Comparing Bonds With Similar Duration

Why Similar Duration Does Not Mean Similar Price Changes

Duration summarizes a bond’s first-order sensitivity to a yield change. If two bonds have the same modified duration, the duration-only estimate predicts the same percentage price change for both, provided the yield shift is the same. But the bonds can still respond differently: convexity captures the curvature of price as yield changes. For bonds with positive fixed cash flows, greater convexity increases the second-order adjustment to the estimated price change.

The comparison is meaningful only when the bonds are measured consistently. In this tutorial, each bond has fixed, positive cash flows on specified annual dates, is valued at the same annual effective yield, and experiences the same small parallel change in that yield. The cash flows do not change when the yield changes. The estimates use duration and convexity; they are not exact repricing.

Definition: For fixed annual cash flows $C_t$ at times $t=1,2,\ldots,n$, valued at annual effective yield $i$, the price is $P(i)=\sum_{t=1}^{n}C_t(1+i)^{-t}$. Macaulay duration is the present-value-weighted average payment time. Modified duration is $D=-P'(i)/P(i)$, and annual-compounding convexity is $C=P''(i)/P(i)$. Thus $D$ measures first-order percentage price sensitivity, while $C$ measures the second-order curvature.

For this annual-yield model, the calculations from cash flows are

D=\frac{\sum_{t=1}^{n}tC_t(1+i)^{-(t+1)}}{P(i)}, \qquad C=\frac{\sum_{t=1}^{n}t(t+1)C_t(1+i)^{-(t+2)}}{P(i)}.

Macaulay duration is $D(1+i)$. In a comparison, use modified duration and convexity calculated with the same yield convention; do not mix annual-yield measures with measures defined for another compounding frequency.

Why Convexity Changes the Estimate

Suppose the yield changes from $i$ to $i+\Delta i$. A second-order Taylor approximation to the price is

P(i+\Delta i)\approx P(i)+P'(i)\Delta i+\frac{1}{2}P''(i)(\Delta i)^2.

Divide by $P(i)$, then use $D=-P'(i)/P(i)$ and $C=P''(i)/P(i)$. This gives the estimated relative price change:

\frac{\Delta P}{P}\approx -D\Delta i+\frac{1}{2}C(\Delta i)^2.

The first term is the duration estimate; the second is the convexity adjustment. For a positive yield shift, the duration term is negative. For a negative yield shift, it is positive. Because the yield change is squared, the convexity adjustment is positive for either direction when convexity is positive. Therefore, with equal duration, the bond with greater convexity has a slightly higher estimated price change in either direction: a smaller estimated loss when yields rise, or a larger estimated gain when yields fall.

“Higher estimated price change” does not mean the bonds have the same price or that their dollar changes are identical. The formula estimates a percentage change. Multiply that estimate by the bond’s own price to obtain its approximate dollar change.

Worked Example: Equal Duration, Different Convexity

Two hypothetical bonds are valued at an annual effective yield of 5%. Bond A pays $110.25 at the end of year 2. Bond B pays $78.75 at the end of year 1 and $31.9070390625 at the end of year 5. Compare their estimated price changes for a 1-percentage-point increase in yield.

Method. Discount each payment at 5%, calculate $D$ and $C$, and then apply the second-order estimate with $\Delta i=0.01$.

Bond A price and duration. Its year-2 payment has present value $110.25/(1.05)^2=100$, so $P_A=100$. Its Macaulay duration is 2 years, and its modified duration is $D_A=2/1.05=1.9047619$.

Bond A convexity. Its only payment is at $t=2$, so $C_A=2(3)\times100/[100(1.05)^2]=6/1.1025=5.4421769$.

Bond B price and duration. Its discounted payments are $78.75/1.05=75$ at $t=1$ and $31.9070390625/(1.05)^5=25$ at $t=5$. Thus $P_B=75+25=100$, and its Macaulay duration is $(1\times75+5\times25)/100=2$ years. Its modified duration is also $D_B=2/1.05=1.9047619$.

Bond B convexity. The weighted sum of $t(t+1)$ is $1(2)\times75+5(6)\times25=150+750=900$. Therefore, $C_B=900/[100(1.05)^2]=9/1.1025=8.1632653$.

