Solving For The Original Price From A Later Book Value

Why Work Backward From Book Value?

A bond’s book value at a later coupon date reflects both the yield used to value the bond and all coupon payments made since purchase. If the original yield and coupon amount are known, that later book value contains enough information to recover the original purchase price. The calculation reverses the same accumulation and coupon-subtraction steps used to move forward.

The key is to reverse the cash-flow order correctly. Over one coupon period, the opening book value first accumulates interest at the original yield; the coupon is then paid, leaving the closing book value immediately after the coupon. Reversing that period therefore requires adding the coupon back before dividing by the accumulation factor.

Definition: Let $B_k$ denote a bond’s book value immediately after the $k$th coupon has been paid, and let $B_0$ be its original price at the initial coupon date. Let $C$ be the level coupon paid at the end of each period, and let $j$ be the effective yield per coupon period used to establish the original book value. Under the level-coupon, fixed-yield model, $$ B_k=B_{k-1}(1+j)-C. $$ The original price is recovered from a later book value by reversing this recurrence.

This model assumes coupon dates are equally spaced, the coupon is level, the yield $j$ remains the original book-value yield for the calculation, and each coupon is paid at the end of its period. The observed book value is immediately after a coupon payment. If a rate is quoted annually but coupons are more frequent, first convert it to an effective yield per coupon period. For example, a nominal annual yield $i^{(2)}$ convertible semiannually gives $j=i^{(2)}/2$. For annual coupons, $j$ is the effective annual yield $i$.

Reversing One Period at a Time

Starting with the recurrence $B_k=B_{k-1}(1+j)-C$, add the coupon amount to both sides and divide by $1+j$. This gives the previous coupon-date book value:

B_{k-1}=\frac{B_k+C}{1+j}.

Apply this calculation repeatedly until the value at time zero is reached. Every reversal has the same order: add back the coupon paid at the end of the period, then undo the interest accumulation by dividing by $1+j$.

Set the period yield.
Convert any quoted rate to the effective yield $j$ for one coupon period. Do not use an annual rate directly if the coupon period is shorter than a year.

Fix the timing convention.
Confirm that the given book value is immediately after a coupon and count how many coupon periods separate it from the original price.

Reverse each period.
Replace the current value $B_k$ with $(B_k+C)/(1+j)$, repeating once per elapsed coupon period.

Check by moving forward.
Starting from the recovered price, apply $B_k=B_{k-1}(1+j)-C$ through the same periods. The calculation should reproduce the given book value.

The repeated reversal also gives a compact expression. After $k$ periods, the later book value consists of the original price accumulated for $k$ periods, less the accumulated value of the $k$ coupons:

B_k=B_0(1+j)^k-C\,s_{\overline{k}|j}, \qquad s_{\overline{k}|j}=\frac{(1+j)^k-1}{j}.

Solving this equation for $B_0$ yields a useful check on the period-by-period method:

B_0=\frac{B_k+C\,s_{\overline{k}|j}}{(1+j)^k} =B_kv_j^k+C\,a_{\overline{k}|j}, \qquad v_j=\frac{1}{1+j}, \qquad a_{\overline{k}|j}=\frac{1-v_j^k}{j}.

Here the annuity factors use the effective rate $j$ per coupon period. The expression says that the original price equals the present value at time zero of the later book value plus the present value of the coupons paid in between. It is an algebraic check on the reversal, not a different yield assumption.

Worked Reversals

Worked Example: Recovering an Annual-Coupon Bond Price

A bond pays annual coupons of $60. Its original effective annual yield is 5%, and its book value immediately after the third coupon is $1,014.78. Recover the original price. Assume the purchase was made at the initial coupon date, so there are three complete annual periods.

Method. The coupon period is one year, so $j=0.05$. Starting at $B_3=1{,}014.78$, reverse three periods using $B_{k-1}=(B_k+60)/1.05$.

First reversal, from $B_3$ to $B_2$: add back the third coupon, then undo one year of yield accumulation:

B_2=\frac{1{,}014.78+60}{1.05} =\frac{1{,}074.78}{1.05} =1{,}023.60.

Second reversal, from $B_2$ to $B_1$:

B_1=\frac{1{,}023.60+60}{1.05} =\frac{1{,}083.60}{1.05} =1{,}032.00.

Third reversal, from $B_1$ to $B_0$:

B_0=\frac{1{,}032.00+60}{1.05} =\frac{1{,}092.00}{1.05} =1{,}040.00.

Check. Move forward from $1,040.00: $1{,}040(1.05)-60=1{,}032$; $1{,}032(1.05)-60=1{,}023.60$; and $1{,}023.60(1.05)-60=1{,}014.78$. The given third book value is reproduced exactly, so the original price is $1,040.00. The method avoids treating a coupon already paid as a future cash flow during reversal.

Worked Example: Converting a Semiannual Yield Before Reversing

A bond pays $75 every six months. Its original yield is quoted as a nominal annual rate of 8% convertible semiannually. Immediately after the third coupon, its book value is $5,165.2272. Recover the original price.

Method. There are three semiannual periods. The effective yield per period is $j=i^{(2)}/2=0.08/2=0.04$, not 8%. Reverse three times, adding back $75 each time.

B_2=\frac{5{,}165.2272+75}{1.04} =\frac{5{,}240.2272}{1.04} =5{,}038.68.

B_1=\frac{5{,}038.68+75}{1.04} =\frac{5{,}113.68}{1.04} =4{,}917.00.

