Deriving The Coupon-Date Book Value Recurrence

Why a Coupon-Date Recurrence Is Useful

A bond’s book value at a coupon date can be found by valuing the remaining payments at the yield used to price the bond. That prospective valuation gives a direct way to move from one coupon date to the next: allow the current book value to accumulate for one period, then account for the payment made at the next date. The result is a recurrence, not a new pricing rule.

The order of operations matters. Interest accumulates on the book value held during the period. The coupon is then paid at the period’s end. At maturity, the final payment includes both the coupon and the redemption amount; the book value after that payment is zero.

Definition: Consider a bond with payments on dates $0,1,\ldots,n$, where each interval is one coupon period. Let $i$ be the effective yield per coupon period, and let $C_j$ be the coupon paid at date $j$. Let $R$ be the redemption amount paid at maturity, date $n$. The book value $B_k$ at date $k$ is the present value, at yield $i$, of the bond’s payments still due, measured immediately after any coupon payment at date $k$. In particular, $B_0$ is the price at date 0, and $B_n=0$ immediately after the final coupon and redemption have been paid.

This setup assumes that the book value is carried at the same yield $i$ used to value the remaining payments, that payments occur at the stated coupon dates, and that $i$ is an effective rate for one coupon period. If a yield is quoted on a different basis, first express the rate applicable to a coupon period. The recurrence below does not use a newly observed market yield to revalue the bond.

Deriving the Recurrence

At date $k$, the next payment occurs one period later. If $k+1<n$, that payment is the coupon $C_{k+1}$, and the value just after that coupon is $B_{k+1}$. From date $k$, both are one period in the future. Their combined value at date $k+1$ is therefore $C_{k+1}+B_{k+1}$.

B_k=\frac{C_{k+1}+B_{k+1}}{1+i}

Multiplying by $1+i$ and rearranging gives the recurrence. Its logic is that the amount held at date $k$ earns one period of interest, and the coupon is then removed at date $k+1$.

Theorem: For $k=0,1,\ldots,n-2$, the coupon-date book values satisfy

B_{k+1}=(1+i)B_k-C_{k+1}.

At the final date, the amount paid is not just the coupon: it is $C_n+R$. Immediately after that payment, no bond cash flows remain, so the terminal relation is

B_n=(1+i)B_{n-1}-(C_n+R)=0.

This final equation is also a useful consistency check. If a proposed initial price and yield are correct for the stated cash flows, repeatedly applying the recurrence and subtracting the final coupon plus redemption must leave zero.

Start with the book value.
Use the price or the prospective value of the remaining payments at the specified yield.

Accumulate for one coupon period.
Multiply the current value by $1+i$, with $i$ measured per coupon period.

Subtract the payment at the next date.
Subtract the coupon at an ordinary coupon date. At maturity, subtract the coupon and redemption together.

Check the terminal value.
After the final payment, the remaining book value must be zero when the starting price and yield are consistent.

Worked Examples

Worked Example: Level Coupons and a Terminal Check

A bond pays a coupon of $80 at the end of each of three annual periods and redeems for $1,000 at the end of period 3. Its effective yield is 10% per year. Find its price, the book values immediately after the first and second coupons, and verify the final balance. Round dollar answers to the nearest cent only at the end.

Method. First find the price as the present value of all scheduled payments. Then apply the recurrence using the unrounded price and intermediate values.

Price at date 0. The cash flows are $80 at dates 1 and 2, and $1,080 at date 3. With $i=0.10$, the discount factor is $v=1/1.10=0.9090909091$. Thus

\begin{aligned} B_0&=80(v+v^2+v^3)+1{,}000v^3\\ &=80(0.9090909091+0.8264462810+0.7513148009)+751.3148009\\ &=80(2.4868519910)+751.3148009\\ &=198.9481593+751.3148009\\ &=950.2629602. \end{aligned}

After coupon 1. The value accumulates at 10% for one year, then the $80 coupon is paid:

B_1=1.10(950.2629602)-80=1{,}045.2892562-80=965.2892562.

After coupon 2.

B_2=1.10(965.2892562)-80=1{,}061.8181818-80=981.8181818.

Final check. At maturity, the payment is the $80 coupon plus the $1,000 redemption:

B_3=1.10(981.8181818)-1{,}080=1{,}080-1{,}080=0.

Rounded to the nearest cent, the price is $950.26, $B_1$ is $965.29, and $B_2$ is $981.82. The zero terminal balance confirms that the recurrence agrees with the specified cash flows and yield. The calculation avoids the mistake of rounding the initial price before carrying it forward.

Worked Example: Different Coupons and Nonpar Redemption

A two-period bond pays $40 at the end of period 1 and $60 at the end of period 2. It redeems for $900 at the end of period 2. The effective yield is 5% per period. Find the initial price and the book value immediately after the first coupon, then verify the maturity balance. Round to cents only at the end.

