The bond price calculator above values a coupon bond the way bonds are actually valued: as the present value of every remaining cash flow, discounted at a required yield, with the discounting done at the coupon frequency rather than annually. It returns the clean price, the accrued interest, the dirty price you would actually settle, the current yield, and both modified duration and convexity so you can see how the price responds to a rate move.
Arb Digital publishes it as the counterpart to our YTM calculator, which solves the inverse problem — given a price, what yield does the bond deliver. The two use the same equation from opposite ends. Pricing from a yield is a direct calculation; solving for the yield from a price requires iteration, because the equation cannot be rearranged for the yield in closed form.
What This Bond Price Calculator Does
It builds the full cash flow schedule, discounts each payment at the periodic rate, and sums the result. Coupon frequency changes both the size of each payment and the number of discounting periods, so it is a real input rather than a cosmetic one — the same annual coupon rate produces a different price at semi-annual and annual frequency.
It also handles settlement between coupon dates, which most simple bond calculators skip. When you buy a bond partway through a coupon period, the seller is owed the interest that has accrued since the last payment. The quoted or clean price excludes that; the dirty price, also called the full or invoice price, includes it and is what actually settles.
Duration and convexity round out the picture. Price alone tells you what the bond is worth today; duration tells you how sharply that value moves when yields move, which is usually the more actionable number. Our bond yield calculator covers the yield measures side, and this page covers the valuation side.
How to Use It
- Enter face value and coupon rate. The coupon rate is annual even when payments are semi-annual; the tool divides it by the frequency.
- Set the coupon frequency. Semi-annual is the default because it covers US Treasuries and most corporate issues.
- Enter years to maturity and your required yield. The yield is nominal annual, compounded at the coupon frequency, which is the market convention.
- Set the accrual days. Zero prices the bond on a coupon date; any other value produces accrued interest and separates clean from dirty.
- Compare the price against face value. Above par means the coupon exceeds the required yield; below par means the reverse.
The Formula and How It's Calculated
The price on a coupon date is P = Σ C ÷ (1 + y÷f)t + F ÷ (1 + y÷f)n, summed over t from 1 to n, where C is the coupon per period, y is the annual required yield, f is the coupon frequency, F is face value and n is the number of remaining periods. The coupon stream is an ordinary annuity, so it can be collapsed to C × [1 − (1 + i)−n] ÷ i with i = y÷f.
Work the defaults through with the accrual days set to zero. A 1,000 face bond with a 5 per cent annual coupon paid semi-annually pays C = 25 per period, and ten years to maturity gives n = 20 periods. A 6 per cent required yield gives i = 0.03. Then 1.0320 = 1.806111, so the discount factor is 0.553676 and the annuity factor is (1 − 0.553676) ÷ 0.03 = 14.877475. The coupons are worth 25 × 14.877475 = 371.94 and the principal is worth 1,000 × 0.553676 = 553.68, for a total price of 925.61. The bond trades at a discount because its 5 per cent coupon is below the 6 per cent yield the market demands.
Between coupon dates the calculation shifts. The dirty price is the coupon-date price grown forward by the elapsed fraction w of the period: Dirty = P × (1 + i)w. Accrued interest is C × w, and the clean price is dirty minus accrued. With 60 days elapsed of a 182-day period, w = 0.32967, so the dirty price is 925.61 × 1.030.32967 = 934.68, accrued interest is 25 × 0.32967 = 8.24, and the clean price is 926.44.
Clean Price, Dirty Price and Why Both Exist
The distinction looks like an accounting nicety and is actually the reason bond prices are quotable at all. A bond's value rises steadily through a coupon period as the next payment approaches, then drops by the coupon amount the moment it is paid. If markets quoted that raw value, every bond's price chart would be a sawtooth, and comparing two bonds at different points in their coupon cycles would be meaningless.
Stripping out accrued interest removes the sawtooth. The clean price moves only in response to yield changes and credit perceptions, which is exactly what a quote should convey. The accrued interest is added back at settlement so the seller receives the interest they earned while holding the bond.
The practical consequences are worth naming. The cash leaving your account is the dirty price, so a purchase costs more than the quote implies — here 934.68 rather than 926.44 per bond. Accrued interest is generally taxable as interest income to the seller rather than as a capital gain, which matters at year end. And the gap widens the further into a coupon period you settle, reaching close to a full coupon just before a payment date.
Why Price and Yield Move in Opposite Directions
A bond's coupon is fixed at issue. If market yields rise afterwards, the only way an existing bond can compete with newly issued paper offering a higher coupon is to sell for less, so its price falls. If market yields fall, the existing bond's above-market coupon becomes valuable and its price rises. That inverse relationship is mechanical, not behavioural.
