Why Prospective and Retrospective Bond Book Values Agree

Why Two Book-Value Methods Give One Answer

A bond’s book value at a coupon date can be calculated in two directions. The prospective method values the coupons and redemption still to come. The retrospective method accumulates the original price to the date and subtracts the coupons already paid, accumulated to that same date. These calculations look different, but under consistent assumptions they must agree: each is a way to measure the value at that date of the same remaining bond cash flows.

The equality is useful both as a calculation method and as a check. In a long bond problem, one method may be easier to apply, while the other can reveal a timing or sign error. The key is to use the same yield and cash-flow schedule throughout.

Definition: Consider a bond purchased at time 0 for price $P$, with level coupon $C$ paid at the end of each year for $n$ years and redemption amount $R$ paid with the final coupon at time $n$. Let $i$ be the effective annual yield used to value the bond and $v=1/(1+i)$. The book value immediately after coupon $t$, for $0\leq t\leq n$, is the value at time $t$ of the bond cash flows not yet paid. At time $n$, “immediately after the coupon” means after the final coupon but before the redemption is paid.

For this tutorial, coupons occur at regular annual intervals, the coupon amount and redemption amount are known, and $i$ is constant and positive. There are no missed payments, fees, taxes, or changes in the valuation yield. The initial price is set using that same yield. These assumptions make the timing and the comparison precise; if dates are irregular or a valuation is between coupon dates, the simple annual-period formulas below do not directly apply.

The Prospective and Retrospective Formulas

At time $t$, the remaining cash flows are $C$ at times $t+1,\ldots,n$ and $R$ at time $n$. Discounting those payments to time $t$ gives the prospective formula. The retrospective formula starts from $P$, accumulates it for $t$ years, and removes each coupon already paid after accumulating that coupon to time $t$.

\begin{aligned} B_t &= C a_{\overline{n-t}|}+Rv^{\,n-t} &&\text{(prospective)},\\ B_t &= P(1+i)^t-C s_{\overline{t}|} &&\text{(retrospective)}. \end{aligned}

Here $a_{\overline{m}|}$ is the present value of an annuity-immediate of $m$ annual payments of 1, and $s_{\overline{m}|}$ is the accumulated value at the end of $m$ years of those payments. Thus, the already-paid coupons are not merely subtracted as $tC$: earlier coupons have accumulated for longer than later coupons. For $t=0$, the retrospective expression is $P$. For $t=n$, the prospective expression is $R$, since only redemption remains just after the final coupon.

Important: The coupon amount $C$ and redemption amount $R$ need not be equal, and neither must equal the face amount. The formulas use the actual cash flows specified for the bond. The yield $i$ is the yield used to establish the book-value basis, not necessarily a later market yield.

Why the Formulas Agree

At issue, the price is the present value of all coupons and redemption, discounted at $i$:

P=C a_{\overline{n}|}+Rv^n =C\sum_{k=1}^{n}v^k+Rv^n.

Accumulate this equation to time $t$. Each coupon due at time $k$ is then represented at time $t$ by $C(1+i)^{t-k}$, and redemption is represented by $R(1+i)^{t-n}$. For coupons paid by time $t$, subtract their accumulated values. What remains is exactly the accumulated value of coupons due after $t$, plus the accumulated value at time $t$ of redemption.

\begin{aligned} P(1+i)^t-C s_{\overline{t}|} &= C\sum_{k=1}^{n}(1+i)^{t-k} +R(1+i)^{t-n} -C\sum_{k=1}^{t}(1+i)^{t-k}\\ &= C\sum_{k=t+1}^{n}(1+i)^{t-k} +R(1+i)^{t-n}\\ &= C\sum_{j=1}^{n-t}v^j+Rv^{n-t}\\ &= C a_{\overline{n-t}|}+Rv^{n-t}. \end{aligned}

The second line follows by cancelling the terms for coupons $1$ through $t$. In the third line, set $j=k-t$, so that $t-k=-j$. The remaining sum is the present value at time $t$ of the future coupons, and the redemption term is its value at time $t$ as well. This proves the equality at every coupon date under the stated assumptions.