Method. Treat the second-date cash flow as one combined payment of $60+$900=$960. Use the prospective value to find the initial price, then use the recurrence.

At date 0, the first coupon is one period away and the final $960 payment is two periods away:

\begin{aligned} B_0&=\frac{40}{1.05}+\frac{960}{1.05^2}\\ &=38.0952380952+\frac{960}{1.1025}\\ &=38.0952380952+870.7482993197\\ &=908.8435374149. \end{aligned}

After the first coupon, the book value is

\begin{aligned} B_1&=1.05(908.8435374149)-40\\ &=954.2857142856-40\\ &=914.2857142856. \end{aligned}

At maturity, the book value accumulates for one more period, and the combined final payment is subtracted:

$$ B_2=1.05(914.2857142856)-960=960-960=0. $$

To the nearest cent, $B_0=$$908.84 and $B_1=$$914.29. The calculation avoids treating the redemption as if it were a separate extra period after the final coupon; both amounts are paid on date 2.

Worked Example: Checking Whether a Price and Redemption Are Consistent

A two-period bond is priced at $1,200. It pays a $50 coupon at the end of each period and has an effective yield of 5% per period. What redemption amount makes the stated price and yield consistent with the recurrence? Verify the price from the cash flows. Round to cents only at the end.

Method. Advance the given price through the first coupon date, then use the terminal condition to determine the final total payment and hence the redemption amount.

After the first period, the book value is

B_1=1.05(1{,}200)-50=1{,}260-50=1{,}210.

Immediately after the maturity payment, the book value must be zero. Therefore, the final coupon plus redemption must equal the amount accumulated from $B_1$:

\begin{aligned} C_2+R&=1.05B_1\\ 50+R&=1.05(1{,}210)=1{,}270.50\\ R&=1{,}270.50-50=1{,}220.50. \end{aligned}

Price check. The date-1 payment is $50, and the date-2 payment is $50+1{,}220.50=\$1{,}270.50$. Their present value is

\begin{aligned} \frac{50}{1.05}+\frac{1{,}270.50}{1.05^2} &=47.6190476190+1{,}152.3809523810\\ &=1{,}200. \end{aligned}

The redemption amount is $1,220.50, and the terminal recurrence gives $1.05(1{,}210)-1{,}270.50=0$. This check avoids inferring the redemption from the final coupon alone: the terminal payment includes both.

Timing, Rate, and Maturity Traps

The recurrence is simple, but a timing error changes the answer. If the book value is measured immediately after a coupon, it is the amount carried during the next period. The next coupon is paid at the end of that period, after interest has accumulated. Therefore the correct order is multiply by $1+i$, then subtract the coupon.

For example, if $B_k=\$1{,}000$, the effective yield is 6% per period, and the next coupon is $40, the correct next value is

B_{k+1}=1.06(1{,}000)-40=1{,}020.

A common incorrect calculation subtracts the coupon first and then accumulates interest:

$$ (1{,}000-40)(1.06)=1{,}017.60. $$

That calculation treats the coupon as if it were removed at the beginning of the period. But the coupon is due at the end, so the full $1,000 earns interest during the period. The correction is to apply the recurrence in the stated date order.

Use a per-period yield. The factor $1+i$ must correspond to the interval between the two coupon dates in the recurrence.
Keep the date convention explicit. Here, $B_k$ is immediately after any coupon at date $k$. A value immediately before a payment is not the same quantity.
Include redemption at maturity. The final subtraction is $C_n+R$, not just $C_n$.
Do not round early. Carry sufficient precision through the recurrence and round the reported dollar amounts only at the end.

Key takeaway: At an ordinary coupon date, the next post-coupon book value is the current book value accumulated for one coupon period, less the coupon paid at the end of that period. At maturity, subtract the final coupon and redemption together; a correctly priced bond then has a zero value after payment.

Check Your Understanding

Apply the recurrence with the book-value date convention stated in each problem.

A bond has a current post-coupon book value of $800, an effective yield of 5% per coupon period, and a $30 coupon at the next date. Find its book value immediately after that coupon. (Answer: $810)
A two-period bond has an initial price of $900, an effective yield of 4% per period, and coupons of $25 at both dates. Find the redemption amount that makes the price and yield consistent. (Answer: $922.44)
A two-period bond is stated to have an initial price of $1,000, an effective yield of 3% per period, a $60 coupon at date 1, and a combined coupon-plus-redemption payment of $1,100 at date 2. Apply the recurrence from the stated price and find the terminal balance. Is the stated price consistent with those cash flows and yield? (Answer: $B_1=\$970$, terminal balance $=-\$100.90$; no, the stated price is inconsistent.)
A bond has three periods, an initial price of $1,000, a 5% effective yield per period, and a $40 coupon at each date. Find $B_1$, $B_2$, and the redemption amount required for a zero terminal balance. (Answer: $B_1=\$1{,}010$, $B_2=\$1{,}020.50$, redemption $=\$1{,}031.53$)

What You'll Learn