Bond Price Calculator

Background.

A bond's coupon is fixed on the day it is issued; the return the market demands is not. That mismatch is what moves bond prices. If a ten-year bond pays 5% a year and investors can now get 6% on similar debt, nobody will pay full face value for the 5% stream. The price drops until the coupons plus the eventual return of principal give a new buyer 6%. When market yields fall below the coupon, the opposite happens and the bond trades above face value, at a premium.

The price worked out here is that break-even figure: the present value of every remaining coupon and of the face value repaid at maturity, each discounted at the market yield. The yield is split to match the coupon schedule, so a 6% annual yield on a bond paying twice a year becomes 3% per half-year, applied over 20 half-years for a ten-year bond. Semiannual coupons are the pattern for US Treasury notes, which pay interest every six months.

Two effects show up quickly once you try a few numbers. A rise in yields hurts long bonds far more than short ones: at a 7% yield, a 5% coupon bond with two years left is worth about 963 per 1,000 of face value, while the same bond with 30 years left falls to about 751. And as maturity approaches, any discount or premium shrinks toward face value, because fewer coupons remain to be under- or over-paid.

The model is deliberately clean. It values the bond on a coupon date, so there is no accrued interest, and assumes every payment arrives on schedule. Calls, default risk, taxes and the shape of the yield curve sit outside it, and the result is an arithmetic valuation, not a market quote or a recommendation to buy or sell.

What is bond price calculator?

A bond's price, in the sense used here, is the present value of its remaining cash flows at a chosen market yield. For a plain fixed-rate bond those cash flows are a coupon every period, equal to the face value times the annual coupon rate divided by the number of payments a year, plus the face value itself returned with the last coupon. The price comes out in whatever currency the face value is entered in. A bond priced below face value trades at a discount, one above it at a premium, and one whose coupon rate equals the market yield trades exactly at par.

Markets quote that price as a percentage of face value: with a face value of 100, the worked example comes to 92.5613, which a dealer screen would show as 92.56. The quoted figure is the clean price. Between coupon dates the buyer also pays the interest the seller has earned since the last coupon; two months into a six-month period on a 5% semiannual bond that is roughly two-sixths of the 25 coupon, about 8.33 per 1,000 of face value. Clean price plus accrued interest is the dirty price that actually changes hands. US Treasuries count the days actual/actual and US corporate bonds use a 30/360 convention, so the accrued amount differs a little between them.

How to use this calculator.

1. 01Enter the face value, the amount repaid at maturity. Market prices are usually quoted per 100 of face value, so entering 100 gives a figure you can compare directly with a quote.
2. 02Enter the annual coupon rate printed on the bond and the market yield you want to price it at, such as the current yield on comparable debt. Either may be 0: a coupon of 0 prices a zero-coupon bond, and a yield of 0 adds up the cash flows without discounting.
3. 03Enter the years to maturity and the coupon payments per year: 2 for US Treasuries and most US corporate bonds, 1 for bonds that pay annually. Years times payments must be a whole number, so 7.5 years works with semiannual coupons but 7.3 years does not, and the total is capped at 10,000 periods.
4. 04Read the price against the face value: below it is a discount, above it a premium. If you are buying between coupon dates, the seller will also charge accrued interest on top of this figure.

The formula.

Price = Σ C/(1 + y/m)^t + F/(1 + y/m)^N, with the sum running from t = 1 to N. F is the face value, m the number of coupon payments a year and N = years × m the number of remaining periods. Each coupon is C = F × c ÷ m, where c is the annual coupon rate, and y/m is the market yield per period. The first term discounts every coupon back to today; the second discounts the face value repaid at the end. The coupons alone can also be written as C × [1 − (1 + y/m)^−N] ÷ (y/m), the ordinary annuity formula, which gives the same total.

Rearranged, the same equation shows where a discount or premium comes from: Price = F + (C − F × y/m) × A, where A = [1 − (1 + y/m)^−N] ÷ (y/m) is the annuity factor. F × y/m is the coupon a newly issued bond selling at par would pay each period, so the bracket is this bond's coupon shortfall or surplus per period, and the price is face value less or plus that gap valued over the remaining life. In the worked example A is 14.877475 and the bracket is 25 − 30 = −5, so the price is 1,000 − 5 × 14.877475, the same 925.61. When c equals y the bracket vanishes and the price is exactly F at any maturity; the longer the bond, the larger A and the further a given gap pushes the price from par.

Two conventions are built in. Every cash flow is discounted at one flat yield, the yield to maturity used in quotes and comparisons; desks that price off the government curve discount each payment at its own spot rate instead, which gives a slightly different figure when short and long rates differ. And the yield compounds at the coupon frequency, the bond-equivalent basis for semiannual payers, so a yield quoted on another basis should be converted before it is entered.

A worked example.

Frequently asked questions.

References& sources.

1. [1]FINRA, Bond Yield and Return.
2. [2]U.S. Treasury, Treasury Notes: Rates and Terms.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]SEC Investor Bulletin, Corporate Bonds.