Bond Price

Present value of a coupon bond: the sum of discounted periodic coupons plus the discounted face value at maturity. Discount rate is the yield to maturity, applied periodically as y/m.

When to use: Use to price any plain-vanilla coupon bond given face value, coupon rate, yield, term, and coupons-per-year. The closed-form combines an annuity (the coupon stream) with a single-sum present value (the face return).

Calculator

Formula

P = \frac{F \cdot CR / m \cdot \left[1 - (1 + y/m)^{-Nm}\right]}{y/m} + \frac{F}{(1 + y/m)^{Nm}}

Variables

Symbol	Name	Description	Unit
BondPrice	Bond Price	Theoretical (clean) price of the bond	$
F	Face Value	Par value paid at maturity	$
CR	Coupon Rate	Annual coupon rate as a decimal (e.g. 0.05 for 5%)	%
y	Yield	Annual yield as a decimal; periodic yield is y/m	%
N	Years to Maturity	Years remaining until the bond matures	years
m	Coupons per Year	Number of coupons paid per year (e.g. 2 for semi-annual)	integer

Real-Life Examples

Example 1: Par Bond at Coupon-Equals-Yield

10-year, 5% coupon, semi-annual, $1,000 face. Market yield equals coupon rate at 5%.

Given

F = $1,000.00CR = 5.0000%y = 5.0000%N = 10.00 yearsm = 2.00

Step-by-Step

1.Periods = 10 × 2 = 20

2.Periodic rate = 0.05 / 2 = 0.025

3.Periodic coupon = 1000 × 0.05 / 2 = 25

4.Annuity PV = 25 × [1 − 1.025⁻²⁰] / 0.025 = 25 × 15.5892 = 389.73

5.Face PV = 1000 / 1.025²⁰ = 1000 × 0.61027 = 610.27

6.P = 389.73 + 610.27 = 1000.00

Result:$1,000.00

When yield equals coupon rate, the bond prices at par. Prices above par signal yield below coupon (premium); prices below par signal yield above coupon (discount).

Example 2: Discount Bond at Higher Yield

Same 10-year, 5% coupon, semi-annual, $1,000 face — but the market now demands a 6% yield.

Given

F = $1,000.00CR = 5.0000%y = 6.0000%N = 10.00 yearsm = 2.00

Step-by-Step

1.Periods = 20, periodic rate = 0.03, periodic coupon = 25

2.Annuity PV = 25 × [1 − 1.03⁻²⁰] / 0.03 = 25 × 14.8775 = 371.94

3.Face PV = 1000 / 1.03²⁰ = 553.68

4.P = 371.94 + 553.68 = 925.61

Result:$925.61

A 100bp rise in yield (5%→6%) drops the price ~7.4%, illustrating duration-driven price risk. Selling the bond before maturity locks in this paper loss; holding to maturity recovers par.

Frequently Asked Questions

This model settles on a coupon date, with level coupons and a positive whole number of periods remaining. Between coupons, move each discount time to the actual settlement date and account for accrued interest; merely adding accrued interest to an unchanged coupon-date valuation is not exact.

Yield is the rate that equates discounted contractual cash flows to price. Realizing that rate as a compounded terminal-wealth return requires the assumed payments and reinvestment at that rate. Actual reinvestment rates can differ.

Yes, provided the periodic discount factor remains positive. The solver verifies that the returned yield reproduces the entered price.

Related Formulas

Zero-Coupon Bond Price

Yield to Maturity (YTM)

Inverse calculation

Macaulay Duration

Modified Duration

Zero-Coupon Bond Price