Bond Pricing: A Simple Present Value Guide

What cash flows determine a bond’s price?

A bond is a loan from investors to a company or government. A basic fixed-rate bond promises periodic interest, called coupons, and repayment of principal at maturity. Face value, par value, and principal describe the maturity amount in this introductory model.

An ordinary annuity is a series of equal payments made at the end of each period. For bond pricing at issuance on an interest date, use the market yield, which is the return investors currently require on comparable bonds. The OpenStax bond valuation explanation also separates coupons from principal. This guide assumes fixed payments, no issuance costs, and no special conversion or redemption features.

Coupon rate vs. market rate: premium or discount?

The coupon rate determines the cash the issuer promises to pay. The market rate determines how investors discount those promised payments. They answer different questions, so don’t swap them.

Exam tip: Questions use different names for the same two rates. The coupon rate may appear as the stated rate or the contract rate. The market rate may appear as the yield or the effective rate.

Compare rates on the same periodic basis. A premium does not increase the face amount repaid at maturity; a discount does not reduce it. The difference is only in what investors pay for the bond on day one.

What is the bond pricing formula?

Let C be the cash coupon each period, F the face value, i the market rate per period, and n the number of coupon periods.

Hypothetical example. Cedar Manufacturing issues $50,000 of four-year bonds with an 8% annual coupon. Interest is paid at each year-end. The market yield is 5% annually. The annual cash coupon is $50,000 × 8% = $4,000.

The premium is $5,318.93. That result is reasonable because the 8% coupon exceeds the 5% market yield. The final year includes both the $4,000 coupon and the $50,000 principal repayment. The annuity already includes the last coupon, so do not add it twice.

The displayed components sum to $55,318.92 because each is rounded separately. The price above is rounded only after combining the full-precision values. Use the rounding instruction supplied in a problem.

Check your answer in Excel or with PV tables

You can check Cedar’s price in Excel with =-PV(0.05,4,4000,50000), which returns 55,318.93. The minus sign turns Excel’s negative result into a positive price. For the semiannual bond in the next section, use =-PV(0.04,6,1500,50000), which returns 47,378.93.

PV tables give a slightly different answer because their factors are rounded. Using the five-decimal factors 3.54595 and 0.82270, Cedar’s price comes to $55,318.80. That 13-cent gap is rounding, not a mistake.

How do you price a bond with semiannual interest?

Semiannual means twice a year. If the stated annual market yield is a nominal rate compounded semiannually, divide it by two. Divide annual cash interest by two and multiply the years by two.

Hypothetical example. A separate $50,000 bond pays a 6% annual coupon semiannually for three years. Its market yield is 8% nominal, compounded semiannually.

This bond sells at a $2,621.07 discount because its coupon rate is below the market yield. If a problem instead supplies an effective annual yield, convert it to a semiannual rate using (1 + annual yield)^1/2 − 1; do not automatically halve it.

Why do bond prices fall when market rates rise?

A fixed-rate bond’s coupons never change. When new bonds offer a higher return, investors will only buy the older bond at a lower price. When market rates fall, the older bond’s coupons look better, so its price rises. In short, bond prices move in the opposite direction from market rates.

Here’s Cedar’s bond, with its $4,000 annual coupon and four-year term, priced at different market yields:

Bond prices are usually quoted as a percentage of face value. Cedar’s $55,318.93 price is a quote of about 110.64, and the $47,378.93 semiannual bond is a quote of about 94.76.

How do you record a bond at its issue price?

The issue price is what goes on the issuer’s books. Any gap between the price and face value is recorded as a premium or discount. That amount is then amortized, meaning spread over the bond’s life, using the effective interest method described in FASB ASC 835-30. Other methods are allowed only if the results aren’t materially different.

Under the effective interest method, each period’s interest expense equals the beginning carrying amount times the market rate at issuance. The carrying amount is face value plus any unamortized premium, or minus any unamortized discount. On the balance sheet, a premium is added to Bonds Payable and a discount is subtracted.

Bonds payable due beyond one year from the balance-sheet date, or beyond the operating cycle if longer, are generally grouped with long-term liabilities on a classified balance sheet. The portion due within that period is generally reported as a current liability.

