Bond Book Values Between Coupon Dates

Why Between-Date Book Values Need a Convention

Bond book values are often calculated at coupon dates, but a question may ask for the carrying amount on a date between them. The balance cannot be advanced correctly until the problem’s yield convention for the fraction of a coupon period is identified. A full coupon period may use a known effective yield, while the fractional-period accrual may be specified as simple interest or as compound interest over a fractional period.

Here, a book value is the bond’s effective-interest carrying amount. The balance is measured immediately after a coupon payment at the start of a period and is carried forward until the next coupon is actually paid. It is not a market quotation. In particular, do not assume that a coupon has been paid merely because some of its period has elapsed.

Definition: Let $B_k$ be a bond’s book value immediately after the coupon at date $k$. Let $C$ be the coupon paid at the end of each regular coupon period, and let $j$ be the effective yield per coupon period. If a fraction $\alpha$, with $0\leq\alpha\leq1$, of the period has elapsed, the between-date book value depends on the stated fractional-period convention. Under simple-interest accrual, it is $B_k(1+j\alpha)$; under fractional-period compound accrual, it is $B_k(1+j)^\alpha$.

The yield $j$ must match the coupon period. For $m$ regular coupon periods per year, an effective annual yield $i$ gives $j=(1+i)^{1/m}-1$. If the stated rate is nominal $i^{(m)}$, convertible $m$ times per year, then $j=i^{(m)}/m$. These are different yield specifications; use the one given rather than treating an annual rate as if it applied to one coupon period.

Coupon-Date Book Values and Carrying Forward

At a regular coupon date, interest is first earned over the period and then the coupon is paid. If the balance just after the previous coupon is $B_k$, the balance immediately before the next coupon, under the effective yield for a full period, is $B_k(1+j)$. Immediately after paying the coupon $C$, the next book value is

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

Between coupon dates, apply only the fractional-period accrual specified by the question. The coupon is a cash flow at its payment date, not a deduction made progressively from the book value. A separate accounting presentation may track accrued coupon interest in a separate account; that does not change the convention for the book value defined here.

Locate the last coupon date.
Use the book value immediately after the coupon paid on that date. Do not start with a value from before that payment.

Find the elapsed fraction.
For a regular period, divide elapsed time by the length of one coupon period, using the time units stated in the problem.

Apply the specified accrual rule.
Use $B_k(1+j\alpha)$ for simple interest, or $B_k(1+j)^\alpha$ for fractional-period compound interest.

Handle the coupon at its payment date.
At the end of a full period, the balance before payment is $B_k(1+j)$. Subtract $C$ only to find the book value immediately after the coupon is paid.

Why the Two Accrual Formulas Work

Under a simple-interest convention, interest on the starting balance $B_k$ for fraction $\alpha$ of a period is $B_kj\alpha$. Adding that interest to the starting balance gives the simple-accrual formula:

B_k+B_kj\alpha=B_k(1+j\alpha).

Under fractional-period compound accrual, a full period has accumulation factor $1+j$. The factor for a fraction $\alpha$ is defined as $(1+j)^\alpha$. This choice is consistent across adjacent parts of the period because

(1+j)^\alpha(1+j)^\beta=(1+j)^{\alpha+\beta}.

In particular, if $\alpha+\beta=1$, the two fractional accumulations together produce the full-period factor $1+j$. This multiplicative property is why the compound convention uses an exponent. Neither fractional-period rule should be substituted for the other without support from the stated yield convention.

Worked Example: Simple Accrual Between Semiannual Coupons

A hypothetical bond has a book value of $1,000 immediately after a coupon payment. It pays $40 every six months and has a nominal annual yield of 8% convertible semiannually. The problem specifies simple-interest accrual between coupon dates. Find the book value three months after the coupon, and check the balance immediately after the next coupon.

Method. Convert the yield to a per-coupon-period rate, use the stated simple-accrual convention for the elapsed fraction, and subtract the coupon only at its payment date.

There are two coupon periods per year, so the effective yield per six-month period is

j=\frac{0.08}{2}=0.04.

Three months is half of a six-month period, so $\alpha=3/6=0.5$. The interest accrued on the book value over that fraction is

$$ 1000(0.04)(0.5)=20. $$

Therefore, three months after the coupon, the book value is

1000+20=1000(1+0.04(0.5))=\$1{,}020.00.

For the check, the balance immediately before the next coupon is $1000(1.04)=\$1{,}040.00$. After the $40 coupon is paid, it is $1040-40=\$1{,}000.00$, consistent with the given yield and coupon. The method avoids subtracting a portion of the coupon from the between-date book value.

