To value a bond, add together two present values: the present value of the stream of coupon interest payments, and the present value of the face amount returned at maturity. Both are discounted at the yield the market currently demands for a bond of similar quality and maturity. When market rates rise, existing bonds with lower coupons lose value; when rates fall, older bonds with higher coupons gain value.1U.S. Securities and Exchange Commission. Interest Rate Risk – When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall Once you have the inputs lined up, the arithmetic is straightforward.
Inputs You Need Before You Start
Five numbers drive every bond valuation. The first four come from the bond’s prospectus, the disclosure document issuers must file under federal securities law 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. This is the principal the issuer pays back 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. How often interest is paid. Semi-annual is standard in the United States, though some bonds pay quarterly or annually.
- Years to maturity. Time remaining until the issuer returns the face value.
- Required rate of return, also called the discount rate. The yield you would demand today given market conditions and the bond’s credit quality, usually based on what comparable bonds are yielding right now.
Before any formula, adjust your annual figures to match the payment frequency. For a semi-annual bond, divide both the coupon rate and the discount rate by two, and multiply the years to maturity by two. A 6% annual coupon on a $1,000 bond becomes a $30 payment every six months. A 4% annual discount rate becomes 2% per period. Ten years to maturity becomes 20 periods. Skipping this step is the most common mistake in bond valuation. If your periods do not match your rates, every number that follows will be wrong.
Present Value of the Coupon Payments
The coupon stream is an ordinary annuity, a series of equal payments arriving at regular intervals. Its 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 fraction in brackets is the annuity factor. It captures the fact that a payment arriving later is worth less today, because you lose the chance to reinvest that money in the meantime.
When the discount rate equals the coupon rate, this piece plus the face-value piece adds up to exactly par. When the discount rate is higher than the coupon rate, the annuity factor shrinks and the present value of the interest stream drops. That shortfall is what pushes the bond’s price below face value.
Present Value of the Face Value
The face value returned at maturity is a single lump sum, so it uses the simpler present-value formula for one future payment:
PV of face value = FV / (1 + r)n
FV is the face value, r is the discount rate per period, and n is the total number of periods. The denominator grows exponentially as maturity extends. A $1,000 payment due in two years at a 3% semi-annual rate discounts to roughly $942; the same $1,000 payment due in 30 years discounts to around $169.
A Worked Example
Consider a bond with a $1,000 face value, a 6% annual coupon, semi-annual payments, 10 years to maturity, and a market yield of 4%.
First, adjust for semi-annual periods. Coupon payment per period: $1,000 × 0.06 / 2 = $30. Discount rate per period: 0.04 / 2 = 0.02. Total periods: 10 × 2 = 20.
Second, work out the present value of the coupon stream. The annuity factor is (1 − (1.02)−20) / 0.02. Start with (1.02)20 = 1.4859, then take the inverse: (1.02)−20 = 0.6730. Subtract from 1 to get 0.3270, and divide by 0.02 to get 16.3514. Multiply by the $30 coupon: $30 × 16.3514 = $490.54.
Third, work out the present value of the face value. Divide $1,000 by (1.02)20 = 1.4859: $1,000 / 1.4859 = $672.97.
Add the two pieces: $490.54 + $672.97 = $1,163.51. The bond is worth about $1,163.51. That is a premium over the $1,000 face value, which makes sense: the 6% coupon beats the 4% market yield, so an investor would pay more than par to lock in that above-market income.
Why the Answer Lands Above, Below, or at Par
The relationship between the coupon rate and the market yield decides where the price sits.1U.S. Securities and Exchange Commission. Interest Rate Risk – When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall
- Premium. The coupon rate is higher than the market yield. Investors pay more than face value for the higher income stream. In the example, the 6% coupon against a 4% yield produced a price of $1,163.51.
- Discount. The market yield is higher than the coupon rate. If the same bond faced an 8% market yield instead, each semi-annual period would use a 4% discount rate. The coupon stream would be worth about $407.71 and the face value about $456.39, for a price around $864.10, well below par.
- Par. The coupon rate exactly matches the market yield. The formula produces a price of exactly $1,000. The bond’s promised return equals what the market demands.
As a bond approaches maturity, its price gradually moves toward face value regardless of where it started. This is sometimes called pull to par. The remaining cash flows shrink until only the final face-value payment is left.
Valuing a Zero-Coupon Bond
A zero-coupon bond pays no periodic interest. It sells at a deep discount and returns the full face value at maturity. Because there are no coupon payments, the annuity portion drops out and the whole value comes from the lump-sum formula:
Price = FV / (1 + r)n
Even with no coupons, the U.S. convention is to discount using semi-annual compounding. A 10-year zero-coupon bond 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 over the bond’s life rather than arriving as cash along the way.
Zero-coupon bonds are highly sensitive to interest rate changes because all of their value sits in one distant payment. There is no coupon income to cushion price swings.
Clean Price and Dirty Price Between Coupon Dates
The formulas above give you the bond’s value on a coupon payment date, right after interest has been paid and none has yet accumulated for the next period. In practice bonds trade every business day, and between coupon dates the seller has earned interest that has not yet been paid out. That earned-but-unpaid interest is accrued interest.
- Clean price. The bond’s value without accrued interest. This is the number quoted on financial websites and trading screens.
- Dirty price, also called invoice price. The clean price plus accrued interest. This is what the buyer actually wires to settle the trade.
Accrued interest depends on days elapsed since the last coupon date and the bond’s day-count convention. U.S. Treasury bonds use actual/actual, counting the real number of days elapsed divided by the real number of days in the coupon period. Corporate bonds typically 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). If a corporate bond pays $30 every six months and 45 days have passed since the last coupon date, the accrued interest is $30 × (45 / 180) = $7.50, and the dirty price is the clean price plus $7.50.
When You Know the Price and Want the Yield
Bond valuation runs the other direction too. If you already know the market price and want the yield it implies, you are solving for yield to maturity: the discount rate that makes the present value of all future cash flows equal the current price. It is effectively the bond’s internal rate of return if you hold to maturity and reinvest coupons 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 you close. For an exact answer, iterate until the calculated price matches the market price, or use a spreadsheet function (typically RATE or YIELD). For semi-annual bonds, solve for the semi-annual rate first and double it to annualize.
If the bond has a call provision letting the issuer redeem it before maturity, the same calculation done to the call date and call price gives yield to call. Yield to call is often more relevant than yield to maturity when rates have fallen and the issuer is likely to refinance.