Estimate the changes. For Bond A, the duration term is $-1.9047619(0.01)=-0.019047619$, and the convexity term is $0.5(5.4421769)(0.01)^2=0.000272109$. Its estimated relative change is $-0.019047619+0.000272109=-0.018775510$, or $-1.877551\%$. The estimated dollar change is $-0.018775510\times100=-\$1.877551$, so the estimated price is about $\$98.122449$.

For Bond B, the duration term is again $-0.019047619$, while the convexity term is $0.5(8.1632653)(0.01)^2=0.000408163$. Its estimated relative change is $-0.019047619+0.000408163=-0.018639456$, or $-1.863946\%$. The estimated dollar change is $-\$1.863946$, giving an estimated price of about $\$98.136054$.

Check. The duration-only estimates are identical because the modified durations match. Bond B has greater convexity, so its estimated loss is smaller. The estimated prices are consistent with that ranking. The calculations use the second-order approximation and are rounded here to six decimal places.

When Durations Are Close but Not Equal

An actual comparison often involves similar, rather than identical, durations. Then both terms matter. A small duration difference may reinforce or counteract the convexity advantage. To compare two bonds, subtract their approximate relative changes:

\left(\frac{\Delta P}{P}\right)_B-\left(\frac{\Delta P}{P}\right)_A \approx -(D_B-D_A)\Delta i+\frac{1}{2}(C_B-C_A)(\Delta i)^2.

This difference is in percentage-change units, not dollars. If the bonds have different prices, compare their dollar estimates separately after multiplying each relative change by its own price.

Worked Example: Similar Durations and Yield Moves in Both Directions

At a 5% annual effective yield, Bond C has present values of $50 at year 1 and $50 at year 3. Bond D has present values of $72.50 at year 1 and $27.50 at year 5. Each has a price of $100. Compare their estimated relative price changes for a 1-percentage-point yield increase and decrease.

Method. Use the present-value amounts as weights to find Macaulay duration and the weighted $t(t+1)$ sum. Convert to modified duration and annual-compounding convexity by dividing each measure by the appropriate power of $1.05$.

Bond C. Its Macaulay duration is $(1\times50+3\times50)/100=2$ years, so $D_C=2/1.05=1.9047619$. Its weighted $t(t+1)$ sum is $1(2)\times50+3(4)\times50=100+600=700$. Thus $C_C=7/1.1025=6.3492063$.

Bond D. Its Macaulay duration is $(1\times72.50+5\times27.50)/100=(72.50+137.50)/100=2.1$ years, so $D_D=2.1/1.05=2.0$. Its weighted $t(t+1)$ sum is $1(2)\times72.50+5(6)\times27.50=145+825=970$. Thus $C_D=9.7/1.1025=8.7981859$.

For a 1-percentage-point increase, $\Delta i=0.01$. Bond C’s estimate is $-1.9047619(0.01)+0.5(6.3492063)(0.0001)=-0.019047619+0.000317460=-0.018730159$, or $-1.873016\%$. Bond D’s estimate is $-2(0.01)+0.5(8.7981859)(0.0001)=-0.020000000+0.000439909=-0.019560091$, or $-1.956009\%$.

For a 1-percentage-point decrease, $\Delta i=-0.01$. The squared term is unchanged, while the duration term changes sign. Bond C’s estimate is $0.019047619+0.000317460=0.019365079$, or $1.936508\%$. Bond D’s estimate is $0.020000000+0.000439909=0.020439909$, or $2.043991\%$.

Check. Bond D has both greater modified duration and greater convexity. For the yield increase, its larger duration produces the larger estimated loss, despite the convexity adjustment. For the decrease, its larger duration and convexity both increase its estimated gain. This comparison shows why convexity alone cannot rank bonds when their durations differ. Results are rounded to six decimal places.

A Small Duration Gap Can Still Matter

When both bonds have positive convexity and one has greater convexity, the convexity term favors that bond for either sign of a yield shift. But a duration gap can be decisive, especially for an upward shift. The next example finds the shift at which the second-order estimates are equal. This is a comparison of approximations, not a guarantee that exact prices will cross at that yield change.

Worked Example: Finding the Estimated Crossover Shift

Two hypothetical bonds each have a price of $250,000 at a common annual effective yield. Bond E has modified duration 4.200 and convexity 24. Bond F has modified duration 4.205 and convexity 31. Estimate their relative price changes for a 0.50-percentage-point yield increase, then find the positive yield shift at which the two second-order estimates are equal.

Method. Apply the approximation to each bond, then set the difference of the estimates equal to zero. All yield changes are expressed as decimals.

