Book Value Immediately Before And After A Coupon

Why the Coupon Date Has Two Book Values

A bond’s book value changes at a coupon date in two distinct steps. First, interest accrues on the amount carried from the previous coupon date. Then the coupon is paid, reducing the amount still owed to the bondholder. The book value immediately before payment and the book value immediately after payment therefore differ by one coupon.

To keep the timing unambiguous, this tutorial uses an annual coupon bond with annual effective yield $i$. The coupon is $C$, redemption value is $R$, and there are $n$ annual coupon periods to maturity. The book value is measured using the original yield, not a newly observed market yield. The methods below concern coupon-date boundaries; they do not calculate a value at a time between coupon dates.

Definition: At coupon date $k$, $B_k^-$ is the bond’s book value immediately before the coupon due at that date is paid. $B_k^+$ is its book value immediately after that coupon is paid. At a date before maturity, the coupon is the only cash flow paid at that instant, so $B_k^-=B_k^++C$. Book value is the value of the bond’s remaining contractual cash flows at the original yield.

This distinction is about timing, not a change in the bond’s promised cash flows. Just before the coupon payment, the coupon due now is still included in the value. Just after payment, that coupon has been discharged and must be excluded. The remaining future payments are the same in both descriptions.

Prospective Valuation at a Coupon Date

At a coupon date before maturity, the book value immediately after the coupon is the present value, at that date, of the later coupons and redemption payment. The coupon just paid is not among those remaining cash flows. Immediately before payment, add the coupon due at that instant.

B_k^+ = \sum_{t=k+1}^{n-1} C v^{t-k} + (C+R)v^{n-k}, \qquad B_k^- = C+B_k^+, \qquad v=\frac{1}{1+i}.

In the sum, each coupon is discounted from its payment date back to date $k$. The final payment includes both the last coupon and redemption. The expression for $B_k^-$ adds the coupon due immediately, so no discount factor applies to that payment.

Timing rule: At a non-maturity coupon date, moving from immediately before to immediately after the coupon means subtracting $C$: $B_k^+=B_k^- - C$. Moving from immediately after one coupon date to immediately before the next means accumulating for one period: $B_{k+1}^-=(1+i)B_k^+$.

The two operations occur in that order. If $B_k^+$ is known, it earns one period of yield up to the next coupon date, giving $B_{k+1}^-$. The coupon is then paid, giving $B_{k+1}^+=B_{k+1}^- - C$. Equivalently, at a non-maturity date,

B_{k+1}^+=(1+i)B_k^+-C.

This is the coupon-date recurrence used in bond book-value calculations. Its inputs must refer to the same yield period: here, one year of accumulation uses the annual effective yield $i$, and one coupon is paid each year.

Why the Recurrence Holds

The recurrence follows from the timing of the cash flows and the yield definition. Between coupon dates, a carried amount of $B_k^+$ accumulates at the annual effective yield $i$. Immediately before the next coupon, its accumulated value is $(1+i)B_k^+$. Paying the coupon reduces the amount carried by exactly $C$. Thus:

B_{k+1}^-=(1+i)B_k^+, \qquad B_{k+1}^+=B_{k+1}^- - C =(1+i)B_k^+-C.

This also agrees with prospective valuation. At date $k$, $B_k^+$ is the present value of all cash flows after date $k$. One period later, just before the next coupon, the value of those same future cash flows has accumulated by $1+i$; the coupon due at that date is still included. Subtracting that coupon leaves exactly the value of the cash flows still outstanding after payment.

Worked Example: A Premium Bond at Its First Coupon Date

A hypothetical bond has redemption value $R=\$1{,}000$, annual coupon $C=\$60$, three years to maturity, and an annual effective yield $i=5\%$. Find the book values immediately before and after the first coupon. Use the prospective method and verify the coupon-date adjustment.

Method. First value the remaining payments immediately after the first coupon. Then add the coupon due at that date to obtain the value immediately before payment. This avoids including the first coupon in the post-payment value.

The discount factor is $v=1/1.05=0.9523809524$. At time 1, the remaining payments are $\$60$ at time 2 and $\$1{,}060$ at time 3. Therefore,

B_1^+=60v+1{,}060v^2 =60(0.9523809524)+1{,}060(0.9070294785) =\$1{,}018.5941.

Immediately before the first coupon, the $\$60$ coupon due at time 1 is still included:

B_1^-=60+B_1^+ =60+1{,}018.5941 =\$1{,}078.5941.

Rounded to cents, $B_1^+=\$1{,}018.59$ and $B_1^-=\$1{,}078.59$. The difference is $\$60.00$, exactly one coupon. As a recurrence check, the initial price is $B_0^+=60v+60v^2+1{,}060v^3=\$1{,}027.2325$. Accumulating gives $1.05(1{,}027.2325)=\$1{,}078.5941=B_1^-$, and subtracting $\$60$ gives $B_1^+$.

Common mistake avoided: Subtracting the coupon from the post-payment value would reverse the timing. The coupon is added to $B_1^+$ to obtain the pre-payment value.

Worked Example: A Discount Bond at an Intermediate Coupon Date

A hypothetical bond has redemption value $R=\$5{,}000$, annual coupon $C=\$200$, two years to maturity, and annual effective yield $i=6\%$. Find its book value immediately before and after the first coupon.

Method. Value the payment remaining after the first coupon, then add the coupon due immediately. Since the yield exceeds the coupon rate, the post-payment book value is below redemption; that observation is a reasonableness check, not a substitute for calculation.

