Why A Par Bond’s Yield Equals Its Coupon Rate

Why Par Connects Coupon Rate and Yield

A bond’s coupon rate describes the contractual interest payments relative to its face amount. Its yield to maturity is the rate that equates the purchase price with the present value of all scheduled coupon and redemption payments, assuming the bond is held to redemption and payments are made as promised. These are different concepts in general. Under a particular set of conditions, however, they coincide: a bond priced at par has a yield equal to its coupon rate when both rates are expressed using the same coupon-period convention.

The qualification about conventions matters. A bond paying coupons twice per year has two coupon periods per year. A quoted nominal annual rate convertible semiannually must be divided by two to obtain its rate per coupon period. The coupon payment and the yield equation must use that same period length.

Definition: A bond is priced at par when its price equals its face amount, also called its redemption value here. The coupon rate per period is the coupon payment for one period divided by the face amount. The yield per period is the rate used to discount cash flows from one coupon period to the next in the yield-to-maturity equation.

Set Up the Bond Price Equation

Assume a bond has face amount $F$, pays a level coupon $C$ at the end of each of $n$ equal periods, and redeems for $F$ at the end of period $n$. It is purchased on a coupon date, so there is no accrued interest in the price. Let $y$ be the yield per coupon period. Under the conventional fixed-cash-flow yield model, the price is the present value of the coupons and redemption:

P=C a_{\overline{n}|y}+F(1+y)^{-n}.

Here $a_{\overline{n}|y}$ denotes the present value of an $n$-payment annuity-immediate when the interest rate per period is $y$. The notation makes the yield assumption explicit: the annuity payments occur once per period, and each period is discounted at $y$.

For a par bond, $P=F$. The coupon rate per period is $C/F$. The key case is $C/F=y$, or equivalently $C=Fy$. Inserting this coupon amount into the price equation gives the identity that explains the result.

Theorem: Suppose a fixed-coupon bond pays $C$ at the end of each of $n$ equal periods, redeems for face amount $F$ at the end of period $n$, and is priced on a coupon date. If $C/F=y$, where $y$ is the yield per coupon period, then its price is $F$. Thus, for a bond with these cash flows priced at par, the coupon rate per period equals the yield per period.

Why the Formula Holds

For $y\ne 0$, the annuity-immediate present-value formula is $a_{\overline{n}|y}=(1-(1+y)^{-n})/y$. If the coupon is $C=Fy$, the present value of the coupons is therefore a fraction of face amount equal to $1-(1+y)^{-n}$. The present value of redemption is the remaining fraction, $(1+y)^{-n}$. Together they equal the face amount.

\begin{aligned} P &=Fy\left(\frac{1-(1+y)^{-n}}{y}\right)+F(1+y)^{-n}\\ &=F\left(1-(1+y)^{-n}\right)+F(1+y)^{-n}\\ &=F. \end{aligned}

The cancellation has a direct interpretation. At this yield, the coupon stream is worth $F$ minus the present value of the redemption payment. Adding the redemption payment’s present value completes the price to exactly $F$. If $y=0$, the matching coupon is also zero; the bond’s only payment is $F$ at redemption, which has present value $F$ at a zero yield. The identity therefore also has a zero-rate limiting case.

This proves that a yield equal to the coupon rate produces a par price. To conclude that the yield of a bond already known to be at par must equal its coupon rate, the conventional yield must be the unique solution of the price equation. For a positive fixed coupon and positive redemption, the price decreases as the positive yield increases; the uniqueness result for conventional bond yields then gives the conclusion. No special assumption about maturity length is needed.

Keeping Periodic and Quoted Rates Consistent

A quoted nominal annual coupon rate and a quoted nominal annual yield can be compared directly when they have the same conversion frequency. Let $m$ be the number of coupon periods per year. Suppose the coupon rate is quoted as a nominal annual rate convertible $m$ times per year, $c^{(m)}$, and the yield is quoted on the same basis, $j^{(m)}$. The coupon payment per period and yield per period are, respectively,

C=F\frac{c^{(m)}}{m}, \qquad y=\frac{j^{(m)}}{m}.

