Why Maturity Is Not the Whole Comparison
A bond’s maturity tells you when its final scheduled payment is due. It does not tell you when, on average, the bond’s value is returned through all its payments. Nor does it, by itself, determine how sensitive the bond’s price is to a small change in yield. Those are related but distinct questions, answered by Macaulay duration and modified duration.
This distinction matters whenever two bonds have different coupon rates or maturities. A bond with a later final payment can have a shorter duration if it returns more value through earlier coupons. For a small yield change, modified duration gives a local estimate of the percentage price change. That estimate is not a statement about the bond’s maturity or a guarantee about its price change for a large yield movement.
The weights in the Macaulay formula are each payment’s present value divided by the bond’s price. They are positive and sum to one when the cash flows are positive and the price is calculated consistently. A large early coupon can therefore pull duration toward earlier times, even if the bond’s final payment is far in the future. Maturity is the time of the last payment; duration averages the times of all payments using their present values as weights.
Why Modified Duration Tracks Local Price Sensitivity
For fixed cash flows discounted at one annual effective yield, the price function is \(P(i)=\sum_t C_t(1+i)^{-t}\). Differentiating each discounted cash flow with respect to the yield gives a direct relationship between price change and modified duration.
Dividing by \(P\) gives the percentage price change per unit change in yield. For a sufficiently small yield change \(\Delta i\), the first-order estimate is
The minus sign reflects the inverse relationship between yield and price for positive fixed cash flows: an increase in yield reduces their present values. The approximation uses the derivative at the current yield, so it is a local estimate. It assumes the cash flows do not change when yield changes and that the same yield shift applies to every payment. It is not an exact repricing formula for a finite yield change, and it does not automatically apply to a bond whose payment amounts or timing change with market conditions.
Comparing Maturity and Duration
A useful comparison starts by putting both bonds on the same basis: use the same valuation date and a stated yield convention, and express payment times in consistent units. The examples below use annual payments and annual effective yields. For another compounding frequency, the duration units and yield conversion must match that frequency.
| Measure | What it describes | What it does not describe |
|---|---|---|
| Maturity | Time until the final scheduled payment | The present-value-weighted timing of all payments |
| Macaulay duration | Present-value-weighted average payment time | The final payment date, unless only one payment remains |
| Modified duration | Local percentage price sensitivity to yield | An exact price change for a finite yield shift |
A higher coupon often increases the share of the bond’s price represented by earlier payments, which can reduce Macaulay duration. A longer maturity often adds later payments and can increase duration. Neither tendency is a ranking rule by itself: the comparison depends on the full set of cash flows and the yield used to discount them.
Worked Example: A Longer-Maturity Bond With Shorter Duration
Two hypothetical bonds each have a face amount of $1,000. Bond Z is a four-year zero-coupon bond. Bond C pays an annual coupon of $200 and returns $1,000 of principal at the end of year 5. Both are valued at an annual effective yield of 10%. Compare their maturities, Macaulay durations, and modified durations.
Method. Discount each payment at 10%, calculate the price and the time-weighted present-value total, then divide to obtain Macaulay duration. This avoids treating maturity as though it were the weighted-average payment time.
For Bond Z, the only payment is $1,000 at time 4. Its price is
Because the only payment occurs at year 4, its Macaulay duration is 4 years. Its modified duration is \(4/1.10=3.636364\) years.
Bond C pays $200 in years 1 through 4 and $1,200 in year 5. The annual discount factors and present values are:
| Time \(t\) | Cash flow \(C_t\) | \(1.10^{-t}\) | Present value \(C_t1.10^{-t}\) | Time-weighted present value \(tC_t1.10^{-t}\) |
|---|---|---|---|---|
| 1 | $200 | 0.909091 | $181.818182 | $181.818182 |
| 2 | $200 | 0.826446 | $165.289256 | $330.578512 |
| 3 | $200 | 0.751315 | $150.262960 | $450.788881 |
| 4 | $200 | 0.683013 | $136.602690 | $546.410759 |
| 5 | $1,200 | 0.620921 | $745.105564 | $3,725.527822 |
The displayed factors and table entries are rounded to six decimal places for readability; the duration calculation uses the unrounded discount factors. Adding the present values gives \(P_C=1{,}379.078652\). Adding the time-weighted present values gives \(5{,}235.124156\). Thus
Check. Bond C matures one year later than Bond Z, but its Macaulay duration is about 3.7961 years, less than Bond Z’s 4 years. Its modified duration is also lower: 3.4510 versus 3.6364 years. The large coupons bring more of Bond C’s present value forward, so its weighted payment time and local yield sensitivity are both lower in this comparison.