Journal entries for a premium: Cedar’s bond

At issuance:

First interest payment:

Interest expense is $55,318.93 × 5% = $2,765.95. The cash paid stays at $4,000, and the $1,234.05 difference reduces the premium.

Year 4 interest is reduced by a penny for rounding so the balance lands exactly on the $50,000 face value.

Journal entries for a discount: the semiannual bond

At issuance:

First interest payment:

Interest expense is $47,378.93 × 4% = $1,895.16, and the carrying amount rises to $47,774.09.

Exam tip: On a premium bond, interest expense is less than the cash paid. On a discount bond, it’s more. Either way, the carrying amount moves toward face value. If your schedule shows the opposite, recheck which rate you multiplied.

Bond price vs. sinking fund: what’s the difference?

A bond’s issue price is a present value. A sinking fund is money accumulated to help repay principal later. Determining whether deposits will cover maturity principal is a future-value question.

For example, five hypothetical year-end deposits of $9,000 earning 4% grow to $48,746.90 on the date of the fifth deposit. If the $50,000 of bonds matures that same day, the shortfall is $1,253.10. Compare both amounts at the maturity date.

For deposits that begin after a delay, review deferred annuity timing and future value. For a valuation based on uncertain amounts, review the expected cash flow approach and discount rates. Keep in mind that a bond’s yield is meant for promised, contractual payments, so don’t use it to discount probability-weighted amounts.

Common bond pricing mistakes

• Using the market rate to calculate the coupon. The cash payment comes from the stated rate, so it doesn’t change with the market.
• Using the coupon rate to discount the payments. That forces the answer to par.
• Leaving out the maturity principal. That skips a major cash flow.

Another error is converting the rate to six months while leaving the period count in years. Write C, i, and n together before calculating. Check the premium or discount direction before accepting the numerical result.

Three practice questions with explained answers

1. Identify the coupon

A $120,000 bond has a 6% annual coupon and pays interest semiannually. What is each cash interest payment?

2. Predict the issue price

A bond pays a 7% annual coupon while comparable bonds yield 9% on the same basis. Will it sell at par, a premium, or a discount?

3. Combine the present values

For a hypothetical bond, the present value of coupons is $12,400 and the present value of principal is $61,600. What is the issue price?

Key takeaways

• A bond’s price is the present value of its coupons plus the present value of its face value.
• The stated rate sets the cash coupon. The market rate does the discounting.
• Coupon above market means a premium. Coupon below market means a discount. Equal rates mean par.
• For semiannual bonds, divide annual cash interest and the nominal annual market rate compounded semiannually by two; multiply the years by two.
• Bond prices move in the opposite direction from market rates.
• After issuance, the issuer amortizes any premium or discount using the effective interest method.

Frequently asked questions

Why are bond coupons an ordinary annuity?

In this model, equal interest payments occur at each period-end, beginning one period after issuance.

Does the coupon change when market rates change?

For a fixed-rate bond, the contractual coupon stays the same. Its present value changes when the required market yield changes.

Is a bond’s issue price always its face value?

No. In the basic model, the price equals face value when the coupon and market rates match. Otherwise, a premium or discount results.

What is the difference between coupon rate and yield?

The coupon rate is fixed in the bond contract and sets the cash interest. The yield is the return investors require at today’s price. Yield to maturity (YTM) is the single rate that makes the present value of a bond’s payments equal its price. In this page’s examples, YTM equals the market rate.

How do you price a zero-coupon bond?

A zero-coupon bond pays no periodic interest, so its price is just the present value of its face value. For example, a $50,000 bond due in five years at a 6% annual market rate is worth $50,000 ÷ 1.06⁵ = $37,362.91.

What happens when a bond sells between interest dates?

The buyer pays the bond’s price plus the interest built up since the last payment date, called accrued interest. For example, a $50,000, 6% bond with semiannual interest sold two months after an interest date carries accrued interest of $50,000 × 6% × 2/12 = $500. This example uses a simplified month-based calculation consistent with a 30/360 convention; actual accrued interest depends on the bond’s day-count convention. A clean price leaves this out, and a dirty price includes it.

How do debt issuance costs affect a bond?

Under ASU 2015-03, debt issuance costs are shown as a direct deduction from the carrying amount of the debt, the same way a discount is shown. You can read a summary in the Journal of Accountancy. This guide leaves them out to keep the pricing math simple.