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.
By AdminDrafted with AI writing toolsUpdated Checked examples: 7
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.
A bond's price is its coupons plus its face value, each discounted at the yield per coupon period. A 10-year, 1,000 bond paying a 5% coupon twice a year is worth 925.61 at a 6% yield, below face value because the coupon is lower than the yield.
Semi-annual payments halve both the coupon and the yield:
The 20 coupons are worth 25 × (1 − 1.03⁻²⁰) ÷ 0.03 = 371.94 today, and the face value is worth 1,000 ÷ 1.03²⁰ = 553.68. Together they give 925.61. In Excel, =PV(3%, 20, -25, -1000) returns the same price.
Only the annual yield changes in these rows; the bond is the same 10-year 5% semi-annual issue.
| Annual yield (%) | Bond price |
|---|---|
| 4 | 1081.76 |
| 6 | 925.61 |
| 7 | 857.88 |
The coupons are fixed, so a higher yield discounts the same payments more heavily and the price falls.
At 5%, equal to the coupon rate, the bond is worth exactly 1,000. A yield below the coupon rate gives a premium; a yield above it gives a discount.
Going from a price to a yield takes trial and error, as with IRR: find the per-period rate that discounts the payments to the price, then multiply by the payments per year.
A price of 925.61 gives 6.000037%, not exactly 6%, because that price is rounded to cents.
That annual figure is the bond-equivalent yield. Two other yields describe the same bond:
Macaulay duration is the present-value-weighted average time to the cash flows: 7.895 years for this bond, the measure Microsoft's DURATION function returns.
Modified duration divides it by 1 + y per period, 7.895 ÷ 1.03 = 7.665, the approximate percentage price change for a one-point move in yield.
DV01 is the price change per 0.01% of yield: 7.665 × 925.61 × 0.0001 = 0.71. Repricing at 6.01% gives 924.90, a fall of 0.71.
For a full one-point rise, duration alone predicts −7.665%, or 854.66. Adding convexity (71.79) gives −7.665% + ½ × 71.79 × 0.01² = −7.31%. The exact price at 7% is 857.88, a fall of 7.32%.
A zero pays only its face value, so its price is F ÷ (1 + y)ⁿ and its Macaulay duration equals its maturity.
A 10-year zero bought at 500 that repays 1,000 yields 2^(1/10) − 1 = 7.18% with annual compounding, and its modified duration is 9.33.
The bond is valued on a coupon date, so there is no accrued interest and the clean and full prices are equal. Between coupon dates, quoted prices usually exclude the interest accrued since the last coupon.
Every payment is assumed to arrive on schedule. Default risk, call features and taxes are left out: the result values the promised payments at the yield you enter.
For an uneven series of cash flows, use the NPV and IRR calculator. The time value of money calculator handles a single lump sum at any compounding.
Checked against: 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; exact decimal arithmetic agrees
Checked against: Microsoft MDURATION documentation (2008-01-01 to 2016-01-01, a coupon date): 5.73567; exact decimal arithmetic 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.
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
This calculator includes 7 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.