The relationship is also convex rather than linear. A one-point fall in yields raises the price by slightly more than a one-point rise lowers it, because the discounting is exponential. Convexity measures the size of that asymmetry, and it works in the bondholder's favour: more convexity means more upside from falling yields and less downside from rising ones, at the same duration.
Three things increase price sensitivity to yields: longer maturity, lower coupon and lower yield. A thirty-year zero-coupon bond has the maximum sensitivity for its maturity because every cash flow sits at the far end; a short high-coupon bond has the least. That is why the same one-point yield move produces a modest loss in a two-year note and a severe one in a long bond, a point the SEC's investor education material on Bonds makes in its description of how bonds work.
Reading Duration Properly
Macaulay duration is the weighted average time until you receive the bond's cash flows, weighted by the present value of each. Modified duration divides that by one plus the periodic yield and converts it into a sensitivity: the approximate percentage price change for a one-percentage-point change in yield.
At the defaults, modified duration comes out near 7.7 years, meaning a one-point rise in yields would cut the price by roughly 7.7 per cent. That estimate is a first-order approximation and degrades for large moves, which is where convexity comes in. The second-order estimate is −Dmod × Δy + ½ × convexity × Δy², and for a two-point move the convexity term is no longer negligible.
Two cautions. Duration assumes a parallel shift in the whole yield curve, which is not how curves usually move; short and long rates move by different amounts and sometimes in different directions. And duration says nothing about credit risk. A corporate bond can lose value because the issuer's creditworthiness deteriorated with no change in rates at all, and no duration figure will warn you about that.
Where Real Bonds Depart From This Model
Four features break the clean arithmetic above. Call and put provisions let the issuer or holder end the bond early, which truncates the cash flow schedule and caps the price appreciation of a callable bond as yields fall. Day-count conventions differ between markets — 30/360, actual/actual, actual/360 — and each produces a slightly different accrued interest figure for identical dates, which is why the tool takes the period length as an input rather than assuming one. Credit risk means the promised cash flows are not certain, and the required yield you type should already embed a spread for that. Reinvestment matters because yield to maturity implicitly assumes every coupon is reinvested at the same yield, which will not happen.
Sovereign issuance illustrates the conventions clearly: TreasuryDirect's page on Treasury Bonds notes that they are issued in 20 or 30-year terms, pay interest every six months until maturity, and carry a rate fixed at auction that does not vary over the life of the bond. For the discounting arithmetic on its own, our present value calculator and compound interest calculator cover the mechanics, and the tax equivalent yield calculator handles comparisons between taxable and tax-exempt issues.
Arb Digital builds calculators and technical content for firms whose audiences will notice if the day-count is wrong.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Discounting annually on a semi-annual bond — the period count and the periodic rate both change, and the resulting price is wrong by a meaningful margin.
- Paying the clean price — settlement is at the dirty price, so budget for the accrued interest as well as the quote.
- Dividing the annual coupon rate twice — enter the annual rate and let the tool split it, rather than entering a per-period rate.
- Using duration for a large yield move — it is a first-order estimate, and convexity becomes material beyond about one percentage point.
- Ignoring a call provision — a callable bond's upside is capped as yields fall, so pricing it to maturity overstates its value.
Related Free Tools From Arb Digital
Solve the inverse problem with the YTM calculator, compare yield measures with the bond yield calculator, discount a single amount with the present value calculator, project growth with the compound interest calculator, or compare municipal and taxable issues with the tax equivalent yield calculator. The full free online tools hub lists every investing tool we publish.
Frequently Asked Questions
Discount every remaining coupon and the face value back to today at the required yield, using the periodic rate and period count implied by the coupon frequency, then add the discounted amounts together.
The clean price excludes interest accrued since the last coupon and is the figure quoted in the market. The dirty price adds that accrued interest and is the amount actually paid at settlement.
Because the coupon is fixed at issue. When market yields rise, an existing bond can only compete with higher-coupon new issues by selling at a lower price, so its value falls until its yield matches the market.
It changes both the size of each payment and the number of discounting periods. The same annual coupon rate produces a different price under annual and semi-annual payment because the cash arrives at different times.
The approximate percentage change in price for a one-percentage-point change in yield. It is a first-order estimate that becomes less accurate for larger yield moves, which is where convexity is needed.
The coupon for the period multiplied by the fraction of the period that has elapsed since the last payment. The exact fraction depends on the day-count convention the market uses for that instrument.
Because its coupon rate exceeds the yield the market currently requires. The premium over face value is what brings the bond's effective return down to the market yield.
No. It prices to maturity with a fixed cash flow schedule. A call provision truncates that schedule and caps price appreciation as yields fall, so the value shown would be too high for a callable issue.
This page explains a standard valuation calculation for educational purposes only. It is not financial or investment advice, no output is a recommendation to buy or sell any security, and bond investments carry risk of loss; consult a qualified financial professional.
