Book Value Between Coupon Dates Under Compound Interest

Why Book Value Between Coupon Dates Needs a Timing Rule

A bond’s book value is often calculated at coupon dates, when the cash-flow pattern has a clear boundary: a coupon is paid, and the value after that payment is the starting point for the next period. Between coupon dates, interest has accumulated under the bond’s original yield, but the next coupon has not yet been paid. An intermediate value must reflect both facts.

In this tutorial, the book value between coupon dates is the value obtained by carrying the preceding coupon-date book value forward under compound interest at the bond’s original yield. It is a theoretical carrying value under that yield assumption; it is not, by definition, a price at which the bond must trade.

Definition: Let $B_k$ be the bond’s book value immediately after the coupon paid at coupon date $k$. Let $j$ be the effective yield per coupon period, and let $s$ be the fraction of a coupon period elapsed since date $k$, with $0<s<1$. Under compound-interest interpolation at the original yield, the book value at the intermediate time is $$ B_{k+s}=B_k(1+j)^s. $$

The exponent $s$ measures elapsed time in coupon periods, not elapsed calendar years. For example, $s=\frac12$ means halfway through one coupon period. The yield $j$ must be the yield for a full coupon period. If a bond has a different quoted yield convention, first express that yield as an effective rate per coupon period, consistently with the earlier material on matching yield conventions to coupon frequency.

No coupon is subtracted in this formula: the next coupon is still in the future. Its timing nevertheless matters. From the intermediate date, the next coupon is due after $1-s$ coupon periods, rather than after a full period. The remaining cash flows must be discounted using those shortened times if the value is checked prospectively.

Why Carrying Forward Gives the Prospective Value

Suppose the bond has $n-k$ coupon payments remaining after date $k$. Let $C$ be the coupon paid each period, and let $R$ be the redemption payment at maturity, in addition to the final coupon. With yield $j$, the coupon-date book value immediately after the coupon at date $k$ is the present value of all later coupons and the redemption payment.

B_k=\sum_{r=1}^{n-k}\frac{C}{(1+j)^r}+\frac{R}{(1+j)^{n-k}}.

At the intermediate time $k+s$, the coupon originally due $r$ periods after date $k$ is now due in $r-s$ periods. Thus its prospective value at the intermediate time is $C/(1+j)^{r-s}$. The redemption payment is due in $n-k-s$ periods. Multiplying the coupon-date expression by $(1+j)^s$ gives exactly these discounting periods:

\begin{aligned} B_k(1+j)^s &=\left(\sum_{r=1}^{n-k}\frac{C}{(1+j)^r} +\frac{R}{(1+j)^{n-k}}\right)(1+j)^s\\ &=\sum_{r=1}^{n-k}\frac{C}{(1+j)^{r-s}} +\frac{R}{(1+j)^{n-k-s}}. \end{aligned}

This establishes the key relationship: under the same compound-interest yield assumption, carrying the previous coupon-date value forward and discounting the remaining cash flows from the intermediate date give the same value. The next coupon is not ignored; its time to payment changes from one full period to $1-s$ periods.

Theorem: Between coupon dates, if book value is carried forward under compound interest at the original effective yield per coupon period $j$, then $B_{k+s}=B_k(1+j)^s$. Equivalently, it is the present value at $k+s$ of the remaining coupons and redemption payment, with each cash flow discounted for its actual remaining fraction of a coupon period. This equivalence assumes the original yield and cash-flow schedule are unchanged.

Worked Valuations

Worked Example: Checking the Next Coupon’s Timing

A hypothetical bond has a yield of 10.25% per coupon period. Immediately after a coupon date, its book value is $1,000. The next coupon, due one period later, is $110.25. At the following coupon date, the total payment, including coupon and redemption, is $1,093.955625. Find the book value halfway through the period, and verify it from the remaining payments.

Method. Carry the coupon-date book value forward for half a period, then independently discount each remaining payment from its actual time to payment. This checks that the upcoming coupon is placed at the correct time.

The period yield is $j=0.1025$, and the elapsed fraction is $s=0.5$. Since $1.1025=1.05^2$, the accumulation factor for half a period is $1.05$. Therefore,

B_{k+0.5}=1{,}000(1.1025)^{0.5}=1{,}000(1.05)=\$1{,}050.

For the prospective check, the next coupon is due in $1-0.5=0.5$ periods. The total payment at the following coupon date is due in $2-0.5=1.5$ periods. The corresponding accumulation factors are $(1.1025)^{0.5}=1.05$ and $(1.1025)^{1.5}=1.1025(1.05)=1.157625$. Thus,

\frac{110.25}{1.05}+\frac{1{,}093.955625}{1.157625} =105+945 =\$1{,}050.

As a consistency check, the coupon-date value from those payments is $110.25/1.1025+1{,}093.955625/1.21550625=100+900=\$1{,}000$, as stated. Both methods give an intermediate value of $1,050. The common mistake this avoids is discounting the next coupon for a full period from the halfway date: it is only half a period away.