B_0=\frac{4{,}917.00+75}{1.04} =\frac{4{,}992.00}{1.04} =4{,}800.00.

Check. The forward values are $4{,}800(1.04)-75=4{,}917$, $4{,}917(1.04)-75=5{,}038.68$, and $5{,}038.68(1.04)-75=5{,}165.2272$. Thus the original price is $4,800.00. Using 8% for each half-year would apply twice the correct period yield and would not match the stated nominal-rate convention.

Why the Reverse Formula Works

The coupon-date recurrence records two operations in a specific order: accumulate the opening book value, then subtract the coupon. Reversing a sequence of operations requires undoing them in reverse order. First undo the subtraction by adding $C$; then undo multiplication by $1+j$ by dividing by $1+j$. This establishes the one-period formula directly:

B_k=B_{k-1}(1+j)-C \quad\Longrightarrow\quad B_k+C=B_{k-1}(1+j) \quad\Longrightarrow\quad B_{k-1}=\frac{B_k+C}{1+j}.

Repeating that one-period identity gives the full formula. For example, reversing two periods gives

B_0 =\frac{\frac{B_2+C}{1+j}+C}{1+j} =\frac{B_2+C}{(1+j)^2}+\frac{C}{1+j}.

For $k$ periods, the later book value is discounted $k$ periods, while each intervening coupon is discounted from its own payment date. The coupon paid after the first period is discounted one period, the second coupon two periods, and so on. Therefore

B_0=\frac{B_k}{(1+j)^k} +\frac{C}{1+j}+\frac{C}{(1+j)^2} +\cdots+\frac{C}{(1+j)^k} =B_kv_j^k+C\,a_{\overline{k}|j}.

This also explains why the reversal should not subtract coupons: the price at time zero must account for coupon cash flows that have already been paid by the later valuation date. From the perspective of that later date, those payments are added back to reconstruct the book value that existed before each coupon.

Worked Example: Using a Forward Check After Reversal

A bond pays annual coupons of $40 and was originally valued at an effective annual yield of 5%. Its book value immediately after the second coupon is $980. Recover the original price, then verify the result by carrying it forward. A market quotation now gives a 7% yield, but the problem asks for the original price under the original book-value basis.

Method. Use the original yield $j=0.05$, not the later market quotation. Reverse two periods:

B_1=\frac{980+40}{1.05} =\frac{1{,}020}{1.05} =971.428571\ldots

B_0=\frac{971.428571\ldots+40}{1.05} =\frac{1{,}011.428571\ldots}{1.05} =963.265306\ldots

Rounded to the nearest cent, the original price is $963.27. To check without carrying rounding errors, use the unrounded value $B_0=963.265306\ldots$. Then $B_1=B_0(1.05)-40=971.428571\ldots$, and $B_2=B_1(1.05)-40=980$. The 7% market yield does not replace the original yield in this reconstruction: the stated book value arose under the original 5% yield.

Common Mistakes and Checks

The most frequent error is to subtract the coupon during reversal. Subtraction is part of the forward recurrence because the coupon leaves the bondholder’s invested value at the coupon date. Reversing that step requires adding the coupon back.

For the first example, the incorrect operation would give

\frac{1{,}014.78-60}{1.05} =\frac{954.78}{1.05} =909.314285\ldots,

which is not $B_2$. The correct operation is $(1{,}014.78+60)/1.05=1{,}023.60$. A forward check exposes the error: $909.314285\ldots(1.05)-60$ does not equal the given later book value.

Using the wrong period rate. A nominal annual rate convertible $m$ times per year must be converted to its per-period rate before applying the recurrence. For a nominal rate $i^{(m)}$, the per-period rate is $i^{(m)}/m$.
Reversing the wrong number of periods. Count coupon intervals from the original price date to the stated book-value date, not simply the number of calendar years mentioned.
Misreading coupon timing. This recurrence treats $B_k$ as the value immediately after the $k$th coupon. A value immediately before a coupon is a different timing point and cannot be inserted without adjusting for that coupon.
Substituting a current market yield. The reconstruction uses the yield underlying the original book-value calculation. A new market yield does not change the historical book value used as the input.
Rounding at every reversal. Keep full precision through the calculation and round the recovered price only at the end, unless a problem explicitly specifies a rounding rule for each period.

Key takeaway: At each coupon date, reverse the recurrence by adding the coupon and dividing by one plus the original effective yield per coupon period. Repeat for every elapsed period, then verify by carrying the recovered price forward to the stated book value. This skill is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the effective yield per coupon period and keep full precision until the final answer.

A bond pays annual coupons of $50. At an effective annual yield of 5%, its book value immediately after the second coupon is $950. Recover the original price. (Answer: $954.65)
A bond pays semiannual coupons of $40. Its nominal annual yield is 6% convertible semiannually, and its book value immediately after the third coupon is $1,000. Recover the original price. (Answer: $1,028.29)
A bond pays annual coupons of $30 and has an original effective annual yield of 4%. Its book value immediately after the fourth coupon is $925.478784. Recover the original price and verify it by carrying the result forward. (Answer: $900.00)
A bond pays annual coupons of $40. Its original effective annual yield was 5%, and its book value immediately after the second coupon is $980. A current market yield is 7%. Which yield should be used to recover the original price, and what is that price? (Answer: Use 5%; $963.27)
A bond pays annual coupons of $60, has an original effective annual yield of 4%, and has a book value of $1,100 immediately after its third coupon. Recover the original price, showing the three successive reversals. (Answer: $1,144.40)

What You'll Learn