Consequently, $C/F=y$ exactly when $c^{(m)}=j^{(m)}$. The par-bond result then says the quoted nominal annual coupon rate equals the quoted nominal annual yield. This direct comparison is not justified if the two quotes use different conversion frequencies; first express both using the same period convention.

The result also relies on the stated cash flows and timing. It is for a bond with level coupons, redemption equal to face amount, and purchase at a coupon date. A different redemption amount, an irregular cash-flow pattern, or a price that includes accrued interest changes the setup. The theorem should not be applied by looking only at the word “par” while ignoring these conditions.

Worked Example: Annual Coupons and a Par Price

A hypothetical bond has face amount $1,000, pays annual coupons for five years, and redeems for $1,000 at the end of year five. Its annual coupon rate is 6%, and it is purchased at a coupon date for $1,000. Find its annual yield to maturity. Use the same annual period for the coupon and yield.

Method. Use the par-bond identity, then verify the present values. The coupon is $C=1{,}000(0.06)=\$60$ per year, and the number of periods is $n=5$. The candidate yield per period is $y=0.06$.

At this rate, $v=1/(1.06)=50/53$, so the price equation gives

\begin{aligned} P &=60a_{\overline{5}|0.06}+1{,}000(1.06)^{-5}\\ &=1{,}000(0.06)\frac{1-(1.06)^{-5}}{0.06} +1{,}000(1.06)^{-5}\\ &=1{,}000\left(1-(1.06)^{-5}\right) +1{,}000(1.06)^{-5}\\ &=\$1{,}000. \end{aligned}

For a numerical check, $(1.06)^{-5}\approx0.747258$. The coupon present value is approximately $1{,}000(1-0.747258)=\$252.74$, and the redemption present value is approximately $1{,}000(0.747258)=\$747.26$. Their sum is $1,000.00, so the annual yield to maturity is 6%. The common mistake avoided here is treating the coupon payment of $60 as the coupon rate; the rate is $60/$1,000, or 6%.

Worked Example: Semiannual Coupons and a Nominal Quote

A hypothetical bond has face amount $5,000, pays coupons semiannually for eight years, and redeems at face amount. Its coupon rate is 4.8% nominal per year convertible semiannually. It is purchased at a coupon date for $5,000. Find its yield, reported as a nominal annual rate convertible semiannually.

Method. First convert the coupon quote to a coupon-period amount and count the periods. There are $m=2$ periods per year and $n=8(2)=16$ periods. The coupon each half-year is $C=5{,}000(0.048/2)=\$120$, so the coupon rate per period is $120/5{,}000=0.024$.

Set the yield per half-year equal to $y=0.024$. The price at that rate is

\begin{aligned} P &=120a_{\overline{16}|0.024} +5{,}000(1.024)^{-16}\\ &=5{,}000(0.024) \frac{1-(1.024)^{-16}}{0.024} +5{,}000(1.024)^{-16}\\ &=5{,}000\left(1-(1.024)^{-16}\right) +5{,}000(1.024)^{-16}\\ &=\$5{,}000. \end{aligned}

The price matches the stated par price, so the periodic yield is 2.4% per half-year. The quoted nominal annual yield is $2(0.024)=0.048$, or 4.8% convertible semiannually. The convention matters: reporting 2.4% as the nominal annual yield would omit one of the two periods per year. The calculation uses the coupon-period rate throughout and converts to the requested annual quote only at the end.

Worked Example: Inferring the Yield From a Coupon Amount

A hypothetical bond has face amount $12,000 and makes quarterly payments for six years. Each coupon is $108, and the bond redeems for $12,000. It is bought at a coupon date for $12,000. Find the yield per quarter and the nominal annual yield convertible quarterly.

Method. Find the coupon rate per period directly from the payment and face amount, then use the par-bond result. There are $n=6(4)=24$ quarterly periods. The coupon rate per quarter is $108/12{,}000=0.009$, or 0.9%.