Common mistake avoided: ranking these bonds by final maturity and concluding that Bond C must have the greater duration or yield sensitivity. Its later last payment does not outweigh the effect of its earlier coupons.
Coupon Differences Can Change the Ranking
The next comparison holds maturity and yield fixed but changes the coupon. This isolates one reason that two bonds ending on the same date can have different durations. The price as well as the time-weighted total must be recalculated for each bond; using the same duration weights for both would be incorrect.
Worked Example: Same Maturity, Different Coupons
Two hypothetical four-year bonds each have a face amount of $1,000 and pay coupons annually. Bond L has a 4% annual coupon rate; Bond H has an 11% annual coupon rate. Both are valued at an annual effective yield of 8%, and each repays principal at maturity. Find and compare their Macaulay and modified durations.
Method. For each bond, discount all four payments at 8%, sum the present values to obtain price, and divide the sum of time-weighted present values by price. This avoids assuming that equal maturity means equal duration.
The discount factors for years 1 through 4 are \(0.925925926\), \(0.857338820\), \(0.793832241\), and \(0.735029853\). Bond L pays $40 in years 1 through 3 and $1,040 in year 4. Bond H pays $110 in years 1 through 3 and $1,110 in year 4.
| Bond and time | Cash flow | Present value | Time-weighted present value |
|---|---|---|---|
| L, year 1 | $40 | $37.037037 | $37.037037 |
| L, year 2 | $40 | $34.293553 | $68.587106 |
| L, year 3 | $40 | $31.753290 | $95.259870 |
| L, year 4 | $1,040 | $764.431047 | $3,057.724188 |
| H, year 1 | $110 | $101.851852 | $101.851852 |
| H, year 2 | $110 | $94.307270 | $188.614540 |
| H, year 3 | $110 | $87.321546 | $261.964638 |
| H, year 4 | $1,110 | $815.883137 | $3,263.532548 |
The table entries are shown to six decimals; the sums use unrounded discount factors. For Bond L, the price is $867.514927 and the time-weighted present-value total is $3,258.608201. For Bond H, these totals are $1,099.363805 and $3,815.963578. Therefore,
Check. Both bonds mature in four years, but the higher-coupon Bond H has the shorter duration. More of its price is represented by earlier coupon payments. At the common yield, its modified duration is also lower, so its first-order percentage price response to a small yield change has smaller magnitude.
Common mistake avoided: comparing only the coupon rates or only the final payment. Duration depends on the present value of every payment relative to the bond’s total price.
Using Duration for a Yield-Sensitivity Comparison
Modified duration makes the comparison in the previous example operational. A yield change must be entered as a decimal: a rise of one percentage point is \(\Delta i=0.01\), while a rise of one-half of one percentage point is \(\Delta i=0.005\). The estimate is a percentage change; multiply by the current price to estimate the dollar change.
Worked Example: Estimating Price Changes for Two Bonds
Use the four-year Bonds L and H from the previous example. Estimate each price change if the annual effective yield rises from 8% to 9%, using modified duration. Then reprice at 9% to check the estimate. Assume each bond’s cash flows remain fixed.
Method. First apply \(\Delta P/P\approx-D_{\text{mod}}\Delta i\), with \(\Delta i=0.01\). Then calculate the exact prices at 9% by discounting the same cash flows at the new yield. This separates a local estimate from a full repricing.