Worked Example: Fractional-Period Compound Accrual

A hypothetical bond has a book value of $1,000 immediately after a coupon payment. Its effective yield is 21% per coupon period, and its coupon is $210 per period. The stated convention is fractional-period compound accrual. Calculate the book value halfway through a period and immediately after the next coupon.

Method. Use the fractional exponent for the half-period balance, then verify the result by carrying the balance through a complete period and applying the coupon.

Here $j=0.21$ and $\alpha=1/2$. Thus the halfway book value is

1000(1.21)^{1/2}=1000(1.1)=\$1{,}100.00,

because $1.1^2=1.21$. For a full period, the balance immediately before the coupon is

1000(1.21)=\$1{,}210.00.

After the $210 coupon is paid, the book value is $1210-210=\$1{,}000.00$. This check confirms that the coupon is deducted at the payment date, not halfway through the period. It also verifies the stated periodic yield and coupon are consistent with the balance returning to $1,000 after payment.

Comparing Conventions on the Same Balance

When a question does not name the fractional-period convention, do not silently choose one and present it as universal. State the convention used or explain that the amount depends on the convention. The difference can be small for a short interval, but an exam question may require a particular method or ask for a comparison.

Worked Example: Detecting a Convention Difference

A bond’s book value immediately after a coupon is $800. Its effective yield is 8% per coupon period, and a quarter of the period has elapsed. Calculate the between-date book value under each of two explicitly stated conventions. The next coupon is $50.

Method. Compute both accruals from the same starting balance and elapsed fraction. Do not subtract the $50 coupon, which is not yet payable.

Under simple-interest accrual, the effective interest is

800(0.08)(0.25)=\$16.00,

so the book value is $800+16=\$816.00$.

Under fractional-period compound accrual, the book value is

800(1.08)^{0.25}\approx 800(1.0194265)=\$815.54.

The conventions differ by about $816.00-815.54=\$0.46$. As a full-period check, either convention gives a pre-coupon balance of $800(1.08)=\$864.00$. After the $50 coupon is paid, the next book value is $864-50=\$814.00$. This example shows why the convention matters even though the two methods agree at the coupon dates.

Common Mistake: Subtracting the Coupon Before It Is Paid

A frequent error is to treat the coupon as if it were deducted from the book value continuously during the coupon period. In the first example, a mistaken calculation might subtract the $20 of coupon that has accrued over three months from the $1,020 carried balance:

1020-20=\$1{,}000.00 \qquad \text{(incorrect for the stated book-value convention).}

That subtraction confuses coupon accrual with a cash payment. The book value under the stated convention is still $1,020.00 at the three-month point. At the next coupon date, first carry the book value to $1,040.00; only then subtract the $40 coupon to obtain the post-payment balance of $1,000.00. Always label whether a balance is immediately before or immediately after a coupon.

Key takeaway: A between-date book value is the previous post-coupon book value carried forward under a specified yield convention. Use the correct per-period yield and elapsed fraction, and deduct the coupon only when the payment occurs. Simple and compound fractional-period conventions agree over a full period but can give different intermediate balances.

Check Your Understanding

For each question, identify the yield convention and whether the requested value is before or after a coupon payment.

A bond has a book value of $1,200 immediately after a coupon. Its nominal annual yield is 6% convertible semiannually. Under simple-interest accrual, find the book value four months into the six-month period. (Answer: $1,224.00)
A bond’s book value immediately after a coupon is $950. The effective yield is 4% per coupon period, and fractional-period compound accrual applies. Find the book value halfway through the period. (Answer: $968.81)
A bond’s post-coupon book value is $2,400. The effective yield is 2.5% per quarter, and simple accrual applies. Find the book value two months later and then the balance immediately after a $55 coupon is paid at the end of the quarter. (Answer: $2,440.00 between dates; $2,405.00 after the coupon)
A bond’s post-coupon book value is $1,500, its effective yield is 12% per coupon period, and one quarter of the period has elapsed. Find the balance under simple accrual and under fractional-period compound accrual. (Answer: $1,545.00 under simple accrual; $1,543.11 under compound accrual)
A bond’s book value immediately after a coupon is $875. Its effective yield is 6% per six-month period, and simple accrual applies. Five months later, a $32 coupon is due. Find the accrued effective interest, the balance immediately before payment, and the balance immediately after payment. (Answer: interest $43.75; before payment $918.75; after payment $886.75)

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