At a quarterly yield of $y=0.009$, the coupon present value is $12{,}000(1-(1.009)^{-24})$, and the redemption present value is $12{,}000(1.009)^{-24}$. Thus,

\begin{aligned} P &=12{,}000(0.009) \frac{1-(1.009)^{-24}}{0.009} +12{,}000(1.009)^{-24}\\ &=12{,}000\left(1-(1.009)^{-24}\right) +12{,}000(1.009)^{-24}\\ &=\$12{,}000. \end{aligned}

This is the purchase price, so the yield per quarter is 0.9%. There are four quarters per year; the nominal annual yield convertible quarterly is $4(0.009)=0.036$, or 3.6%. The common mistake avoided is multiplying by six, the number of years, rather than by four, the number of coupon periods per year.

Common Convention Trap

Suppose the semiannual bond in the second example is incorrectly assigned a yield of 4.8% per half-year. That is not the periodic rate corresponding to a 4.8% nominal annual yield convertible semiannually; the corresponding rate is $0.048/2=0.024$. At the incorrectly chosen period rate, the price equation would give

\begin{aligned} P_{\text{wrong}} &=120a_{\overline{16}|0.048}+5{,}000(1.048)^{-16}\\ &=\frac{120}{0.048}\left(1-(1.048)^{-16}\right) +5{,}000(1.048)^{-16}\\ &=2{,}500+2{,}500(1.048)^{-16}\\ &<\$5{,}000. \end{aligned}

The incorrect rate does not reproduce the par price. It treats 4.8% as the rate for each half-year, rather than for a full year divided between two periods. The correction is to align the periods: the coupon rate is 2.4% per half-year, the yield is 2.4% per half-year, and the matching nominal annual quotes are both 4.8% convertible semiannually.

When to Use the Result

For an ordinary fixed-coupon bond with redemption equal to face amount, the result is a useful check on a yield calculation. If the bond is explicitly at par and the coupon and yield conventions match, the yield does not require numerical iteration: it is the coupon rate on that same convention. Conversely, if the coupon rate per period is known, substituting it into the price equation must give the face amount.

This relationship is aligned with the SOA Exam FM (Financial Mathematics) syllabus. Apply it only after checking the payment frequency, redemption amount, price date, and rate convention. In particular, a coupon rate stated per year cannot be inserted as though it were automatically the rate per coupon period.

Key takeaway: With level end-of-period coupons, redemption equal to face amount, and purchase at a coupon date, $C/F=y$ makes the coupon present value and redemption present value add to $F$. Therefore, at par, the yield per coupon period equals the coupon rate per coupon period; matching nominal annual quotes are equal when they use the same conversion frequency.

Check Your Understanding

For each problem, identify the coupon-period rate before stating the yield. Assume each bond is purchased at a coupon date and redeems for its face amount.

A bond with face amount $2,000 pays annual coupons at a 3.5% annual coupon rate for six years. It is priced at $2,000. Find its annual yield to maturity. (Answer: 3.5% per year)
A bond with face amount $8,000 pays semiannual coupons at a 6% nominal annual coupon rate convertible semiannually for ten years. It is priced at par. Find its yield per half-year and its nominal annual yield convertible semiannually. (Answer: 3% per half-year; 6% nominal annual)
A quarterly bond has face amount $15,000, pays a 4.8% nominal annual coupon rate convertible quarterly, and matures in three years. It is priced at par. Find its yield per quarter and nominal annual yield convertible quarterly. (Answer: 1.2% per quarter; 4.8% nominal annual)
A semiannual bond has face amount $2,000 and pays $45 every six months. It is priced at par and has 12 remaining coupon periods. Find the yield per period and the nominal annual yield convertible semiannually. (Answer: 2.25% per half-year; 4.5% nominal annual)
A quarterly bond has face amount $10,000 and pays $175 per quarter for five years, then redeems for $10,000. It is purchased at par. Find the yield per quarter and the nominal annual yield convertible quarterly. (Answer: 1.75% per quarter; 7% nominal annual)

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