For Bond L, the estimated percentage change is \(-3.4780(0.01)=-0.034780\), or \(-3.4780\%\). Its estimated dollar change is
For Bond H, the estimated percentage change is \(-3.2139(0.01)=-0.032139\), or \(-3.2139\%\). Its estimated dollar change is
For the check, the 9% discount factors for years 1 through 4 are \(0.917431193\), \(0.841679993\), \(0.772183480\), and \(0.708425211\). For Bond L, the four present values are $36.697248, $33.667200, $30.887339, and $736.762219. Their sum is $838.014006, so the exact price change is \(838.014006-867.514927=-\$29.500921\), or \(-\$29.50\) to the cent.
For Bond H, the four present values are $100.917431, $92.584799, $84.940183, and $786.351984. Their sum is $1,064.794397, so the exact price change is \(1{,}064.794397-1{,}099.363805=-\$34.569408\), or \(-\$34.57\) to the cent.
Check. Both exact prices fall, as expected when the yield rises. The duration estimates give the direction and a reasonable local comparison, but they are not the exact repriced changes. Bond H has the smaller percentage decline estimated by modified duration, yet its larger starting price means its estimated dollar decline is larger.
Common mistake avoided: treating a percentage sensitivity as a dollar change, or treating a one-percentage-point yield shift as \(1\) rather than \(0.01\). Apply modified duration to the decimal yield change, then multiply the estimated percentage change by the bond’s price.
How to Make a Sound Comparison
When comparing bonds by duration rather than maturity, use a consistent sequence:
Identify the yield convention, payment frequency, and time units. A duration stated in years cannot be compared directly with one stated in coupon periods without conversion.
Discount every scheduled cash flow using the yield assumption for that bond. Duration weights are based on present values, not undiscounted cash flows.
Divide the sum of payment time times present value by the bond price. Compare this weighted payment horizon with the final maturity separately.
Under annual compounding, divide Macaulay duration by \(1+i\). Use \(-D_{\text{mod}}\Delta i\) for the approximate percentage price change.
Say whether the comparison concerns weighted payment time, percentage sensitivity, or estimated dollars. They are different comparisons and need not rank bonds identically.
For example, a lower modified duration indicates a smaller estimated percentage price response for a small yield movement, under the stated fixed-cash-flow model. It does not necessarily mean a smaller dollar loss: the starting prices may differ. It also does not establish which bond is preferable overall; that depends on the investor’s liabilities, risk tolerance, and other features not represented by duration.
Common Comparison Traps
- Equating maturity with duration. Maturity is the final payment time; duration incorporates all payments.
- Using undiscounted cash flows as weights. Macaulay duration weights by present value at the stated yield.
- Comparing Macaulay durations as though they were modified durations. For annual compounding, the conversion depends on \(1+i\); different yields can affect the comparison.
- Using the wrong yield-change units. A one-percentage-point change is 0.01, not 1.
- Calling a duration estimate an exact repricing. The derivative gives local sensitivity; calculate new present values when an exact price at a new yield is required.
Check Your Understanding
For each problem, use the stated annual compounding convention and keep payment times in years.
- A six-year zero-coupon bond is valued at an annual effective yield of 4%. Find its Macaulay and modified durations. (Answer: \(D_M=6.0000\) years; \(D_{\text{mod}}=5.7692\) years.)
- A three-year bond with face amount $1,000 pays a 10% annual coupon and is valued at a 10% annual effective yield. Find its price and Macaulay duration. (Answer: price $1,000.00; \(D_M=2.7355\) years.)
- A bond has modified duration 4.2 years. Estimate its percentage price change for a yield decrease of 0.30 percentage points. (Answer: approximately \(+1.26\%\).)
- Bond A is a five-year zero-coupon bond. Bond B is a four-year bond with positive annual coupons and the same face amount. Both have the same positive annual effective yield. Is it possible for Bond A to have greater Macaulay duration than Bond B? State the condition that decides. (Answer: Yes; it does if Bond B’s present-value-weighted average payment time is less than five years, as it must be when its positive payments occur only by year 4.)
- Two fixed-cash-flow bonds have modified durations of 3.5 and 4.0 years, respectively, at their current yields. For the same small yield increase, which has the larger estimated percentage price decline, and can that alone determine which has the larger dollar decline? (Answer: The bond with modified duration 4.0 has the larger estimated percentage decline; no, compare starting prices as well to determine estimated dollar declines.)