Bond price, YTM, duration and convexity calculator

92.56price per 100 face value

Macaulay duration: 7.895 years
Modified duration: 7.665 – a change of 1 percentage point in the yield moves the price by about 7.67%
Convexity: 71.7854
Price value of a basis point: 0.0709 per 100 face value

If the yield rises by 100 basis points to 7.00%, duration alone estimates −7.665%, duration with convexity −7.306%, and exact repricing gives −7.318%.

Price after the yield changes, from 92.5613
Yield change	Duration only	With convexity	Exact repricing
−100 bp to 5.00%	99.6561 (+7.665%)	99.9884 (+8.024%)	100.0000 (+8.037%)
+100 bp to 7.00%	85.4664 (−7.665%)	85.7986 (−7.306%)	85.7876 (−7.318%)

Exact priceDuration estimateDuration and convexity

Chart as a table

Price against yield
Yield	Exact	Duration only	With convexity
3.00%	117.1686	113.8459	116.8359
3.75%	110.3440	108.5247	110.2066
4.50%	103.9909	103.2036	103.9511
5.25%	98.0742	97.8824	98.0693
6.00%	92.5613	92.5613	92.5613
6.75%	87.4223	87.2401	87.4270
7.50%	82.6297	81.9190	82.6665
8.25%	78.1582	76.5978	78.2797
9.00%	73.9841	71.2767	74.2667

Cash flows, present values and weights at 6.00%
Period (years)	Cash flow	Present value	Weight	Years × weight
1 (0.5)	2.5000	2.4272	2.62%	0.0131
2 (1)	2.5000	2.3565	2.55%	0.0255
3 (1.5)	2.5000	2.2879	2.47%	0.0371
4 (2)	2.5000	2.2212	2.40%	0.048
5 (2.5)	2.5000	2.1565	2.33%	0.0582
6 (3)	2.5000	2.0937	2.26%	0.0679
…	12 more periods, each paying the coupon of 2.5000
19 (9.5)	2.5000	1.4257	1.54%	0.1463
20 (10)	102.5000	56.7518	61.31%	6.1313
Total		92.5613	100.00%	7.895

How it is calculated

Coupon periodsN = years × coupons per year = 10 × 2 = 20
Coupon per periodPMT = face × coupon rate ÷ coupons per year = 100.00 × 5% ÷ 2 = 2.5000
Yield per periody = YTM ÷ coupons per year = 6.00% ÷ 2 = 3.00%
PricePMT × (1 − (1 + y)^−N) ÷ y + face × (1 + y)^−N = 2.5000 × 14.877475 + 100.00 × 0.553676 = 92.56
Macaulay durationΣ (k ÷ 2) × PV of each flow ÷ price = 730.7709 ÷ 92.5613 = 7.895 years
Modified durationD_Mac ÷ (1 + y) = 7.894997 ÷ 1.03 = 7.665
ConvexityΣ CF × k(k + 1) ÷ (2² × (1 + y)^k+2) ÷ price = 71.7854
Price value of a basis point (PVBP)D_mod × price × 0.0001 = 7.665 × 92.5613 × 0.0001 = 0.0709
Duration estimate, yield +100 bp−D_mod × Δy = −7.665046 × 0.01 = −7.665%
Convexity adjustment½ × C × Δy² = 0.5 × 71.7854 × 0.01² = +0.3589%
Exact repricing at 7.00%The same flows discounted at 7.00% a year: 85.7876 = −7.318%

BA II Plus and HP 12C keystrokes for this price

Key facts

Macaulay duration is the average time to a bond’s cash flows, weighted by their present values; for a zero-coupon bond it equals the years to maturity.
Modified duration = Macaulay duration ÷ (1 + yield ÷ coupons per year): a 1 percentage point rise in yield lowers the price by about that many per cent.
ΔP ÷ P ≈ −modified duration × Δy + ½ × convexity × Δy². For the 5% bond at 6% and a rise of 100 basis points: −7.67% with duration alone, −7.31% with convexity, −7.32% exactly.
PVBP = modified duration × price × 0.0001: 0.0709 per 100 of face value for the same bond.
Excel’s DURATION function returns Macaulay duration and MDURATION modified duration, both for a face value of 100.

