How a bond is priced
A bond pays a fixed coupon every period and returns its face value at maturity. Its price is what all those payments are worth today, discounted at the yield the market asks for. When the yield is higher than the coupon rate, the bond sells below face value, at a discount. When the yield is lower, it sells above face value, at a premium.
This bond calculator has two parts. The first assumes today is a coupon date: enter any four values and it finds the fifth, the price, the face value, the yield to maturity, the time to maturity or the coupon. The second prices a bond bought between coupon dates, with the clean price, the dirty price and the accrued interest.
The bond price formula
Price = C / (1 + i) + C / (1 + i)2 + … + (C + F) / (1 + i)n, or in short Price = C × (1 − (1 + i)−n) / i + F / (1 + i)n, where C is the coupon per period, F the face value, i the yield per period and n the number of coupon periods left. For a semiannual bond, C is half the annual coupon, i is half the yield and n is twice the years.
Example
A 3-year bond with a $100 face value and a 5% annual coupon pays $5 a year. At a 6% yield it costs $5 / 1.06 + $5 / 1.062 + $105 / 1.063 = $97.3270, a discount because the coupon is below the yield. With semiannual coupons of $2.50, the price is $97.2914.
Bond price by yield and term
The price of a bond with a $100 face value and a 5% annual coupon:
| Yield | 3 years | 10 years | 30 years |
|---|---|---|---|
| 3% | $105.6572 | $117.0604 | $139.2009 |
| 4% | $102.7751 | $108.1109 | $117.2920 |
| 5% | $100.0000 | $100.0000 | $100.0000 |
| 6% | $97.3270 | $92.6399 | $86.2352 |
| 7% | $94.7514 | $85.9528 | $75.1819 |
At a 5% yield the bond trades at face value. The longer the term, the more the price moves when the yield changes, which is why long bonds carry more interest rate risk.
Yield to maturity and current yield
- Yield to maturity (YTM): the rate that makes the payments worth the price. It has no closed formula, so the calculator finds it by trial. For the example bond at $97.3270 it is 6%.
- Approximate YTM: (C + (F − P) / n) / ((F + P) / 2) = 5.971%, close to the exact value.
- Current yield: the annual coupon divided by the price, $5 / $97.3270 = 5.137%. It ignores the gain at maturity.
- Coupon rate: the annual coupon divided by the face value, 5%. It never changes.
Duration: how much the price moves
Duration measures how sensitive a bond's price is to the yield. The Macaulay duration is the average time until the payments arrive, weighted by their value today: 2.857 years for the example bond. The modified duration is Macaulay duration ÷ (1 + i) = 2.696, so a 1% rise in the yield cuts the price by about 2.70%, to $94.7034. The exact price at 7% is $94.7514.
Zero-coupon bonds
A zero-coupon bond pays no coupons, only the face value at maturity, so its price is F / (1 + i)n. A 3-year zero with a $100 face value at a 6% yield costs $83.9619. To use the calculator, set the coupon to 0. Treasury bills and STRIPS work this way; the present value calculator does the same math for any future amount.
Clean price, dirty price and accrued interest
Between coupon dates the seller has earned part of the next coupon. The buyer pays it to the seller as accrued interest: accrued interest = coupon per period × days since the last coupon ÷ days in the period. The dirty price is what the buyer actually pays; the clean price is the dirty price minus the accrued interest, and it is the price bonds are usually quoted at.
For example, a $1,000 bond with a 5% semiannual coupon pays $25 every six months. 90 days into a 180-day period, the accrued interest is $25 × 90 ÷ 180 = $12.50.
| Day count | Used for |
|---|---|
| 30/360 | U.S. corporate and municipal bonds |
| Actual/Actual | U.S. Treasury notes and bonds |
| Actual/360 | Money market instruments |
| Actual/365 | Some non-U.S. bonds |
Types of bonds and taxes
- Treasury bonds and notes: issued by the U.S. government; the interest is free of state and local income tax.
- Municipal bonds: issued by states and cities; the interest is usually free of federal income tax.
- Corporate bonds: issued by companies; they pay more for the credit risk, and the interest is fully taxable.
- TIPS: Treasury bonds whose face value rises with inflation.
To compare a tax-free municipal bond with a taxable one, use the tax-equivalent yield = muni yield ÷ (1 − tax rate). A 3.5% muni in the 24% bracket is worth as much as a 4.605% taxable bond. For a fixed-rate deposit with no price risk, see the CD calculator, and for other time value of money problems the finance calculator.