Book Value Versus Market Price

Why a Bond Has Two Values

A bond can have a carrying book value and a market price at the same time. These values answer different questions. Book value tracks the bond using the yield established when it was purchased or issued. Market price reflects the yield investors require now. If that required yield changes, market price can change even though the bond’s promised coupons and redemption amount have not changed.

This distinction matters whenever a bond is held after its purchase date. Book value is a measure on the original yield basis; market price is the amount implied by current market conditions. Neither should be substituted for the other without stating which question is being answered.

Definition: For a conventional bond valued on a coupon date, its book value is the value of its remaining contractual cash flows calculated using the bond’s original yield. Its market price is the value of those same remaining cash flows calculated using the yield currently required by investors. Both values use the same coupon and redemption payments; the yield basis is what differs.

The examples here assume annual coupon payments and annual effective yields, valuation on a coupon date, and no default. Prices are quoted without accrued interest. These assumptions keep the comparison focused: between coupon dates, accrued interest and clean-versus-dirty price conventions also have to be handled consistently.

The Two Yield Bases

Suppose a bond has $n$ remaining annual periods. Let $C_k$ be the payment in period $k$, including the redemption payment in the final period. Let $i_0$ be the original annual yield and $i_m$ the current market yield. Under the assumptions above, the values are

\text{Book value}=\sum_{k=1}^{n}\frac{C_k}{(1+i_0)^k}, \qquad \text{Market price}=\sum_{k=1}^{n}\frac{C_k}{(1+i_m)^k}.

The original yield $i_0$ is not necessarily the coupon rate. It is the yield that matched the bond’s purchase price to its promised cash flows when the bond was acquired. The current yield $i_m$ is the rate investors require at the later valuation date.

For a bond with redemption amount $R$, level annual coupon $K$, and $n$ remaining periods, the cash flows are $K$ in each of the first $n-1$ periods and $K+R$ in the final period. Substituting those payments gives

\text{Value at yield }j = K a_{\overline{n}|\,j}+Rv^n, \qquad v=\frac{1}{1+j}.

Here the subscript $j$ indicates that the annuity is valued at annual effective rate $j$. Using $j=i_0$ gives the book value; using $j=i_m$ gives the market price. This is a valuation comparison, not a claim that the original yield must remain the rate available to investors.

Worked Example: A Premium Bond When Market Yields Rise

A hypothetical bond has redemption value $1,000, annual coupons of $80, and four years to maturity when issued. Its original annual effective yield is 6%. After two coupons have been paid, investors require a 9% annual effective yield for comparable bonds. Find the book value and market price at that coupon date, and compare them.

Method. There are two remaining payments of $80, followed by $1,080 in the second year. Discount both payments at the original yield for book value and at the current yield for market price.

Book value at 6%. The first remaining coupon has present value $80/1.06=75.4717$. The final payment has present value $1{,}080/1.06^2=1{,}080/1.1236=961.1962$. Therefore,

\text{Book value}=75.4717+961.1962=\$1{,}036.67.

Market price at 9%. The corresponding present values are $80/1.09=73.3945$ and $1{,}080/1.09^2=1{,}080/1.1881=909.0144$. Thus,

\text{Market price}=73.3945+909.0144=\$982.41.

The market price is $\$1{,}036.67-\$982.41=\$54.26$ below book value. The book value is above the $1,000 redemption amount, consistent with the coupon rate exceeding the original yield. The market price is below redemption because the current required yield exceeds the 8% coupon rate. Check: both calculations discount the same two payments, and the larger discount rate produces the smaller price.

Why the Values Move in Opposite Directions

The comparison follows directly from the cash-flow valuation formula. For any positive payment $C_k$ due in $k$ periods, its present value at yield $j$ is $C_k/(1+j)^k$. If $j$ increases, the denominator increases, so that payment’s present value decreases. Adding the present values of all positive bond payments preserves this direction.

In particular, compare the market yield $i_m$ with the original yield $i_0$. If $i_m>i_0$, every remaining payment is discounted more heavily in the market-price calculation than in the book-value calculation, so market price is below book value. If $i_m<i_0$, market price is above book value. If the yields are equal, the two values are equal.

Result: For a conventional bond with positive remaining cash flows valued on the same date and with the same price convention:

If the current market yield is greater than the original yield, market price is less than book value.
If the current market yield is less than the original yield, market price is greater than book value.
If the yields are equal, market price equals book value.

This result concerns market price relative to book value. It does not, by itself, determine whether either value is above or below redemption. The coupon rate and the relevant yield help determine that separate comparison.