How is the result calculated?

Price and yield. On a coupon date the price is the present value of the remaining coupons and the face value, discounted at the yield per period $y$ – the yield to maturity divided by the $m$ coupons a year (NYU Stern, time value of money (secondary source)) (RBI, bond sensitivity (secondary source)). A price above the face value means a yield below the coupon rate, and the other way round. For the yield from a price there is no formula: the calculator searches for the yield at which the discounted flows equal the price.

P = \sum_{k=1}^{n}\frac{CF_k}{(1+y)^k}\qquad D_{\mathrm{Mac}} = \frac{1}{P}\sum_{k=1}^{n}\frac{k}{m}\,\frac{CF_k}{(1+y)^k}\qquad D_{\mathrm{mod}} = \frac{D_{\mathrm{Mac}}}{1+y}

Duration. Macaulay duration is the average time to the cash flows in years, each weighted by its share of the price (IMF, duration definitions (secondary source)); for a zero-coupon bond it is the time to maturity (RBI, bond sensitivity (secondary source)). Modified duration divides it by one plus the yield per period and estimates the percentage price change for a change in yield, with the cash flows held fixed (RBI, bond sensitivity (secondary source)) (OSFI, duration and value changes).

Convexity. The price–yield curve bends, so a straight duration line misses part of the change; the convexity term corrects for it and, for a bond with fixed positive payments, raises the estimate whether yields rise or fall (IMF, convexity and curve assumptions). With $\Delta y$ the change in the yield per year as a decimal (100 basis points = 0.01):

C = \frac{1}{P}\sum_{k=1}^{n}\frac{k(k+1)\,CF_k}{m^2\,(1+y)^{k+2}}\qquad \frac{\Delta P}{P} \approx -D_{\mathrm{mod}}\,\Delta y + \tfrac12\,C\,(\Delta y)^2

The convexity here is in years squared and is used with the factor ½. The price value of a basis point (PVBP) is the price change for one basis point, $D_{\mathrm{mod}} \times P \times 0.0001$ (CME Group, basis-point value (secondary source)) (OSFI, duration and value changes). To show how good the estimates are, the calculator also reprices the bond exactly at the new yields: the same cash flows, discounted again.

Worked example

1. A 5% bond at a 6% yield

Face value 100, a 5% coupon paid twice a year and 10 years to maturity: 20 coupons of 2.50, then the 100 back. At a yield of 6% a year, or 3% per half-year:

P = 2.50 \times \frac{1 - 1.03^{-20}}{0.03} + \frac{100}{1.03^{20}} \approx 92.56

Weighting each payment’s time by its share of the price gives a Macaulay duration of 7.895 years – less than 10, because the coupons arrive earlier. Modified duration is 7.895 ÷ 1.03 = 7.665 and convexity 71.79. For a rise of 100 basis points to 7%:

\frac{\Delta P}{P} \approx -7.665 \times 0.01 + \tfrac12 \times 71.79 \times 0.01^2 \approx −7.31\%

Duration alone gives −7.67%; repricing at 7% gives 85.79, a change of −7.32%. If the yield falls by 100 basis points to 5%, the yield equals the coupon and the price is exactly 100.00: +8.04%, against +7.67% from duration alone and +8.02% with convexity. The PVBP is 0.0709 per 100 of face value.

2. A zero-coupon bond

A bond that pays only 100 in 7 years, priced at a 4% annual yield, costs 75.99. With a single payment, all the weight sits on year 7, so the Macaulay duration is 7.00 years. Modified duration is 7 ÷ 1.04 = 6.7308, and convexity is $7 \times 8 / 1.04^2 \approx 51.78$ . A 5% coupon bond with the same yield and maturity would have a shorter duration, because part of its value comes back earlier.

3. The yield and duration from a price

A bond with a face value of 1,000, a 4% annual coupon and 6 years to maturity is bought for 950. The yield that makes the discounted coupons and repayment equal the price is 4.985% – above the coupon rate, because the bond trades below its face value. At that yield, Macaulay duration is 5.4352 years and modified duration 5.1771. The PVBP is 0.49 per 1,000 of face value. For a rise of 50 basis points, duration alone estimates a price of 925.41; with convexity the estimate is 925.80, the same as exact repricing (925.80) to two decimal places.