For $\Delta i=0.005$, Bond E’s duration term is $-4.200(0.005)=-0.021$, and its convexity term is $0.5(24)(0.005)^2=12(0.000025)=0.000300$. Its estimated relative change is $-0.020700$, or $-2.0700\%$.

Bond F’s duration term is $-4.205(0.005)=-0.021025$, and its convexity term is $0.5(31)(0.005)^2=15.5(0.000025)=0.0003875$. Its estimated relative change is $-0.0206375$, or $-2.06375\%$.

The estimated dollar changes are $-0.020700(250{,}000)=-\$5{,}175$ for Bond E and $-0.0206375(250{,}000)=-\$5{,}159.375$ for Bond F. Thus F’s estimated loss is smaller by $\$15.625$, despite its slightly greater duration.

For an arbitrary positive shift, subtract E’s estimate from F’s: $- (4.205-4.200)\Delta i+0.5(31-24)(\Delta i)^2=-0.005\Delta i+3.5(\Delta i)^2$. Set this equal to zero. Factoring gives $\Delta i(-0.005+3.5\Delta i)=0$. The nonzero solution is $\Delta i=0.005/3.5=0.001428571$, approximately $0.142857\%$, or $14.3$ basis points.

Check. The 0.50-percentage-point increase exceeds the estimated crossover shift, and F has the better estimate, as the calculation confirms. This crossover is only a feature of the second-order estimates. Results are rounded to the displayed precision.

Common Comparison Traps

Comparing convexity without duration. Greater convexity favors a bond in the second-order adjustment, but a duration difference may dominate. Compare the complete estimates when durations are not equal.
Entering a percentage-point change as a whole number. A 1-percentage-point shift is $0.01$, not $1$. The convexity term uses the square of the decimal change: $0.01^2=0.0001$.
Using the unscaled convexity sum. In the annual effective yield model, the denominator includes $(1+i)^2$. For example, the sum 9 in Worked Example 1 gives convexity $9/(1.05)^2$, not 9.
Confusing relative and dollar changes. A relative change of $-1.8\%$ is not a dollar loss of $\$1.80$ for every bond. Multiply by that bond’s own price to obtain the estimated dollar change.
Treating an estimate as an exact price. The duration-convexity formula omits higher-order terms. Its accuracy depends on the size of the shift and the bond’s price curvature.

Key takeaway: Under a common small yield shift, compare bonds using $-D\Delta i+\frac{1}{2}C(\Delta i)^2$. Equal duration makes the convexity comparison especially clear; when durations differ, evaluate both terms before ranking the estimated price changes.

Check Your Understanding

For each question, use the annual-yield duration-convexity approximation unless the question asks for a different comparison.

Two bonds have the same modified duration, 3.10. Their convexities are 18 and 26. Estimate each relative price change for a 0.25-percentage-point yield decrease. Which bond has the greater estimated gain? (Answer: The estimates are $+0.780625\%$ and $+0.783125\%$, respectively; the bond with convexity 26 has the greater estimated gain.)
Bond A has $D=5.40$, $C=40$; Bond B has $D=5.35$, $C=46$. Estimate their relative price changes for a 0.30-percentage-point yield increase. Which has the smaller estimated loss? (Answer: A: $-1.6020\%$; B: $-1.5843\%$; Bond B has the smaller estimated loss.)
Two bonds have the same modified duration, 2.75. Their convexities are 12 and 17. Estimate their relative price changes for a 1-percentage-point increase and decrease. (Answer: For the increase, $-2.6900\%$ and $-2.6650\%$; for the decrease, $+2.8100\%$ and $+2.8350\%$.)
Bond B’s modified duration exceeds Bond A’s by 0.012, while its convexity exceeds A’s by 4.8. According to the second-order estimates, at what positive yield shift are their estimated relative changes equal? For a larger positive shift, which bond has the better estimate? (Answer: $0.005$, or 0.50 percentage points (50 basis points); for a larger positive shift, Bond B has the better estimate.)
Bond G has $D=4.00$, $C=20$, and price $80,000. Bond H has $D=4.00$, $C=28$, and price $120,000. Estimate their dollar price changes for a 0.50-percentage-point yield increase. (Answer: G: approximately $-\$1{,}580$; H: approximately $-\$2{,}358$. H has the smaller estimated percentage loss, but its larger price produces the larger dollar loss.)

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