Bond Price Calculator

What a bond price actually is

A bond price is a present value, nothing more. You own a claim to a fixed schedule of payments — a coupon every six months for most US issues, then the face value back at maturity — and the price is what that schedule is worth today at the return you demand for holding it. Change the demanded return and the price changes; the cash flows never do.

That single sentence explains the two facts every fixed-income desk works from. First, price and yield move in opposite directions: if you insist on a higher return from an unchanged stream of payments, the only lever you have is to pay less for it. Second, a bond's price is not an opinion about the issuer's business the way a share price is. It is arithmetic applied to a promise, plus a judgement about whether that promise will be kept. The judgement lives entirely inside the yield you type in.

The yield you supply here is the market's required yield for this credit quality and this maturity — roughly the Treasury yield for the same term plus a credit spread. If you already know the price and want the yield instead, invert the problem with the yield to maturity calculator.

The formula, term by term

You discount each cash flow by the number of periods until you receive it. With m coupons a year, the periodic yield is i = y/m and the number of periods to the k-th coupon is simply k. So the k-th coupon of C/m dollars is worth (C/m) ÷ (1 + i)^k today, and the face value F arriving at period N is worth F ÷ (1 + i)^N.

Because every coupon is identical, the coupon leg is an ordinary annuity and collapses to a closed form: (C/m) × (1 − (1 + i)^−N) ÷ i. That is what the calculator evaluates, which is why it stays fast even on a 30-year bond with 60 periods. The one case the closed form cannot handle is i = 0, where the fraction is 0/0; the code substitutes (C/m) × N, which is the correct limit.

Why divide the annual yield by the frequency rather than compound it? Because bond markets quote yields on a nominal basis. A 6% semiannual bond-equivalent yield means 3% every six months, not 6% compounded annually. The effective annual rate is 1.03² − 1 = 6.09%. Both numbers are correct; they answer different questions, and mixing them is the most common source of a price that is off by a few tenths of a point.

Why a whole number of periods? The formula above prices the bond on a coupon date, when exactly N periods remain. Real settlement almost never falls on a coupon date, which is where accrued interest comes in.

Clean price, dirty price and accrued interest

Dealers quote the clean price, but the buyer wires the dirty price — also called the full or invoice price. The difference is accrued interest: the seller held the bond for part of the current coupon period and is entitled to that part of the coupon, even though the issuer will pay the whole coupon to whoever owns the bond on the record date.

Accrued interest is a simple pro-rata share, not a compounded amount: AI = (C/m) × d/T, where d is the days from the last coupon to settlement and T is the days in the full coupon period. The convention that fixes d and T is the day count. US corporates and municipals use 30/360, which treats every month as 30 days and every year as 360, so T is always exactly 180 for a semiannual bond. US Treasury notes and bonds use actual/actual, where T is the real number of calendar days between coupon dates — 181, 182, 184 depending on the months involved. The Municipal Securities Rulemaking Board sets out the required calculations for municipal dollar prices and accrued interest in MSRB Rule G-33, and the Treasury's own pricing and accrual formulas are published in 31 CFR Part 356, Appendix B.

The dirty price itself is the coupon-date price carried forward by the fraction of a period elapsed: P × (1 + i)^d/T. Subtract the simple-interest accrual and you get the clean price. Because the price grows by compound interest while the accrual grows by simple interest, a clean price computed this way dips slightly below par mid-period even on a bond trading exactly at its coupon rate. That sawtooth is real and it is why clean prices, not dirty prices, are the quoted convention: clean prices are almost flat when nothing has changed.

How to read the price you get

Start with the price as a percentage of par, because that is how the market talks. A bond at 87.538 is quoted as "87 and a half" and everyone in the conversation knows what a point is worth: one point is 1% of face, so $10 on a $1,000 bond and $10,000 on a million of face. Dollar prices are for confirmations; percentage prices are for trading.

