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.
Actualizado Ejemplos verificados: 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 | Años | Coupon | Capital | 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.
Fuente de comprobación: 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%
Fuente de comprobación: Microsoft DURATION documentation (settlement 2018-07-01 is a coupon date, maturity 2048-01-01): 10.9191453; Python decimal agrees
Fuente de comprobación: Microsoft MDURATION documentation (2008-01-01 to 2016-01-01, a coupon date): 5.73567; Python decimal price 94.382992
Fuente de comprobación: 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.
La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 7. Por ejemplo, «10-year 5% semi-annual bond at 6%» se comprueba con 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).
Solo para planificar. Prestamistas, autoridades fiscales y mercados aplican sus propios redondeos, comisiones y reglas; confirma las cifras con ellos antes de comprometerte.
Esta calculadora incluye 7 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.
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