Duration Coefficient Calculator
The duration coefficient is the signed form of modified duration: it states, with its sign attached, how much a bond’s price should move for a change in yield. Enter a Macaulay duration and a yield to maturity, or build the duration from the bond’s own terms, and this calculator returns the modified duration, the duration coefficient, and the predicted percentage price change for a yield shift you choose. It is the standard bond-math definition used by the other calculators with this name, worked end to end with the payments-per-year step most of them skip.
The prediction is a straight-line estimate. It grades small yield moves well and larger moves loosely, because the real price-yield relationship curves (convexity). See the limits section below.
Quick Answer
Quick answer: a bond with a Macaulay duration of 6 years and a yield to maturity of 5% has a modified duration of 5.7143 and a duration coefficient of -5.7143. The negative sign is the point: if yields rise 1%, the price is predicted to fall by about 5.71%. The calculator is pre-filled with these numbers, so one click on Calculate reproduces them, and the From Bond Terms mode builds the same figures from a bond’s face value, coupon and maturity when you do not know its Macaulay duration.
Key takeaways
- The duration coefficient is modified duration with its sign attached: coefficient = -1 x modified duration.
- Modified duration = Macaulay duration / (1 + YTM / payments per year). The payments-per-year step is the one most simple tools leave out.
- Predicted price change = coefficient x yield shift. A coefficient of -5.7143 and a 1% yield rise give a predicted fall of 5.71%.
- The estimate is a straight line drawn on a curved relationship, so treat big shifts as approximate: convexity bends the real answer.
- Quick Answer
- How to Use the Duration Coefficient Calculator
- The Formula: From Macaulay Duration to a Signed Coefficient
- Worked Example 1: A Macaulay Duration of 6 at a 5% Yield
- Worked Example 2: Building the Coefficient From Bond Terms
- Reading Your Result: What the Size of the Coefficient Tells You
- Macaulay, Modified and the Coefficient: Three Numbers, Three Jobs
- What Pushes a Coefficient Up or Down
- The Honest Limits: Convexity, Curve Shape and Options
- What Competing Calculators Miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to Use the Duration Coefficient Calculator
Use the first mode when a factsheet, broker or analytics page already gives you the bond’s Macaulay duration. Use the second when all you have is the bond itself: its face value, coupon rate, maturity and yield. Both modes end at the same three numbers.
Pick a mode. From Macaulay Duration if you know it, From Bond Terms if you do not. The bond-terms mode discounts every coupon and the final principal at the yield you enter, so it also shows the bond’s price at that yield.
Enter the yield as a percentage and set payments per year. Type 5 for a 5% yield, and pick 2 for the semi-annual payments most US bonds make. Frequency matters: the same duration at a 4% yield gives a slightly larger coefficient on semi-annual payments than on annual ones, because each period’s yield is smaller.
Set the yield shift you want priced. One point is the textbook case, but any shift works: 0.25 for a single central-bank move, -1 for a one-point fall. Enter 0 if you only want the coefficient itself.
Read the sign, not just the size. The coefficient is negative for every ordinary fixed-rate bond, because prices fall when yields rise. The predicted price change already carries the direction: negative for a yield rise, positive for a yield fall.
The Formula: From Macaulay Duration to a Signed Coefficient
Three steps connect the bond’s cash-flow timing to a price prediction. Each step has one job, and the calculator prints the result of every step so nothing hides inside a single black-box number.
Step 1 (only needed from bond terms): Macaulay duration = Sum of (time of each payment x present value of that payment) / bond price. It is the cash-flow-weighted average waiting time, in years.
Step 2: Modified duration = Macaulay duration / (1 + YTM / payments per year).
Step 3: Duration coefficient = -1 x modified duration, and predicted price change (%) = duration coefficient x yield shift (%).
Macaulay duration answers a timing question: how long, on average, until the bond pays you back? Modified duration converts that timing into sensitivity: roughly how many percent the price moves per 1% move in yield. The coefficient then attaches the direction. ThisMatter’s duration and convexity formulas set out the same chain, and the Corporate Finance Institute’s duration guide works the same definitions with bond examples.
