Retrospective Book Value From Original Price

Why Calculate Book Value Retrospectively?

A bond’s book value at a coupon date can be calculated from the cash flows still to come. It can also be calculated from the other end of the transaction: start with the original purchase price, accumulate it at the yield used to price the bond, and subtract the accumulated value of coupons already paid. This second method is called a retrospective valuation.

The method is useful when the original price and yield are known, and the question asks for the value after one or more coupons have been paid. The key is to keep the timing consistent: at a coupon date, this tutorial defines book value as the value immediately after that date’s coupon has been paid. Thus, the coupon paid on the valuation date is included among the payments subtracted.

Definition: The retrospective book value at a coupon date is the original purchase price accumulated to that date at the bond’s original yield, less the accumulated value at that date of all coupons paid on or before it. It is an accounting value at the original yield, not necessarily the bond’s current market price.

The Retrospective Formula

Suppose a bond is purchased at time 0 for price $P_0$. It pays a level coupon of $C$ at the end of each period, and the effective yield per coupon period at purchase is $i$. For a valuation immediately after coupon $k$, assume that $k$ full coupon periods have elapsed, the first $k$ coupons have been paid, and the bond’s cash flows and original price are consistent with yield $i$. The retrospective book value is

B_k=P_0(1+i)^k-C\left[(1+i)^{k-1}+(1+i)^{k-2}+\cdots+(1+i)+1\right]

The bracketed expression is the accumulated value at time $k$ of one unit paid at the end of each of the first $k$ periods. In standard actuarial notation, it is $s_{\overline{k}|}$ at rate $i$. Therefore,

B_k=P_0(1+i)^k-C s_{\overline{k}|}

Here, $P_0(1+i)^k$ is the original price accumulated for $k$ periods. A coupon paid at time $r$, where $1\le r\le k$, accumulates for $k-r$ periods, giving it value $C(1+i)^{k-r}$ at time $k$. Summing these accumulated coupons produces $C s_{\overline{k}|}$. Subtracting them leaves the value carried forward for the bond’s remaining obligation.

Important condition: Use the yield per coupon period. If coupons are semiannual and the stated yield is a nominal annual rate convertible semiannually, convert it to the corresponding half-year rate before using the formula. The formula also depends on the valuation being immediately after the $k$th coupon; a value immediately before that coupon has a different coupon-payment count.

Worked Retrospective Valuations

Worked Example: A Discount Bond After One and Two Coupons

A two-year bond with redemption value $1,000 pays an annual coupon of $100. It was purchased for $784, and its original effective annual yield is 25%. Find its book value immediately after the first coupon and immediately after the second coupon. Confirm that the given price is consistent with the stated yield.

Method. Use the retrospective formula, taking one year as one coupon period. First confirm the original price by discounting the bond’s two payments:

\frac{100}{1.25}+\frac{1,100}{1.25^2} =80+704 =784

The purchase price is consistent with the 25% yield. Immediately after the first coupon, $k=1$, so the accumulated price is $784(1.25)=980$, and the first $100 coupon is subtracted:

B_1=784(1.25)-100=980-100=\$880

Immediately after the second coupon, the original price has accumulated for two years. The first coupon has accumulated for one year, while the second coupon has just been paid and has accumulated for zero years:

\begin{aligned} B_2 &=784(1.25)^2-100(1.25)-100\\ &=1,225-125-100\\ &=\$1,000 \end{aligned}

The result at year 2 equals the $1,000 redemption value, as expected immediately after the final coupon and just before redemption. The calculation avoids treating both coupons as if they had accumulated for the same length of time.

Worked Example: A Premium Bond’s Book Value

A two-year bond has a $1,000 redemption value, pays an annual coupon of $150, and has an original effective annual yield of 10%. Find its original price and its book value immediately after the first coupon. Report money to the nearest cent.

Method. Calculate the original price from the bond’s promised cash flows at the original yield. The payments are $150 at year 1 and $1,150 at year 2:

\begin{aligned} P_0 &=\frac{150}{1.10}+\frac{1,150}{1.10^2}\\ &=\frac{1,500}{11}+\frac{115,000}{121}\\ &=\frac{131,500}{121}\\ &=1,086.7768595\ldots \end{aligned}

The bond’s original price is therefore $1,086.78 to the nearest cent. For the value immediately after the first coupon, accumulate this unrounded price for one year, then subtract the $150 coupon just paid:

\begin{aligned} B_1 &=P_0(1.10)-150\\ &=\frac{131,500}{121}(1.10)-150\\ &=\frac{13,150}{11}-150\\ &=1,045.454545\ldots\\ &=\$1,045.45 \end{aligned}

The original price exceeds the $1,000 redemption value because the coupon rate is higher than the yield. After one coupon, the book value remains above redemption, but it has moved closer to it. The calculation uses the original yield, not a new rate inferred from market conditions at year 1.

