Calculate a bond's price from its yield to maturity, or the YTM from its price, with current yield, Macaulay and modified duration and convexity.
At a 6% yield the bond is worth $925.61, a discount to its $1,000.00 face value. A one-percentage-point rise in the yield would cut the price by about 7.67%.
| Period | Years | Coupon | Principal | Discount factor | Present value |
|---|---|---|---|---|---|
| 1 | 0.5 | $25.00 | $0.00 | 0.9709 | $24.27 |
| 2 | 1 | $25.00 | $0.00 | 0.9426 | $23.56 |
| 3 | 1.5 | $25.00 | $0.00 | 0.9151 | $22.88 |
| 4 | 2 | $25.00 | $0.00 | 0.8885 | $22.21 |
| 5 | 2.5 | $25.00 | $0.00 | 0.8626 | $21.57 |
| 6 | 3 | $25.00 | $0.00 | 0.8375 | $20.94 |
| 7 | 3.5 | $25.00 | $0.00 | 0.8131 | $20.33 |
| 8 | 4 | $25.00 | $0.00 | 0.7894 | $19.74 |
| 9 | 4.5 | $25.00 | $0.00 | 0.7664 | $19.16 |
| 10 | 5 | $25.00 | $0.00 | 0.7441 | $18.60 |
| 11 | 5.5 | $25.00 | $0.00 | 0.7224 | $18.06 |
| 12 | 6 | $25.00 | $0.00 | 0.7014 | $17.53 |
Valued on a coupon date, so there is no accrued interest and the clean and full prices are equal.
Fabozzi's convexity measure in years². A 1-point yield rise changes the price by about −D_mod × Δy + ½ C × Δy².
A bond's price is the present value of its coupons and its face value, each discounted at the yield to maturity for the coupon period. With semi-annual coupons, a 5% coupon on 1,000 pays 25 every six months and the annual yield is halved per period, the bond-equivalent convention. Solving the same equation for the rate turns a market price into a yield. Duration is the present-value-weighted average time until the cash flows arrive; modified duration and convexity estimate how far the price moves when the yield changes.
With the defaults, a 10-year 5% semi-annual bond with a face value of 1,000, priced to yield 6%, is worth 925.61: a discount, because its coupon is below the yield. Its Macaulay duration is 7.895 years and its modified duration 7.665, so a 0.01% rise in yield lowers the price by about 0.71.
The price assumes settlement on a coupon date, so there is no accrued interest. Credit risk, call features and taxes are not modeled.
Checked against: Python decimal (prec 50): cash flows discounted term by term; Fabozzi convexity Σt(t+1)PV/((1+y)²·P·f²); current yield 50/925.6126 = 5.4018278%
Checked against: Microsoft DURATION documentation (settlement 2018-07-01 is a coupon date, maturity 2048-01-01): 10.9191453; Python decimal agrees
Checked against: Microsoft MDURATION documentation (2008-01-01 to 2016-01-01, a coupon date): 5.73567; Python decimal price 94.382992
Checked against: A bond whose coupon rate equals its yield prices at par (definition)
Discount every coupon and the face value at the yield per period and add them. A 10-year 5% bond paying 25 twice a year, at a 6% yield (3% per half-year), is worth 371.94 for its 20 coupons plus 1,000 ÷ 1.03^20 = 553.68 for the face value: 925.61 in total. Excel's =PV(3%, 20, -25, -1000) gives the same price.
Current yield is the annual coupon divided by the price; yield to maturity also counts the gain or loss to face value at maturity and the timing of every payment. A 5% bond bought at 925.61 has a current yield of 50 ÷ 925.61 = 5.40% but a yield to maturity of 6%, because the buyer also gains 74.39 when the bond repays 1,000.
The coupons are fixed, so a higher yield discounts the same payments more heavily. The default 10-year 5% bond is worth 1,081.76 at a 4% yield, exactly 1,000 at 5%, 925.61 at 6% and 857.88 at 7%. When the yield equals the coupon rate the bond trades at par; above it, at a discount; below it, at a premium.
Modified duration is the approximate percentage change in price for a one-percentage-point change in yield. The default bond's modified duration is 7.665, so a rise from 6% to 7% should cut the price by about 7.67%; the actual fall, from 925.61 to 857.88, is 7.32%, because convexity cushions large moves. Macaulay duration, 7.895 years here, is the weighted average time to the cash flows.
Every six months. The US Treasury issues notes with terms of 2, 3, 5, 7 and 10 years and bonds with terms of 20 and 30 years, and both pay interest semi-annually, which is why this page defaults to two coupons a year. Choose annual, quarterly or monthly coupons for bonds that pay on another schedule.
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “10-year 5% semi-annual bond at 6%” is checked against Python decimal (prec 50): cash flows discounted term by term; Fabozzi convexity Σt(t+1)PV/((1+y)²·P·f²); current yield 50/925.6126 = 5.4018278%.
Microsoft Excel DURATION function; Microsoft Excel MDURATION function; Fabozzi — Bond Markets, Analysis, and Strategies, ch. 2 and 4 (pricing, duration, convexity).
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
7 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.
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