Bond Price and Yield Calculator

What this calculator answers

What return a bond gives if you buy it at a given price and hold it to maturity, or what the bond is worth if the market wants a given yield. Enter the face value, coupon rate, how often coupons are paid and the time left, then either the price or the yield. You get the yield to maturity in two conventions, the current yield, whether the bond trades at a premium or a discount, its duration, and a table of every payment with its value today.

How to use it

• Find: Yield from price when you know what the bond costs, Price from yield when you know the yield you want. Switching carries the answer across, so both directions describe the same bond.
• Face value: the amount repaid at maturity, also called par value, for example $1,000.
• Annual coupon rate: the interest rate printed on the bond, per year, in percent. Type 4.25 for 4.25%. Enter 0 for a zero-coupon bond.
• Coupons paid: how many coupons a year. Twice a year (semiannual) is the most common for corporate and government bonds, and US Treasury notes and bonds pay every six months.
• Time to maturity: years and months until the bond repays its face value, counted from a coupon date. It must be a whole number of coupon periods: with coupons twice a year, 7 years 6 months works but 7 years 3 months doesn’t.
• Price (when finding the yield): the price for one bond on a coupon date, without accrued interest. Keep the toggle on % of the face value to type a quote such as 96.5, or switch to $ amount to type dollars.
• Yield to maturity (when finding the price): the yearly yield as bonds are quoted, for example 5.5. It can be negative.

The Try buttons load the other cases worked through below. To compare two bonds, press Save for comparison, change the inputs and save again: each saved bond shows how its price, yields and duration differ from the first.

How to calculate yield to maturity

The yield to maturity is the rate that makes the present value of the bond’s remaining payments equal its price. Each coupon and the final face value are discounted back to today at the yield per period $j$:

• $P$ is the price in dollars.
• $F$ is the face value, repaid at maturity.
• $f$ is the number of coupons a year, and $C = F \times \text{coupon rate} \div f$ is each coupon.
• $n$ is the number of coupon periods left: years to maturity × $f$.
• $j$ is the yield per coupon period.

$j$ can’t be isolated on one side of this equation, so the calculator solves it numerically with Brent’s method, starting from a range of yields that is sure to contain the answer and narrowing it to within 0.0000000001 percentage points per period. One Newton step then refines it, so the present values in the cash-flow table add up to the price to the cent even for a very large bond. A price always has exactly one yield: the price falls steadily as the yield rises.

The yearly figure is quoted bond-equivalent: the yield per period times the number of coupons a year. This is how US Treasury notes and bonds are priced (a yield based on semiannual payments). The effective annual yield, $(1 + j)^f - 1$, counts the compounding within the year, so it is the one to line up against an APY on a savings account or CD.

Current yield vs. yield to maturity

Current yield is the coupon income for a year divided by the price. It ignores the gain or loss you make when the bond repays its face value, so it differs from the yield to maturity whenever the price isn’t equal to face value. It tells you the income a bond pays for each dollar invested today; the yield to maturity tells you the whole return if you hold it to the end.

A quick check by hand

A rough estimate of the yield to maturity divides the average yearly income by the average amount invested: (annual coupon + (face value − price) ÷ years) ÷ ((face value + price) ÷ 2). The calculator shows it in the steps next to the exact answer. It comes within 0.011 points of the exact yield in the worked example below but misses the zero-coupon example by 0.090 points, so treat it as a sanity check only.

Worked example: a 4.25% bond quoted at 96.50

A $1,000 bond pays a 4.25% coupon twice a year and matures in 7 years 6 months. It is quoted at 96.50, which means 96.5% of face value.

1. Price in dollars: 96.5% × $1,000 = $965.00.
2. Coupon per period: C = $1,000 × 0.0425 ÷ 2 = $21.25, or $42.50 a year.
3. Periods left: n = 7.5 years × 2 = 15.
4. Solve $965.00 = $21.25 × (1 − (1 + j)^−15) ÷ j + $1,000 × (1 + j)^−15. The answer is j = 0.02405726, or 2.405726% per 6 months.
5. Check: the 15 coupons are worth $264.94 today and the $1,000 at maturity $700.06, which add up to $965.00.
6. Yield to maturity: 2.405726% × 2 = 4.811% a year.
7. Effective annual yield: 1.02405726² − 1 = 4.869%.
8. Current yield: $42.50 ÷ $965.00 = 4.404%.
9. The shortcut gives ($42.50 + $35.00 ÷ 7.5) ÷ ($1,965 ÷ 2) = 4.801%, 0.011 points below the exact answer.

In a spreadsheet, =RATE(15, 21.25, -965, 1000) * 2 gives the same 4.811%.

How to calculate a bond’s price from its yield

Discount each payment at the yield per period and add them up. For the same bond at a 5.5% yield to maturity:

1. Yield per period: j = 0.055 ÷ 2 = 0.0275.
2. The coupons are worth $21.25 × (1 − 1.0275^−15) ÷ 0.0275 = $258.33.
3. The face value is worth $1,000 × 1.0275^−15 = $665.69.
4. Price: $258.33 + $665.69 = $924.02, or 92.402% of face value.

In a spreadsheet, =PV(0.055/2, 15, -21.25, -1000) returns $924.02. At that price the current yield is 4.599% and the effective annual yield 5.576%.

