Closed-Form Book Value Relative To Redemption Value

Why Compare Book Value With Redemption Value?

A bond’s book value on a coupon date is the value of its remaining promised payments at the yield used to price the bond. Comparing that value with the redemption amount gives a direct measure of the remaining premium or discount. A useful closed-form expression avoids calculating the coupon annuity and the discounted redemption amount separately.

The key comparison is between the coupon paid each period and the interest earned each period on the redemption amount at the yield. Their difference determines both the sign and the size of the book-value difference. This relationship applies at a coupon date under a level-coupon, fixed-yield model; it is not a formula for a market price after the market yield changes.

Definition: Let $R$ be a bond’s redemption amount, $C$ its coupon payment each period, $i$ the effective yield per coupon period, and $n$ the number of coupon periods remaining at a coupon date. The book value $B_n$ is the present value at yield $i$ of the remaining coupons and redemption payment. For $i \ne 0$, the difference from redemption value is $$ B_n-R=(C-iR)a_{\overline{n}|}, $$ where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$.

The rate $i$ must be measured per coupon period, and $C$ must be the coupon amount for that same period. For example, if there is one coupon each year, use the effective yield per year and the annual coupon. If a problem gives a different coupon frequency, first identify the yield per coupon period consistent with the bond’s pricing model. The formula does not itself convert a quoted annual rate into a periodic yield.

The number $n$ counts remaining coupon payments, not the periods that have already elapsed. At a coupon date immediately after a coupon has been paid, the next coupon is one period away. In the model here, the remaining cash flows are $C$ at each of the next $n$ coupon dates and $R$ in addition to the final coupon at the last date.

Deriving the Closed-Form Difference

The prospective-value method writes book value as the present value of the remaining coupons plus the present value of redemption. Subtracting $R$ gives the difference from redemption value. The derivation uses only the standard annuity-immediate and discount-factor formulas.

Write the prospective value.
There are $n$ coupons of $C$, followed by redemption of $R$ at the final coupon date, so $B_n=Ca_{\overline{n}|}+Rv^n$.

Subtract redemption value.
Subtract $R$ from both sides: $B_n-R=Ca_{\overline{n}|}+R(v^n-1)$.

Rewrite the redemption term.
Since $a_{\overline{n}|}=(1-v^n)/i$, it follows that $v^n-1=-ia_{\overline{n}|}$. Therefore $R(v^n-1)=-iRa_{\overline{n}|}$.

Factor the annuity value.
Substitution gives $B_n-R=Ca_{\overline{n}|}-iRa_{\overline{n}|}=(C-iR)a_{\overline{n}|}$, as required.

This derivation also explains why the formula is useful: the entire premium or discount is the present value of a level amount, $C-iR$, paid for each of the remaining periods. The amount $iR$ is the interest for one period on the redemption amount at the bond’s yield. No assumption that the coupon equals interest on the current book value is needed for this derivation.

Sign test: Because $a_{\overline{n}|}$ is positive when $i>0$ and $n>0$, the sign of $B_n-R$ is the sign of $C-iR$. If $C>iR$, book value is above redemption value; if $C<iR$, it is below; and if $C=iR$, it equals redemption value at that coupon date.

Applying the Formula

The main calculation is to find the periodic interest amount $iR$, subtract it from the coupon $C$, and multiply by the annuity factor for the remaining periods. Add the resulting difference to $R$ if the question asks for book value. If the difference is negative, the book value is below redemption value; the negative sign should be retained until the comparison is interpreted.

Worked Example: Finding a Premium From the Remaining Term

A hypothetical bond has a redemption amount of $R=\$1{,}000$, pays an annual coupon of $C=\$60$, and has four annual coupon periods remaining. The effective annual yield is $i=5\%$. Find the book value and its difference from redemption value at the coupon date.

Method. Use the closed-form difference, with the annual coupon and annual yield both measured per year. The common mistake to avoid is using the original term rather than the four periods that remain.

First calculate the discount factor for four periods:

v^4=\frac{1}{(1.05)^4}=0.822702475

Then calculate the annuity factor:

a_{\overline{4}|}=\frac{1-v^4}{0.05} =\frac{1-0.822702475}{0.05} =3.545950504

Interest on redemption value for one period is $iR=0.05(1{,}000)=\$50$. Thus $C-iR=60-50=\$10$, and

B_4-R=(60-50)(3.545950504)=\$35.45950504.

Therefore $B_4=1{,}000+35.45950504=\$1{,}035.45950504$, or $\$1{,}035.46$ to the nearest cent. As a check, the prospective value is $60(3.545950504)+1{,}000(0.822702475)=212.75703024+822.702475=\$1{,}035.45950524$, which agrees to the displayed precision.

Worked Example: Identifying a Discount and Its Sign

A hypothetical bond has $R=\$1{,}000$, an annual coupon of $C=\$80$, an effective annual yield of $i=10\%$, and three annual coupon periods remaining. Find its book value and say whether it is above or below redemption value.

Method. Compare $C$ with $iR$ before calculating the book value. Here $iR=0.10(1{,}000)=\$100$, which exceeds the coupon by $\$20$; the difference must therefore be negative.

