To value a bond, add two present values together: the present value of every coupon payment the bond will pay, plus the present value of the face amount returned at maturity. Both are discounted at the yield the market currently demands for a bond of that credit quality and maturity.1U.S. Securities and Exchange Commission. Interest Rate Risk – When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall That is the whole calculation. Everything else is arithmetic and unit conversion.
What You Need Before You Start
Five numbers drive the calculation. The first four come from the bond’s prospectus, the disclosure document issuers must file before selling bonds to the public.2Office of the Law Revision Counsel. 15 USC 77j – Information Required in Prospectus
- Face value, also called par value. The principal the issuer repays at maturity. Most bonds use $1,000.
- Coupon rate. The annual interest rate stated on the bond, expressed as a percentage of face value.
- Payment frequency. Most U.S. bonds pay semi-annually; some pay quarterly or annually.
- Years to maturity. How long until the face value comes back.
- Required rate of return, also called the discount rate. The yield you would demand today, generally read off what comparable bonds are currently yielding.
Before any formula, convert everything to the payment period. If the bond pays semi-annually, divide the coupon rate by two, divide the discount rate by two, and multiply years to maturity by two. A 6% annual coupon on a $1,000 bond becomes a $30 payment every six months. A 4% annual market yield becomes 2% per period. Ten years becomes 20 periods. This is the single most common place bond valuations go wrong. If your rate and your period count are on different clocks, every number that follows will be off.
The Two-Part Formula
The coupon payments are an ordinary annuity, a series of equal payments arriving at regular intervals. Their present value is:
PV of coupons = C × [(1 − (1 + r)−n) / r]
C is the coupon payment per period, r is the discount rate per period, and n is the total number of periods. The bracketed expression is called the annuity factor. It reflects that a payment further in the future is worth less today, because money in hand now can be reinvested.
The face value is a single lump sum paid once at maturity, so it uses the standard present-value formula for a single future payment:
PV of face value = FV / (1 + r)n
The denominator grows exponentially with n, which is why long-dated bonds react so violently to rate changes. A $1,000 payment due in two years at a 3% semi-annual rate is worth about $942 today; the same payment 30 years out is worth about $169.
The bond’s value is those two numbers added together. That is it.
A Worked Example, Step by Step
Take a bond with a $1,000 face value, a 6% annual coupon paid semi-annually, 10 years to maturity, and a current market yield of 4%.
Adjust to the payment period first. Coupon per period: $1,000 × 0.06 / 2 = $30. Discount rate per period: 0.04 / 2 = 0.02. Number of periods: 10 × 2 = 20.
Present value of the coupon stream. Compute (1.02)20 = 1.4859. Take its inverse: (1.02)−20 = 0.6730. Subtract from 1: 1 − 0.6730 = 0.3270. Divide by 0.02: 0.3270 / 0.02 = 16.3514. That is the annuity factor. Multiply by the $30 coupon: $30 × 16.3514 = $490.54.
Present value of the face value. Divide $1,000 by 1.4859: $1,000 / 1.4859 = $672.97.
Add them: $490.54 + $672.97 = $1,163.51. That is the bond’s value. It sits above the $1,000 face value because the 6% coupon beats the 4% market yield, and buyers will pay extra to lock in that above-market income.
What the Answer Tells You
The relationship between the coupon rate and the market yield tells you immediately, without any arithmetic, whether the answer will land above, below, or at face value.1U.S. Securities and Exchange Commission. Interest Rate Risk – When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall
- Premium bond. Coupon rate exceeds market yield. Price is above face value. The example above, at $1,163.51, is a premium bond.
- Discount bond. Market yield exceeds coupon rate. If the same bond faced an 8% market yield, each semi-annual period would use a 4% discount rate, the coupon stream would come to about $407.71, the face value to about $456.39, and the bond would price near $864.10.
- Par bond. Coupon rate equals market yield. The calculation lands at exactly $1,000.
As a bond gets closer to maturity, its price drifts toward face value regardless of where it started. The remaining cash flows shrink until only the final face-value payment is left. Traders call this the pull to par.