Worked Applications

Worked Example: A Premium Bond After Its Second Coupon

A bond has annual coupons of $60, a redemption amount of $1,000 after four years, and an effective annual yield of 5%. It is priced at the yield at time 0. Calculate its book value immediately after the second coupon using both methods. State whether the book value is above or below redemption.

Method. First find the issue price from all four coupons and redemption. Then calculate the value after coupon 2 independently from future cash flows and from the accumulated issue price. This checks that the coupon timing is consistent.

Issue price. At $i=0.05$, $v=1/1.05$. The four-year annuity factor is $a_{\overline{4}|}=3.545950504$, and $v^4=0.822702475$. Therefore,

$$ P=60(3.545950504)+1000(0.822702475) =212.757030+822.702475 =1035.459505. $$

Prospective value after coupon 2. Two coupons remain, and redemption is due at the end of year 4, two years after the valuation date. Since $a_{\overline{2}|}=1.859410431$ and $v^2=0.907029478$,

$$ B_2=60(1.859410431)+1000(0.907029478) =111.564626+907.029478 =1018.594104. $$

Retrospective value after coupon 2. The issue price accumulates for two years. The first coupon accumulates for one year and the second for zero years, so $s_{\overline{2}|}=1+1.05=2.05$.

\begin{aligned} B_2 &=1035.459505(1.05)^2-60(2.05)\\ &=1141.594104-123\\ &=1018.594104. \end{aligned}

Both methods give $1,018.594104, or $1,018.59 to the nearest cent. The coupon rate is $60/1000=6\%$, above the 5% yield, so the bond was issued at a premium and its book value is still above the $1,000 redemption amount at this date. The check is the agreement of the two independently computed values. The mistake to avoid is subtracting two coupons without accumulating the first one to time 2.

Worked Example: A Discount Bond Must Remain Below Redemption Here

A bond pays $40 annually for five years and redeems for $1,000 with its fifth coupon. The effective annual yield is 5%. Find the issue price and the book value immediately after coupon 2 by both methods. Compare the book value with redemption.

Method. Use the yield to value the issue cash flows, then calculate the remaining three coupons and redemption prospectively. For the retrospective check, accumulate the price for two years and subtract the two accumulated coupons.

The required factors are $a_{\overline{5}|}=4.329476670$, $v^5=0.783526167$, $a_{\overline{3}|}=2.723248030$, and $v^3=0.863837599$. The issue price is

$$ P=40(4.329476670)+1000(0.783526167) =173.179067+783.526167 =956.705234. $$

After coupon 2, three coupons and redemption remain. The prospective value is

$$ B_2=40(2.723248030)+1000(0.863837599) =108.929921+863.837599 =972.767520. $$

For the retrospective value, $s_{\overline{2}|}=2.05$, so

\begin{aligned} B_2 &=956.705234(1.05)^2-40(2.05)\\ &=1054.767520-82\\ &=972.767520. \end{aligned}

The methods agree at $972.77 to the nearest cent. The coupon rate is $40/1000=4\%$, below the 5% yield, so the bond was issued at a discount. Its book value here is below the $1,000 redemption amount because the discount has not yet been fully amortized. A common mistake is to say that book value exceeds redemption just because it has increased from its issue price; increasing does not imply that it has crossed redemption.

Worked Example: Redemption Need Not Equal Face Amount

A bond pays $100 at the end of year 1 and $1,200 at the end of year 2; the second payment consists of a $100 coupon and $1,100 redemption. The effective annual yield is 10%, and the bond is priced at that yield. Calculate the book value immediately after the first coupon by both methods.

Method. Treat the full payment at year 2 as coupon plus redemption when finding the issue price. At time 1, the prospective value is the discounted value of the year-2 payment.

Since $v=1/1.10$ and $v^2=1/1.21$, the issue price is

P=100v+1200v^2 =\frac{100}{1.10}+\frac{1200}{1.21} =90.909091+991.735537 =1082.644628.

