Duration-Only And Convexity-Corrected Estimates

Why Compare the Two Estimates?

When a bond’s yield changes, its price generally changes in the opposite direction. Modified duration estimates the size of that change using a straight-line approximation. Convexity adds a second-order adjustment for the curvature of the price-yield relationship. Comparing the two estimates shows how much the curvature term contributes for a particular bond and yield shock.

The calculations below use a bond with fixed cash flows and a single annual effective yield $y$. The yield is positive, and all cash flows occur at integer year-ends. The shock $\Delta y$ is a change in the annual yield expressed as a decimal, not as a number of percentage points. These estimates assume the cash flows do not change when yield changes. They are local approximations, not substitutes for pricing a bond at a new yield.

Definition: For fixed cash flows $C_t$ at times $t=1,\ldots,n$, the bond price at annual effective yield $y$ is $P(y)=\sum_{t=1}^{n} C_t(1+y)^{-t}$. Modified duration is $D_{\text{mod}}=-P'(y)/P(y)$, and convexity is $C=P''(y)/P(y)$. For a small yield change $\Delta y$, the duration-only estimate is $\Delta P/P\approx-D_{\text{mod}}\Delta y$. The duration-convexity estimate is $\Delta P/P\approx-D_{\text{mod}}\Delta y+\tfrac12 C(\Delta y)^2$.

How to Calculate and Compare the Estimates

Under the annual-compounding model, the price derivatives give these cash-flow formulas:

D_{\text{mod}} = \frac{\sum_{t=1}^{n} tC_t(1+y)^{-(t+1)}}{P(y)} \qquad\text{and}\qquad C = \frac{\sum_{t=1}^{n}t(t+1)C_t(1+y)^{-(t+2)}}{P(y)}.

The duration-only estimate retains the first-order term. The convexity-corrected estimate adds $\tfrac12 C(\Delta y)^2$, the second-order term. Multiplying either estimated percentage change by the current price gives the estimated dollar price change; adding that dollar change to the current price gives the estimated new price.

Find the current price and sensitivity measures.
Calculate $P(y)$, $D_{\text{mod}}$, and $C$ using the bond’s cash flows and current yield.

Express the yield shock as a decimal.
For example, a rise of 40 basis points is $\Delta y=0.004$; a fall of 40 basis points is $\Delta y=-0.004$.

Calculate both percentage estimates.
Use $-D_{\text{mod}}\Delta y$ for duration alone. Add $\tfrac12 C(\Delta y)^2$ for the corrected estimate.

Convert to dollars and compare.
Multiply each percentage change by the current price. The difference between the estimates is the dollar contribution of the convexity term.

Why the Convexity Term Appears

The price function is differentiable for positive yields. A second-order Taylor expansion around the current yield writes the price at $y+\Delta y$ as its current value, plus its first derivative times $\Delta y$, plus half its second derivative times $(\Delta y)^2$, with higher-order terms omitted. Subtracting the current price and dividing by that price gives:

\frac{P(y+\Delta y)-P(y)}{P(y)} \approx \frac{P'(y)}{P(y)}\Delta y +\frac{1}{2}\frac{P''(y)}{P(y)}(\Delta y)^2 = -D_{\text{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2.

This derivation explains two useful features. First, the duration term changes sign when the yield shock changes sign. Second, the convexity term contains a squared shock, so it has the same sign for an upward or downward shock when convexity is positive. For a conventional fixed-cash-flow bond with positive cash flows, convexity is positive; in that case the correction raises the estimated percentage price change relative to the duration-only estimate.

Worked Example: Comparing Both Estimates for a Par Bond

A two-year bond with face value $100 pays a $10 coupon at the end of each year and returns $100 principal with its second coupon. Its annual effective yield is 10%. Estimate its price after a 100-basis-point increase in yield using duration alone and using duration plus convexity. Quantify the convexity contribution.

Method. Calculate the current price, modified duration, and convexity from the annual cash flows, then apply both approximations to the same shock. This comparison avoids treating a percentage-point change as a decimal yield change.