Then read the sign of the premium or discount. A bond prices above par whenever its coupon rate exceeds the required yield, and below par whenever it does not. The size of the gap tells you how far the market has moved since the bond was issued, and the maturity tells you how long you will live with it. That gap does not persist: it is amortised or accreted away, so a premium bond's price grinds down toward 100 and a discount bond's grinds up, even if yields never move again. This pull to par is a real component of return and it is the reason a premium bond's current yield always overstates what you will actually earn.

Finally, look at how steep the price–yield curve is around your yield. The chart the calculator draws is not a straight line — it is convex, bowing upward. A 100 basis point fall in yield gains you slightly more than a 100 basis point rise costs you, and that asymmetry grows with maturity. Quantify it with the Macaulay duration calculator and the modified duration price change calculator rather than eyeballing the curve.

One judgement this calculator deliberately does not make: whether your required yield is right. Pricing a single-B corporate at the 10-year Treasury yield will return a beautiful, precise, useless number.

Price per $100 of par for a 5% semiannual-coupon bond

Read across a row to see how much more a long bond moves for the same yield change: at an 8% yield the 5-year loses 12.17 points from par while the 20-year loses 29.69. That difference is duration.

Mistakes that produce a wrong bond price

• Using the coupon rate as the discount rate. The coupon rate sets the cash flows; the required yield discounts them. They are equal only for a bond priced at par.
• Quoting the dirty price. Dealers quote clean. If you compare a dirty price against a screen quote you will think the bond is cheap by roughly one accrued coupon.
• Assuming 180 days for a Treasury. Treasuries accrue on actual/actual, so the coupon period is the real day count between coupon dates. Using 180 misprices the accrual by up to a few cents per $100.
• Ignoring the call. Most corporate and municipal bonds are callable. A bond trading well above par is usually priced to the call date, not to maturity — use the yield to call calculator and price to the worst outcome.
• Comparing a municipal yield to a taxable one. Municipal coupons are generally exempt from federal income tax, so their yields are not comparable to corporate yields until you gross them up with the tax-equivalent yield calculator.
• Forgetting that this model assumes no default. Discounted cash flow pricing takes every promised payment as certain. Credit risk enters only through the yield you choose, which is a blunt instrument for a bond that might not pay.

What this model leaves out, and what to use instead

This is a single-yield model: every cash flow is discounted at the same rate. Real term structures are not flat, and an arbitrage-free valuation discounts each cash flow at the spot rate for its own maturity. For an option-free bond the two approaches give prices that are close, and the single yield that reproduces the spot-curve price is precisely the yield to maturity. The single-yield form remains the market's language because it compresses a whole curve into one quotable number.

Embedded options break the model outright. A callable bond gives the issuer the right to redeem early, which caps the price on rallies and produces negative convexity; a putable bond does the opposite. Neither is captured by discounting a fixed schedule, and pricing them properly needs a lattice or a Monte Carlo model with a volatility assumption. Floating-rate notes are a different problem again, since their coupons reset — they price near par by construction and are analysed on discount margin rather than yield.

Two nearby tools cover the simpler special cases. A bond with no coupons at all is handled directly by the zero-coupon bond price calculator, which is this formula with the annuity term deleted. And if you only want the income-per-dollar-invested view rather than a full valuation, the current yield calculator gives you coupon divided by price in one step — useful, but silent about the pull to par that dominates returns on deeply discounted or premium bonds.

Key terms

Par (face) value

The principal the issuer repays at maturity and the base on which the coupon is calculated. Coupon payments do not change when the market price does.

Clean price

The quoted price, excluding accrued interest. Expressed as a percentage of par in the market, in dollars on a confirmation.

Dirty (full, invoice) price

Clean price plus accrued interest — the amount that actually settles.

Day count convention

The rule for counting days in the accrual: 30/360 for US corporates and municipals, actual/actual for Treasuries, actual/360 for many money-market instruments.

Pull to par

The tendency of a bond's price to converge on face value as maturity approaches, holding the yield constant. It amortises a premium and accretes a discount.

Basis point

One hundredth of a percentage point. Yields move in basis points; prices move in points and thirty-seconds.