Fixed Incomeintermediate

Bond Pricing

A bond’s price is the present value of its promised cash flows, the stream of coupons plus the face value, each discounted at the market interest rate required by investors. When the coupon rate equals the market rate the bond prices at par. When the coupon rate exceeds the market rate the bond sells at a premium above face value, and when it falls short the bond sells at a discount below face value. The price therefore adjusts so that any bond, whatever its fixed coupon, offers the return the market currently demands. This single principle, discounting future cash flows, underlies every fixed-income valuation.

Try it yourself

Bond price vs yield

A bond's price is the present value of its coupons plus face, discounted at the market yield. Move the yield and watch price trace the downward-sloping, convex curve — price moves inversely to yield.

Price 100.00Current yield 5.00%Coupon income 5.00/yr

parcoupon = yield, price = face

Coupon rate5.0%

Yield to maturity5.0%

Years to maturity5 yr

Coupon frequency

Face value fixed at 100, so price reads as a percent of face.

Why it matters

A bond is just a basket of future cash payments, and its fair price today is what those payments are worth now. The discount rate is the market’s required return, the going rate for money of similar risk and maturity. If a bond’s fixed coupon is generous relative to that going rate, investors bid its price up above face value until the richer coupon is offset. If the coupon is stingy, the price falls below face until the discount makes up the difference. The coupon is locked, so the price does all the adjusting.

Formulas

Bond price as present value

P = \sum_{t=1}^{N} \frac{C}{(1+y)^{t}} + \frac{F}{(1+y)^{N}}

Each coupon

C

and the face value

F

are discounted at the per-period market yield

y

over

N

periods. For semi-annual bonds, use the semi-annual coupon and half the annual yield.

Premium, par, and discount condition

c > y \Rightarrow P > F,\quad c = y \Rightarrow P = F,\quad c < y \Rightarrow P < F

Comparing the coupon rate

c

with the market yield

y

tells you immediately whether the bond trades at a premium, at par, or at a discount.

Worked examples

Scenario

A three-year bond pays an annual coupon of A$80 on A$1,000 face value. The market yield is ten percent. Find the price and state whether it is a premium or discount.

Solution

Discount each cash flow at ten percent. The coupons are worth $\tfrac{80}{1.1}+\tfrac{80}{1.21}+\tfrac{80}{1.331}=72.73+66.12+60.11=198.96$ . The face value is worth $\tfrac{1000}{1.331}=751.31$ . The price is $P=198.96+751.31=950.27$ , about A$950. Because the eight percent coupon is below the ten percent market yield, the bond sells at a discount.

NoteThe price sits below face value precisely because the coupon rate is less than the required yield, matching the discount condition.

Common mistakes

✗A bond always sells for its face value. Price equals face value only when the coupon rate equals the market yield. Otherwise the bond trades at a premium or a discount.
✗A premium bond is a better investment than a discount bond. Premium and discount only reflect how the fixed coupon compares with the market yield. Both are priced to offer the same market return for equivalent risk.
✗Rising market rates raise bond prices because bonds pay more. Coupons are fixed at issue, so when market rates rise the price must fall to deliver the higher required return.
✗You discount coupons at the coupon rate. Cash flows are discounted at the market yield investors require, not at the coupon rate printed on the bond.

Revision bullets

•Bond price is the present value of coupons plus face value
•Discount at the market yield, not the coupon rate
•Coupon rate above market yield gives a premium price
•Coupon rate below market yield gives a discount price
•Coupon rate equal to market yield gives a par price
•The fixed coupon means price adjusts to deliver the required return

Quick check

A bond carries a five percent coupon while the market yield for similar bonds is seven percent. The bond will most likely trade at

What discount rate is used to find a bond’s fair price?

Connected topics

Bond Basics Bond Yield Yield Curve Duration DDM

Sources

Brailsford, Heaney & Bilson (2015), Ch. on bond valuation
Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
Develops bond price as the present value of coupons and face value and the premium/discount logic.
Bodie, Kane & Marcus (2021), Ch. 14
Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
Presents the bond pricing formula and the relationship between coupon rate, yield, and price.

How to cite this page

Dr. Phil's Quant Lab. (2026). Bond Pricing. Derivatives Atlas. https://phucnguyenvan.com/concept/im-bond-pricing

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