Valuing A Bond Purchase Between Coupon Dates

Why a Purchase Between Coupon Dates Needs Care

A bond’s coupon-date book value is defined at a coupon date, but a purchase can occur between two coupon dates. The key is to move the book value forward from the most recent coupon date using the bond’s original yield and the elapsed fraction of a coupon period. That accumulated amount is the carrying value under the fractional-period compound-interest model.

A transaction may also specify an accrued-coupon amount to be included or excluded from the settlement. That amount is a separate adjustment: it is not automatically part of the carrying-value calculation. The problem’s stated price or settlement convention determines how to use it.

Definition: Let $B_k$ be the bond’s book value immediately after the coupon at coupon date $k$. Let $j$ be the original yield per coupon period, and let $f$ be the fraction of the next coupon period that has elapsed, where $0 \leq f < 1$. Under fractional-period compound interest, the carrying value at the purchase date is $$ B(t)=B_k(1+j)^f. $$ The elapsed fraction $f$ must be measured in coupon periods. Any separately specified accrued-coupon adjustment is accounted for independently, according to the transaction convention.

How to Value the Purchase Date

Begin with the book value immediately after the most recent coupon payment, not with the value immediately before that coupon. The coupon has already been paid at that date, so it is not included again in the value being accumulated. Then determine the original yield per coupon period and the fraction of a period from that date to the purchase date.

Locate the preceding coupon date.
Use the book value immediately after the coupon paid on that date.

Express the original yield per coupon period.
Use the yield that determined the bond’s original book values, matched to the coupon frequency.

Measure the elapsed fraction.
Divide the elapsed time since the preceding coupon date by the length of one coupon period, using the time convention stated in the problem.

Accumulate and handle adjustments separately.
Calculate $B_k(1+j)^f$. Add or subtract an accrued-coupon amount only if the requested price or settlement convention calls for it.

If the yield is quoted as a nominal annual rate convertible $m$ times per year and coupons are paid $m$ times per year, the yield per coupon period is $j=i^{(m)}/m$. The fractional exponent applies to that period yield. It is not generally correct to multiply the period yield by $f$ and use $1+fj$; that would use simple interest over the partial period rather than the compound-interest model.

Why the Fractional-Period Formula Holds

The result follows from valuing the remaining cash flows at the purchase date. Suppose the preceding coupon date is time 0, the purchase occurs $f$ coupon periods later, and the remaining payments occur at coupon periods $1,2,\ldots,n$. Let $C_r$ be the payment at period $r$, including redemption in the final payment. At the preceding coupon date, the book value is the present value of those remaining payments:

B_k=\sum_{r=1}^{n} C_r(1+j)^{-r}.

At the purchase date, the same payments are respectively $1-f,2-f,\ldots,n-f$ periods away. Discounting each payment from its new time to the purchase date gives:

\begin{aligned} B(t) &=\sum_{r=1}^{n} C_r(1+j)^{-(r-f)}\\ &=(1+j)^f\sum_{r=1}^{n} C_r(1+j)^{-r}\\ &=(1+j)^f B_k. \end{aligned}

Thus, accumulating the preceding coupon-date book value gives the same value as discounting all remaining cash flows from the purchase date, provided the same yield and fractional-period compound-interest model apply. The coupon already paid at time 0 is excluded from those remaining cash flows.

Worked Example: A Par Bond Purchased Halfway Through a Coupon Period

A hypothetical bond has a redemption value of $1,000, pays annual coupons of $40, and has two years remaining immediately after an annual coupon. Its original effective annual yield is 4%. An investor purchases it six months after that coupon. Find the carrying value at the purchase date. Use fractional-period compound interest and round the final amount to the nearest cent.

Method. First find the book value immediately after the coupon, then accumulate it for one-half of a year. The coupon rate and yield are both 4% of redemption value, so the coupon-date book value is par. This can also be verified from the remaining cash flows:

\begin{aligned} B_k &=40\left(\frac{1-(1.04)^{-2}}{0.04}\right)+1{,}000(1.04)^{-2}\\ &=40\left(\frac{1-0.9245562}{0.04}\right)+1{,}000(0.9245562)\\ &=40(1.886095)+924.5562\\ &=75.4438+924.5562\\ &=1{,}000.0000. \end{aligned}

Six months is $f=6/12=0.5$ of an annual coupon period. The original yield per annual period is $j=0.04$, so:

\begin{aligned} B(t) &=1{,}000(1.04)^{0.5}\\ &=1{,}000(1.0198039)\\ &=1{,}019.8039\\ &=\$1{,}019.80. \end{aligned}

As a check, the next $40 coupon is six months away and the final $1,040 payment is eighteen months away. Discounting both at 4% gives $40(1.04)^{-0.5}+1{,}040(1.04)^{-1.5}=\$1{,}019.80$, agreeing with the accumulated book value. The common mistake avoided here is using $1+0.04(0.5)$, which would apply simple interest to the half-year rather than the specified fractional compound-interest model.

Carrying Value and Accrued Coupon Are Different Amounts

The carrying value measures the bond’s value under the original yield model. An accrued-coupon amount, when a problem specifies one, is a separate transaction adjustment based on the stated coupon-accrual convention. Do not assume that it must always be added to the carrying value: a question may ask only for carrying value, may state an all-in settlement that includes an adjustment, or may use another quoted-price convention.

