Bond Discount Formula: Price the Promised Cash Flows

Identify a bond discount rather than a retail discount

A bond trading at a discount has a price below its face value under the comparison being made. That is different from applying a shop's percentage discount to a listed price. The site's Discount Calculator does not value coupons or a maturity payment. For a simple fixed-coupon bond valued on a coupon date, price is the present value of future coupon payments plus the present value of principal repaid at maturity. The schedule, required yield and compounding convention determine the arithmetic.

Set up an annual-coupon example

Assume face value of 1,000, an annual coupon rate of 4%, three years remaining and a required annual yield of 6%. Each annual coupon is 40. Payments occur at the ends of years one, two and three, with the 1,000 principal also repaid at year three. This illustration assumes the stated payments occur, excludes accrued interest between coupon dates and has no call feature or transaction costs. The 4% coupon defines the payment amount; the 6% yield is the rate used to discount the future cash flows.

Discount each payment to the valuation date

The price is 40/1.06+40/1.06²+1,040/1.06³, approximately 946.539761. Rounded to two decimals, that is 946.54. The discount below face value is 1,000−946.539761, approximately 53.46, or about 5.3460% of face value. Enter the first two 40 coupons and the final combined 1,040 payment separately in the Present Value Calculator, with rate 6 and years 1, 2 and 3 respectively. The tool values one future amount at a time; it does not accept the complete bond schedule in one run.

Use the coupon-annuity version of the formula

The same price can be written as C×[1−(1+y)^(-n)]/y+F/(1+y)^n, where C is coupon per period, y the yield per period, n the remaining payment count and F face value. The coupon term is an ordinary-annuity present value and the final term discounts principal. If there are two coupons per year, align the coupon amount, yield convention and payment count with half-year periods. Do not keep annual n while halving only the yield. An actual market quotation can also use day-count and settlement conventions beyond this simplified coupon-date example.

Compare coupon rate, current yield and yield to maturity

The coupon rate here is 40/1,000=4%. Current yield based on the modeled price is 40/946.539761, approximately 4.2259%. The assumed yield to maturity is 6%, which also reflects the difference between the purchase price and the principal repaid at maturity through the discounted cash-flow equation. These are different measures. Dividing the annual coupon by price does not generally produce yield to maturity. Likewise, the 5.3460% discount to face value is a price ratio, not an annual investment return.

Explain the direction of the price effect

With this unchanged payment schedule, a required yield equal to the 4% coupon rate produces a price of 1,000 on the stated coupon-date basis. A required yield above the coupon rate lowers the present value below face value; a lower yield raises it above face value. This relationship follows from discounting the same positive cash flows at different rates. It does not establish that a particular market bond is mispriced. Credit risk, optional features, settlement details and other terms can affect the relevant yield and cash-flow expectations.

Keep the model's limitations with the answer

This example does not determine a live market price, future resale price, default probability or tax treatment. A zero-coupon bond has no intermediate coupon stream and uses only the discounted maturity amount. A callable, floating-rate or irregular-payment bond needs a different schedule or model. Preserve the face value, coupon frequency, payment dates, valuation date and yield convention so another reader can reproduce the result. Use the local calculator to check component present values, while obtaining actual security terms and quotations from the relevant issuer or market records.

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