Annual-Coupon Bond Pricing Workshop

Introduction

An annual-coupon bond has a simple payment pattern: one coupon arrives at the end of each year, and the redemption payment arrives with the final coupon. Pricing it correctly still requires careful attention to three amounts that need not be equal: the bond’s face value, the annual coupon, and the redemption amount. In particular, a coupon rate is applied to face value, not automatically to the redemption amount.

This tutorial develops a consistent method for pricing annual-coupon bonds when the yield is quoted as an effective annual rate. The examples include bonds redeemed at face value and at a nonpar amount, as well as a quotation per 100 of face value. The bond is valued on a coupon date, with the next coupon due one year later; there are no stub periods or accrued interest in these examples.

Definition: For a bond with face value $F$, annual coupon rate $r$, $n$ remaining annual coupon payments, redemption amount $R$, and effective annual yield $i$, the annual coupon is $C=Fr$. The redemption amount $R$ is paid at the end of year $n$, together with the final coupon. Face value is the amount used to calculate the coupon; it does not have to equal the amount redeemed.

Annual-Coupon Price Formula

Assume the bond makes level annual coupons at times $1,2,\ldots,n$, measured in years from the valuation date. Assume the effective annual yield $i$ is constant for all these cash flows, and that the bond has no other payments or embedded features affecting the promised cash flows. Under these assumptions, each payment $k$ years from the valuation date is discounted by $v^k$, where $v=1/(1+i)$.

P=\sum_{k=1}^{n}Cv^k+Rv^n =C\,a_{\overline{n}|}+Rv^n, \qquad a_{\overline{n}|}=\sum_{k=1}^{n}v^k.

Here $P$ is the bond’s price. The annuity factor can also be written as $a_{\overline{n}|}=(1-v^n)/i$ when $i\ne 0$. If $i=0$, define $a_{\overline{n}|}=n$, which agrees with its sum definition because then $v=1$. Thus the sum definition works for both zero and nonzero yields.

The formula separates the present value of the coupons from the present value of redemption. It remains valid when $R\ne F$, provided the redemption amount is included as stated in the bond’s cash flows. Do not replace $R$ with face value unless the bond is actually redeemed at face value.

Why the Formula Holds

The price is the sum of the present values of the promised payments. Each of the first $n-1$ payments is a coupon $C$. At time $n$, the investor receives both the final coupon $C$ and redemption $R$. Therefore, the cash flows are $C,C,\ldots,C,C+R$ at times $1,2,\ldots,n$, and discounting each at the effective annual yield gives

P=Cv+Cv^2+\cdots+Cv^n+Rv^n =C\left(\sum_{k=1}^{n}v^k\right)+Rv^n.

The bracketed sum is $a_{\overline{n}|}$ by definition, so $P=C\,a_{\overline{n}|}+Rv^n$. When $i\ne0$, the geometric-series identity gives $\sum_{k=1}^{n}v^k=v(1-v^n)/(1-v)$. Since $v=1/(1+i)$, $1-v=i/(1+i)=iv$, and the sum simplifies to $(1-v^n)/i$. At $i=0$, the geometric-series quotient is not defined, but the original sum is $n$.

Identify the contract amounts.
Record face value $F$, coupon rate $r$, number of annual periods $n$, and redemption amount $R$. Compute the coupon as $C=Fr$.

Match the yield to the payment interval.
For an effective annual yield and annual coupons, use $i$ directly once per year. Do not divide it by the number of coupons.

Discount coupons and redemption.
Use $C\,a_{\overline{n}|}$ for the coupons and $Rv^n$ for redemption, or discount each cash flow separately.

Check the requested quotation.
If the question asks for price per 100 of face value, divide the total price by $F$ and multiply by 100.

Worked Pricing Examples

Worked Example: Par Redemption with an Annual Yield

A bond has face value \$2,000, an annual coupon rate of 6%, four years remaining, and redemption at face value. Find its price at an effective annual yield of 5%.

Method. First calculate the coupon from face value: $C=2{,}000(0.06)=\$120$. The annual effective yield applies once to each annual payment, so $v=1/1.05=0.9523809524$. Its successive powers are $v^2=0.9070294785$, $v^3=0.8638375986$, and $v^4=0.8227024748$.

The four-period annuity factor is the sum of these four discount factors: $a_{\overline{4}|}=0.9523809524+0.9070294785+0.8638375986+0.8227024748=3.5459505043$. The coupon present value is $120(3.5459505043)=\$425.5140605$.

Redemption is \$2,000 at the end of year 4, so its present value is $2{,}000(0.8227024748)=\$1{,}645.4049496$. Therefore, $P=425.5140605+1{,}645.4049496=\$2{,}070.9190101$, or \$2,070.92 to the nearest cent.

Check. The coupon rate of 6% exceeds the 5% yield, and redemption equals face value, so a price above face value is consistent. This check supports the result but does not replace the present-value calculation.