For example, in a convention that quotes a full price as a clean quoted amount plus accrued coupon, the clean amount is not the same thing as the accumulated carrying value. Read the requested quantity carefully and follow the convention stated. The calculation below explicitly says to add the separate accrued amount to obtain the total specified purchase amount.

Worked Example: Adding a Separately Specified Coupon Adjustment

A hypothetical bond has a book value of $996.40 immediately after a coupon date. Its original yield is 2% per quarter. A purchase occurs two-thirds of a quarter after that coupon. The transaction terms separately specify an accrued-coupon adjustment of $24 multiplied by the elapsed fraction, and define the total purchase amount as carrying value plus that adjustment. Find both amounts, rounding to the nearest cent.

Carrying value. Here $j=0.02$ and $f=2/3$:

\begin{aligned} B(t) &=996.40(1.02)^{2/3}\\ &=996.40(1.0132880\ldots)\\ &=1{,}009.6414366\ldots\\ &=\$1{,}009.64. \end{aligned}

Separate accrued-coupon adjustment. The problem defines this amount as $24 times the elapsed fraction:

24\left(\frac{2}{3}\right)=\$16.00.

Total purchase amount under the stated terms.

\$1{,}009.6414366\ldots+\$16.00 =\$1{,}025.6414366\ldots =\$1{,}025.64.

The results are a carrying value of $1,009.64 and a specified total purchase amount of $1,025.64. The calculation avoids treating the $16 adjustment as part of the fractional-yield accumulation; it is added only because the problem explicitly defines the total that way.

Worked Example: Converting a Time Interval to a Fraction of a Coupon Period

A hypothetical bond has a book value of $1,025 immediately after a coupon date. The original yield is 1.5% per quarter. The purchase date is 20 days after that coupon date, and the problem uses a 60-day coupon period. The transaction also specifies a separate accrued-coupon adjustment equal to $18 times the fraction of the coupon period elapsed, to be added to carrying value. Find the carrying value and total purchase amount, to the nearest cent.

Method. Convert elapsed time to coupon periods before applying the yield. The elapsed fraction is $f=20/60=1/3$, not $20/365$, because the specified yield is per quarter and the problem gives a 60-day coupon period.

\begin{aligned} B(t) &=1{,}025(1.015)^{1/3}\\ &=1{,}025(1.0049752\ldots)\\ &=1{,}030.0996\ldots\\ &=\$1{,}030.10. \end{aligned}

The separately specified adjustment is:

18\left(\frac{1}{3}\right)=\$6.00.

Therefore, under the stated settlement convention, the total is:

\$1{,}030.0996\ldots+\$6.00 =\$1{,}036.0996\ldots =\$1{,}036.10.

The check is that the time fraction is one-third of the stated coupon period, and the total exceeds the carrying value by exactly the specified $6 adjustment. Using a year fraction instead would mismatch the quarter-period yield and the time measurement.

Common Exam Traps

Starting from the wrong side of a coupon date. Use the book value immediately after the most recent coupon, unless the problem explicitly gives a different starting value. A coupon already paid is not among the remaining cash flows.
Using the wrong time fraction. Measure elapsed time as a fraction of one coupon period. A stated 20 days in a 60-day period is $1/3$, not $20/365$.
Using simple interest for a fractional period. Under the compound model the factor is $(1+j)^f$, not $1+fj$.
Changing the yield at the purchase date. The calculation uses the original yield associated with the book value, not a newly observed market yield.
Automatically adding accrued coupon. The carrying value and a separately specified coupon adjustment are distinct. Add or subtract the adjustment only when the requested transaction amount and stated convention require it.
Rounding too early. Keep the accumulated value unrounded through any subsequent addition, then round the requested dollar amount to cents.

Key takeaway: For a purchase between coupon dates, accumulate the book value immediately after the preceding coupon by $(1+j)^f$, where $j$ is the original yield per coupon period and $f$ is the elapsed fraction of that period. Treat any separately specified accrued-coupon amount according to the transaction convention; it is not automatically part of the carrying value.

Check Your Understanding

For each problem, use fractional-period compound interest and keep any stated coupon adjustment separate until asked for the total. Round dollar amounts to the nearest cent.

A bond’s book value immediately after a coupon is $980. Its original yield is 2% per coupon period, and the purchase occurs halfway through the period. Find its carrying value. (Answer: $989.75)
A bond has a preceding coupon-date book value of $1,200, an original quarterly yield of 1.5%, and a purchase date two months into a three-month coupon period. Find its carrying value. (Answer: $1,211.97)
A bond’s preceding coupon-date book value is $1,040. Its original monthly yield is 0.8%. A purchase occurs 15 days into a 30-day period. A transaction convention specifies a separate $12 monthly coupon adjustment multiplied by the elapsed fraction and added to carrying value. Find the carrying value and total. (Answer: Carrying value $1,044.15; total $1,050.15)
A bond’s preceding coupon-date book value is $1,000, and its original quarterly yield is 2%. The carrying value on a purchase date is reported as $1{,}000(1.02)^{1/2}$, without rounding. What fraction of a coupon period has elapsed? (Answer: $1/2$ of a coupon period)

What You'll Learn