Book Value Versus Market Price After a Yield Change

Why Two Values Can Apply to the Same Bond

A bond’s book value and market price answer different questions. Book value tracks the bond using the yield established at its original purchase or valuation. Market price asks what the remaining promised cash flows are worth at the yield currently required by investors. If that required yield changes, the two values can differ even when the bond’s coupon and redemption amount have not changed.

To compare them cleanly, this tutorial values bonds immediately after a coupon payment. There is then no accrued coupon to include, and the remaining cash flows are the same for both calculations. Between coupon dates, the same distinction applies, but the valuation date and whether a quoted price is clean or dirty must be handled consistently.

Definition: At a coupon date, the book value is the present value of the bond’s remaining contractual cash flows discounted at its original yield per coupon period. The market price is the present value of those same remaining cash flows discounted at the current market yield per coupon period. These values agree if the yields are equal, but need not agree after a yield change.

Assume a noncallable bond with fixed, positive coupon payments and a fixed redemption amount. Let $C$ be each coupon, $R$ the redemption amount, and $n$ the number of coupon periods remaining just after a coupon date. Let $i$ be the original yield per coupon period and $j$ the new market yield per coupon period. Under these assumptions, the two values are:

B_n = C a_{\overline{n}|\,i} + Rv_i^n \qquad\text{and}\qquad M_n = C a_{\overline{n}|\,j} + Rv_j^n, \quad\text{where }v_y=\frac{1}{1+y}.

Here $B_n$ denotes book value and $M_n$ market price. Equivalently, each value is the sum of each remaining cash flow discounted from its payment date to the valuation date. If coupon amounts vary, discount each coupon separately rather than using the level-coupon annuity expression.

Why a Yield Change Separates the Values

The distinction follows directly from present value. A future payment $X_t$, due in $t$ coupon periods, has value $X_t/(1+y)^t$ at yield $y$. The book value uses $y=i$, while the market price uses $y=j$. Subtracting the two values gives the following comparison:

M_n-B_n = \sum_{t=1}^{n} X_t \left[ \frac{1}{(1+j)^t}-\frac{1}{(1+i)^t} \right].

For a bond with positive remaining cash flows, if $j>i$, then $(1+j)^t>(1+i)^t$ for every $t>0$. Therefore each bracketed term is negative and $M_n<B_n$. A rise in the required yield lowers market price relative to book value. If $j<i$, each bracketed term is positive and $M_n>B_n$. This sign comparison relies on positive future cash flows; it does not automatically apply to cash-flow streams containing negative amounts.

Book value is not reset just because the market yield changes. It continues to reflect the original yield under the stated carrying-value convention. Market price, by contrast, responds to the new yield. For a level-coupon bond, the original-yield book value also has a useful recurrence: after a coupon is paid, the next book value equals the prior book value accumulated for one period at the original yield, less the coupon just paid.

B_{\text{next}}=B_{\text{current}}(1+i)-C.

To see why, immediately after the current coupon date, the next coupon and the value just after that next coupon have present value $C/(1+i)+B_{\text{next}}/(1+i)$. This must equal $B_{\text{current}}$. Multiplying by $1+i$ and rearranging gives the recurrence. It is a way to check the original-yield carrying schedule, not a way to calculate the market price at a changed yield.

Worked Comparisons at a Coupon Date

Worked Example: Annual Bond Repriced After a Yield Decrease

A bond has face value $R=\$1{,}000$, annual coupons of $\$80$, and four years to maturity when purchased. Its original annual effective yield is 10%. Immediately after the second coupon is paid, the market yield for the remaining cash flows becomes 5% per year. Find the book value and market price on that date, and verify that the book value is consistent with the original purchase price.

Method. There are two annual coupons remaining. Discount those coupons and the redemption amount at the original yield for book value, and at the new yield for market price. The date is immediately after a coupon, so no coupon is accrued.

Original purchase price. At 10%, four coupons and the redemption amount were valued as follows:

P_0 =80a_{\overline{4}|\,0.10}+1000(1.10)^{-4} =80(3.169865)+1000(0.683013) =\$936.60.

The book value two years later can be found prospectively from the two remaining cash flows:

B_2 =\frac{80}{1.10}+\frac{1080}{(1.10)^2} =72.727273+892.561983 =\$965.29.

As a check, rolling the initial price forward for two years at the original yield and subtracting the two coupons gives $936.602691(1.10)^2-80[(1.10)+1]=965.289256$, or $\$965.29$ to the nearest cent.

Market price at 5%. Use the same remaining cash flows, but discount them at 5%:

M_2 =\frac{80}{1.05}+\frac{1080}{(1.05)^2} =76.190476+979.591837 =\$1{,}055.78.

The market price is $\$90.49$ above book value. This is consistent with the yield decrease: the bond’s remaining positive payments are discounted less heavily at 5% than at the original 10%.

Common mistake avoided: Using 10% for both values would give the book value twice, not the market price after the yield change. The market calculation must use 5%, while the carrying-value calculation must retain 10%.

Worked Example: Convert Nominal Yields Before Valuing

A bond has face value $\$1{,}000$, pays $\$25$ every six months, and has two semiannual periods remaining immediately after a coupon date. Its original yield is $5\%$ nominal convertible semiannually. The current market yield is $4\%$ nominal convertible semiannually. Find its book value and market price.