What do the results mean for you?

Modified duration is a quick rule of thumb: if yields rise by 1 percentage point, the price falls by about that many per cent. The example bond, with a modified duration of 7.67, loses about 7.67% – a little less in fact, which is the convexity at work. The estimate is reliable for small changes; for large ones, use the exact repricing the calculator shows (RBI, bond sensitivity (secondary source)). PVBP turns the same idea into money: multiply it by the number of 100s of face value you hold.

Duration grows with the time to maturity and falls as the coupon rises, because higher coupons bring more of the value back earlier. The table shows both effects, computed with the same calculator:

Macaulay duration in years at a 6% yield, coupons paid twice a year
Years to maturity	0% coupon	3% coupon	6% coupon	9% coupon
2	2.00	1.95	1.91	1.88
5	5.00	4.65	4.39	4.19
10	10.00	8.50	7.66	7.13
30	30.00	16.54	14.25	13.31

A zero-coupon bond’s duration equals its maturity. For coupon bonds, duration rises much more slowly than maturity: at a 6% yield even a 30-year bond with a 6% coupon has a Macaulay duration of about 14.25 years.

Which duration and convexity mistakes are common?

Macaulay duration in the price-change formula. The estimate uses modified duration; Macaulay duration is a time in years.
The wrong compounding. For half-yearly coupons, divide Macaulay duration by one plus half the yield, not by one plus the whole yield.
Duration left in periods. Weighting by coupon periods gives a duration in periods; with half-yearly coupons, divide it by 2 to get years (RBI, bond sensitivity (secondary source)).
Basis points as whole numbers. A change of 50 basis points is 0.005 in the formula, not 50 or 0.5.
Mixing convexity conventions. Some texts include the ½ in the convexity figure or quote it in periods squared, so the same bond can show a convexity half as large or, with half-yearly coupons, four times as large. Use each figure with its own formula.
Confusing the spreadsheet functions. Excel’s DURATION returns Macaulay duration and MDURATION modified duration, both for a face value of 100 (Microsoft, Excel DURATION function) (Microsoft, Excel MDURATION function).

How do the results compare with a spreadsheet?

Microsoft’s DURATION example is an 8% bond with half-yearly coupons at a 9% yield, settled on 1 July 2018 and maturing on 1 January 2048; the function returns 10.9191453 (Microsoft, Excel DURATION function). Both dates are coupon dates, so the bond has 29.5 years to run, and this calculator gives a Macaulay duration of 10.9191453 years. The MDURATION example, the same bond with 8 years to run, returns 5.736 (Microsoft, Excel MDURATION function); the calculator gives 5.7357. Between coupon dates the spreadsheet also counts accrued interest and days, which this calculator leaves out.

What are the limits of duration and convexity?

Fixed cash flows. Modified duration assumes the payments do not change when rates move. For callable bonds or mortgage-backed securities use effective duration from a pricing model; prepayment can even make convexity negative (OSFI, duration and value changes) (San Francisco Fed, cash flows and options (secondary source)). The bond duration and convexity guide works through an effective duration example.
One yield for every payment. The estimate moves the bond’s own yield in parallel; a change in the shape of the yield curve is not captured (IMF, convexity and curve assumptions).
Coupon dates only. Prices between coupon dates, accrued interest and day counts are not included.

Can you use these numbers in the exam?

This calculator supports study for the CFA® Program Level I, run by CFA Institute (CFA Institute, Level I exam), and the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP) (GARP, FRM Exam information). Our tools complement the exam bodies’ own material; they do not replace it. Which calculators you may use in the exam is set by CFA Institute for the CFA® Program and by GARP for the FRM® Exam, so check their current calculator rules; our page on which calculators are allowed in the exams sets out the models with their sources. The link under the result opens the online financial calculator with the keys for the price or yield; duration and convexity you work out with the formulas above.

BA II Plus is a trade mark of Texas Instruments, HP 12C a trade mark of HP and Excel a trade mark of Microsoft. None of these companies is connected with this page.