Current price. The cash flows are $C_1=10$ and $C_2=110$, and $y=0.10$:

P(0.10)=\frac{10}{1.10}+\frac{110}{1.10^2} =9.090909+90.909091 =100.

Modified duration. The Macaulay duration is the present-value-weighted average payment time:

D_{\text{Mac}} = \frac{1(9.090909)+2(90.909091)}{100} =1.909091. \qquad D_{\text{mod}}=\frac{D_{\text{Mac}}}{1.10}=1.735537.

Convexity. Using the annual-compounding cash-flow formula:

C= \frac{\frac{1(2)(10)}{1.10^3}+\frac{2(3)(110)}{1.10^4}}{100} = \frac{15.026296+450.788880}{100} =4.658152.

A 100-basis-point increase means $\Delta y=+0.010$. The duration-only percentage estimate is:

-D_{\text{mod}}\Delta y =-(1.735537)(0.010) =-0.01735537 =-1.735537\%.

The convexity contribution is:

\frac12 C(\Delta y)^2 =\frac12(4.658152)(0.010)^2 =0.000232908 =0.023291\%.

Therefore the corrected percentage estimate is $-1.735537\%+0.023291\%=-1.712246\%$. The duration-only dollar change is $$100(-0.01735537)=-$1.73554$, giving an estimated price of $98.26446. The corrected dollar change is $$100(-0.01712246)=-$1.71225$, giving an estimated price of $98.29 when rounded to the nearest cent. The convexity term raises the estimated price by $$100(0.000232908)=$0.02329$, or about 2.33 cents. As a check, the corrected estimate still indicates a price decrease, consistent with a yield increase.

Direction of the Shock and the Size of the Correction

For a positive-convexity bond, the correction is upward whether yields rise or fall. That does not mean the estimated price change is always positive: the duration term may be larger in magnitude. A yield increase typically produces a price decrease, but the positive convexity term makes that decrease less severe than duration alone predicts. A yield decrease typically produces a price increase, and the correction makes that increase larger.

The correction depends on the square of the yield shock. Doubling the shock quadruples the second-order term, while reversing the shock’s direction leaves that term unchanged. This is why keeping the signed shock in the duration term and squaring it only in the convexity term matters.

Worked Example: A Yield Decrease for a Zero-Coupon Bond

A three-year zero-coupon bond pays $1,331 at maturity. Its annual effective yield is 10%, and its current price is $1,000. Estimate the price change for a 200-basis-point decrease in yield using both methods, and find the dollar contribution of convexity.

Method. A zero-coupon bond has one cash flow at time 3. Its Macaulay duration is therefore 3 years. The calculation avoids weighting nonexistent intermediate coupons.

Current price check.

P(0.10)=\frac{1331}{1.10^3} =\frac{1331}{1.331} =1000.

For a zero-coupon bond, $D_{\text{Mac}}=3$, so $D_{\text{mod}}=3/1.10=2.727273$. Its convexity is:

C=\frac{3(4)}{1.10^2}=\frac{12}{1.21}=9.917355.

The yield falls by 200 basis points, so $\Delta y=-0.020$. The duration-only estimate is:

-D_{\text{mod}}\Delta y =-(2.727273)(-0.020) =0.05454545 =5.454545\%.

The convexity contribution is $\tfrac12(9.917355)(0.020)^2=0.001983471$, or $0.198347\%$. The corrected estimate is therefore $0.05652893=5.652893\%$. Duration alone gives an estimated increase of $$1{,}000(0.05454545)=$54.54545$, for an estimated price of $1,054.55. Adding the convexity term gives an estimated increase of $$1{,}000(0.05652893)=$56.52893$, for an estimated price of $1,056.53, rounded to the nearest cent. The second-order term contributes $$1{,}000(0.001983471)=$1.98$, rounded to the nearest cent. Both estimates predict an increase because yields fell, and the positive convexity correction makes the corrected increase larger.

