Why Timing and Signs Matter
A bond book value can be numerically plausible and still be wrong because it belongs to the wrong instant. The coupon date has two distinct values: one immediately before the coupon is paid and another immediately after. Confusing them can shift a correct result by exactly one coupon. A separate but related error is to reverse the coupon and interest signs, or to use an annual yield as though it were a rate per coupon period.
The reliable check is to put the calculation on a timeline. First accrue the previous after-coupon book value at the yield for one coupon period; that gives the value immediately before the next coupon. Then subtract the coupon to obtain the value immediately after it. This tutorial focuses on diagnosing those steps, not on rebuilding a full bond amortization table.
The superscripts are timing labels, not powers. The minus sign in \(B_k^{-}\) means “before”; it does not mean a negative book value. Likewise, \(B_k^{+}\) means “after.” At the coupon instant, the coupon is still included in the value just before payment and is no longer included just after payment.
Deriving the Two-Step Recurrence
The recurrence follows from the yield-based accumulation and payment sequence. Between coupon dates, no coupon is paid, so the prior after-coupon book value accumulates for one period at rate \(j\). At the coupon date, the issuer pays \(C\), reducing the bond's value by that amount.
Multiply the previous after-coupon value by \(1+j\): \(B_k^{-}=B_{k-1}^{+}(1+j)\).
Subtract the cash payment from the just-before value: \(B_k^{+}=B_k^{-}-C\).
\(B_k^{+}-B_{k-1}^{+}=jB_{k-1}^{+}-C\). The change equals interest accrued minus the coupon paid.
This last form is a useful sign check. If the bond is selling at a premium, its coupon typically exceeds the interest earned on its book value, so the book value falls over the period. For a discount bond, the coupon is typically less than the interest earned, so the book value rises. These are pattern checks, not substitutes for using the stated values and the recurrence.
Worked Example: Correcting a Before-and-After Mislabel
Worked Example: Correcting a Before-and-After Mislabel
A bond has a book value of \(\$970\) immediately after a coupon date. The yield is \(5\%\) per coupon period, and the next coupon is \(\$40\). A calculation reports \(\$978.50\) as the book value immediately before the next coupon. Diagnose the error and find both correctly timed values.
Method. Apply the two steps in time order. The value before the next coupon is found by accruing the \(\$970\) for one period: $$ B_k^{-}=970(1.05)=\$1{,}018.50. $$
Now subtract the coupon to obtain the value after payment: $$ B_k^{+}=1{,}018.50-40=\$978.50. $$
The reported \(\$978.50\) is not an incorrect arithmetic result; it is the correct after-coupon value mislabeled as the before-coupon value. A different incorrect procedure—subtracting the coupon first and then accruing—would give $$ (970-40)(1.05)=930(1.05)=\$976.50, $$ which is neither correct value. The order matters because the coupon is paid at the end of the period, after the book value has accrued. The correct values are \(\$1{,}018.50\) immediately before payment and \(\$978.50\) immediately after payment. As a check, their difference is exactly the \(\$40\) coupon.
Signs: Interest Accrued Versus Coupon Paid
A frequent error is to add the coupon in the book value recurrence. The coupon is cash received by the bondholder, but it is cash paid by the issuer and removed from the bond after payment. In the recurrence for the bond's book value, it is therefore subtracted. Interest accrual has the opposite sign: it increases the value from one coupon date to the next before the payment.
Worked Example: Finding the Correct Signs for a Premium Bond
Immediately after a coupon, a bond's book value is \(\$1{,}040\). Its yield is \(4\%\) per coupon period, and the next coupon is \(\$50\). Find the book values immediately before and after the next coupon, and explain the direction of change.
Step 1: Calculate interest for the period. $$ 1{,}040(0.04)=\$41.60. $$
Step 2: Accrue before payment. $$ B_k^{-}=1{,}040+41.60=\$1{,}081.60. $$
Step 3: Subtract the coupon. $$ B_k^{+}=1{,}081.60-50=\$1{,}031.60. $$
The change from the previous after-coupon value is \(\$1{,}031.60-\$1{,}040=-\$8.40\), which also equals interest minus coupon: \(\$41.60-\$50=-\$8.40\). The book value falls because the coupon exceeds the interest accrued on the book value. A common wrong calculation adds the coupon, producing \(\$1{,}131.60\) after payment; that treats a payment out of the bond as an addition to its value. The before-minus-after difference here is \(\$50\), as required.
Use a Yield Rate for the Coupon Period
The rate in the recurrence must be the effective yield for one coupon period. If a yield is quoted as a nominal annual rate convertible \(m\) times per year and the bond pays coupons \(m\) times per year, the period rate is the nominal rate divided by \(m\). If instead an effective annual yield \(i\) is given and there are \(m\) equal coupon periods per year, the effective rate per period is \((1+i)^{1/m}-1\). These conversions are not interchangeable: use the convention stated in the problem.
Using an annual rate as the period rate effectively compounds it once per coupon period, which usually overstates the accrual. Conversely, dividing an effective annual rate by the number of periods is generally not the exact effective period rate. Always write the period rate beside the timeline before calculating.
Worked Example: Avoiding an Annual-versus-Period Rate Error
A bond pays coupons semiannually. Its yield is quoted as \(9\%\) nominal per year convertible semiannually. Its book value immediately after a coupon is \(\$1{,}500\), and its next semiannual coupon is \(\$72\). Find the book values immediately before and after that coupon.
Step 1: Convert the quoted yield to the coupon-period rate. There are two coupon periods per year, so $$ j=\frac{0.09}{2}=0.045. $$ The rate for this calculation is \(4.5\%\) per half-year, not \(9\%\) per half-year.
