximate percentage change, then state whether the estimate is an increase or decrease.
A frequent exam trap is entering a percentage-point movement as though it were a decimal rate change. For a rise of 0.75 percentage point, the correct input is \(\Delta i=0.0075\). Substituting \(0.75\) would make the duration contribution 100 times too large and the convexity contribution 10,000 times too large. Another trap is writing the duration contribution as \(+D_{\mathrm{mod}}\Delta i\). That reverses the basic price-yield direction for a positive-duration bond.
Use the formula only with the quantities it defines. In particular, the approximation shown is a percentage change, not a dollar change. It also uses modified duration, not Macaulay duration, under the stated annual effective yield convention. A bond whose cash flows depend on yield does not satisfy the fixed-cash-flow assumption used to justify the displayed derivatives.
Check Your Understanding
For each problem, use the duration-convexity approximation and keep yield changes in decimal form.
- A fixed-cash-flow bond has annual effective yield 3%, \(D_{\mathrm{mod}}=3.5\), and \(C=18\). Estimate its percentage price change for a yield rise of 2.00 percentage points. (Answer: \(-6.64\%\))
- A conventional bond has annual effective yield 6%, \(D_{\mathrm{mod}}=7.2\), and \(C=64\). Estimate the percentage price change for a yield fall of 1.00 percentage point. (Answer: \(+7.52\%\))
- A bond has \(D_{\mathrm{mod}}=4.4\) and \(C=40\). Estimate its percentage price changes for yield shocks of \(+1.50\) and \(-1.50\) percentage points. State the difference between the two estimates. (Answer: \(-6.15\%\) and \(+7.05\%\); the estimates differ by 13.20 percentage points.)
- Two bonds have the same modified duration, \(5.5\), but convexities \(25\) and \(65\). For a yield rise of 2.00 percentage points, find each approximate percentage price change and their difference. (Answer: \(-10.50\%\) and \(-9.70\%\); the second estimate is 0.80 percentage points higher.)
- A bond has \(D_{\mathrm{mod}}=4\), \(C=30\), and experiences a yield rise of 2.50 percentage points. Calculate the duration contribution, the convexity adjustment, and the total estimated percentage price change. (Answer: duration contribution \(-10.00\%\); convexity adjustment \(+0.9375\%\); total \(-9.0625\%\), or \(-9.06\%\) to two decimal places.)