To value a bond, you add the present value of its future coupon payments to the present value of the principal you’ll receive at maturity, discounting both streams at the bond’s yield to maturity. That’s the whole method. The rest is knowing what each input means, running the arithmetic, and recognizing a few variants of the basic formula.
The Four Inputs
Every bond valuation rests on four numbers.
Face value (par value) is the principal the issuer repays at maturity. For corporate bonds this is almost always $1,000, and $1,000 is the standard reference point for pricing even when Treasury securities use a different figure.
Coupon rate is the annual interest the issuer pays, stated as a percentage of face value. A 6% coupon on a $1,000 bond is $60 a year. In the U.S., most bonds split that into two semi-annual payments, so a 6% coupon pays $30 every six months.1TreasuryDirect. Understanding Pricing and Interest Rates Treasury notes and bonds follow the same semi-annual schedule.2eCFR. 31 CFR 356.30 – When Does the Treasury Pay Principal and Interest on Securities
Maturity is the date the issuer repays principal. The time remaining determines how many payment periods go into the calculation.
Yield to maturity (YTM) is the discount rate. It represents the total annual return you’d earn buying the bond at its current price and holding to maturity, collecting every coupon on the way. YTM reflects prevailing interest rates, the issuer’s creditworthiness, and market risk appetite. When someone quotes “the yield” on a bond, they almost always mean YTM.
The Formula
A coupon-paying bond throws off two kinds of cash flow: a stream of equal interest payments and a single principal repayment at the end. You value each one separately and add them.
The coupon payments form an ordinary annuity. Their present value is:
PV of coupons = PMT × [(1 − (1 + r)−n) / r]
The principal repayment is a single lump sum:
PV of face value = FV / (1 + r)n
In both formulas, r is the YTM per period and n is the total number of periods. For a U.S. bond paying semi-annually, you divide the annual YTM by two to get r, and multiply years to maturity by two to get n.
Bond value = PV of coupons + PV of face value.
A Worked Example
Take a bond with a $1,000 face value, a 6% coupon, 5 years to maturity, and a 5% YTM.
- PMT: $1,000 × 6% ÷ 2 = $30 per period
- r: 5% ÷ 2 = 0.025
- n: 5 × 2 = 10 periods
Present value of the coupons: $30 × [(1 − (1.025)−10) / 0.025] = $30 × 8.7521 = $262.56.
Present value of the face value: $1,000 / (1.025)10 = $1,000 / 1.2801 = $781.20.
Add them: $262.56 + $781.20 = $1,043.76.
That’s the price the formula says this bond should trade at today.
Checking Your Answer Against Par
Before you calculate anything, you can tell whether the price should land above, below, or at $1,000. Bond prices and yields move in opposite directions, which produces three scenarios:
- At par ($1,000). The coupon rate equals the YTM. The bond pays exactly the market rate, so its price sits at face value.
- Premium (above $1,000). The coupon rate is higher than the YTM. The fixed payments beat what comparable new bonds offer, and investors pay extra to get them.
- Discount (below $1,000). The coupon rate is lower than the YTM. The price drops below par so that the built-in gain at maturity makes up the shortfall.
The example above has a 6% coupon against a 5% YTM, so a premium price was predictable. The $1,043.76 result passes the sanity check. If your math produced a number below $1,000 for a coupon-above-market bond, something went wrong in the arithmetic.
Valuing Zero-Coupon Bonds
Zero-coupon bonds, including U.S. Treasury STRIPS, pay no periodic interest. The entire return comes from buying at a discount and receiving face value at maturity. The annuity half of the formula drops out:
Price = FV / (1 + r)n
A 10-year zero with a $1,000 face value and a 3% YTM compounded semi-annually uses r = 0.015 and n = 20:
Price = $1,000 / (1.015)20 = $1,000 / 1.3469 = $742.47.
You’d pay about $742 today for $1,000 in ten years. Because there are no coupons cushioning the price, zero-coupon bonds move more sharply than coupon bonds of the same maturity when yields change.
Clean Price vs. Dirty Price
The formula produces the dirty price, also called the full price: the full economic value of the bond at that moment, including interest that has built up since the last coupon date. Bond quotes on trading screens and in the financial press show the clean price, which strips the accrued interest back out.
The amount you actually pay a seller is the dirty price. Accrued interest compensates the seller for holding the bond through part of a coupon period:
Accrued interest = (days since last coupon / days in coupon period) × coupon payment
Corporate bonds typically use a 30/360 day-count convention, treating each month as 30 days and the year as 360. Treasury bonds use actual/actual. The bond’s terms specify which applies. When you buy between coupon dates, you pay the clean price plus accrued interest, and on the next coupon date you receive the full coupon, which reimburses the portion you advanced to the seller.
Callable Bonds: Value to the Call Date
A callable bond lets the issuer redeem it early at a stated call price. Issuers call bonds when rates have fallen far enough to make refinancing worthwhile, so the risk falls on the investor exactly when a high-coupon bond is most valuable.
To value a callable bond at the call date rather than maturity, substitute the call date for maturity and the call price for face value:
Price = PMT × [(1 − (1 + r)−n) / r] + Call Price / (1 + r)n
Here n is the number of periods until the first call date. The discount rate that makes this equation equal the bond’s current market price is the yield to call (YTC).
Investors compare YTC with YTM and focus on whichever is lower. That worst-case number is the yield to worst, and it’s the minimum return you can expect assuming the issuer acts in its own interest. When YTC comes in below YTM, the issuer has every reason to call, and YTC becomes the realistic measure of what you’ll earn.
Running the Math in a Spreadsheet
Hand calculations build intuition. For everyday use, Excel’s PRICE function returns the clean price per $100 of face value and handles settlement dates, day-count conventions, and payment frequencies automatically.3Microsoft. PRICE Function The syntax is:
PRICE(settlement, maturity, rate, yld, redemption, frequency, [basis])
“Rate” is the annual coupon rate, “yld” is the YTM, “redemption” is the amount per $100 face value paid at maturity (usually 100), and “frequency” is 1 for annual, 2 for semi-annual, or 4 for quarterly. The optional “basis” argument sets the day-count convention: 0 for 30/360, common in corporate bonds, or 1 for actual/actual used by Treasuries.
The companion YIELD function reverses the calculation: given a current price, it returns the YTM. Financial calculators from HP and Texas Instruments offer the same functionality through their time-value-of-money keys. Brokerage platforms show calculated prices and yields directly, but working through the formula at least once is what lets you tell whether the number on the screen is reasonable.
A Note on What Valuation Doesn’t Tell You
The formula gives you a pre-tax price. The after-tax return on a corporate bond, a Treasury, or a municipal bond can differ substantially because interest from each is treated differently under federal and state tax law, and bonds issued at a discount can generate taxable income before any cash is paid. Valuation and after-tax return are separate calculations, and the price produced by the formulas above is the starting point for the first, not the second.