Why the Final Book Value Matters
A bond’s book value is the value assigned to its remaining contractual cash flows under the yield used to establish the bond’s carrying value. As coupons are paid, the book value changes. At maturity, however, no coupon payments remain after the final coupon has been paid, and the only remaining contractual payment is redemption. The book value at that point must therefore equal the redemption amount.
That conclusion is simple to state, but the last coupon creates an important timing distinction. Immediately before the final coupon is paid, both the coupon and redemption are due. Immediately after the coupon is paid, only redemption remains. This tutorial uses that distinction to prove the maturity result from both prospective valuation and the coupon-date recurrence.
This timing convention makes the recurrence precise. At time \(k\), for \(0 \leq k < n\), the remaining payments are the coupons at times \(k+1,\ldots,n\) and the redemption amount at time \(n\). At time \(n\), the coupon has just been paid, so the remaining payment is redemption alone.
The Prospective Formula at Maturity
Under the stated assumptions, each future payment is discounted at the fixed annual yield \(i\). Write \(v=1/(1+i)\). For \(0 \leq k \leq n\), the prospective book value is
where the sum is zero when \(k=n\). Equivalently, using the standard annuity-immediate notation, \(B_k=C a_{\overline{n-k}|}+Rv^{n-k}\), with \(a_{\overline{0}|}=0\). Substituting \(k=n\) gives
This is a direct prospective proof: immediately after the last coupon, there are no future coupons to include, and redemption is due without any remaining period of discounting. The proof depends on the timing convention. It does not say that the value immediately before the final coupon is \(R\); immediately before that coupon, the coupon and redemption are both due.
Why the Coupon-Date Recurrence Gives the Same Result
The recurrence says that a book value grows for one year at the yield rate and then the coupon is subtracted. To justify it, start with the prospective formula at time \(k\), and write \(m=n-k\), the number of periods remaining:
Since \(1+i=1/v\), multiplying the value by \(1+i\) shifts every discounted future payment one period closer to the valuation date. Subtracting the coupon just paid at the next coupon date removes that coupon from the remaining cash flows. Algebraically,
Thus, for each \(k=0,\ldots,n-1\),
At the last step, \(k=n-1\), only one period remains. The prospective value one year earlier is \(B_{n-1}=(C+R)v\), because the final coupon and redemption are both payable at time \(n\). Applying the recurrence gives
The recurrence and prospective formula agree because both value the same remaining cash flows at the same yield. The final subtraction of \(C\) matters: the value grows to the total amount due at the final date, \(C+R\), and then the final coupon is paid, leaving \(R\).
At a coupon date, discount every coupon still due and the redemption amount to that date.
Multiply the current book value by \(1+i\), using the same fixed yield as in the valuation.
The resulting value is the book value immediately after that coupon date.
The amount just before the final coupon is \(C+R\); after subtracting \(C\), the remaining book value is \(R\).
Worked Examples
Worked Example: Two Annual Periods and a Final Recurrence Check
A bond pays an annual coupon of $40, has redemption amount $1,000, and matures in two years. Its fixed effective annual yield is 5%. Find the issue-date book value and verify the book values through maturity. Use unrounded intermediate values and report money to the nearest cent.
Method. First use prospective valuation at time 0. Then apply the coupon-date recurrence. This checks that the final value is obtained by subtracting the final coupon from the amount due at maturity.
The discount factor is \(v=1/1.05=0.9523809524\). The cash flows are $40 at the end of year 1 and $1,040 at the end of year 2. Therefore,
Thus \(B_0=\)$981.41. At the end of year 1, after the first coupon is paid,
At the end of year 2, after the final coupon is paid,
The tiny excess is only from displaying rounded intermediate figures; using the unrounded values gives \(B_2=\)$1,000 exactly. The prospective check is \(B_2=R=\)$1,000, since after the final coupon no coupons remain. The common mistake this avoids is stopping at \(B_1(1.05)=\)$1,040 and calling that the final book value; that amount is before the final coupon is paid.
Worked Example: A Discount Bond Over Four Years
A bond pays an annual coupon of $24, has a redemption amount of $1,000, and has four years remaining. Its fixed effective annual yield is 3%. Find \(B_0\), \(B_2\), and \(B_4\). Carry unrounded values through the calculations and report money to the nearest cent.