Worked Example: Three Annual Coupons and a Maturity Check

A three-year bond pays an annual coupon of $100 and redeems for $1,000. Its original effective annual yield is 20%. Find its original price and its retrospective book value immediately after the second coupon. Then check the book value immediately after the third coupon. Report money to the nearest cent.

Method. First calculate the original price by discounting each payment at the stated yield. The year-3 payment includes both the $100 coupon and the $1,000 redemption:

\begin{aligned} P_0 &=\frac{100}{1.20}+\frac{100}{1.20^2}+\frac{1,100}{1.20^3}\\ &=\frac{250}{3}+\frac{625}{9}+\frac{137,500}{216}\\ &=\frac{213,125}{270}\\ &=789.3518518\ldots\\ &=\$789.35 \end{aligned}

For the value immediately after coupon 2, the original price accumulates for two periods. Coupon 1 accumulates for one period; coupon 2 has just been paid:

\begin{aligned} B_2 &=P_0(1.20)^2-100(1.20)-100\\ &=\frac{213,125}{270}(1.44)-120-100\\ &=1,136.6666667\ldots-220\\ &=916.6666667\ldots\\ &=\$916.67 \end{aligned}

For the maturity check immediately after coupon 3, all three coupons are subtracted, each accumulated to year 3. Their accumulated value is $100(1.20^2+1.20+1)=364$. The original price accumulated to year 3 is:

\begin{aligned} P_0(1.20)^3 &=\frac{213,125}{270}(1.728)\\ &=1,364 \end{aligned}

It follows that $B_3=1,364-364=\$1,000$, the redemption value. This check confirms both the coupon timing and the accumulated-price calculation.

A Reliable Calculation Sequence

For a bond with equal coupons and a coupon-date valuation, the following sequence keeps the time periods and payment count explicit.

Set the time unit.
Identify one coupon period and express the original yield as an effective rate per period.

Count coupons paid.
At the instant immediately after coupon $k$, exactly $k$ coupons have been paid.

Accumulate the original price.
Calculate $P_0(1+i)^k$, using the original purchase yield $i$.

Accumulate the coupons.
Use $C s_{\overline{k}|}$, or add the individual coupon accumulation terms if that makes the timing clearer.

Subtract and check.
Subtract accumulated coupons from accumulated price. If $k$ is the final coupon date, the result should equal the redemption value.

Common Mistakes and Exam Traps

Subtracting coupons without accumulating them. For the first example at time 2, the incorrect calculation $784(1.25)^2-2(100)=1,025$ treats the first coupon as if it had no time to earn interest. The correct coupon accumulation is $100(1.25)+100=225$, so $B_2=1,225-225=\$1,000$.
Omitting the coupon paid on the valuation date. Immediately after coupon $k$, that coupon has already been paid, even though it accumulates for zero periods. Its term is $C(1+i)^0=C$.
Using the wrong rate period. With semiannual coupons, $k$ counts half-year periods, not years. Use the yield per half-year period and count each coupon date as one period.
Replacing the original yield with a current market yield. Retrospective book value carries the original price and coupons at the original yield. A change in current market yield does not alter this calculation.
Rounding too early. Carry the original price and accumulated amounts at full available precision, then round the final dollar value. Early rounding can create a small discrepancy, especially after several periods.

Key takeaway: At a coupon date immediately after payment, retrospective book value equals the original price accumulated at the original yield, less the accumulated value of every coupon paid through that date. The coupon on the valuation date is included with zero accumulation time.

Check Your Understanding

For each problem, assume book value is measured immediately after the stated coupon payment. Use the yield per coupon period and report dollar values to the nearest cent.

A bond was purchased for $920 and has an effective annual yield of 8%. It pays an annual coupon of $40. Find its book value immediately after the second coupon. (Answer: $989.89)
A bond was purchased for $1,025 and has a yield of 3% per half-year period. It pays a $24 coupon every half-year. Find its book value immediately after coupon 3. (Answer: $1,045.86)
A bond was purchased for $1,140 and has an effective annual yield of 6%. It pays an annual coupon of $80. Find its book value immediately after coupon 4. (Answer: $1,089.25)
A bond has an effective annual yield of 4%, pays an annual coupon of $50, and has a book value of $1,000 immediately after coupon 3. Use retrospective valuation to find its original purchase price. (Answer: $1,027.75)
A bond was purchased for $980 and has a yield of 2.5% per half-year period. It pays a $35 coupon every half-year. Find its book value immediately after coupon 4. (Answer: $936.40)

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