Price of a fixed-coupon bond.
P = C × [1 − (1 + r)^(−n)] / r + F / (1 + r)^n
The price of a bond equals the present value of all future coupon payments plus the present value of the face value (par) repaid at maturity. Each coupon payment is discounted by the periodic market yield, and the face value is discounted over all remaining periods. The formula below assumes a fixed-coupon bond with equal periodic payments:
When the coupon rate equals the market yield, the bond prices at par. When the market yield exceeds the coupon rate, the bond prices at a discount; when below, at a premium.
Step 1 – Periodic coupon: C = (6% × $1,000) / 2 = $30 Step 2 – Periodic yield: r = 8% / 2 = 4% = 0.04 Step 3 – Total periods: n = 5 × 2 = 10 Step 4 – PV of coupons: 30 × [1 − (1.04)^(−10)] / 0.04 = 30 × [1 − 0.67556] / 0.04 = 30 × 8.1109 = $243.33 Step 5 – PV of face value: 1,000 / (1.04)^10 = 1,000 / 1.48024 = $675.56 Step 6 – Bond Price: P = $243.33 + $675.56 = $918.89
Result: Bond Price ≈ **$918.89**
The bond prices at $918.89, which is below the $1,000 face value — a discount bond. This makes intuitive sense: the bond's coupon rate (6%) is lower than the market's required yield (8%), so investors will only purchase the bond if they can buy it cheaply enough that the total return (coupons + capital gain to par at maturity) equals the 8% market rate. The $81.11 discount compensates the buyer for the below-market coupon rate.
A bond is a debt instrument where the issuer (corporation, government, municipality) borrows money from the investor and promises to pay periodic interest (coupon payments) and return the principal (face value) at a specified maturity date.
Par, Premium, and Discount
Inverse Relationship — Price vs. Yield Bond prices and yields move in opposite directions. When market interest rates rise, existing bonds with lower coupons become less attractive, so their prices fall. When rates fall, existing higher-coupon bonds become more valuable and their prices rise. This is the most fundamental concept in fixed-income investing.
Yield to Maturity (YTM) YTM is the annualized total return an investor earns if the bond is purchased at the current market price and held until maturity, assuming all coupons are reinvested at the same rate. It is the internal rate of return (IRR) of the bond's cash flows. Solving for YTM algebraically is not possible in closed form — it requires iterative numerical methods (such as Newton-Raphson) or a financial calculator.
Duration and Interest Rate Risk Duration measures a bond's sensitivity to interest rate changes. A bond with longer maturity or lower coupon rate has higher duration and greater price volatility for a given yield change. Modified Duration ≈ (% price change) / (% yield change).
Credit Risk and Yield Spread The required market yield reflects not just the risk-free rate but also a credit spread compensating investors for the issuer's default risk. Higher-rated (AAA) bonds trade at lower yields; junk bonds (below BB) offer higher yields to compensate for risk.
| Type | Key Feature | |---|---| | Zero-coupon | No periodic payments; issued at deep discount | | Fixed-rate coupon | Regular, equal coupon payments | | Floating-rate | Coupon adjusts with benchmark rate | | Convertible | Can be converted into equity | | Callable | Issuer may redeem before maturity |
Results from this calculator are mathematical estimates. Actual bond prices may differ due to accrued interest, bid-ask spreads, tax treatment, and real-time market conditions. Consult a licensed financial advisor before making investment decisions.
Rising inflation typically pushes market interest rates higher, which causes existing bond prices to fall (since the nominal coupon payments are worth less in real terms and new bonds offer higher nominal yields). Inflation-protected bonds like US TIPS adjust their principal value with the CPI to preserve purchasing power.
Modified Duration = Macaulay Duration / (1 + r/m), where m is periods per year. It estimates the percentage change in bond price for a 1% (100 basis point) change in yield. For example, a bond with a modified duration of 7 will lose approximately 7% of its price if yields rise by 1%.
Use yield to maturity (YTM) as a standardized, apples-to-apples comparison metric. YTM accounts for price paid, coupon income, and the time value of money across all periods to maturity. Also consider duration (interest rate risk), credit rating, and tax treatment when comparing bonds.
Regardless of whether a bond trades at a premium or discount, its price converges toward the face value (par) as the maturity date approaches — a phenomenon known as 'pull to par.' The present value of the face value increases as the discounting period shortens, dominating the price calculation.
Current yield = Annual Coupon Payment / Current Market Price. It is a simple measure of income return but ignores the capital gain or loss from buying at a discount or premium and holding to maturity. YTM is a more comprehensive measure that captures total return including price appreciation or depreciation.
You need four inputs: (1) the bond's face value (par value), (2) the annual coupon rate, (3) the years remaining to maturity and coupon frequency, and (4) the current annual market yield (required rate of return). The calculator handles the discounting automatically.
The coupon rate is fixed at issuance and determines the dollar amount of each periodic interest payment. The yield to maturity (YTM) reflects the bond's total annualized return if purchased at the current market price and held to maturity. YTM changes every time the market price changes; the coupon rate does not.
A bond sells below par (at a discount) when the prevailing market interest rates are higher than the bond's coupon rate. Investors demand a lower purchase price so that the effective yield — combining coupon income plus the capital gain from buying below par and receiving full par at maturity — equals the higher market rate.
Yes. Set the coupon rate to 0%. The formula simplifies to P = F / (1 + r)^n, giving the deeply discounted price of a bond that pays no periodic interest and returns only the face value at maturity. US Treasury STRIPS are a common example.
Yes. Select 'semi-annual' as the coupon frequency. The calculator divides the annual coupon rate and annual yield by 2, and doubles the number of years to get total periods, which is the standard US bond market convention.
The clean price is the bond price without accrued interest — the number displayed by this calculator and quoted in most markets. The dirty price (invoice price) adds the interest accrued since the last coupon date. When you actually buy a bond between payment dates, you pay the dirty price.
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