What this tool covers

Last updated

30 October 2026

Based on

The formulas for bond price, yield to maturity, duration and convexity (see the sources)
Your own bond data; the default values are illustrative

Includes

Price from the yield to maturity, or the yield from the price, on a coupon date, with 1, 2, 4 or 12 coupons a year
Macaulay and modified duration, convexity and the price value of a basis point
The price after a fall and a rise in yield: duration estimate, estimate with convexity and exact repricing
A price–yield chart, the cash flows with their weights and the worked steps

Not included

Prices between coupon dates, accrued interest and day-count conventions
Callable, putable and floating-rate bonds, and effective duration from a pricing model
Changes in the shape of the yield curve, credit spreads and taxes

Frequently asked questions

How do you calculate a bond’s price from its yield to maturity?

Discount each coupon and the face value at the yield per period – the yield to maturity divided by the coupons per year – and add them up. A 5% bond with half-yearly coupons and 10 years to maturity costs 92.56 per 100 at a 6% yield.

How is the yield to maturity calculated from the price?

There is no formula: the calculator searches for the yield at which the discounted coupons and face value equal the price. A 1,000 bond with a 4% annual coupon and six years to maturity, bought for 950, yields 4.98%.

What is the difference between Macaulay and modified duration?

Macaulay duration is the present-value-weighted average time to the cash flows, in years. Modified duration divides it by one plus the yield per period and estimates the percentage price change for a change in yield: 7.89 years and 7.67 for the 5% bond at 6%.

What does convexity add to the duration estimate?

Duration draws a straight line, but the price–yield curve bends. The convexity term, ½ × convexity × Δy², corrects for the bend; for a bond with fixed positive payments it raises the estimated price whether yields rise or fall.

Why does the estimate differ from the exact price change?

Duration and convexity are approximations around the starting yield, and the error grows with the size of the change. For the 5% bond and a rise of 100 basis points, the estimate with convexity is −7.31% and exact repricing −7.32%.

What is the price value of a basis point?

It is the price change for a change of one basis point (0.01 percentage points) in yield: modified duration × price × 0.0001. For the 5% bond at 6% it is 0.0709 per 100 of face value.

Can I use this online calculator in the exam?

No. It is a study aid that runs in a web page. Which calculators you may use in the exam is set by CFA Institute for the CFA® Program and by GARP for the FRM® Exam, so check their current calculator rules.

Does the calculator save my numbers?

No. The calculation runs in your browser. Your numbers appear only in the page address after the # sign, so you can share a result; they are not sent to us.

Sources

Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on the Time Value of Money. Accessed 29 October 2026.
Reserve Bank of India (educational primer, secondary source): Government Securities Market in India — A Primer: full price; §27, duration, PV01 and convexity. Accessed 29 October 2026.
International Monetary Fund (methods overview, secondary source): A Primer on Managing Sovereign Debt-Portfolio Risks — Macaulay and modified duration (Working Paper 2018/074). Accessed 29 October 2026.
Office of the Superintendent of Financial Institutions: Mortgage Insurer Capital Adequacy Test (2025) — §§5.1.1.4–5.1.1.8: modified and effective duration; dollar fair value change. Accessed 29 October 2026.
International Monetary Fund: Financial Soundness Indicators: Compilation Guide — Appendix VI, §§50–55: convexity and the term structure. Accessed 29 October 2026.
CME Group (market education, secondary source): How Can You Measure Risk in Treasuries? — Value of a basis point and modified duration. Accessed 29 October 2026.
Federal Reserve Bank of San Francisco (research overview, secondary source): Measuring Interest Rate Risk for Mortgage-Related Assets. Accessed 29 October 2026.
Microsoft Support: DURATION function – Macaulay duration for an assumed par value of 100, the weighted average of the present value of the cash flows; example: 8% coupon, 9% yield, half-yearly coupons, 1 July 2018 to 1 January 2048 gives 10.9191453. Accessed 30 October 2026.
Microsoft Support: MDURATION function – modified Macaulay duration for an assumed par value of 100; example: 8% coupon, 9% yield, half-yearly coupons, 1 January 2008 to 1 January 2016 gives 5.736. Accessed 30 October 2026.
CFA Institute: CFA Program Level I exam: format and topic weights. Accessed 26 October 2026.
Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.

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