Zero-Coupon Bonds
A zero-coupon bond pays no periodic interest. It sells at a deep discount and returns the full face value at maturity. With no coupons, the annuity term disappears and the entire valuation collapses to the single lump-sum formula:
Price = FV / (1 + r)n
The U.S. convention is still to discount using semi-annual compounding even though nothing is being paid semi-annually. A 10-year zero with a $1,000 face value and a 5% annual market yield is worth $1,000 / (1.025)20 = $1,000 / 1.6386 = $610.27. The $389.73 gap between purchase price and face value is the investor’s return, accruing implicitly rather than arriving as cash.
Zeros are extremely sensitive to rate changes because all their value rides on one distant payment, with no coupon income to cushion the swing.
Clean Price vs. Dirty Price
The formula gives you the bond’s value on a coupon payment date, the moment right after interest has just been paid. Bonds trade every business day, though, and between coupon dates the seller has earned interest that has not yet been paid out. That is accrued interest, and it changes what the buyer actually pays.
- Clean price. The value from the formula, without accrued interest. This is what quote screens show.
- Dirty price, sometimes called the invoice price. The clean price plus accrued interest. This is what the buyer wires at settlement.
The accrued amount depends on the day-count convention. U.S. Treasury bonds use actual/actual, counting real elapsed days over real days in the coupon period. Corporate bonds usually use 30/360, treating every month as 30 days and every year as 360. The formula is Accrued Interest = Coupon Payment × (Days Since Last Payment / Days in Coupon Period). For a corporate bond paying $30 every six months, 45 days into the period, the accrued interest is $30 × (45 / 180) = $7.50, and the dirty price is the clean price plus $7.50.
Solving for Yield Instead of Price
Sometimes the market price is what you know, and you want the yield. That is yield to maturity, or YTM. Instead of picking a discount rate and computing a price, you pick a price and solve for the discount rate that makes the two present values add up to it. YTM is effectively the bond’s internal rate of return if you hold it to maturity and reinvest every coupon at that same rate.
There is no clean algebraic solution. A common approximation is:
YTM ≈ [Annual Coupon + (Face Value − Price) / Years to Maturity] / [(Face Value + Price) / 2]
The approximation gets close. The exact answer requires iteration or a built-in spreadsheet function like RATE or YIELD. For semi-annual bonds, solve for the per-period rate and double it.
Yield to call, or YTC, uses the same procedure but substitutes the call date for the maturity date and the call price for the face value. Callable bonds let the issuer redeem the bond early, typically when rates have fallen and the issuer can refinance cheaper. If you paid a premium for a callable bond, an early call cuts short the time you have to collect that above-market coupon, and the calculation should reflect that shorter horizon.
How Much the Price Will Move When Rates Move
Knowing what a bond is worth today is half the story. Knowing how much that value will change if rates shift is the other half, and duration is the standard measure.
Macaulay duration is the weighted-average time until you receive the bond’s cash flows, with each payment weighted by its present value as a share of the total price. A high-coupon 10-year bond has shorter Macaulay duration than a 10-year zero, because more of the money comes back sooner.
Modified duration divides Macaulay duration by (1 + yield per period) and turns it into a direct price-sensitivity estimate: a bond with modified duration of 7 will lose roughly 7% of its value for a one percentage point rise in yield. The relationship is approximately linear for small moves and drifts off for large ones, which is where convexity comes in as a second-order correction.
Bonds Where the Standard Formula Needs Adjustment
The two-part present-value calculation assumes fixed, known cash flows. Two common bond types break that assumption.
Treasury Inflation-Protected Securities adjust their principal based on changes in the Consumer Price Index. When inflation rises, the face value of a TIPS rises, and the coupon payments — calculated as a fixed percentage of the adjusted principal — rise with it. At maturity, you receive the inflation-adjusted principal or the original face value, whichever is greater.3TreasuryDirect. TIPS – TreasuryDirect Valuing a TIPS requires projecting future inflation to estimate what the adjusted cash flows will be, then discounting at the real yield.
Floating-rate notes pay a coupon that resets to a reference rate. U.S. Treasury FRNs mature in two years, pay quarterly, and use the 13-week Treasury bill rate as their index, with the rate resetting weekly and a fixed spread added at auction.4TreasuryDirect. Floating Rate Notes (FRNs) – TreasuryDirect Because the coupon tracks current rates, FRNs trade very close to par and their price barely moves when rates change. The present-value framework still applies, but the future coupons have to be estimated from forward rates rather than read off the bond.