Method. The cash flows occur every six months, so convert both nominal annual yields to rates per half-year before discounting. For a nominal rate convertible twice per year, the period rate is the nominal rate divided by 2.

The original yield per period is $0.05/2=0.025$, or 2.5%. The current market yield per period is $0.04/2=0.020$, or 2%. The remaining payments are $\$25$ in six months and $\$1{,}025$ in one year.

B_2 =\frac{25}{1.025}+\frac{1025}{(1.025)^2} =24.390244+975.609756 =\$1{,}000.00.

The book value is exactly par because each remaining coupon is 2.5% of the face amount per half-year, matching the original yield per half-year. At the new 2% period yield:

M_2 =\frac{25}{1.02}+\frac{1025}{(1.02)^2} =24.509804+985.197999 =\$1{,}009.71.

The market price is $\$9.71$ above book value because the market yield per period has fallen from 2.5% to 2%. Both calculations use half-year cash-flow periods; using an annual rate directly in the half-year exponents would mix time units.

Common mistake avoided: Do not discount semiannual payments at 5% and 4% per half-year. Those are nominal annual rates; the correct period rates are 2.5% and 2%.

Worked Example: A Premium Bond After a Yield Increase

A bond has face value $\$1{,}000$, annual coupons of $\$120$, and five years to maturity at issue. Its original annual effective yield is 8%. Immediately after the second coupon, the market yield rises to 10% per year. Calculate the book value and market price, then explain why the market price remains above face value.

Method. Three annual coupons remain, followed by the $\$1{,}000$ redemption payment. Use 8% for book value and 10% for market price.

B_3 =120a_{\overline{3}|\,0.08}+\frac{1000}{(1.08)^3} =120(2.577097)+793.832241 =\$1{,}103.08.

At the new market yield:

M_3 =120a_{\overline{3}|\,0.10}+\frac{1000}{(1.10)^3} =120(2.486852)+751.314801 =\$1{,}049.74.

The market price is $\$53.35$ below book value, as expected when the yield rises. It is still above the $\$1{,}000$ face value because the 12% annual coupon rate exceeds the new 10% market yield: the remaining coupons are relatively large compared with the return now required. A yield increase lowers the price here, but does not necessarily push it below face value.

Common mistake avoided: Concluding that every yield increase makes a bond sell below face value confuses the direction of a price change with whether the resulting price is at a premium or discount. The premium or discount depends on the coupon relative to the yield as well as the remaining term.

Checks and Limits of the Comparison

Before finalizing a two-value calculation, check the valuation date, remaining payments, rate units, and discount rate used for each value. A compact audit is to write both cash-flow expressions side by side before substituting numbers.

Fix the date.
State whether valuation is immediately before or after a coupon. Here the formulas use a date immediately after a coupon payment.

List the remaining cash flows.
Include each remaining coupon and the redemption amount at its payment date. Do not include a coupon that has already been paid.

Use the correct yield for each value.
Book value uses the original yield; market price uses the new yield. Convert nominal rates to the payment-period rate first.

Check direction and units.
For positive remaining cash flows, a higher market yield should produce a lower market price relative to book value. Confirm that the answer is in the stated currency and rounded only at the end.

This comparison assumes that the promised cash flows are unchanged and that the stated yields are appropriate for discounting them. If default risk, liquidity, taxes, or expected cash flows also change, a single yield change may not explain the full difference in observed market price. For a valuation between coupon dates, accrued interest and the price quotation convention must also be treated consistently; comparing a clean quoted price directly with a dirty carrying value would not be an apples-to-apples comparison.

Key takeaway: At one valuation date, book value and market price use the same remaining bond cash flows but different yields. Keep the original yield for book value, use the changed yield for market price, and compare like dates and price conventions.

Check Your Understanding

For each question, calculate book value using the original yield and market price using the changed yield, both immediately after the stated coupon date. Round dollar amounts to the nearest cent.

A $\$1{,}000$ bond pays annual coupons of $\$60$. One year remains, its original annual effective yield is 5%, and the new market yield is 8%. Find both values. (Answer: Book value $\$1{,}009.52$; market price $\$981.48$.)
A $\$2{,}000$ bond pays $\$60$ every six months and has two semiannual periods remaining. Its original yield is $8\%$ nominal convertible semiannually; the new yield is $6\%$ nominal convertible semiannually. Find both values. (Answer: Book value $\$1{,}962.28$; market price $\$2{,}000.00$.)
A $\$1{,}000$ bond pays annual coupons of $\$90$, with two annual periods remaining. Its original annual effective yield is 6%, and the new market yield is 8%. Find both values and state which is larger. (Answer: Book value $\$1{,}055.00$; market price $\$1{,}017.83$; book value is larger.)
A bond has face value $\$1{,}000$, annual coupons of $\$50$, and one year remaining. Its original annual effective yield is 5%; the new market yield is 4%. Find both values and the market price’s percentage premium over book value. (Answer: Book value $\$1{,}000.00$; market price $\$1{,}009.62$; premium $0.96\%$.)

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