Pricing a Bond on a Coupon Date

Introduction

A bond’s coupon date is a natural valuation point: the scheduled payment for that date has just been made, and every remaining cash flow falls an exact whole number of coupon periods in the future. The valuation then reduces to discounting the remaining coupons and the redemption payment to the date being considered.

The main exam risks are not complicated arithmetic. They are counting the wrong number of payments, using an annual yield as though it were a coupon-period yield, or including a coupon that has already been paid. A clear timeline and a stated yield convention prevent these errors.

Definition: In this tutorial, a bond is valued immediately after the scheduled coupon payment on a coupon date. Its remaining scheduled coupons and redemption amount are discounted to that date. The coupon paid on the valuation date is not a future cash flow and is excluded. The bond is assumed to have known, fixed cash flows and to be valued using one yield per coupon period.

The Coupon-Date Pricing Formula

Let $F$ be the bond’s face amount, $c$ its annual coupon rate, and $m$ the number of coupon payments per year. Assume the coupon rate is stated as a nominal annual rate payable in $m$ equal installments. Each coupon is

C=\frac{Fc}{m}.

Let $n$ be the number of coupon payments still to be made after the valuation date, and let $R$ be the redemption amount paid with the final coupon. Let $j$ be the yield rate for one coupon period. Define the coupon-period discount factor $w=(1+j)^{-1}$. In the formulas below, $a_{\overline{n}|}$ is evaluated at rate $j$.

Theorem: For a bond with $n$ remaining equal coupons $C$, redemption amount $R$ at the $n$th payment, and yield $j$ per coupon period, its value immediately after a coupon date is

P=C a_{\overline{n}|}+Rw^n, \qquad a_{\overline{n}|}=\frac{1-w^n}{j}, \qquad w=\frac{1}{1+j}.

This formula assumes $j\ne 0$. If $j=0$, the value is $P=nC+R$, since no cash flow is discounted.

The first term is the present value of the remaining coupons. The second is the present value of the redemption amount, which is paid at the same time as the last coupon. The face amount $F$ helps determine the coupon, but it is not automatically the amount paid at maturity: use the stated redemption amount $R$.

Why the Formula Holds

Number the remaining coupon dates $1,2,\ldots,n$. The coupon at date $k$ is worth $Cw^k$ at the valuation date, and the redemption payment is worth $Rw^n$. Adding these discounted cash flows gives the bond value:

P=Cw+Cw^2+\cdots+Cw^n+Rw^n =C\sum_{k=1}^{n}w^k+Rw^n.

The coupon discount factors form a geometric series. For $j\ne0$, $w=1/(1+j)$, so $1-w=j/(1+j)=jw$. Therefore,

\sum_{k=1}^{n}w^k =\frac{w(1-w^n)}{1-w} =\frac{1-w^n}{j} =a_{\overline{n}|}.

Substituting this sum into the cash-flow total gives $P=C a_{\overline{n}|}+Rw^n$. The formula is thus a compact way to add the present values of the individual scheduled payments; it does not change which payments belong in the calculation.

Matching the Yield to the Coupon Period

The rate $j$ in the pricing formula must measure interest over one coupon period. If the yield is a nominal annual rate $i^{(m)}$ convertible $m$ times per year and coupon payments occur $m$ times per year, then $j=i^{(m)}/m$. If the yield is instead an effective annual rate $i$, then the equivalent coupon-period rate is $j=(1+i)^{1/m}-1$. Use the yield convention given in the problem; do not divide an effective annual rate by $m$.

Once $j$ is identified, count $n$ in coupon periods, not in years. For example, three years of remaining semiannual coupons means six remaining periods. If the valuation date is immediately after a coupon payment, that payment is excluded, even if the date is also the bond’s issue anniversary.

Worked Example: Semiannual Coupons and a Nominal Yield

A bond has face amount $1,000, a nominal annual coupon rate of 6% payable semiannually, and four years remaining until maturity. It redeems at face amount. Its nominal annual yield is 8% convertible semiannually. Find its value immediately after a coupon payment. Give the answer to the nearest cent.

Method and timing. Use the coupon-date present-value formula. Four years of semiannual payments means $n=4(2)=8$ remaining coupons. The coupon per period is

C=\frac{1000(0.06)}{2}=30.

The yield and coupon periods match, so the yield per period is

j=\frac{0.08}{2}=0.04, \qquad w=\frac{1}{1.04}=0.96153846.

The eighth-period discount factor and annuity factor are

w^8=(1.04)^{-8}=0.730690, \qquad a_{\overline{8}|}=\frac{1-0.730690}{0.04}=6.732750.

The redemption amount is $1,000. Hence,

$$ P=30(6.732750)+1000(0.730690) =201.9825+730.690 =932.6725. $$

Rounded to the nearest cent, the value is $932.67. As a check, the coupon rate of 6% is below the yield of 8%; with redemption at face amount, a discount below $1,000 is consistent with that relationship. This method avoids the common mistake of using four periods just because four years remain.

