How to Price a Bond (With a Calculator)

Discounting coupons and principal, step by step.

A bond's price is the present value of every cash flow it promises. That sounds abstract until you write it down: every coupon payment and the final return of face value gets discounted back to today at the market's required rate of return, and the sum of those discounted cash flows is what the bond is worth right now. A bond pricing calculator automates the arithmetic, but walking through the formula by hand is the fastest way to understand why bond prices move and what drives them. This post does exactly that — one formula, one worked example, and the three rules that follow from it.

The bond pricing formula

The price of a plain fixed-rate bond is:

Price = Σ C / (1+y)^t + F / (1+y)ⁿ

where

• C is the coupon payment per period
• y is the yield per period (the discount rate)
• t is the period index, running from 1 to n
• F is the face value (par), returned at maturity
• n is the total number of periods

The sigma (Σ) just means you sum the discounted coupon terms across all periods and then add the discounted face value. That final term, F / (1+y)ⁿ, is the present value of the principal repayment. Everything else is coupon math.

Semi-annual convention

Most bonds — US Treasuries, corporate bonds, agency debt — pay coupons twice a year. The formula above handles this with three adjustments: halve the annual coupon rate to get the per-period coupon, halve the annual yield to get the per-period discount rate, and double the number of years to get the number of periods.

For a 3-year bond paying semi-annually, n = 6, not 3. For a 5% annual coupon on $1,000 face, C = $25 per period, not $50. For a 4% annual yield, y = 2% per period. This is not optional convention — it changes the price materially, especially at longer maturities — so the inputs to any bond price calculator must match the bond's actual payment frequency.

Worked example: 5% coupon, 3-year, 4% yield

Take a bond with a 5% annual coupon, $1,000 face value, 3 years to maturity, and a market yield to maturity of 4%. Using semi-annual convention: C = $25, y = 2%, n = 6.

The six discounted coupon payments are:

• Period 1: 25 / 1.02¹ = $24.51
• Period 2: 25 / 1.02² = $24.03
• Period 3: 25 / 1.02³ = $23.56
• Period 4: 25 / 1.02⁴ = $23.09
• Period 5: 25 / 1.02⁵ = $22.64
• Period 6: 25 / 1.02⁶ = $22.19

Sum of coupons: $140.02. Discounted principal: 1,000 / 1.02⁶ = $887.97. Total price: $1,027.99.

The bond trades above $1,000. That is not a coincidence — it follows directly from the relationship between the coupon rate and the yield, which is the next section.

Par, premium, and discount bonds

The price relative to face value tells you everything about how the bond's coupon rate compares to the market's current required return — that required return being the yield to maturity.

• Coupon rate = yield: Each cash flow is discounted at exactly the rate it earns, so the price equals face value. The bond trades at par ($1,000).
• Coupon rate > yield: The bond pays more than the market requires. Investors bid it up above par. This is a premium bond — as in the example above, where a 5% coupon beats a 4% yield and the price lands at $1,027.99.
• Coupon rate < yield: The bond pays less than the market requires. No one will pay full price for below-market cash flows, so it trades below par — a discount bond. Flip the example: a 3% coupon on the same bond at a 4% yield would price near $972.

The intuition is simple: when market rates rise above what a bond pays, the bond becomes less attractive, so its price must fall to compensate. When market rates fall, existing bonds paying higher coupons become valuable and their prices rise.

Clean price vs. dirty price

The formula above produces what the market calls the dirty price — also called the full price or invoice price. It includes accrued interest: the coupon that has been building up since the last payment date but has not yet been paid. When a bond trades between coupon dates, the buyer compensates the seller for the fraction of the coupon period that has elapsed.

What you see quoted on most screens is the clean price, which strips out accrued interest. Clean price = dirty price − accrued interest. Settlement calculations and actual payment use the dirty price. The difference matters most mid-period and is smallest just after a coupon payment, when accrued interest has reset to near zero.

Price and yield move in opposite directions

The denominator of every term in the pricing formula contains (1+y)^t. Raise y and every denominator gets larger, so every present value shrinks and the price falls. Lower y and every denominator gets smaller, so every present value rises and the price climbs. This inverse relationship is mechanical, not market sentiment — it is baked into the arithmetic of discounting.

The sensitivity of price to yield is measured by duration. Longer-maturity bonds and lower-coupon bonds have higher duration and therefore larger price swings for a given change in yield. A 30-year zero-coupon bond will move far more per basis point of yield change than a 2-year bond paying a rich coupon — because the zero's single cash flow is discounted over a much longer horizon, amplifying the effect of any rate move.

Understanding this relationship is what makes a bond price calculator more than a number machine. Change the yield input by 100 basis points and watch the price move — that delta is exactly what you are managing when you hold or hedge a fixed-income position.