Bond Price Calculator

Bond price = Σ [ coupon × (1 + i)^−k ] + face value × (1 + i)^−n (k = 1 … n; i = yield ÷ periods per year; n = years × periods per year)

Face value

The amount repaid at maturity and the base the coupon is calculated from, which must be greater than zero

Coupon rate

The annual rate the bond pays on its face value; it is split evenly across the payment periods of the year rather than accrued by actual days

Yield to maturity

The annual rate you discount every payment at — the return the bond offers at the price you are being asked to pay

Years to maturity

Whole years remaining; the number of payments is this multiplied by the number of coupons paid each year

Coupon frequency

How many times a year the coupon is paid: annually, twice a year, quarterly or monthly

Coupon payment

Face value times the coupon rate divided by the number of payments a year — the amount per period, not per year

Current yield

The annual coupon divided by the price, which is not the yield to maturity because it ignores the gain or loss at maturity

Premium or discount

The price minus the face value: positive when the bond costs more than it repays, negative when it costs less

Use it to check a price you have been quoted, to see what a change in the yield would do to a bond you already hold, or to understand why the same bond appears at two different prices in two places. It is also the arithmetic behind a fact that surprises people: the price of a bond you already own moves in the opposite direction to its yield, so an investor holding to maturity is unaffected by a rate rise while one who has to sell is not. Price it at several yields and the pattern becomes concrete — the reference table below walks the same bond from a zero yield, where it is worth face value plus every coupon it will ever pay, down through the point where the yield equals the coupon rate and the price is exactly par, and on to an eight percent yield where the price is deep in discount territory.

1. 1,000 face value, 4% coupon, 10 years, priced at a 5% yield

   1. Coupon each period: 1,000 × 4% ÷ 2 = 20
   2. Periods: 10 years × 2 = 20, and the yield per period is 5% ÷ 2 = 2.5%
   3. Present value of the 20 coupons: 20 × (1 − 1.025⁻²⁰) ÷ 0.025 = 311.78
   4. Present value of the face value: 1,000 × 1.025⁻²⁰ = 610.27
   5. Price: 311.78 + 610.27 = 922.05, a discount of 77.95
   6. Current yield: 40 ÷ 922.05 = 4.34%

   This is the page's default and it is the case where the yield is above the coupon rate, so the bond sells below its face value. The discount is not a penalty, it is the mechanism that raises your return from the 4 percent the coupon pays to the 5 percent the market is offering: you pay 922.05 for a stream worth 1,000 at maturity, and the 77.95 you gain at the end makes up the difference. Note that the coupon payment is 20, which is the half-yearly amount, and that the 4.34 percent current yield is above the 4 percent coupon rate because the price you paid is below par.

2. The same bond priced at par: yield equal to the coupon rate

   1. Coupon each period: 1,000 × 4% ÷ 2 = 20
   2. Yield per period: 4% ÷ 2 = 2%
   3. Present value of the 20 coupons: 20 × (1 − 1.02⁻²⁰) ÷ 0.02 = 327.03
   4. Present value of the face value: 1,000 × 1.02⁻²⁰ = 672.97
   5. Price: 327.03 + 672.97 = 1,000.00 exactly

   When the yield equals the coupon rate the two present values are forced to add up to the face value, and this row is the axis the whole table turns on — above it the bond costs more than it repays and below it costs less. It is also the row that shows the current yield is not the same thing as the yield to maturity: both are 4 percent here, but at any other price they diverge, because the current yield ignores the gain or loss you make when the bond is redeemed.

3. Paying twice a year instead of once changes the price slightly

   1. Coupon each period: 1,000 × 4% ÷ 1 = 40, paid ten times
   2. Yield per period: 5% ÷ 1 = 5%
   3. Present value of the ten coupons: 40 × (1 − 1.05⁻¹⁰) ÷ 0.05 = 308.87
   4. Present value of the face value: 1,000 × 1.05⁻¹⁰ = 613.91
   5. Price: 308.87 + 613.91 = 922.78, a discount of 77.22

   Same bond, same yield, same maturity, one number changed. Paying the coupon once a year instead of twice gives 922.78 rather than 922.05, because money received later is discounted further. The gap is small in this case, but the coupon payment column is not: it reads 40 here against 20 in the semi-annual case, and both are the amount per period. That is the single most misread figure on this page, which is why the label says so.

