Carrying Book Value At The Original Yield

What It Means to Carry Book Value

A bond’s book value at a later date is not automatically the price for which the bond could be sold on that date. For book-value calculations, the relevant yield is the yield established when the bond was acquired. That yield is used to carry the bond’s value through time, even if market yields subsequently change.

This distinction matters whenever a problem supplies both an original yield and a later market yield. The original yield determines the book-value calculation. The market yield may be relevant to a separate market-price calculation, but it does not replace the original yield in the book-value recurrence.

Definition: Consider a bond purchased at a coupon date. Let $B_k$ be its book value immediately after the $k$th coupon, $C$ its coupon paid each period, $R$ its redemption value, and $j$ the yield per coupon period established at acquisition. The book value $B_k$ is the present value, at yield $j$, of the bond’s cash flows remaining after that coupon. Thus, for $k<n$, $$ B_k=C a_{\overline{n-k}|}+R v^{n-k}, \qquad v=\frac{1}{1+j}. $$ Here $n$ is the total number of coupon periods from acquisition to maturity. The yield $j$ must be stated per coupon period.

At maturity, immediately after the final coupon has been paid, no bond cash flows remain and the book value is zero under this after-coupon convention. Immediately before the final coupon and redemption, the value of the cash flow due at that date is $C+R$. Stating whether a value is immediately before or after a coupon is therefore essential.

Why the Coupon-Date Recurrence Works

The prospective definition gives the book value directly by discounting all remaining cash flows. The same value can be carried forward one period at a time. The recurrence is useful for schedules and makes clear which yield governs each period.

Theorem: If $B_{k-1}$ is the book value immediately after coupon $k-1$, then the book value immediately after coupon $k$ is $$ B_k=(1+j)B_{k-1}-C. $$ The coupon $C$ is subtracted because it is paid at date $k$. Use the original yield per coupon period $j$, not a later market yield.

To justify the recurrence, immediately after coupon $k-1$, the remaining cash flows consist of the coupon $C$ at date $k$, followed by the cash flows whose value at date $k$, immediately after that coupon, is $B_k$. Discounting those amounts back one period at $j$ gives $B_{k-1}$. Rearranging yields the recurrence:

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

This relation is also a useful check on a prospective calculation. If the recurrence and the present value of the remaining cash flows do not agree, check the number of remaining periods, the coupon timing, and the yield period.

Worked Book-Value Calculations

Worked Example: Carrying a Discount Bond to Its Second Coupon

A three-year bond with redemption value $1,000 pays annual coupons of $60. It is purchased at a coupon date to yield 8% effective annually. Find its book value immediately after the second coupon. Show the carrying calculation and verify it prospectively.

Method. There is one year remaining after the second coupon. The original yield is 8% per annual coupon period, so the value after that coupon is the present value of the final $60 coupon plus the $1,000 redemption.

First, the acquisition price is the present value of three coupons and the redemption:

B_0=60a_{\overline{3}|}+1000v^3 =60\left(\frac{1-(1.08)^{-3}}{0.08}\right)+1000(1.08)^{-3} =948.46.

Now apply the recurrence using $j=0.08$ and $C=60$:

$$ B_1=948.45806(1.08)-60=964.33470, $$

$$ B_2=964.33470(1.08)-60=981.48148. $$

For the prospective check, immediately after the second coupon only the final coupon and redemption remain, one year later:

B_2=(60+1000)(1.08)^{-1} =\frac{1060}{1.08} =981.48148.

Answer. Rounded to the nearest cent, the book value immediately after the second coupon is $981.48. The recurrence and prospective value agree. The trap is to report $1,000 because that is the redemption value; one year remains, so the final $1,060 payment must still be discounted at the original yield.

Worked Example: A Changed Market Yield and an Unchanged Carrying Yield

A four-year bond has redemption value $1,000, annual coupons of $50, and an original yield of 4% effective annually. It was purchased at a coupon date. Find its book value immediately after the second coupon. If the market yield at that time is 6% effective annually, also calculate the market value of the remaining payments.

Method. First find the acquisition price and carry it forward at the original 4% yield. Then, separately, value the remaining cash flows at the 6% market yield. The two calculations answer different questions.

The acquisition price is:

B_0=50a_{\overline{4}|}+1000v^4 =50\left(\frac{1-(1.04)^{-4}}{0.04}\right)+1000(1.04)^{-4} =1036.30.

Carry the book value forward, subtracting the $50 coupon after each period:

$$ B_1=1036.29895(1.04)-50=1027.75091, $$

$$ B_2=1027.75091(1.04)-50=1018.86095. $$

Check this result by valuing the two remaining coupons and the redemption at the original 4% yield:

B_2=50a_{\overline{2}|}+1000v^2 =50\left(\frac{1-(1.04)^{-2}}{0.04}\right)+1000(1.04)^{-2} =1018.86.

The market value at the 6% yield is instead:

50(1.06)^{-1}+1050(1.06)^{-2} =47.16981+934.49626 =981.67.

Answer. The book value after the second coupon is $1,018.86; the market value at the changed 6% yield is $981.67. The difference is not an inconsistency: the first calculation carries the acquisition yield, while the second prices the remaining payments at the current market yield. Do not insert 6% into the book-value recurrence.