Step 2: Accrue to just before the coupon. $$ B_k^{-}=1{,}500(1.045)=\$1{,}567.50. $$ The interest accrued is \(\$1{,}567.50-\$1{,}500=\$67.50\).
Step 3: Subtract the semiannual coupon. $$ B_k^{+}=1{,}567.50-72=\$1{,}495.50. $$ The change is \(\$67.50-\$72=-\$4.50\), confirming that the book value falls by \(\$4.50\). If \(9\%\) were incorrectly used as the half-year rate, the calculation would give \(\$1{,}635\) before and \(\$1{,}563\) after. Those values result from using twice the correct period yield. The correct values differ by the stated \(\$72\) coupon.
Check a Value by Discounting the Remaining Payments
At a coupon date, the book value immediately after payment can also be checked by discounting the cash flows that remain after that payment, using the same yield per coupon period. Do not include the coupon just paid: it is no longer due. For example, if two coupons remain and the final payment includes redemption, the after-coupon value is the present value of the next coupon and the final coupon plus redemption.
This gives an independent timing check. A value immediately before a coupon includes that coupon; a value immediately after it does not. Discounting the same list of payments for both instants, without changing the time origin or the payments included, is a common source of one-coupon errors.
Worked Example: Checking the Cash Flows Included at a Coupon Date
A bond has redemption value \(\$1{,}000\), pays a \(\$40\) coupon each period, and has a yield of \(5\%\) per period. Immediately after a coupon date, two payments remain: a \(\$40\) coupon one period later and a final \(\$1{,}040\) payment two periods later. Find the after-coupon book value. Then find the book value immediately before the next coupon and immediately after it.
Step 1: Discount only payments still due at the after-coupon date. The discount factor per period is \(v=1/1.05=0.952380952\ldots\). Thus $$ B^{+}=40v+1{,}040v^2 =\frac{40}{1.05}+\frac{1{,}040}{1.05^2} =38.095238\ldots+943.310657\ldots =\$981.405895\ldots. $$ The after-coupon book value is \(\$981.41\), rounded to the nearest cent.
Step 2: Accrue one period to just before the next coupon. $$ B_{\text{next}}^{-}=981.405895\ldots(1.05) =\$1{,}030.476190\ldots =\$1{,}030.48. $$
Step 3: Subtract the coupon just paid. $$ B_{\text{next}}^{+}=1{,}030.476190\ldots-40 =\$990.476190\ldots =\$990.48. $$
As a check, after that coupon only the final \(\$1{,}040\) payment remains one period later, whose value is \(1{,}040/1.05=\$990.476190\ldots\), agreeing with the after-coupon recurrence. The common mistake is to include the \(\$40\) coupon just paid in the cash flows remaining at the original after-coupon date. The stated two remaining payments are the next coupon and the final payment, not the coupon at the valuation instant.
A Practical Diagnostic Sequence
When a book value calculation looks suspicious, identify the exact time being valued before changing any arithmetic. Then check the cash-flow timing, the yield convention, and the signs in that order. This prevents a correction to one part of the calculation from concealing a separate error elsewhere.
- Mark the valuation instant. Record whether the value is just before or just after a coupon.
- State the period yield. Convert the quoted yield according to its convention and the coupon frequency.
- Accrue first. Multiply the previous after-coupon value by \(1+j\) to obtain the before-coupon value.
- Subtract the payment. Subtract \(C\) once to obtain the after-coupon value.
- Check independently. Confirm that before minus after equals \(C\), and, where the remaining payments are given, check the after-coupon value by discounting those payments.
These checks are particularly useful when a problem gives a value with a timing label and asks for an error diagnosis. A reported amount might be arithmetically correct but attached to the wrong instant, as in the first worked example. In that case, identify the mislabel and state both correctly timed values; do not describe the issue as an arithmetic error that the numbers do not support.
Check Your Understanding
For each problem, keep the coupon-period rate and the valuation instant explicit.
- A bond's book value immediately after a coupon is \(\$820\). The yield is \(3\%\) per coupon period and the next coupon is \(\$25\). Find the values immediately before and after the next coupon. (Answer: \(\$844.60\) before; \(\$819.60\) after.)
- A bond has an after-coupon book value of \(\$1{,}025\), a yield of \(4\%\) per period, and a coupon of \(\$48\). A report gives \(\$1{,}018\) immediately before payment. Diagnose the report and give the correctly timed values. (Answer: \(\$1{,}066.00\) before and \(\$1{,}018.00\) after; the reported after-coupon value is mislabeled as before.)
- Immediately after a coupon, a bond with redemption value \(\$2{,}000\) has two payments remaining: \(\$70\) one period later and \(\$2{,}070\) two periods later. The yield is \(3\%\) per period. Find the after-coupon book value to the nearest cent. (Answer: \(\$2{,}019.13\).)
- A bond pays semiannual coupons and has a yield quoted at \(9\%\) nominal convertible semiannually. Its after-coupon book value is \(\$1{,}500\), and its next coupon is \(\$72\). Find the before- and after-coupon values. (Answer: \(\$1{,}567.50\) before and \(\$1{,}495.50\) after.)
- A quarterly coupon bond has an after-coupon book value of \(\$980\). Its effective annual yield is \(10.3813\%\), and its next coupon is \(\$30\). Use the effective quarterly yield to find the values immediately before and after the next coupon. (Answer: The quarterly yield is \(2.5\%\); \(\$1{,}004.50\) before and \(\$974.50\) after.)
The timing, sign, and rate-conversion checks in this tutorial are aligned with the SOA Exam FM (Financial Mathematics) syllabus.