Bond Price Calculator

A bond price calculator turns a coupon rate, a market yield and a maturity date into the dollar value of a bond today. The price is simply the present value of every payment still to come. A $1,000 bond with a 5% coupon paid semiannually and 10 years left, priced to yield 6%, is worth $925.61 - a discount of $74.39, because its fixed coupon is below what the market now demands.

Two sister tools sit on either side of this one. The Bond Yield Calculator runs the same relationship backwards: give it a price and it returns the current yield and yield to maturity. The Present Value Calculator handles the general case of discounting any future cash flow at a chosen rate. Use this page when you know what yield you need and want to see what the bond is worth; use the yield page when you have seen a quoted price and want to know what return it implies.

How a bond price is calculated

Every bond is a stream of fixed coupon payments plus one lump-sum repayment of face value at maturity. Pricing it means discounting all of that back to today at the market yield:

where C is the coupon paid each period (face value × coupon rate ÷ payments per year), y is the market yield per period (annual yield ÷ payments per year), n is the number of periods remaining (years × payments per year), and F is the face value repaid at maturity. The first term is an annuity: the present value of the coupon stream. The second term is the present value of the principal. Nothing else is required, which is why bond pricing is deterministic arithmetic rather than a forecast.

Worked example: a 5% bond priced to yield 6%

Take a $1,000 face value corporate bond with a 5% annual coupon paid in two installments, 10 years to maturity, and a market yield of 6%. Break it into the four inputs the formula needs:

• Coupon per period C: $1,000 × 5% ÷ 2 = $25.00, paid twice a year.
• Yield per period y: 6% ÷ 2 = 3% or 0.03.
• Periods n: 10 years × 2 = 20 payments left.
• Face value F: $1,000, repaid with the last coupon.

The present value of the 20 coupons is $25 × (1 − 1.03⁻²⁰) ÷ 0.03 = $371.94. The present value of the $1,000 face value is $1,000 ÷ 1.03²⁰ = $553.68. Adding them gives a price of $925.61, or 92.56% of par. The bond trades at a $74.39 discount.

The rest of the picture follows from that price. The current yield is $50 of annual coupon divided by $925.61, which is 5.40%. Over the ten years you collect $500 in coupons and $1,000 at maturity, so your total dollar return is $1,500 minus the $925.61 you paid, or $574.39. The Macaulay duration is 7.89 years and the modified duration is 7.67, meaning a further 1 percentage point rise in yields would cost you roughly 7.7% of the price.

Price of a $1,000 bond with a 5% coupon

The table below prices the same 5% semiannual-coupon bond at a range of market yields and maturities. Read down a column to see how a single bond reprices as rates move, and read across a row to see how much more the price swings when there are more years left.

Two things stand out. The 5% row is flat at exactly par: when the market yield equals the coupon rate, maturity is irrelevant and the bond is worth face value whatever its term. And the rows spread apart as you move right: an 8% yield costs a 2-year bond only $54.45 of value but takes $339.35 off a 30-year bond. That widening is the whole story of interest-rate risk.

Premium, discount and par explained

Three labels describe where a price sits relative to face value, and each one is determined by a single comparison:

• Discount (price below par): the market yield is above the coupon rate. You are compensated for the low coupon with a capital gain at maturity.
• Par (price equals face value): the market yield equals the coupon rate. Coupon rate, current yield and yield to maturity all coincide.
• Premium (price above par): the market yield is below the coupon rate. The above-market coupon is paid for upfront, and you take a small capital loss as the price pulls back to par.

A premium is not a bad deal and a discount is not a bargain. Both prices are set so that the total return to maturity equals the market yield. What differs is the shape of that return: a discount bond delivers more of it as principal at the end, and a premium bond delivers more of it as income along the way.

Second worked example: a premium bond

Now flip the relationship. Take a $1,000 bond with a 6% coupon paid semiannually, 8 years to maturity, and a market yield of just 4.5%. The coupon per period is $30, the yield per period is 2.25%, and there are 16 periods left. Running the formula gives a price of $1,099.84, or 109.98% of par. You pay a $99.84 premium for the privilege of collecting $30 twice a year instead of the $22.50 a new bond at 4.5% would pay.

The current yield is $60 divided by $1,099.84, or 5.46%, which looks better than the 4.5% market yield until you remember the ending. Over eight years you collect $480 in coupons and get back $1,000 on a bond you paid $1,099.84 for, so the $99.84 premium is written off against your income. That is exactly why current yield overstates the return on a premium bond and understates it on a discount bond.