Worked Example: A Discount Bond Can Rise Without Reaching Par

A hypothetical four-year bond has redemption value $1,000 and annual coupons of $40. Its original annual effective yield is 8%. After two years, two annual payments remain. At that date, the current required yield has fallen to 5%. Find book value and market price, then explain how market price compares with both book value and redemption value.

Method. The remaining payments are $40 in one year and $1,040 in two years. Use 8% for book value and 5% for market price.

Book value at 8%. The first payment is worth $40/1.08=37.0370$, and the second is worth $1{,}040/1.08^2=1{,}040/1.1664=891.6324$. Therefore,

\text{Book value}=37.0370+891.6324=\$928.67.

Market price at 5%. The two present values are $40/1.05=38.0952$ and $1{,}040/1.05^2=1{,}040/1.1025=943.3107$. Therefore,

\text{Market price}=38.0952+943.3107=\$981.41.

Market price is $\$981.41-\$928.67=\$52.74$ above book value because the current yield is below the original yield. It is still below the $1,000 redemption amount because the current 5% yield remains above the 4% coupon rate. Check: the price rises relative to book value when the yield falls, but that does not require the bond to trade above redemption.

Equal Yields and a Changed Market Yield

The two values coincide when the yield basis has not changed. This equality is useful as a check: if the cash flows, valuation date, and price convention are identical, using the same yield in both calculations must give the same result. A later change in market yield changes market price, not the original yield basis used to state book value.

Worked Example: Book Value Stays on the Original Basis

A hypothetical three-year bond has redemption value $1,000 and annual coupons of $60. Its original annual effective yield is 8%. At the date when two coupons have been paid, one year remains. Calculate the book value and market price if the current yield is first 8%, then 4%.

Method. The sole remaining payment is $1,060. With one year remaining, its value is the payment divided by $1+j$.

Book value. Use the original yield of 8%:

\text{Book value}=\frac{1{,}060}{1.08}=\$981.48.

Market price at a current yield of 8%. Since the current and original yields match,

\text{Market price}=\frac{1{,}060}{1.08}=\$981.48.

Market price at a current yield of 4%. The book value continues to use 8%, while the market price uses 4%:

\text{Market price}=\frac{1{,}060}{1.04}=\$1{,}019.23.

Thus book value remains $\$981.48$ on the original yield basis, while the lower current yield raises market price to $\$1{,}019.23$. Check: at equal yields the values agree; at a lower market yield the market price is higher.

Common Exam Trap: Using the Current Yield for Book Value

A common error is to calculate both values using the current yield and then label one result “book value.” That erases the defining distinction. Book value uses the original yield even when market conditions have changed; market price uses the current required yield.

For example, in the first worked example, discounting the remaining payments at 9% gives $\$982.41$. Calling that the book value is wrong: 9% is the current market yield, not the original 6% yield. The corrected book value discounts the same payments at 6%, giving $\$1{,}036.67$, while $\$982.41$ remains the market price.

Another trap is comparing prices based on inconsistent dates or conventions. A clean price excludes accrued interest, while a dirty price includes it. Do not compare one with the other as if the difference were caused only by a yield change. The examples use coupon dates so accrued interest does not obscure the yield-basis comparison.

Key takeaway: Identify the question before selecting a yield. “What is the carrying value on the original basis?” calls for the original yield. “What price reflects investors’ required return now?” calls for the current market yield.

Check Your Understanding

For each problem, use annual effective yields and value the bond on a coupon date. Round dollar answers to the nearest cent.

A bond has one remaining payment of $1,070. Its original yield is 5%, and the current market yield is also 5%. Find book value and market price. (Answer: Book value = market price = $1,019.05.)
A bond with $1,000 redemption value pays annual coupons of $40. Two payments remain. The original yield is 6% and the current yield is 8%. Find both values. (Answer: Book value = $963.33; market price = $928.67.)
A bond with $1,000 redemption value pays annual coupons of $90, and two payments remain. The original yield is 6%, and the current yield is 7%. Find both values and state which is larger. (Answer: Book value = $1,055.00; market price = $1,036.16; book value is larger.)
A bond has one remaining payment of $1,030. Its original yield is 4%, and its current yield is 3%. Find book value and market price, and compare them. (Answer: Book value = $990.38; market price = $1,000.00; market price is larger.)
A bond has two remaining annual payments: $50 in one year and $1,050 in two years. Its original yield is 8%, and its current yield is 10%. Find the two values and explain their ordering. (Answer: Book value = $946.50; market price = $913.22; market price is lower because the current yield is higher.)

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