Bond Prices & Yields
Build a by discounting its future coupons and principal, one step at a time.
Build a bond price, one step at a time
Stage 1 · One future payment
What is 1,000 received later worth today?
A bond's final principal is a future payment. First, move YTM and watch what that one payment is worth today.
At t₀, $956.94 can grow to 1,000 in 1 year at 4.5%.
If $956.94 grows at 4.5% for 1 year, it reaches about 1,000. Discounting reverses that growth.
The principal does not disappear. At a higher required yield, a smaller amount today can grow to the same future payment.
This backward-growth idea becomes the present-value formula below: PV × (1+r)ⁿ = FV, so PV = FV / (1+r)ⁿ.
What did you just see?
determines scheduled payments; required yield determines what they are worth today. Holding cash flows fixed, a higher yield lowers price.
Formula variables
Face is principal repaid at maturity; is the nominal annual coupon rate; is the nominal annual required yield; m is payments per year; Years is time remaining; N = Years × m is the number of payments. Coupon per period = Face × Coupon Rate ÷ m; discount rate per period = YTM ÷ m.
How coupon rate and YTM affect price
All else equal, a higher raises each payment and price; a higher required yield discounts the same payments more and lowers price. When the two rates match, a standard fixed-coupon bond prices at par.
Discount, par, and premium bonds
A trades below face when required yield generally exceeds ; a par bond trades at face when the rates match; a premium bond trades above face when required yield is generally lower.
Why bond prices usually fall as yields rise
If new bonds offer a higher required return, an older bond's fixed payments look less attractive. Its price generally needs to fall to compete.
Formally, largely fixed future payments are worth less today when discounted at a higher periodic YTM/m. This comparison holds cash flows, credit, and other assumptions constant.
Clean price and dirty price
The price quoted in a market is not always the amount paid at settlement. Clean price excludes interest accrued since the last coupon date; dirty price includes it.
Dirty price = clean price + accrued interest. This page's pricing model assumes valuation on a coupon date, so accrual is zero. For settlement between coupons, use Clean & Dirty Price to calculate accrual and compare quotes and settlement amounts in consistent units.
How zero-coupon bonds work
A zero-coupon bond pays no regular interest. It is usually purchased below and pays full face at maturity; the difference is the main source of return.
For positive YTM, price = face ÷ (1+YTM/m)^(years×m). With all cash flow at maturity, Macaulay duration equals time to maturity, often making it more rate sensitive. At negative YTM, price can exceed face.
Another example
- Enter 1,000 face value, a 5% coupon rate, 6% required yield, five years remaining, and semiannual coupons.
- Each coupon is 1,000 × 5% ÷ 2 = 25 for 10 periods. Discount each coupon and the final principal at 3% per period.
- Coupon PV is about 213.26 and principal PV about 744.09, totaling about 957.35. Below 1,000 face value, it is a discount bond.
Show formal definition and formula
Bond price is the theoretical present value of future coupons and principal at the investor's required yield.
Price equals the present value of each coupon and final face value discounted at the required yield per period; N is remaining payment periods.
Coupon is payment per period; Face is maturity principal; m is payments per year and N remaining periods. YTM is a nominal annual yield; YTM/m is the periodic discount rate. With fixed coupons and principal, a higher yield lowers present value. Valuation is on a coupon date, excluding default and call options.
Common mistakes
Do not confuse coupon rate with required yield or treat this theoretical price as a market dirty price including accrued interest.
Frequently asked questions
Why does a higher required yield lower price?
The same scheduled payments are discounted at a higher rate, reducing their present value.
Can I price a zero-coupon bond?
Yes. Set coupon rate to 0%; price then equals the discounted face value at maturity.
Does this include accrued interest?
This tool assumes valuation on a coupon date, when accrued interest is zero and clean and dirty prices coincide. For settlement between coupons, use Clean & Dirty Price to calculate accrual. This page excludes taxes, fees, default, and early redemption.
How do I infer yield from a market price?
Use the YTM calculator and enter the known market price.
You can now explore…
- Value coupons and principal separately
- Why yield and price move oppositely
Term reference8 terms
Open a term when you need a reminder. Underlined words in the article work too.