Worked Example: Carrying Forward a Discounted Book Value

A hypothetical bond has a book value of $960 immediately after a coupon date. Its original effective yield is 10.25% per coupon period. Find its book value after half a coupon period. Also calculate the amount by which it has increased.

Method. Apply compound accumulation to the prior book value. Do not subtract the next coupon, because it has not yet been paid.

Here $B_k=\$960$, $j=0.1025$, and $s=0.5$. Again, $(1.1025)^{0.5}=1.05$. Therefore,

B_{k+0.5}=960(1.05)=\$1{,}008.

The increase since the preceding coupon date is $\$1{,}008-\$960=\$48$. As a check, carrying the intermediate value backward by half a period gives $\$1{,}008/1.05=\$960$. The result is a book value under the stated yield assumption; the calculation alone does not say that the bond’s market price is $1,008.

The common mistake this avoids is treating “half a period” as half of the annual rate or as a coupon payment. The rate used here is the effective yield for one whole coupon period, and the book value is accumulated for the elapsed fraction of that period.

Worked Example: Comparing Two Times Within the Same Period

A hypothetical bond has a book value of $800 just after a coupon date and an original yield of 10.25% per coupon period. Find its book value after one-quarter of a period and after three-quarters of a period. Round each value to the nearest cent.

Method. Use the same prior coupon-date value and period yield at both times, changing only the elapsed fraction. Since $1.1025=1.05^2$, the quarter-period factor is $(1.1025)^{1/4}=\sqrt{1.05}$, approximately $1.0246950766$.

After one-quarter of a period,

B_{k+0.25}=800(1.1025)^{0.25} \approx 800(1.0246950766) =\$819.7561 \approx \boxed{\$819.76}.

After three-quarters of a period, the accumulation factor is $(1.1025)^{0.75}=1.05\sqrt{1.05}$, approximately $1.0759298304$. Thus,

B_{k+0.75}=800(1.1025)^{0.75} \approx 800(1.0759298304) =\$860.7439 \approx \boxed{\$860.74}.

The values increase as more time elapses at a positive yield, and each can be carried backward to $800 by dividing by its corresponding accumulation factor. The common mistake this avoids is applying the three-quarter-period factor to the one-quarter-period calculation, or measuring elapsed time from the next coupon date instead of the preceding one.

Boundary Timing and Common Traps

As $s$ approaches 1, the carried value approaches $B_k(1+j)$. This is the value immediately before the next coupon payment under the carrying convention. The coupon is then paid; immediately after that payment, the value is reduced by the coupon amount. If the next coupon-date value is $B_{k+1}$, the coupon-date relationship is $B_k(1+j)=C+B_{k+1}$. Thus, the value just before payment and the value just after payment refer to different instants and should not be confused.

The intermediate formula is appropriate when the book value is being carried under the original yield and compound accumulation is assumed within the coupon period. It does not by itself determine a transaction price, account for a change in yield, or replace the actual coupon cash flow. If the problem specifies a different interpolation rule, use that rule rather than assuming compound-interest interpolation.

Using simple interest instead of compound interest. For elapsed fraction $s$, $B_k(1+sj)$ is generally not $B_k(1+j)^s$. It gives a different result except in special cases.
Subtracting the next coupon too early. The coupon is included as a future payment in the prospective valuation, but it is not subtracted from the carried book value before its payment date.
Using the wrong time unit. The fraction $s$ is measured in coupon periods. A fraction of a year must first be related to the coupon-period length.
Discounting the next coupon for a full period. From time $k+s$, the next coupon is due in $1-s$ periods.

Key takeaway: For an intermediate book value under compound-interest interpolation, start with the book value just after the preceding coupon and accumulate it at the original effective yield per coupon period for the fraction elapsed. For a prospective check, discount the next coupon for $1-s$ periods and later cash flows for their respective remaining times.

Check Your Understanding

For each problem, use the stated compound-interest carrying convention and the original yield. Round monetary answers to the nearest cent where needed.

A bond’s book value immediately after a coupon date is $1,250. The effective yield is 4% per coupon period. Find the book value halfway through the period. (Answer: $1,274.75)
A bond’s book value just after a coupon date is $875, and its yield is 10.25% per coupon period. Find its value after half a period. (Answer: $918.75)
A bond has book value $720 after a coupon date and a yield of 33.1% per coupon period. Find its book value one-third of a period later. (Answer: $792.00)
A hypothetical bond has an effective annual yield of 12.36% and pays coupons semiannually. Its book value just after a coupon date is $2,000. Find its book value one-quarter of a year later. (Answer: $2,059.13)
A bond with a 10.25% effective yield per coupon period has a coupon-date book value of $1,000. Its next coupon is $110.25, due one period later, and its total payment at the following coupon date is $1,093.955625. Find the value halfway through the period by discounting the remaining payments from that date. (Answer: $1,050.00)

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