Method. Use prospective valuation for the initial and intermediate values, then verify the last two recurrence steps. Because the coupon is below 3% of redemption, the initial book value is below redemption, but the maturity proof does not depend on whether the bond is at a premium or discount.
Here \(v=1/1.03=0.9708737864\). The issue-date value is
So \(B_0=\)$977.70. The first recurrence step is
Then
Therefore \(B_2=\)$988.52. Continue to maturity:
The unrounded model value is \(B_4=\)$1,000; the displayed discrepancy is rounding in the intermediate figures. Independently, the prospective formula at time 4 has no future coupons and gives \(B_4=1000\). The check also confirms that a discount bond does not remain below redemption at maturity.
Worked Example: Three Periods and the Final-Date Cash Flows
A hypothetical infrastructure bond pays an annual coupon of $50, has a redemption amount of $900, and matures in three years. The fixed effective annual yield is 4%. Find \(B_0\) and use the recurrence to establish \(B_3\). Report money to the nearest cent.
Method. The final coupon and redemption together form the last cash flow when viewed from time 0. After that coupon is paid, only redemption remains. First calculate the prospective value, then advance with the recurrence.
The final payment at time 3 is \(50+900=\)$950, and \(v=1/1.04=0.9615384615\). Thus
Hence \(B_0=\)$938.85. The recurrence gives
The unrounded values give \(B_3=\)$900, the redemption amount. A final prospective check gives \(B_3=900\) directly. The mistake to avoid is treating $950, the total due before the final coupon, as the book value after that coupon.
Timing Traps and Practical Checks
The maturity result is an accounting consequence of which payments remain, not a statement that every measure associated with the bond equals redemption at every moment. In particular, distinguish the following two values at the final coupon date:
| Timing at maturity | Payments still due at that instant | Value under the fixed-yield convention |
|---|---|---|
| Immediately before the final coupon | Final coupon \(C\) and redemption \(R\) | \(C+R\) |
| Immediately after the final coupon, before redemption | Redemption \(R\) only | \(R\) |
A common incorrect calculation is to apply the recurrence but omit the final coupon subtraction:
That equation gives the amount accumulated immediately before the final coupon is paid, not \(B_n\) under the definition used here. The correction is
Another error is to keep applying the recurrence after the redemption payment while still treating the bond as an outstanding obligation. The theorem concerns the book value immediately before redemption; after redemption has been paid, the bond has no remaining cash flows and no outstanding book value under this convention.
The proof also relies on its assumptions. If the bond defaults, changes its payment schedule, or is valued using a yield different from the one used to establish its book value, the stated recurrence need not describe the assigned book value. Rounding each period to cents can also produce a small residual at maturity. The exact result is obtained from unrounded values; a rounded amortization schedule may require a final-period reconciliation.
Check Your Understanding
For each problem, use the stated timing convention: \(B_k\) is immediately after the coupon at time \(k\) has been paid, and \(B_n\) is immediately before redemption.
- A one-year bond pays a $60 coupon and $1,000 redemption. Its effective annual yield is 6%. Find \(B_0\) and verify \(B_1\). (Answer: \(B_0=\)$1,000.00; \(B_1=\)$1,000.00)
- A two-year bond pays $35 annually, redeems for $1,000, and has a 4% effective annual yield. Find \(B_0\), then use the recurrence to find \(B_1\) and \(B_2\). (Answer: \(B_0=\)$990.57; \(B_1=\)$995.19; \(B_2=\)$1,000.00)
- A four-year bond has annual coupon $20, redemption $800, and effective annual yield 2%. What is its book value immediately after the fourth coupon and immediately before redemption? State the recurrence’s final-step calculation. (Answer: \(B_4=\)$800.00; the amount before subtracting the final coupon is $820.00)
- A bond has a $45 annual coupon, $1,000 redemption, and 5% effective annual yield. One year remains after the coupon at time 2. Find \(B_2\) and \(B_3\), and verify \(B_3\) prospectively. (Answer: \(B_2=\)$995.24; \(B_3=\)$1,000.00)
- A bond’s final coupon is $32 and redemption is $1,200. Immediately before the final coupon is paid, what amount is due? What is the book value immediately after that coupon and before redemption? (Answer: $1,232.00 is due; the book value after the coupon is $1,200.00)