Why some sites show this number as positive. Two presentation habits exist for the same quantity. Some calculators print the coefficient as a positive magnitude (5.7143) and put the direction in a sentence underneath (“prices fall if rates rise”). Others print it signed (-5.7143) so the price-change formula works directly: coefficient x shift. This page prints the signed coefficient as the main result and the positive modified duration right below it, so both readings are on screen at once and neither can be mistaken for the other.
A note on this page’s older definition. An earlier version of this calculator divided a bond’s actual price move by the move duration predicted, and called that ratio the duration coefficient. That ratio is not what the term means in bond math, and no standard source uses it. The tool was rebuilt on 2026-10-21 to compute the standard quantity defined above, so the number this page reports now matches the number every other duration coefficient calculator reports.
Worked Example 1: A Macaulay Duration of 6 at a 5% Yield
The pre-filled first mode is the standard textbook case, and it is the same worked example the best-known calculator with this name publishes. A corporate bond has a Macaulay duration of 6 years and a yield to maturity of 5%, with annual payments.
Adjust for yield. Modified duration = 6 / (1 + 0.05 / 1) = 6 / 1.05 = 5.7143.
Attach the sign. Duration coefficient = -1 x 5.7143 = -5.7143.
Price the shift. For a yield rise of 1 percentage point, predicted price change = -5.7143 x 1 = -5.71%. A $1,000 position loses about $57 on the rate move alone. For a fall of half a point, the same coefficient predicts +2.86%.
Run the neighbouring cases and the pattern shows itself. At a 4% yield, a Macaulay duration of 6.5 gives a modified duration of 6.25, the worked example on the standard modified-duration references. A Macaulay duration of 6.87 at a 3% yield gives 6.67, the example worked in the 5MinuteFinance duration course. Lower yields divide by a smaller number, so the same bond gets more sensitive as yields fall: that is duration extending as rates drop, and it is built into the formula rather than added later.
Worked Example 2: Building the Coefficient From Bond Terms
Now the same calculation with no duration handed to you. The bond: $1,000 face value, 5% annual coupon paid semi-annually ($25 every six months), 10 years to maturity, and a yield to maturity of 4%. These are the pre-filled values in the From Bond Terms mode.
| Piece | Value | How it is worked out |
|---|---|---|
| Coupon per period | $25.00 | $1,000 x 5% / 2 payments |
| Number of payments | 20 | 10 years x 2 per year |
| Yield per period | 2.00% | 4% / 2 |
| Price (sum of discounted cash flows) | $1,081.76 | Each $25 coupon and the final $1,025 discounted at 2% per period |
| Macaulay duration | 8.0809 years | Sum of (time x PV) = $8,742.03, divided by the $1,081.76 price |
| Modified duration | 7.9225 | 8.0809 / (1 + 0.04 / 2) |
| Duration coefficient | -7.9225 | -1 x modified duration |
Price the scenario the mode is pre-filled with, a yield rise of 0.75 points: predicted change = -7.9225 x 0.75 = -5.94%, which takes the estimated price from $1,081.76 to about $1,017.48. The premium over face value exists because the 5% coupon beats the 4% yield, and the long wait for that premium is exactly why this bond’s coefficient is large.
A short bond checks the other end of the scale. A 2-year bond paying $20 every six months on $1,000 face, at the same 4% yield, prices at par ($1,000.00), with a Macaulay duration of 1.9419 years, a modified duration of 1.9039 and a coefficient of -1.9039: a 1% yield rise costs it only about 1.9%. Same yield, same payment rhythm, a quarter of the risk, purely because the cash comes back sooner.
