Bond Calculator

Two calculators sit above. The first handles a fixed-rate coupon bond issued or traded on the coupon date: leave any one field blank and it solves for that one, whether that is the price, the yield, the coupon rate, or the face value. The second prices a bond bought between coupon dates, returning the clean price, the dirty price, and the accrued interest under the day-count convention you pick.

Both are built for fixed-rate coupon bonds, which is the bulk of the market. Neither models the other things that move a real quote: credit quality, liquidity, supply and demand, embedded options. Treat the output as the arithmetic of the cash flows, then adjust for the market.

What a bond is

A bond is a loan you make to an issuer, packaged as a tradable security. A government or company needs money, sells bonds to raise it, and agrees in writing to pay interest on a schedule and return the principal on a set date. The investor is the lender; the issuer is the borrower.

They come in several flavours to suit both sides: government bonds, municipal bonds, corporate bonds, and high-yield (junk) bonds, among others. Against stocks, bonds are the lower-risk end of the portfolio, which is why they attract investors who want income without watching the value swing. That said, "bond" covers a wide risk range. A U.S. Treasury and a junk corporate issue are both bonds, and only one of them is treated as close to risk-free. The difference shows up in the yield, which is the market's price for taking on the issuer's credit risk and the time you are locked in.

The main types

• Government bonds. Issued by national treasuries. U.S. Treasuries come as bills (under a year), notes (2 to 10 years), and bonds (20 to 30 years), and are backed by the federal government, which is why they anchor the low-risk end of the market.
• Municipal bonds. Issued by states, cities, and their agencies to fund public projects. The interest is often exempt from federal income tax and sometimes from state tax too, so a lower headline yield can still beat a taxable bond once tax is counted.
• Corporate bonds. Issued by companies to raise capital. They pay more than government debt because a company can fail, and how much more depends on the credit rating.
• High-yield bonds. Corporate debt rated below investment grade, commonly called junk. Higher coupons in exchange for a real chance of default.
• Zero-coupon bonds. No periodic interest at all. They sell well below face value and pay the full amount at maturity, with the discount standing in for the coupons.

The parts of a bond

• Face value. Also called par value. The amount the issuer repays at maturity, and the base the coupon is calculated on. Corporate bonds are commonly issued at $1,000 par.
• Maturity date. When the principal comes back. Maturities run from under a year to 30 years and beyond, and "time to maturity" is whatever is left of that from today.
• Coupon rate. The interest rate the issuer pays on the face value. It can be fixed, floating, or zero, as with zero-coupon bonds. Both calculators above assume it is fixed.
• Coupon frequency. How often the interest is paid: annual, semi-annual, quarterly, or monthly. Semi-annual is the U.S. norm.
• Yield. The return an investor expects, quoted annually. The figure used above is the current yield, which divides the annual coupon by the bond's market price rather than its face value, so it moves whenever the price does.
• Price. What the bond trades for. It is the present value of every future coupon plus the principal, discounted at the rate the market currently demands.

Around these sit the features that shape valuation without appearing in the formula: who the issuer is, whether the bond is callable or puttable, its credit rating, the covenants attached, and how easily it can be sold.

How a bond price is calculated

A bond's price is the sum of the present values of everything it will pay you. Discount each coupon and the final principal back to today at the required yield, then add them up:

Price = Σ [ C ÷ (1 + r)^t ] + F ÷ (1 + r)^N

Take a $1,000 bond paying a 5% coupon semi-annually, maturing in 10 years, when the market demands a 6% yield. Convert everything to periods first: the coupon per period is 5% of $1,000 ÷ 2 = $25, the number of periods is 10 × 2 = 20, and the rate per period is 6% ÷ 2 = 3%.

Price = $25 × 14.8775 + $1,000 ÷ 1.03²⁰ = $925.61

The bond sells below par because its 5% coupon is worse than the 6% the market wants. That is the whole mechanism behind bond prices moving inversely to yields: when required yields rise, existing bonds with lower coupons have to get cheaper to compete. Doing this by hand means twenty separate discountings, which is what the first calculator above is for. To value a single future sum instead, see the Present Value Calculator.

Clean price, dirty price, and accrued interest

The formula above quietly assumes you are buying on a coupon date. In practice most trades settle somewhere between coupon dates, and the money has to be split fairly between the seller who held the bond for part of the period and the buyer who will collect the whole next coupon.

Accrued interest is the seller's share: the interest built up since the last coupon payment but not yet paid out. It is the coupon for the period scaled by the fraction of that period already elapsed.

Clean price excludes accrued interest. It is the number quoted in the market and in the press, because it strips out the sawtooth effect of interest building up and resetting, which makes two bonds comparable regardless of where each sits in its coupon cycle.

Dirty price, also called the invoice price, is what the buyer actually pays. The two are linked by one line:

dirty price = clean price + accrued interest

So the quoted price is not the cheque you write. Buy between coupon dates and you reimburse the seller for the interest they earned but never received, then collect it back in full at the next coupon. The second calculator above reports all three figures along with the number of days accrued.

Day-count conventions

Accrued interest depends on counting days, and the bond market has more than one way of counting them. The convention decides both how many days have elapsed and how many days the year is assumed to hold.

• 30/360 (bond basis). Every month is treated as 30 days and every year as 360. It makes manual calculation simple and is standard for U.S. corporate, agency, and municipal bonds.
• Actual/360. Real calendar days in the period, but a 360-day year. Common in money market instruments such as commercial paper and short-term CDs.
• Actual/365. Real days over a fixed 365-day year, usually ignoring the extra day in leap years. Used for some government bonds outside the U.S. and in parts of the swaps market.
• Actual/Actual. Real days over the real length of the year. The most accurate of the four, and the convention for U.S. Treasury securities.

The differences are small in ordinary cases, often a few cents on a $1,000 bond, but they are not zero: in the worst case the conventions can disagree by up to six days of accrued interest. Which one applies is set by the bond, not by preference, so match the setting above to the security you are pricing.

What else moves the price

The formula prices the cash flows. The market prices the risk around them, and four things do most of that work.

Credit quality. A downgrade raises the yield investors demand, which pushes the price down even though the coupon has not changed. Ratings agencies publish the grades, but prices usually move before the announcement does.

Duration. Longer bonds swing harder when rates move, because more payments sit further out and get discounted more heavily. A 30-year bond and a 2-year bond facing the same one-point rate rise do not lose the same percentage.

Embedded options. A callable bond lets the issuer repay early, which they will do when rates fall, capping your upside exactly when you would most want to keep the coupon. That risk is priced in as a higher yield.

Liquidity and inflation. A bond few people trade sells at a discount for the inconvenience, and expected inflation erodes what fixed payments will buy, so higher expected inflation lifts required yields across the board.

How these calculators work

The first solves whichever field you leave empty. For price it discounts the coupons and principal directly. For yield there is no closed-form answer, so it searches by bisection, narrowing the range until the price implied by the rate matches the price you entered. Coupon rate and face value are rearrangements of the same present-value equation.

The second builds the actual coupon schedule backward from the maturity date, finds the last coupon before settlement and the next one after it, and measures the elapsed fraction of that period under your chosen convention. Each remaining cash flow is discounted over a fractional first period to give the dirty price; accrued interest comes off that to give the clean price. To compare a bond's return against other places you could put the money, the Investment Calculator and the CD Calculator are the useful neighbours, and the Interest Rate Calculator solves a rate from a known payment stream.