Worked Example: Redemption Below Face Value

A bond has face value \$1,200, an annual coupon rate of 4%, three years remaining, and a stated redemption amount of \$1,100. Find its price at an effective annual yield of 5%.

Method. The coupon is based on face value: $C=1{,}200(0.04)=\$48$. With $i=0.05$, $v=1/1.05=0.9523809524$, $v^2=0.9070294785$, and $v^3=0.8638375986$. Thus $a_{\overline{3}|=0.9523809524+0.9070294785+0.8638375986=2.7232480295$.

The coupon present value is $48(2.7232480295)=\$130.7159054$. The redemption present value is $1{,}100(0.8638375986)=\$950.2213585$. Adding the two components gives $P=130.7159054+950.2213585=\$1{,}080.9372639$, or \$1,080.94 to the nearest cent.

Check. The coupon rate is below the yield, and the redemption amount is below face value. Both features reduce value relative to face value; the calculated price is indeed below \$1,200.

Worked Example: Nonpar Redemption and Price per 100

A bond has face value \$2,500, an annual coupon rate of 8%, two years remaining, and redemption of \$2,400. The effective annual yield is 10%. Find the total price and the price per \$100 of face value.

Method. The annual coupon is $C=2{,}500(0.08)=\$200$. The discount factor is $v=1/1.10=0.9090909091$, with $v^2=0.8264462810$. The coupon present value is $200(v+v^2)=200(1.7355371901)=\$347.1074380$.

At year 2, the investor receives the final \$200 coupon and \$2,400 redemption. Equivalently, the redemption component alone has present value $2{,}400(0.8264462810)=\$1{,}983.4710744$. Therefore, $P=347.1074380+1{,}983.4710744=\$2{,}330.5785124$, or \$2,330.58 to the nearest cent.

The price per \$100 of face value is $(2{,}330.5785124/2{,}500)(100)=\$93.2231405$, or \$93.22 per \$100 to the nearest cent. This quotation is a scaled price, not a change to the bond’s face value or its cash flows.

Check. The coupon rate is below the yield, and redemption is below face value, so a price below face value is expected. The result, \$2,330.58 compared with \$2,500 face value, has the expected direction.

Common Mistake: Calculating the Coupon from Redemption

In the second example, the coupon rate is 4% and face value is \$1,200, so the coupon is \$48. A tempting but incorrect calculation is $1{,}100(0.04)=\$44$, using the redemption amount instead. That error would give $44(2.7232480295)+1{,}100(0.8638375986)=\$1{,}070.04$, rather than the correct \$1,080.94. The \$44 coupon does not follow from the stated terms.

The correction is to keep the roles separate: $C=Fr$ uses face value, while $R$ is used only for the redemption cash flow. Another common error is to discount the final coupon for $n-1$ years or to omit it entirely. The final coupon is paid at time $n$, just like the redemption amount, so both are discounted by $v^n$.

Interpreting the Price

A useful check is to compare coupon rate, yield, and redemption amount. For a bond redeemed at face value, a coupon rate above the yield generally means the coupons are relatively attractive and the price is above face value; a coupon rate below the yield generally means the price is below face value. When redemption is nonpar, the difference between $R$ and $F$ also affects price, so a coupon-rate comparison alone is not decisive.

These checks help catch arithmetic or input errors, but the present-value equation remains the pricing method. They assume the stated cash flows are fixed and that the yield is applied consistently to each annual period. If payment dates are not annual coupon dates, or if a bond has additional features that change the cash flows, this annual-coupon formula with integer-year discounting does not by itself handle those complications.

Key takeaway: For annual coupons valued on a coupon date at an effective annual yield, calculate the coupon from face value, discount each annual payment once per year, and use the stated redemption amount at maturity. Then apply any per-100 quotation only after finding the total price.

Check Your Understanding

For each problem, identify the annual coupon and discount the stated cash flows using the effective annual yield. Round monetary answers to the nearest cent unless the requested quotation specifies otherwise.

A bond has face value \$1,000, a 5% annual coupon rate, two years remaining, and redemption at face value. Find its price at an effective annual yield of 4%. (Answer: \$1,018.86)
A bond has face value \$1,500, a 3% annual coupon rate, three years remaining, and redemption of \$1,600. Find its price at an effective annual yield of 5%. (Answer: \$1,504.69)
A bond pays no coupons and redeems for \$1,500 in two years. Find its price at an effective annual yield of 25%. (Answer: \$960.00)
A bond has face value \$3,000, a 6% annual coupon rate, two years remaining, and redemption of \$2,800. At an effective annual yield of 5%, find the price per \$100 of face value. (Answer: \$95.81 per \$100)
A bond has face value \$800, a 10% annual coupon rate, three years remaining, redemption of \$900, and an effective annual yield of 10%. Find its total price. (Answer: \$875.13)

What You'll Learn