Reading Your Result: What the Size of the Coefficient Tells You
The sign is settled (negative, for plain fixed-rate bonds), so the information sits in the size. The calculator’s verdict badge uses the same three bands as this table.
| Coefficient size | Band | What usually sits behind it |
|---|---|---|
| 0 to -3 | Lower sensitivity | Short maturities, high coupons, or both. A 1% yield move changes the price by under 3%. |
| -3 to -7 | Moderate sensitivity | The typical intermediate bond or balanced bond fund. Rate risk is real but absorbable. |
| Below -7 | High sensitivity | Long maturities and low coupons. A 1% move swings the price hard; hedging and position size do the work here. |
For the 10-year bond from Worked Example 2, the chart prices four shifts with its coefficient of -7.9225. Read it as a ruler: every quarter-point of yield rise costs about 1.98% of price.
Macaulay, Modified and the Coefficient: Three Numbers, Three Jobs
The three figures get tangled because they are built from each other. Keep the jobs separate and the formulas stop looking alike.
| Figure | Question it answers | Units | For the 6-year, 5% bond |
|---|---|---|---|
| Macaulay duration | How long until the cash flows repay me, on average? | Years | 6.0000 |
| Modified duration | How many percent does the price move per 1% of yield move? | % per 1% shift | 5.7143 |
| Duration coefficient | Same sensitivity, with the direction attached, ready to multiply by a shift | Signed % per 1% shift | -5.7143 |
One practical consequence: portfolio work runs on the coefficient form. A position’s expected move is coefficient x shift x position value, position by position, and the results add. Fidelity’s guide to what duration is walks the same Macaulay-to-modified path from the investor’s side, and PIMCO’s understanding duration primer states the price rule in the same terms this calculator multiplies out.
What Pushes a Coefficient Up or Down
Three levers set the number before any rate move happens, and all three work through Macaulay duration first.
Coupon size. A higher coupon returns cash earlier, shortens the weighted average wait, and lowers the coefficient. Two 10-year bonds at the same yield can carry very different risk for this reason alone. If you are comparing two bonds and only know their coupons, run the coupon rate calculator first to see the cash-flow difference that drives the duration gap.
Maturity. A longer maturity pushes more value into the distant final payment, lengthens the wait, and raises the coefficient. The effect fades as maturity grows: going from 2 to 10 years multiplies risk roughly fourfold in the examples above, while going from 20 to 30 years adds far less.
The yield itself. A lower yield discounts distant cash flows less, lengthening Macaulay duration, and also divides by a smaller (1 + YTM / m). Both effects push the same way, so bonds get more rate-sensitive in low-yield environments, which is why duration risk surprised so many holders when yields backed up from near zero.
The Honest Limits: Convexity, Curve Shape and Options
The coefficient is a straight line fitted to a curve, and the article that hides that is overselling the tool. Three limits matter in practice.
Convexity. The true price-yield relationship bends: prices fall a little less than the line predicts when yields rise, and gain a little more when yields fall. For a quarter-point move the bend is negligible; for a two-point move it is the difference between a rough guide and a wrong number. FINRA’s bond investor hub covers the underlying price-yield relationship for readers who want the ground rules first.
Parallel shifts only. The formula prices one yield moving the whole curve by the same amount. Real curves twist: short rates can rise while long rates fall. A portfolio spread across maturities needs the shift applied maturity by maturity, not once.
Embedded options. Callable and prepayable bonds change their own cash flows when yields move, which breaks the fixed-cash-flow maths underneath Macaulay duration. For those, the profession uses a repricing-based measure instead: the effective duration calculator prices the bond at shifted yields and measures the move directly. And once a rate move has repriced a bond you hold, the return you actually keep is a tax question as much as a price question, which is the figure the after-tax yield calculator works out.
What Competing Calculators Miss
Miss 1: the payments-per-year step. The best-known page with this exact title divides Macaulay duration by (1 + YTM) and stops. That is the annual-payments version of the formula. Most real bonds pay semi-annually, and the missing / m term quietly understates the coefficient for every one of them. This calculator asks for payments per year in both modes.
