How to calculate a bond's coupon payments and price
Take the bond in the answer above: $100,000 of face value, a coupon of 5.00% paid twice a year, 10 years to maturity and a market yield of 6.00%.
- Multiply the face value by the coupon rate to get the coupon for a whole year, then divide by the number of coupons a year to get each payment. $100,000 × 5.00% = $5,000 a year $5,000 ÷ 2 = $2,500 a coupon
- Each half-year from 1 to 19 pays a coupon of $2,500. The last payment, at maturity in half-year 20, is the coupon plus the face value, which makes 20 coupons in all. $2,500 + $100,000 = $102,500 at maturity 20 × $2,500 = $50,000 in coupons
- The price is what those payments are worth today at the market yield: each is divided by (1 + 6.00% ÷ 2) once for every coupon period until it is paid. The coupons are an annuity and the face value a single sum, so the textbook price is: $2,500 × (1 − (1 + 6.00% ÷ 2)^−20) ÷ (6.00% ÷ 2) + $100,000 ÷ (1 + 6.00% ÷ 2)^20 = $92,561.26
- This calculator discounts one period at a time from maturity: it divides by (1 + 6.00% ÷ 2), rounds to the cent, and adds each coupon on its date. One period before maturity, the last payment of $102,500 is worth this, plus the coupon paid on that date: $102,500 ÷ (1 + 6.00% ÷ 2) = $99,514.56 $99,514.56 + $2,500 = $102,014.56
- Carried back to today the same way, the payments are worth $92,561.26, the same figure the formula gives.
- The price is below the face value because the 6.00% yield is above the 5.00% coupon: part of the return is the gap between the price paid and the face value repaid at maturity. $100,000 − $92,561.26 = $7,438.74 below face value
That is where the $2,500 coupons and the price of $92,561.26 in the answer come from.
Price and yield are one number seen from two sides
A bond promises a fixed set of payments: a coupon every period and the face value at the end. The price is what those promises are worth today at some rate of return, and the yield to maturity is the rate of return implied by a price. Fix either one and the other follows, which is why this page runs in both directions and why the answer to "what is this bond worth" is always "at what yield".
The relationship is the one thing worth taking away: price and yield move in opposite directions, always. When market rates rise, a bond promising yesterday's coupon has to get cheaper before anyone will buy it, and when rates fall the same bond becomes dearer. Nothing about the bond itself has changed.
Above par, below par
A bond trading above its face value is on a premium and one trading below is on a discount, and both are simply the market reconciling a fixed coupon with a different prevailing yield. A buyer at a premium gets back less at maturity than they paid, and the extra coupon income makes up the gap; a buyer at a discount gets back more, which is part of their return.
This is where the current yield misleads. A year of coupons over the price paid is a real and useful number, but it counts only the income and not the pull towards face value. On a bond bought at a premium the current yield always flatters the return, and on one bought at a discount it always understates it. The yield to maturity counts both, which is why it is the headline here.
What "on a coupon date" means and why it matters
This page assumes settlement falls on a coupon date with a whole number of periods left to run. Real trades settle between coupon dates, and then the buyer owes the seller the interest accrued since the last one; the price with that added is what actually changes hands. Working out accrued interest needs a day-count convention, and those differ by market and by instrument.
Rather than pick one convention and let it look authoritative, the page states the assumption next to the answer. For a bond paying twice a year on a coupon date, the formula used here is the one the United States Treasury sets out in its own appendix for exactly that case, and the worked example printed there is one of this project's tests. The other coupon frequencies divide the yield the same way, which is the market convention rather than anything that appendix says.
Common questions
What should I put for coupons a year?
What the issuer actually pays. Twice a year is the convention for United States Treasury notes and bonds and for most corporate issues there; once a year is usual for many European government bonds. It matters more than it looks, because the yield is divided by that number to get the rate for one period.
Is the yield here comparable to the yield I see quoted?
If the quote is a yield to maturity on the same coupon frequency, yes. Be careful with quotes on a different basis: an annual effective figure and a semiannual nominal one describe the same bond with different numbers, which is why both appear on this page. Yields quoted as a bond equivalent, or on a bill using a discount basis, are different conventions again.
Why is there no accrued interest, call date or credit rating?
Each would need something this page does not have. Accrued interest needs a day-count convention and two dates. A call feature turns one certain stream into several possible ones, and the yield to worst that follows is a different calculation. Credit risk is not arithmetic at all. The page lists what it leaves out beside the answer rather than implying it modelled them.
Does this work for a zero coupon bond?
Yes: set the coupon rate to nothing and the price becomes the face value discounted over the periods you gave, which is exactly what a zero coupon bond is.
Is anything I type stored?
No. The calculation happens on the server as part of rendering the page, nothing is written down, and the access log for this site drops the query string precisely so that the numbers you enter are never recorded.
Sources
No figure on this page is read from a table, so there is nothing to source. What these documents define is the convention behind the arithmetic, and the regulated measures this calculator deliberately does not compute:
How this is worked out
The figures are arithmetic on the numbers you type, and nothing here is a forecast of what an investment will actually do.
The price is the present value of every coupon and the face value, discounted one coupon period at a time with the annual yield divided by the number of coupons a year inside the same division. For a bond paying twice a year that is the formula the United States Treasury sets out in its own appendix for a security priced on a coupon date, and the worked example printed there is one of this calculator's tests. For the other coupon frequencies the same division is a convention rather than a rule anybody published. The yield is the same calculation run backwards, by halving a bracket until two neighbouring rates are left.
What this page does not model, in full:
- Accrued interest between coupon dates, and the day-count convention behind it
- Call and put features, and any early redemption
- Credit risk, in other words the chance of not being paid
- Tax of every kind, including tax on interest and on a gain
- An odd first or last period, where money does not land on the grid
- Inflation, so every figure is in today's money at face value