Worked Example: Effective Annual Yield and Nonpar Redemption

A bond has face amount $5,000, pays coupons quarterly at a nominal annual coupon rate of 5.4%, and will make six more quarterly payments. It redeems for $5,200 with the final coupon. The effective annual yield is $(1.015)^4-1$. Find the value immediately after a coupon date, to the nearest cent.

Method and rate conversion. There are $n=6$ periods. The coupon per quarter is

C=\frac{5000(0.054)}{4}=67.50.

The stated effective annual yield corresponds to a quarterly rate of $j=0.015$, because $(1+j)^4-1=(1.015)^4-1$. Thus $w=1/1.015$, and

w^6=(1.015)^{-6}=0.914542, \qquad a_{\overline{6}|}=\frac{1-0.914542}{0.015}=5.697200.

Discount the coupons and the $5,200 redemption separately:

\begin{aligned} \text{Coupon value}&=67.50(5.697200)=384.561,\\ \text{Redemption value}&=5200(0.914542)=4755.6184,\\ P&=384.561+4755.6184=5140.1794. \end{aligned}

The value is $5,140.18 to the nearest cent. This exceeds the $5,000 face amount in part because the redemption amount itself is $5,200. The common mistake is to discount that redemption at the sixth period but use face amount instead of the contractual redemption amount.

Worked Example: Counting Payments After an Earlier Coupon

A bond with face amount $2,000 pays coupons semiannually at a nominal annual coupon rate of 4.8%. It matures five years after issue and redeems at face amount. Find its value immediately after the coupon paid on the second anniversary of issue, if the effective annual yield is 5.0625%. Give the answer to the nearest cent.

Method and payment count. The bond has three years left after the second anniversary. With two coupons per year, that is $n=3(2)=6$ remaining coupons. The coupon amount is

C=\frac{2000(0.048)}{2}=48.

Since $1.050625=(1.025)^2$, the yield per half-year is $j=0.025$. Therefore,

w^6=(1.025)^{-6}=0.862297, \qquad a_{\overline{6}|}=\frac{1-0.862297}{0.025}=5.508120.

Using redemption $R=2000$,

\begin{aligned} \text{Coupon value}&=48(5.508120)=264.38976,\\ \text{Redemption value}&=2000(0.862297)=1724.594,\\ P&=264.38976+1724.594=1988.98376. \end{aligned}

The value is $1,988.98 to the nearest cent. It is below face amount because the coupon rate of 4.8% is below the effective annual yield of 5.0625%. The coupon paid on the second anniversary is not included: the valuation is immediately after that payment.

Timing Traps and Model Limits

The phrase “on a coupon date” should be paired with a precise timing convention. Here the value is immediately after that date’s coupon has been paid. If a problem instead asks for value immediately before the payment, that coupon has not yet been paid and is due at once; it must be treated as a current cash flow. The two values differ by the coupon amount when all else is unchanged.

For the preceding example, incorrectly adding the just-paid $48 coupon would give

P_{\text{wrong}}=1988.98+48=2036.98.

That calculation values the post-payment bond as though the payment were still due. The correction is to use only the six future coupons and the future redemption amount, giving $1,988.98. A quick timeline check is useful: at time zero, immediately after the coupon, the first remaining coupon is one full half-year away.

The formula applies when the cash flows and the yield convention used for discounting are specified. It does not by itself determine an appropriate market yield, or account for uncertain payments, changing rates, or embedded options that can alter the scheduled cash flows. Those features require assumptions beyond this fixed-cash-flow calculation.

Fix the valuation instant.
For the convention used here, stand immediately after the coupon payment on the valuation date.

Count future payments.
Translate remaining years into coupon periods and exclude any coupon already paid.

Find the coupon and period yield.
Calculate the coupon per payment and convert the stated yield to one coupon-period rate.

Discount the two cash-flow components.
Use $C a_{\overline{n}|}$ for coupons and $Rw^n$ for redemption.

Check the result.
Confirm that the redemption amount, payment count, and premium-or-discount direction fit the problem.

A reliable coupon-date price is an equation of value at a clearly stated instant. Once the cash flows are counted correctly, the annuity factor handles the regular coupons and the discount factor handles the final redemption. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume valuation immediately after the coupon payment on the stated date. Round dollar answers to the nearest cent.

A bond with face amount $1,000 pays annual coupons at 4% and has three annual coupons remaining. It redeems at face amount, and its effective annual yield is 5%. Find its value. (Answer: $972.77)
A $4,000 bond pays quarterly coupons at a nominal annual rate of 3.6%. Five quarterly payments remain, and redemption is $4,100. The nominal annual yield is 4% convertible quarterly. Find its value. (Answer: $4,075.73)
A bond with face amount $2,500 pays semiannual coupons at a nominal annual rate of 6%. Four coupons remain, and redemption is $2,400. Its effective annual yield is 8.16%. Find its value. (Answer: $2,323.77)
A $1,000 bond pays semiannual coupons at a nominal annual rate of 5% and matures eight years after issue at face amount. Immediately after the coupon paid at the end of year three, its nominal annual yield is 6% convertible semiannually. Find its value. (Answer: $957.35)

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