Premium, discount and par bonds

A bond’s price sits above or below face value depending on whether its coupon rate is above or below the yield the market asks for. For a bond that pays coupons, the three yields then always line up in the same order:

A discount bond’s yield to maturity is higher than its coupon rate because you also gain the difference between the price and face value by maturity. A premium bond’s is lower because you lose the premium. The result shows which case applies to your bond and why.

Why bond prices fall when yields rise

A bond’s payments are fixed when it is issued. If new bonds start paying more, nobody will pay full price for yours, so its price drops until its yield matches what the market now wants. When market yields fall, the same fixed payments become more valuable and the price rises. The calculator’s Price if yields change table reprices your bond at yields 0.5, 1 and 2 points either side of yours.

Duration: how much the price moves when yields change

Duration puts a number on that sensitivity. Macaulay duration is the average time until you receive the bond’s payments, each weighted by its share of the price. Modified duration is Macaulay duration divided by (1 + j), and it is roughly the percentage the price changes for a 1-point change in yield.

The example bond has a Macaulay duration of 6.48 years and a modified duration of 6.33. So if yields rose 1 point, to 5.811%, duration predicts a fall of about 6.33%. Repricing the bond exactly gives $906.16, a fall of $58.84 or 6.10%. If yields fell 1 point, the price would rise 6.57% to $1,028.37. Duration draws a straight line, while the real price follows a curve that bends upward (convexity), so the estimate overstates falls and understates rises, more so for bigger moves. The chart under the table shows that curve with your bond marked on it.

Longer maturities and lower coupons mean higher durations, and a zero-coupon bond’s Macaulay duration equals its time to maturity.

Zero-coupon bonds

A zero-coupon bond pays nothing until maturity, so its price is just the face value discounted: $P = F \div (1 + j)^n$. A $1,000 zero maturing in 10 years at a 4% yield, compounded twice a year like coupon bonds, costs $1,000 ÷ 1.02^20 = $672.97. Its Macaulay duration is exactly 10 years, and a 1-point rise in yield would cut its price 9.32%, to $610.27. If you want a yield compounded once a year instead, set coupons to annual: at 4% the price is $1,000 ÷ 1.04^10 = $675.56.

Can yield to maturity be negative?

Yes, when the price is more than everything the bond will still pay. A $1,000 bond with a 0.5% coupon paid twice a year and 2 years left pays four $2.50 coupons and the face value, $1,010.00 in total. At a price of 102 ($1,020.00) the buyer loses money by holding to maturity, and the yield to maturity is −0.494%. The calculator accepts negative yields and says so instead of showing an error.

Reading the result

• The headline is what you chose to find: the yield to maturity (bond-equivalent) or the price, with the other one in the line under it.
• The note under it says whether the bond is at a discount, a premium or par (or has a negative yield) and why its yields differ.
• The tiles give the effective annual yield, current yield, coupon, the discount or premium, duration, and what a 1-point rise in yields would do to the price.
• How this was calculated repeats the formulas with your numbers, including the shortcut estimate and the spreadsheet formula.
• Cash flows and their present values lists every payment, its discount factor and its value today. The present values add up to the price, and each payment’s share of the price is the weight Macaulay duration uses. The table downloads as a CSV file.
• Price if yields change compares the exact new price with the duration estimate.
• Yield to maturity at other prices (when finding the yield) solves the yield again at prices 1, 2 and 5 points either side of yours, useful before placing an order at a limit price.
• Continue in the APY Calculator carries the yield and the coupon frequency across, so you can see the same yield as an APY next to a savings account or CD.

What this calculator doesn’t cover

• Accrued interest and settlement between coupon dates. A buyer between coupon dates also pays the seller the interest earned since the last coupon, on top of the quoted price (the price without it is often called the clean price, and the total the dirty price). That needs the settlement date and a day-count convention, which this calculator doesn’t model. Its prices are for a coupon date, when no interest has accrued, so a broker’s yield for a trade between coupon dates can differ from this one.
• Calls. A callable bond may be repaid early; its yield to call uses the call date and call price instead of the maturity date and face value.
• Default and credit risk. The yield assumes every payment arrives in full and on time. A high yield often reflects doubt that it will.
• Taxes, commissions and markups, which lower the return you keep.
• Floating-rate and inflation-linked bonds, whose payments change over time.
• US savings bonds (EE and I bonds). They can’t be sold or transferred to anyone else, so they have no market price and aren’t valued this way. TreasuryDirect shows what they are worth.

The result is an educational estimate, not a quote, an offer or investment advice.

Common mistakes

• Typing a quote as dollars. A quote of 96.5 means 96.5% of face value, $965.00 for a $1,000 bond, not $96.50. The calculator points this out if a dollar price looks like a quote.
• Treating current yield as the return. On a discount or premium bond, current yield leaves out the gain or loss at maturity.
• Comparing a bond-equivalent yield with an APY. A 4.811% bond-equivalent yield compounded twice a year is 4.869% effective; compare that figure with a CD’s APY.
• Using a price with accrued interest in it. The “dirty” price someone pays between coupon dates includes interest the seller earned, which raises the price and lowers the yield you calculate.
• A time to maturity that doesn’t match the coupon dates. With coupons twice a year, 7.3 years is 14.6 periods; the calculator asks for a whole number and suggests the nearest, 15 periods (7 years 6 months).
• Reading duration as exact. It is a straight-line estimate. For moves of 1 or 2 points on long bonds, look at the exact prices in the table.