The Par-Price Condition for Coupon Bonds

What “At Par” Means

A bond is priced at par when its price equals its face value. The coupon rate alone does not determine whether this happens: it must be compared with the required yield using the same period basis. For example, a semiannual coupon amount must be compared with the required yield per half-year, not directly with an annual effective yield.

The par-price condition is useful as both a valuation result and a quick check on a bond-pricing calculation. This tutorial develops it from the coupon-bond price formula and applies it to different payment frequencies.

Definition: Let $F$ be a bond’s face value, $n$ its number of remaining coupon periods, $C$ the coupon paid at the end of each period, and $j$ the required yield per coupon period. The coupon rate per period is $C/F$. The bond is at par if its price $P$ equals $F$. Coupon rate and yield must be expressed on the same period basis to compare them.

Assume the bond pays the same coupon at the end of each period, repays face value with its final coupon at the end of period $n$, and has no default or other payments in the valuation model. The required yield $j$ is constant over all periods, and $j>-1$, so the discount factor $v=1/(1+j)$ is positive. The price is the present value of the coupons plus the present value of face value:

P=C\,a_{\overline{n}|j}+Fv^n.

For $j\ne 0$, the present-value annuity factor is $a_{\overline{n}|j}=(1-v^n)/j$. When $j=0$, this expression would divide by zero; the correct value is $a_{\overline{n}|0}=n$, since the $n$ unit payments are then not discounted. Thus the pricing formula is well-defined at zero yield as well.

Deriving the Par-Price Condition

Suppose first that $j\ne 0$. If the coupon per period equals $Fj$, substituting $C=Fj$ into the price formula gives:

\begin{aligned} P &=Fj\,a_{\overline{n}|j}+Fv^n\\ &=Fj\left(\frac{1-v^n}{j}\right)+Fv^n\\ &=F(1-v^n)+Fv^n\\ &=F. \end{aligned}

The cancellation has a simple interpretation. The present value of the coupons is $F(1-v^n)$, and the present value of the final face-value payment is $Fv^n$. Together they add to face value. The condition is not merely sufficient: if $P=F$, the same pricing formula implies $C=Fj$ for $j\ne0$.

The zero-yield case must be checked separately. If $j=0$, then $v=1$ and $P=nC+F$. This price equals $F$ exactly when $C=0=Fj$. Therefore the condition remains valid at zero yield, even though the annuity-factor expression involving division by $j$ cannot be used there.

Theorem: Under the stated level-coupon bond assumptions, $P=F$ if and only if $C=Fj$, including the case $j=0$. Equivalently, the bond is at par when its coupon rate per payment period equals its required yield per payment period.

For an annual-coupon bond, the coupon rate and yield are both annual when stated as effective annual rates. For a bond paying $m$ coupons per year, quoted nominal annual rates convertible $m$-thly must be divided by $m$ to obtain their per-period rates. A nominal annual coupon rate $c^{(m)}$ gives a coupon $C=Fc^{(m)}/m$, while a nominal annual required yield $y^{(m)}$ gives $j=y^{(m)}/m$. They match when $c^{(m)}=y^{(m)}$. If the required yield is instead an effective annual rate $i$, convert it to a per-period rate using $j=(1+i)^{1/m}-1$; do not compare $i$ directly with a per-period coupon rate.

Worked Applications

Worked Example: An Annual-Coupon Bond at Par

A hypothetical bond has face value $F=\$1{,}000$, a 6% annual coupon rate, and five years remaining. Its required yield is 6% effective annually. Find the coupon, price the bond, and verify whether it is at par. Assume annual coupons paid at each year-end and repayment of face value with the final coupon.

Method. First put the coupon and yield on an annual basis. The coupon per year is $C=0.06(\$1{,}000)=\$60$, and the required yield per year is $j=0.06$. There are $n=5$ annual periods. Since $C/F=60/1{,}000=0.06=j$, the par condition predicts a price of $\$1{,}000$.

To confirm by discounting, $v=1/1.06=0.9433962$, and $v^5=0.7472582$. The annuity factor is:

a_{\overline{5}|0.06} =\frac{1-0.7472582}{0.06} =4.2123633.

The coupon present value is $\$60(4.2123633)=\$252.7418$. The face-value present value is $\$1{,}000(0.7472582)=\$747.2582$. Therefore $P=\$252.7418+\$747.2582=\$1{,}000.0000$, or $\$1{,}000.00$ to the nearest cent. The component values add to face value, as predicted.

Common mistake avoided: comparing the 6% coupon rate with a yield expressed on a different period basis. Here both rates are annual, so the direct comparison is valid.

Worked Example: Matching Semiannual Coupon and Yield Rates

A hypothetical bond has face value $F=\$5{,}000$, pays coupons semiannually for six years, and has a quoted nominal annual coupon rate of 8% convertible semiannually. Its required yield is a nominal annual 8% convertible semiannually. Find the coupon per half-year and verify the price. Assume payments occur at the end of each half-year.

Method. There are $n=6(2)=12$ periods. The coupon per period is $C=\$5{,}000(0.08/2)=\$200$. The required yield per period is $j=0.08/2=0.04$. Thus $C/F=\$200/\$5{,}000=0.04=j$, so the par condition gives $P=\$5{,}000$.