The prospective value just after the first coupon is

B_1=100v+1100v =\frac{1200}{1.10} =1090.909091.

Retrospectively, the price accumulates one year and the first $100 coupon is then subtracted:

$$ B_1=1082.644628(1.10)-100 =1190.909091-100 =1090.909091. $$

Both methods give $1,090.91 to the nearest cent. This example emphasizes that redemption is the final principal payment, not necessarily the face amount. The error to avoid is discounting only the $1,100 redemption at year 2 and omitting the final $100 coupon.

How to Use the Equality as a Check

When both methods are available, organize the calculation around the valuation date and the cash-flow timing. The following sequence makes the reconciliation explicit.

Fix the date and convention.
Specify that book value is immediately after coupon $t$, and identify which coupons have already been paid.

Establish the issue price.
Discount every coupon and redemption payment from its actual scheduled date using the stated yield $i$.

Calculate prospectively.
Discount only the coupons after $t$ and the unpaid redemption amount to time $t$.

Calculate retrospectively.
Accumulate $P$ to time $t$, then subtract each paid coupon accumulated to time $t$.

Reconcile before rounding.
The two values should agree using unrounded factors. Round only the final reported amount.

A disagreement is usually a sign of inconsistent timing or inputs, not a different definition of book value. Check whether the coupon at time $t$ was included in the prospective cash flows after it had already been paid, whether past coupons were accumulated for the correct number of periods, and whether the same yield was used in the price and both balance calculations.

Common Traps and Scope

Subtracting coupons as simple totals. Retrospective value subtracts $C s_{\overline{t}|}$, not $tC$, because paid coupons are accumulated to the valuation date.
Mixing “before coupon” and “after coupon.” Immediately after coupon $t$, that coupon is no longer a future payment. Immediately before it, the cash-flow set is different.
Using a changed market yield midway through the calculation. The equality concerns a consistent book-value basis. A later change in market yield does not retroactively change the yield used to establish this book value.
Assuming book value is always above or below redemption. The direction depends on the coupon relative to the yield and on the date. A premium bond’s book value generally moves toward redemption from above; a discount bond’s moves toward redemption from below under the fixed-yield schedule.
Applying annual-period formulas to irregular dates. These expressions assume annual coupons and valuation at a coupon date. Irregular coupon dates or a date between coupons require a cash-flow valuation with the actual timing.

Key takeaway: At a coupon date, prospective book value is the value of unpaid coupons and redemption. Retrospective book value is the accumulated issue price less the accumulated value of coupons already paid. When the original price and both calculations use the same cash flows, dates, and yield, the two values are equal.

Check Your Understanding

For each problem, take the book value immediately after the specified coupon unless the question says otherwise. Round final dollar amounts to the nearest cent.

A bond pays $250 annually for three years and redeems for $5,000 with its final coupon. Its effective annual yield is 5%, and its issue price is set at that yield. Find the book value immediately after coupon 1 using the prospective method. (Answer: $5,000.00)
A bond pays $30 annually for four years and redeems for $1,000 with its final coupon. The effective annual yield is 4%. Find its book value immediately after coupon 1 prospectively. (Answer: $972.25)
A bond pays $70 annually for three years and redeems for $1,000 with its final coupon. The effective annual yield is 6%. Find its book value immediately after coupon 2 and state whether it is above or below redemption. (Answer: $1,009.43, above redemption)
A bond pays $40 annually for three years and redeems for $900 with its final coupon. The effective annual yield is 5%. Find its issue price and its book value immediately after coupon 1 by both methods. (Answer: Issue price $886.38; both book-value methods give $890.70)
A bond pays $50 annually for four years and redeems for $1,000 with its final coupon. Its effective annual yield is 4%. A student’s retrospective calculation after coupon 2 subtracts $2(50)$ from the accumulated issue price. Identify the error and give the correct retrospective deduction. (Answer: The coupons must be accumulated to time 2; subtract $50s_{\overline{2}|}=50(2.04)=\$102.00$, not $100.00)

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