After the first coupon, only the final payment of $\$5{,}200$ remains, one year later. With $v=1/1.06=0.9433962264$,

B_1^+=5{,}200v =\frac{5{,}200}{1.06} =\$4{,}905.6604.

The coupon due at the first date is $\$200$, so:

B_1^-=200+B_1^+ =200+4{,}905.6604 =\$5{,}105.6604.

Rounded to cents, the values are $B_1^+=\$4{,}905.66$ and $B_1^-=\$5{,}105.66$. The first coupon reduces the book value by $\$200$. For a recurrence check, the initial price is $200v+5{,}200v^2=\$4{,}816.6607$; multiplying by $1.06$ gives $\$5{,}105.6604$, the value immediately before the coupon.

Common mistake avoided: Using $5{,}200v$ for the value immediately before payment would omit the coupon due now. It is the post-payment value; the pre-payment value is $\$200$ larger.

Worked Example: Checking the Values at a Later Coupon Date

A hypothetical bond has redemption value $R=\$1{,}000$, annual coupon $C=\$40$, four years to maturity, and annual effective yield $i=6\%$. Calculate the values immediately after and immediately before the second coupon, using prospective valuation.

Method. At time 2, the cash flows remaining after the coupon are $\$40$ at time 3 and $\$1{,}040$ at time 4. Discount those payments to time 2, then add the time-2 coupon for the pre-payment value.

Here $v=1/1.06=0.9433962264$ and $v^2=0.8899964419$. Thus,

B_2^+=40v+1{,}040v^2 =40(0.9433962264)+1{,}040(0.8899964419) =\$963.3321.

The coupon due at time 2 is $\$40$, so:

B_2^-=40+B_2^+ =40+963.3321 =\$1{,}003.3321.

Rounded to cents, $B_2^+=\$963.33$ and $B_2^-=\$1{,}003.33$. Again, the difference is the coupon. The post-payment value is below the $\$1{,}000$ redemption value because the yield exceeds the coupon rate. The pre-payment value being slightly above $\$1{,}000$ is not a contradiction: it includes the $\$40$ coupon due immediately.

Common mistake avoided: Comparing the pre-payment value directly with redemption value without accounting for the coupon due now can lead to a false premium-or-discount conclusion.

The Maturity Date Requires One Extra Timing Check

At maturity, the last coupon and redemption value are both due at the same date. Immediately before those payments, the value of the promised payments due then is $C+R$. Once both have been paid, no contractual payments remain, so the book value immediately after all maturity payments is zero.

B_n^-=C+R, \qquad B_n^+=0 \quad\text{after both the final coupon and redemption are paid.}

Do not apply the non-maturity formula $B_k^-=B_k^++C$ at maturity if $B_n^+$ is defined after both final payments: the redemption payment must also be included. Alternatively, one may describe the value just after the final coupon but before redemption as $R$, but that is an intermediate instant within the maturity-date payments and must be stated explicitly. Consistent timing is more important than the label used for that instant.

Common Timing Errors

The most reliable check is to write down which cash flows have already been paid and which remain. For a non-maturity coupon date, the coupon is included in the pre-payment value and excluded from the post-payment value. For the next coupon date, first accumulate the post-payment amount and only then subtract the next coupon.

Reversing the coupon adjustment: The correct relation is $B_k^-=B_k^++C$, not $B_k^+-C$.
Counting a coupon twice: Do not include the coupon just paid in a prospective calculation of $B_k^+$.
Discounting a payment due now: The coupon in $B_k^-$ is due immediately, so it is added without a discount factor.
Mixing timing conventions at maturity: State whether the value is before all maturity payments or after both coupon and redemption have been paid.
Accumulating after payment instead of before it: The recurrence from one post-payment value to the next is $B_{k+1}^+=(1+i)B_k^+-C$.

Key takeaway: At a non-maturity coupon date, the pre-payment book value equals the post-payment book value plus the coupon due. Between coupon dates, the post-payment amount accumulates at the yield to become the next pre-payment amount. Identify the cash flows on each side before calculating.

Check Your Understanding

For each problem, distinguish the cash flows due immediately from those remaining afterward. Round money to the nearest cent.

A hypothetical bond has $R=\$1{,}000$, annual coupon $C=\$50$, three years to maturity, and annual effective yield $i=4\%$. Find the values immediately after and immediately before the first coupon. (Answer: $B_1^+=\$1{,}018.86$; $B_1^-=\$1{,}068.86$.)
A hypothetical bond has $R=\$2{,}000$, annual coupon $C=\$120$, four years to maturity, and annual effective yield $i=5\%$. Find the values immediately after and immediately before the second coupon. (Answer: $B_2^+=\$2{,}037.19$; $B_2^-=\$2{,}157.19$.)
A hypothetical bond has $R=\$5{,}000$, annual coupon $C=\$300$, four years to maturity, and annual effective yield $i=6\%$. Find the values immediately after and immediately before the second coupon. (Answer: $B_2^+=\$5{,}000.00$; $B_2^-=\$5{,}300.00$.)
A hypothetical bond has redemption value $R=\$1{,}000$ and a final annual coupon of $\$80$. State the book value immediately before the final coupon and redemption, and immediately after both payments. (Answer: $B_n^-=\$1{,}080.00$; $B_n^+=\$0.00$.)
At a non-maturity coupon date, a bond has $B_k^-=\$1{,}245$ and an annual coupon of $\$75$. The annual effective yield is $4\%$. Find $B_k^+$, then find the book value immediately after the preceding coupon date. (Answer: $B_k^+=\$1{,}170.00$; preceding post-payment value $=\$1{,}197.12$.)

What You'll Learn