How the coupon rate moves the price

Holding the market yield at 6% and maturity at 10 years, here is what different coupon rates are worth on a $1,000 bond with semiannual payments:

The price moves in a perfectly straight line with the coupon rate - each extra percentage point of coupon adds $74.39 to the price of this bond, because it adds the same present value of extra cash flow every time. The current yield column, by contrast, curves: it rises from 0% to 6.96% but never reaches the coupon rate on a premium bond and always sits below it on a discount bond.

Duration: how far the price moves when rates change

Duration is the standard measure of a bond's interest-rate sensitivity. Macaulay duration is the present-value-weighted average number of years until you get your money; modified duration divides that by one plus the yield per period and estimates the percentage price change for a 1 percentage point move in yield. This table starts every bond at par (a 5% coupon priced to yield 5%) and then raises the yield to 6%:

Notice that duration is always shorter than maturity for a coupon bond - the 30-year bond has a Macaulay duration of only 15.84 years because half its value arrives as coupons long before maturity. Notice too that the actual price change is a little smaller than modified duration predicts (7.44% versus 7.79% on the 10-year). That gap is convexity: the price-yield curve bends, so bonds lose slightly less when rates rise and gain slightly more when they fall than the straight-line duration estimate suggests.

Annual, semiannual or quarterly coupons

Almost all U.S. corporate, Treasury and municipal bonds pay coupons twice a year, which is why semiannual is the default here. The frequency matters for two reasons. First, it changes the per-period arithmetic: a 5% coupon on a $1,000 bond is one $50 payment a year, two $25 payments, or four $12.50 payments. Second, it changes the price a little. For our 10-year 5% bond at a 6% yield, the price is $926.40 with annual coupons, $925.61 with semiannual coupons, and $925.21 with quarterly coupons. Getting money sooner is worth something, but discounting at a smaller per-period rate offsets most of it, so the spread here is only $1.19. Always price a bond at its actual payment frequency so the figure is comparable with a broker quote.

How to use this bond price calculator

Five inputs are all you need, and the result updates as you type:

1. Face value (par): what the issuer repays at maturity. Most U.S. corporate and municipal bonds use $1,000; Treasury notes and bonds are also quoted on a $1,000 basis. Enter your total position size if you want the price of the whole holding rather than one bond.
2. Coupon rate: the fixed annual rate printed on the bond, always applied to face value. A 5% coupon on a $1,000 bond pays $50 a year no matter what the bond costs.
3. Market yield: the return investors currently demand for a bond of this credit quality and maturity. This is the input that does the work, and the one you should stress-test.
4. Years to maturity: how long until the face value is repaid. Half-years are fine.
5. Coupon frequency: annual, semiannual or quarterly. Semiannual matches most U.S. bonds.

Read the price at the top, both in dollars and as a percentage of par so you can compare it directly against a broker quote. Then check the premium or discount badge, the split between the present value of coupons and the present value of the principal, and the two tables showing how the price would react to different yields and different maturities.

Who this calculator is for

• Individual bond buyers checking whether a quoted price is fair for the yield they are being offered.
• Retirees building a bond ladder who need to value each rung at today's rates before committing cash.
• Finance and accounting students working through present-value problem sets and needing a reliable check on their arithmetic.
• CFA and Series 7 candidates drilling the premium, par and discount relationships until they are automatic.
• Anyone holding bonds who wants to understand why a statement shows an unrealized loss on a bond that will still pay back every dollar at maturity.

Key bond terms explained

• Face value (par): the principal amount repaid at maturity, and the base for every coupon calculation.
• Coupon rate: the fixed annual interest rate set at issue. It never changes over the life of a conventional bond.
• Market yield (yield to maturity): the discount rate that makes the present value of the remaining cash flows equal the price. It moves constantly with rates and credit conditions.
• Current yield: annual coupon divided by price. An income measure only, blind to the gain or loss at maturity.
• Clean price: the price excluding accrued interest, which is how bonds are quoted.
• Dirty price: the clean price plus accrued interest, which is what actually settles.
• Duration: the sensitivity of the price to a change in yield, expressed in years (Macaulay) or as a percentage per point (modified).
• Convexity: the curvature of the price-yield relationship, which makes price gains from falling yields slightly larger than the losses from equal rises.
• Call feature: the issuer's right to redeem the bond early at a set price, which caps how far a premium bond's price can rise.