Miss 2: no way in if you lack a Macaulay duration. Two-input tools assume a number many holders cannot find for a bond they own. The From Bond Terms mode here builds Macaulay duration from face value, coupon, maturity and yield, which is how the figure is defined in the first place.
Miss 3: the sign convention is left as a trap. Across the top results for this term, one tool insists the coefficient can never be negative, another prints negative table values with no explanation of why its rival shows positive ones. Both are describing the same quantity with different sign habits. This page prints the signed coefficient and the positive modified duration side by side and states the convention in the formula section, so a result can be compared against any other source without a sign error.
Miss 4: no scenario output. A coefficient alone still leaves the reader to multiply. The yield-shift input finishes the job: the predicted percentage price change for the shift you actually care about, plus the estimated new price in bond-terms mode.
Frequently Asked Questions
It is modified duration with its sign attached: duration coefficient = -1 x modified duration, where modified duration = Macaulay duration / (1 + YTM / payments per year). A coefficient of -5.7143 means a 1% rise in yield is predicted to cut the bond’s price by about 5.71%, and a 1% fall is predicted to raise it by about the same.
Bond prices and yields move in opposite directions. The minus sign records that direction inside the number itself, so the prediction formula needs no extra sign handling: predicted price change = coefficient x yield shift. A rise in yield multiplied by a negative coefficient gives a negative price change, which is what actually happens.
Both presentations describe the same quantity. Sites that print it positive are reporting the magnitude, which equals the modified duration, and state the direction in words. This calculator prints the signed coefficient as the main result and the positive modified duration directly underneath it, so a result can be checked against either convention without a sign error.
They are the same measurement with different packaging. Modified duration is stated as a positive sensitivity: percent of price move per 1% of yield move. The duration coefficient is that same figure negated, so it can be multiplied straight by a signed yield shift to get a signed predicted price change. If you know one, you know the other.
Use the From Bond Terms mode. Enter the face value, coupon rate, years to maturity, yield to maturity and payments per year, and the calculator discounts every cash flow, works out the Macaulay duration from the definition, and continues to the coefficient. That is the mode to use for a bond you hold when no factsheet publishes its duration.
It is a straight-line approximation, and it is close for small yield moves: a quarter point either way is graded well. For large moves the real price-yield relationship curves (convexity), so actual falls are usually a little smaller than predicted and actual gains a little larger. Treat a two-point prediction as a guide to the size of the move, not a quote.
Yes, in the From Macaulay Duration mode, with one adjustment. Funds publish modified or effective duration directly rather than Macaulay duration. Enter the published modified duration as the Macaulay figure and set the yield shift you want priced: the coefficient is simply the negative of that published duration, because the division step has already been done by the fund. The result estimates the fund’s NAV move for your shift.
Yes, through the (1 + YTM / payments per year) divisor. More frequent payments make each period’s yield smaller, so the divisor shrinks and the modified duration and coefficient grow slightly. At a 4% yield, a Macaulay duration of 8.0809 gives a modified duration of 7.7701 with annual payments and 7.9225 with semi-annual payments. Tools that skip the frequency input bake in the annual answer for every bond.
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References
- Fidelity Investments, “What is duration?”: why duration gauges a bond’s sensitivity to rate changes, and how coupon and maturity move it. Fidelity: What Is Duration?. Accessed October 2026.
- PIMCO, “Understanding Duration”: the rule that a bond’s price changes by roughly its duration for each 1% move in yield, stated for investors. PIMCO: Understanding Duration. Accessed October 2026.
- FINRA, “Bonds” investor hub: bond pricing basics, including the inverse relationship between bond prices and interest rates. FINRA: Bonds. Accessed October 2026.
- ThisMatter, “Duration and Convexity”: the Macaulay, modified and effective duration formulas and the convexity correction to the straight-line rule. ThisMatter: Duration and Convexity. Accessed October 2026.
- Corporate Finance Institute, “Duration”: Macaulay and modified duration definitions, formulas and worked bond examples. CFI: Duration. Accessed October 2026.