Yield Periods and Coupon Frequency

The rate in the recurrence is a rate per coupon period. For an annual-coupon bond, an effective annual yield is used directly as $j$. For a bond paying coupons twice a year, a nominal annual yield convertible semiannually must first be converted to its semiannual rate: if the original quote is $i^{(2)}$, then $j=i^{(2)}/2$. Each recurrence step then represents one half-year.

Worked Example: Semiannual Carrying at the Original Yield

A three-year bond has redemption value $1,000 and pays $32 every six months. Its original yield is 6% nominal per year convertible semiannually. It was purchased at a coupon date. Find its book value immediately after the fourth coupon. At that date, the market yield is 8% nominal per year convertible semiannually; identify which yield belongs in the book-value calculation.

Method. There are six coupon periods in three years. The original yield per period is $j=0.06/2=0.03$, and the coupon per period is $32. Calculate the acquisition price, then apply the recurrence four times.

The acquisition price is:

B_0=32a_{\overline{6}|}+1000v^6 =32\left(\frac{1-(1.03)^{-6}}{0.03}\right)+1000(1.03)^{-6} =1010.83.

Applying the original 3% per-period yield:

\begin{aligned} B_1&=1010.83438(1.03)-32=1009.15941,\\ B_2&=1009.15941(1.03)-32=1007.43419,\\ B_3&=1007.43419(1.03)-32=1005.65722,\\ B_4&=1005.65722(1.03)-32=1003.82694. \end{aligned}

After four coupons, two coupon periods remain. The prospective check at the original yield is:

B_4=32a_{\overline{2}|}+1000v^2 =32\left(\frac{1-(1.03)^{-2}}{0.03}\right)+1000(1.03)^{-2} =1003.83.

Answer. The book value immediately after the fourth coupon is $1,003.83. The stated market yield of 8% nominal convertible semiannually corresponds to 4% per half-year, but it is not used to carry the book value. Using the market yield here would calculate a market value instead.

Reading a Book-Value Schedule Correctly

A carrying schedule records the movement in book value from one coupon date to the next. For each period, interest at the original yield is applied to the opening book value, and the coupon is then subtracted. Algebraically,

B_k-B_{k-1}=jB_{k-1}-C.

The amount $jB_{k-1}$ is the interest component under this carrying convention. When the coupon exceeds that amount, the book value decreases; when the coupon is smaller, it increases. This period-by-period behavior follows from the recurrence and the original yield, not from the direction in which the market yield moves.

Information in the problem	Use in book-value calculation
Original yield per coupon period	Use as $j$ in the recurrence and in the prospective present value
Coupon paid each period	Subtract $C$ after applying interest for the period
Redemption value and remaining coupons	Include all remaining payments in the prospective check
Later market yield	Do not substitute for the original yield when carrying book value

Fix the time unit.
Count coupon periods and express the original yield per coupon period.

Choose the requested value date.
Confirm whether the value is immediately before or immediately after the coupon.

Carry or value prospectively.
Use $B_k=(1+j)B_{k-1}-C$, or discount all remaining cash flows at $j$.

Check the result.
Compare the recurrence with the prospective value, and keep any market-price calculation separate.

Common Exam Traps

Using the new market yield to carry book value. This changes the question from carrying value at the acquisition yield to market valuation.
Using an annual quote for a semiannual step. Convert the yield to a rate per half-year before applying one recurrence step.
Forgetting the coupon subtraction. The recurrence is $(1+j)B_{k-1}-C$, not $(1+j)(B_{k-1}-C)$; the coupon is paid at the end of the period.
Leaving out a remaining coupon. A prospective check must include every coupon still due as well as the redemption payment.
Confusing redemption value with book value. Before maturity, the remaining payments are valued at the original yield; book value need not equal redemption value.

Key takeaway: Carry book value forward with the yield established at acquisition, using the rate per coupon period. A later market yield belongs to a market-value calculation, not to the book-value recurrence.

The distinction between a carrying value and a market value is a standard part of bond analysis and is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In a multi-step problem, write down the requested value and the relevant yield before calculating; that single check prevents many rate-substitution errors.

Check Your Understanding

For each problem, identify the yield per coupon period before calculating.

A four-year bond has redemption value $2,000, annual coupons of $100, and an original effective annual yield of 4%. Find its book value immediately after the first coupon. (Answer: $2,055.50)
A three-year bond has redemption value $1,000 and annual coupons of $60. It was purchased at a coupon date to yield 8% effective annually. Find the book value immediately after the second coupon. (Answer: $981.48)
A four-year bond has redemption value $1,000 and semiannual coupons of $20. Its original yield is 5% nominal per year convertible semiannually. Find the book value immediately after the third coupon. (Answer: $976.77)
A bond’s book value immediately after a coupon is $1,040. Its original yield is 5% per annual coupon period, and the next annual coupon is $70. The market yield has changed to 3%. Find the book value immediately after the next coupon. (Answer: $1,022.00)
A bond’s book value immediately after its second annual coupon is $1,015. Its original yield is 6% effective annually, and the annual coupon is $40. If the market yield is now 8%, find the book value immediately after the third coupon. (Answer: $1,035.90)

Information in the problem	Use in book-value calculation
Original yield per coupon period	Use as \(j\) in the recurrence and in the prospective present value
Coupon paid each period	Subtract \(C\) after applying interest for the period
Redemption value and remaining coupons	Include all remaining payments in the prospective check
Later market yield	Do not substitute for the original yield when carrying book value

What You'll Learn