What changes the result the most

• The gap between coupon and yield: this alone decides premium versus discount. Every percentage point of gap is worth about $74 on our 10-year example bond.
• Years to maturity: the same yield change moves a 30-year price roughly seven times as far as a 2-year price.
• Coupon size: low-coupon and zero-coupon bonds have longer durations and swing much harder for a given yield move.
• Coupon frequency: a small effect, typically well under 1% of the price, but worth matching to the real bond.
• Face value: the price scales exactly in proportion, so a $10,000 position is simply ten times the $1,000 price.

Clean price, dirty price and accrued interest

This calculator returns the clean price on a coupon date, which is the convention for quotes. In reality you rarely buy on a coupon date. Interest accrues day by day between payments and belongs to whoever held the bond during that stretch, so a buyer pays the clean price plus accrued interest. On a $1,000 bond with a $25 semiannual coupon, 45 days into a 182-day coupon period, accrued interest is $25 × 45 ÷ 182 = $6.18. Add that to the clean price and you get the dirty or invoice price that settles in your account. Different markets count days differently - Treasuries use actual/actual, most corporate and municipal bonds use 30/360 - which shifts the accrued figure slightly. None of this changes the underlying value of the bond; it only splits the next coupon fairly between seller and buyer.

Taxes on bond interest

Price and yield here are pre-tax, and the tax treatment differs sharply by issuer. Interest on U.S. Treasury securities is subject to federal income tax but exempt from state and local income tax, which matters most in high-tax states. Interest on corporate bonds is fully taxable as ordinary income at every level. Interest on many municipal bonds is exempt from federal income tax, and often from state tax as well when you buy bonds issued in your own state. Because of this, a lower-yielding municipal bond can beat a higher-yielding corporate bond after tax. There are also rules for original issue discount and market discount on bonds bought below par, which can turn part of your gain into ordinary income. Run the pre-tax comparison here, then check IRS Publication 550 or a tax adviser for your own situation.

Practical tips

• Compare the quote, not the dollars. Convert your calculated price to a percentage of par and put it next to the broker's quote. A quote of 92.56 and a price of $925.61 are the same thing on a $1,000 bond.
• Stress-test the yield. Move the market yield up and down a point before you buy. If the resulting price swing would worry you, the maturity is too long for your horizon.
• Match maturity to your need for the money. Price risk only becomes a real loss if you sell early. Hold to maturity and you get par regardless of what happened in between.
• Watch callable premium bonds. If a bond is trading well above par and the issuer can call it, your realistic return may be the yield to call, not the yield to maturity.
• Check the credit, not just the math. The formula assumes every payment arrives. That assumption is far safer for a Treasury than for a low-rated corporate issuer.

Limitations and assumptions

• It prices a plain vanilla fixed-coupon bond on a coupon date. It does not model floating-rate notes, inflation-linked securities such as TIPS, sinking funds, or convertible features.
• It ignores call and put options. A callable bond trading at a premium is usually worth less than the formula says, because the issuer can take the high coupon away.
• It assumes no default: every coupon and the full face value arrive on schedule.
• It uses a single flat yield for all cash flows rather than discounting each payment at its own point on the yield curve.
• It excludes accrued interest, day-count conventions, taxes, commissions and bid-ask spreads, all of which affect what you actually pay or receive.

How it compares to related calculators

This page answers "what is this bond worth at a given yield?" If your question is different, another tool fits better:

• To go from a known price to the return it implies, use the Bond Yield Calculator.
• To discount any single future amount or cash flow, use the Present Value Calculator or the Future Value Calculator.
• To value an uneven stream of project cash flows, use the NPV Calculator or the IRR Calculator.
• To solve for any one of rate, term, payment or value in a general time-value problem, use the TVM Calculator.
• To compare a bond against a bank alternative, use the CD Calculator.
• To measure income on a stock instead, use the Dividend Yield Calculator.

Sources

• U.S. Department of the Treasury - TreasuryDirect: Treasury bonds, coupon payments and maturity.
• U.S. Securities and Exchange Commission - Investor.gov: Bonds.
• U.S. Securities and Exchange Commission - Investor Bulletin: Interest rate risk when investing in bonds.
• Internal Revenue Service - Publication 550, Investment Income and Expenses.