The three-period discount factor and annuity factor are

v^3=\frac{1}{(1.10)^3}=0.751314801, \qquad a_{\overline{3}|}=\frac{1-0.751314801}{0.10}=2.486851991.

Now apply the difference formula:

B_3-R=(80-100)(2.486851991) =-\$49.73703982.

Thus $B_3=1{,}000-49.73703982=\$950.26296018$, or $\$950.26$ to the nearest cent. The sign check confirms a discount: the coupon is less than interest on redemption value at the yield. As a separate check, $80(2.486851991)+1{,}000(0.751314801)=198.94815928+751.314801=\$950.26296028$, agreeing to the displayed precision.

Worked Example: Solving for a Coupon From the Difference

A hypothetical bond has a redemption amount of $\$1{,}000$, four annual coupon periods remaining, and an effective annual yield of $5\%$. Its book value at a coupon date is $\$50$ above redemption value. Find the annual coupon amount.

Method. Rearrange $B_n-R=(C-iR)a_{\overline{n}|}$ to solve for $C$. The common mistake here is to treat the full premium as the coupon amount; the premium is instead the present value of the excess of the coupon over $iR$.

Using $a_{\overline{4}|}=3.545950504$ and $iR=0.05(1{,}000)=\$50$, rearrangement gives

C=iR+\frac{B_4-R}{a_{\overline{4}|}} =50+\frac{50}{3.545950504} =50+14.100591 =\$64.100591\text{ (approximately)}.

The coupon is approximately $\$64.10$ to the nearest cent. Check the unrounded result: $C-iR$ is approximately $\$14.100591$, and $14.100591(3.545950504)$ is approximately $\$50.00$, the stated difference. If the coupon is restricted to cents, using $\$64.10$ gives a difference of $14.10(3.545950504)=\$49.997902$, which rounds to $\$50.00$ to the nearest cent.

Assumptions, Limits, and Exam Traps

The expression relies on a specific bond cash-flow and valuation model. It applies when the coupon is level, the redemption amount is paid with the final coupon, the bond is valued at a coupon date, and all remaining cash flows are discounted at the same effective yield per coupon period. The yield is held fixed for this calculation. If the market yield changes, the formula can value the remaining cash flows at that new yield, but that is a new valuation—not a change to a previously established book value under the original yield.

It does not directly apply to a bond with irregular remaining cash flows, a changing coupon, or a valuation between coupon dates. Those cases require a cash-flow model that reflects their timing and amounts. Also, when $i=0$, the expression $a_{\overline{n}|}=(1-v^n)/i$ is undefined as written. In that special case, the prospective value is $B_n=nC+R$, so $B_n-R=nC$.

A frequent exam error is reversing the subtraction and writing $(iR-C)a_{\overline{n}|}$ for $B_n-R$. In the discount example above, that incorrect expression gives $+\$49.74$ and would imply a book value of about $\$1{,}049.74$. The correct subtraction is $C-iR=80-100=-20$, so the book value is below redemption value by about $\$49.74$, not above it.

Another trap is mixing units. If a coupon is paid twice each year, an annual coupon total cannot be inserted as $C$ while using a yield per half-year and a number of half-year periods. The coupon amount, yield period, and remaining-period count must all describe the same payment interval. A quick reasonableness check is to establish the expected sign from $C-iR$ before doing the annuity calculation.

Key takeaway: At a coupon date, calculate $C-iR$ using the coupon and yield per period, then multiply by the annuity factor for the remaining periods. The result is $B_n-R$; its sign identifies a premium or discount, and adding it to $R$ gives book value.

Check Your Understanding

For each problem, assume level coupons, redemption with the final coupon, and a fixed effective yield per coupon period. Round dollar answers to the nearest cent.

A bond has $R=\$2{,}000$, $C=\$120$ annually, $i=5\%$ effective annually, and five annual periods remaining. Find $B_n-R$ and the book value. (Answer: $B_n-R\approx \$86.59$; book value $\approx \$2{,}086.59$.)
A bond has $R=\$800$, $C=\$32$ annually, $i=5\%$ effective annually, and three annual periods remaining. Is it at a premium or discount, and what is its book value? (Answer: Discount; $B_n-R\approx-\$21.79$, so book value $\approx \$778.21$.)
A bond has $R=\$2{,}500$, $C=\$100$ per period, $i=4\%$ per period, and eight periods remaining. Find its book value. (Answer: $C-iR=100-100=0$, so $B_n-R=0$ and book value is $\$2{,}500.00$.)
A bond has $R=\$1{,}500$, $i=4\%$ per period, and five periods remaining. Its book value is $\$75$ above redemption value. Find the coupon per period. (Answer: $C=60+75/a_{\overline{5}|}\approx \$76.85$ per period.)
A bond has $R=\$1{,}000$, $C=\$45$ per period, $i=5\%$ per period, and ten periods remaining. Find its book value and identify the sign of the difference. (Answer: $B_n-R=-5a_{\overline{10}|}\approx-\$38.61$; book value $\approx \$961.39$, a discount.)

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