For a numerical confirmation, $v^{12}=1/(1.04)^{12}\approx0.6245971$. Hence:

a_{\overline{12}|0.04} =\frac{1-0.6245971}{0.04} \approx9.385073.

The coupon present value is approximately $\$200(9.385073)=\$1{,}877.01$. The face-value present value is approximately $\$5{,}000(0.6245971)=\$3{,}122.99$. Their sum, rounded to the nearest cent, is $\$1{,}877.01+\$3{,}122.99=\$5{,}000.00$. The calculation agrees with the par condition.

Common mistake avoided: using 8% as the yield for each half-year. That would mix a quoted nominal annual yield with a per-period yield. The correct yield per half-year is 4%.

Worked Example: Finding the Coupon Rate That Gives Par

A hypothetical bond has face value $F=\$10{,}000$, three years remaining, and annual coupons. The required yield is 5% effective annually. What annual coupon and coupon rate make the bond sell at par? Verify the price using the present-value formula.

Method. For annual periods, $j=0.05$, and the par condition requires $C=Fj$. Thus $C=\$10{,}000(0.05)=\$500$ per year, which is an annual coupon rate of $\$500/\$10{,}000=5\%$. There are $n=3$ payments.

To verify, $v^3=1/(1.05)^3\approx0.8638376$. The annuity factor is:

a_{\overline{3}|0.05} =\frac{1-0.8638376}{0.05} \approx2.723248.

The coupon present value is approximately $\$500(2.723248)=\$1{,}361.62$. The face-value present value is approximately $\$10{,}000(0.8638376)=\$8{,}638.38$. The rounded components sum to $\$1{,}361.62+\$8{,}638.38=\$10{,}000.00$, confirming the par price to the nearest cent.

Common mistake avoided: treating the coupon rate as an amount. The coupon rate is 5%; the annual coupon amount is $\$500$, obtained by multiplying that rate by face value.

Period Basis and Calculation Checks

The most important practical check is to identify the number of coupon periods and express the coupon and required yield per one of those periods. A rate quoted “per year” may be effective annually or nominally convertible at a specified frequency; those descriptions are not interchangeable. The bond’s payment frequency determines the number of discounting periods, while the stated yield convention determines how to convert the required return into a per-period rate.

For example, with semiannual coupons and an effective annual required yield $i$, the yield per half-year is $j=(1+i)^{1/2}-1$. It is not generally $i/2$. By contrast, a nominal annual yield convertible semiannually is divided by two. The coupon must be handled consistently: an annual coupon rate quoted as nominal and payable semiannually is divided by two to obtain the coupon rate per half-year.

A useful check is to compare $C/F$ with $j$, not to compare an unconverted annual quote with a coupon amount. If they are equal, the price formula must reduce to face value. At zero yield, do not use an annuity formula that divides by $j$: use $a_{\overline{n}|0}=n$, so $P=nC+F$. These checks are aligned with the SOA VEE Accounting and Finance syllabus.

Identify the payment period.
Count the remaining coupon payments to determine $n$.

Convert both rates.
Find the coupon $C$ and required yield $j$ for one coupon period, using the stated rate conventions.

Test the equality.
Compare $C/F$ with $j$. Equality is the par-price condition.

Verify if needed.
Use $P=C\,a_{\overline{n}|j}+Fv^n$, applying $a_{\overline{n}|0}=n$ when $j=0$, and check whether the result is $F$.

Key takeaway: A level-coupon bond is priced at face value precisely when the coupon rate and required yield are equal per coupon period. Convert quoted rates to the same basis before applying the condition, and handle zero yield separately in the annuity formula.

Check Your Understanding

For each problem, identify the payment-period basis before applying the par-price condition.

A hypothetical bond has face value $\$2{,}000$, annual coupons, a 4% annual coupon rate, and a 4% effective annual required yield. It has seven years remaining. Is it at par, and what is its price? (Answer: Yes; $\$2{,}000$.)
A hypothetical bond with face value $\$8{,}000$ pays semiannually for ten years. Its coupon rate and required yield are both nominal annual rates of 6% convertible semiannually. Find the coupon per period, the required yield per period, and the price. (Answer: Coupon $\$240$; per-period yield 3%; price $\$8{,}000$.)
A four-year annual-coupon bond has face value $\$1{,}000$, annual coupons of $\$50$, and a 5% effective annual required yield. Does it meet the par condition, and what is its price? (Answer: Yes; $\$1{,}000$.)
A hypothetical bond has face value $\$5{,}000$ and annual coupons at a 3% rate. Its required yield is 4% effective annually. Is the par-price condition met? (Answer: No; the coupon rate is 3%, while the per-period required yield is 4%.)
A hypothetical three-year bond has quarterly coupons. Its coupon rate and required yield are both nominal annual rates of 4% convertible quarterly, and its face value is $\$4{,}000$. Give the number of coupon periods, the coupon per period, and the price. (Answer: 12 periods; $\$40$ per quarter; $\$4{,}000$.)
A hypothetical bond pays semiannually, has a nominal annual coupon rate of 8% convertible semiannually, and is evaluated at an 8% effective annual required yield. Is it at par? (Answer: No; the coupon rate per half-year is 4%, while the required yield per half-year is $(1.08)^{1/2}-1\approx3.923\%$.)

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