Worked Example: A Smaller Shock for a Coupon Bond

A two-year bond with face value $100 pays a 5% annual coupon and has an annual effective yield of 5%. Estimate its price after the yield decreases by 50 basis points using duration alone and with convexity. Report the second-order contribution in dollars.

Method. Start from the two annual cash flows, calculate the two sensitivity measures, and use the signed yield change $\Delta y=-0.005$. The common error to avoid is entering a 50-basis-point change as $0.5$ rather than $0.005$.

The cash flows are $5 at year 1 and $105 at year 2. The current price is:

P(0.05)=\frac{5}{1.05}+\frac{105}{1.05^2} =4.761905+95.238095 =100.

The Macaulay duration is $\{1(4.761905)+2(95.238095)\}/100=1.952381$, so $D_{\text{mod}}=1.952381/1.05=1.859410$. Convexity is:

C= \frac{\frac{2(5)}{1.05^3}+\frac{6(105)}{1.05^4}}{100} = \frac{8.638376+518.302564}{100} =5.269409.

The duration-only percentage estimate is $-(1.859410)(-0.005)=0.00929705$, or $0.929705\%$. The convexity contribution is $\tfrac12(5.269409)(0.005)^2=0.000065868$, or $0.006587\%$. The corrected estimate is $0.00936292=0.936292\%$. Thus duration alone gives an estimated price of $100(1+0.00929705)=$100.92971$; the corrected estimate is $100(1+0.00936292)=$100.93629$, or $100.94 to the nearest cent. The second-order dollar contribution is $100(0.000065868)=$0.00659$, less than one cent. The estimated price rises under both methods, as expected for a yield decrease.

Common Mistakes and Checks

A basis-point conversion error can overwhelm the estimate. For example, incorrectly using $\Delta y=1$ for a 100-basis-point shock would give a duration term of $-1.735537$, or $-173.5537\%$, in the first example. The correct conversion is $100$ basis points $=0.010$, giving $-1.735537\%$. Likewise, the convexity term must use $(\Delta y)^2$: it is $\tfrac12 C(0.010)^2$, not $\tfrac12 C(0.010)$.

Keep the shock’s sign in the duration term. A yield decrease has $\Delta y<0$, so $-D_{\text{mod}}\Delta y$ is positive.
Square the shock only in the convexity term. The sign of $\Delta y$ does not change that term’s sign.
Do not mix units. Use a decimal yield change in the formulas, then label the resulting relative change as a percentage if multiplying by 100.
Keep the comparison on the same basis. Both estimates use the same current price and the same yield shock. Their difference is the convexity contribution, not a separate yield effect.

The duration-convexity estimate is most naturally used for a relatively small change in a single yield applied to the bond’s fixed cash flows. It omits terms of order three and higher, so it remains an estimate. For this tutorial’s purpose, the key comparison is the calculated first-order change versus that change plus the explicit second-order correction.

Check Your Understanding

For each problem, use the stated annual effective yield and calculate both approximations. Round estimated prices to the nearest cent.

A two-year zero-coupon bond pays $108.16 at maturity and currently yields 4% per year. Estimate its price after a 100-basis-point yield increase using duration alone and with convexity. (Answer: duration-only estimate $98.08; convexity-corrected estimate $98.10.)
A one-year bond has a single payment of $210 and an annual effective yield of 5%, giving a current price of $200. Estimate its price after a 50-basis-point yield decrease using both methods. (Answer: duration-only estimate $200.95; convexity-corrected estimate $200.96.)
A two-year bond with face value $100 pays an 8% annual coupon and has an annual effective yield of 8%. Estimate its price after a 100-basis-point yield increase using duration alone and with convexity. (Answer: duration-only estimate $98.22; convexity-corrected estimate $98.24.)
A three-year zero-coupon bond has a current price of $500 and an annual effective yield of 5%. Estimate its price after a 50-basis-point yield increase. State the dollar contribution of convexity. (Answer: duration-only estimate $492.86; convexity-corrected estimate $492.93; convexity contribution approximately $0.07.)

What You'll Learn