This is the clean version of bond pricing, and real settlement is messier in three specific ways. It prices a bond whose next coupon is a full period away, so it does not handle accrued interest — the seller is owed the part of the current coupon that has already elapsed, and a real invoice adds that on top. It assumes every remaining coupon is a full period, so a bond with an odd first or last coupon, which is common for new issues, is not modelled. It prices straight bonds only, so a bond that the issuer can redeem early or that you can put back has a price that depends on those options as well as on the yield. Beyond those, the calculation uses whole years and evenly spaced periods rather than actual calendar dates, so it does not match the day-count conventions used to settle real trades to the cent. It ignores tax on the coupon and on the capital gain, ignores transaction costs, and says nothing about credit risk — the yield you type is taken as given, and a bond that might not be repaid should be priced at a higher one.

How do I calculate the price of a bond?

Discount every payment the bond will make back to today at the yield to maturity and add them up. For a 1,000 face value bond paying 4 percent twice a year with ten years left, priced at a 5 percent yield: the twenty coupon payments of 20 are worth 311.78 today, the 1,000 face value is worth 610.27, and the price is 922.05.

Why is the price below the face value?

Because the yield you are pricing at is above the coupon rate the bond pays. The bond only pays 4 percent on its face value, so nobody will give you 1,000 for it when 5 percent is available elsewhere; the price falls until the coupon plus the gain you make at maturity adds up to a 5 percent return. The difference between the price and the face value is the discount, and the same mechanism in reverse produces a premium when the coupon rate is above the yield.

What is the difference between current yield and yield to maturity?

Current yield is the annual coupon divided by the price you pay, so on the default case it is 40 divided by 922.05, which is 4.34 percent. Yield to maturity is the rate you would earn if you held the bond to the end, and it accounts for the 77.95 you gain when a 922.05 bond repays 1,000. On a bond priced at par the two are equal; when the bond is at a discount the current yield is the lower of the two, and it is the higher of the two when the bond is at a premium.

Is the coupon payment shown per year or per period?

Per period. A 1,000 face value bond with a 4 percent coupon pays 40 a year in total, and that 40 is split across however many payments the year holds: 20 twice a year, 10 four times a year, 40 once a year. The field on this page is the amount of a single payment, which is why the same 4 percent bond shows 20 in the semi-annual case and 40 in the annual one.

Why does the payment frequency change the price?

Because money paid earlier can be reinvested earlier, so a bond that pays you sooner is worth slightly more at the same yield. On the default bond, paying twice a year gives 922.05 and paying once a year gives 922.78, a difference of 73 cents; quarterly and monthly payments push the price a little lower again, to 921.68 and 921.43. The effect is small at these rates but it grows with the coupon and the maturity.

What happens if the yield to maturity is zero?

Every discount factor becomes 1 and the price is simply the sum of everything the bond will ever pay: the face value plus every coupon. On a 1,000 face value bond paying 4 percent twice a year with ten years left, that is 1,000 plus twenty payments of 20, which is exactly 1,400. This case is worth knowing because the formula written as a closed form has to special-case it, and the arithmetic on this page does not need to.

• 31 CFR Part 356, Appendix B — Formulas and Tables: the present value factors and the price formula for Treasury notes and bonds — U.S. Department of the Treasury rule, via the Electronic Code of Federal Regulations (United States)
• Understanding Pricing and Interest Rates — why the price of a note or bond moves in the opposite direction to its yield — U.S. Department of the Treasury, Bureau of the Fiscal Service (TreasuryDirect), United States
• Current Yield — Investor.gov glossary (the ratio of the interest paid on a bond to its market price) — U.S. Securities and Exchange Commission, Investor.gov (United States)