Last updated: 2026-05-19

Bond price

A bond's price is the sum of all its future cash flows discounted to today — the periodic coupon payments and the face value repayment at maturity. Given the required yield to maturity (YTM), the exact fair value of any fixed-rate bond can be determined, whether it trades above par (premium), below par (discount), or at par. This calculator performs that discounting and displays the split between the present value of the coupon stream and the present value of the face value.

How bond pricing works

A fixed-rate bond promises two streams of cash:

1. Coupon payments — a series of equal periodic payments equal to (face value × annual coupon rate / payments per year).
2. Face value (par) repayment — the lump sum returned to the holder at maturity.

The price of the bond is the combined present value of both streams, discounted at the yield to maturity:

where:

• C = face value × annual coupon rate / m (coupon per period)
• r = YTM / m (yield per period)
• N = years to maturity × m (total periods)
• m = coupon payments per year
• F = face value

Worked example

Consider a 10-year US Treasury bond:

• Face value: $1,000
• Annual coupon rate: 4.5 %
• Yield to maturity: 5.2 %
• Coupon frequency: semiannual (m = 2)

Step 1 — per-period values:

• Coupon per period: C = $1,000 × 4.5 % / 2 = $22.50
• Yield per period: r = 5.2 % / 2 = 2.6 %
• Total periods: N = 10 × 2 = 20

Step 2 — PV of coupons (annuity): $PV_{\text{coupons}} = 22.50 \times \frac{1 - (1.026)^{-20}}{0.026} \approx 22.50 \times 15.4429 \approx \$347.47$

Step 3 — PV of face value: $PV_{\text{face}} = \frac{1{,}000}{(1.026)^{20}} \approx \frac{1{,}000}{1.6709} \approx \$598.48$

Step 4 — bond price: $P = 347.47 + 598.48 = \$945.95$

The bond trades at a discount of $54.05 vs par because the coupon rate (4.5 %) is below the required yield (5.2 %). An investor buying at $945.95 earns exactly 5.2 % annualized if held to maturity.

Why bonds trade at a premium or discount

The relationship between coupon rate, YTM, and price is central to fixed-income pricing:

Intuition: A bond's coupon is fixed at issuance. If interest rates rise after issuance, new bonds offer higher coupons. To compete, existing bonds must fall in price so that their total return (fixed coupon + price appreciation to par) matches the new higher yield. Conversely, if rates fall, existing bonds with above-market coupons become more valuable, so their price rises above par.

Zero-coupon bonds

A zero-coupon bond pays no periodic interest. The entire return comes from purchasing the bond at a deep discount and receiving the full face value at maturity. Setting coupon rate = 0 in the formula reduces to:

$P = \frac{F}{(1+r)^N}$

For example, a $1,000 zero-coupon bond maturing in 10 years at a 6 % YTM (semiannual discounting) would price at $1,000 / (1.03)^20 ≈ $553.68 — roughly 55 cents on the dollar.

Duration and maturity sensitivity

The longer the maturity, the more sensitive the bond's price is to changes in interest rates. This is because more of the bond's value sits in distant cash flows, which are discounted over more periods. The formal measure of this sensitivity is duration — the weighted average time until a bond's cash flows are received, expressed in years, which approximates the percentage price change for a 1 % shift in yields:

• A 2-year bond might have a duration of ~1.9 years → a 1 % rate rise causes roughly a 1.9 % price decline.
• A 10-year bond might have a duration of ~8 years → the same 1 % rate rise causes roughly an 8 % price decline.
• A zero-coupon bond's duration equals its maturity — it is the most rate-sensitive structure for a given term.

Investors who expect rates to fall seek longer-duration bonds to maximize capital gains. Those expecting rates to rise prefer shorter maturities to limit price losses.

Assumptions and limitations

This calculator computes the clean price — the theoretical fair value assuming:

• Flat yield curve — the same YTM is applied to every future cash flow regardless of when it occurs. In reality, the yield curve usually slopes upward.
• Settlement on a coupon date — no accrued interest is modeled. The actual dirty price paid in the market includes accrued coupon since the last payment.
• No default risk — the issuer is assumed to pay all coupons and the face value in full. Corporate bonds command a credit spread above the risk-free rate to compensate for default probability.
• No embedded options — callable or putable bonds require additional yield-spread adjustments not captured here.