Diagnosing Premium And Discount From Book Value Patterns

Reading a Bond’s Book-Value Pattern

A sequence of bond book values is more than a record of how an investment is carried from one coupon date to the next. With the coupon amount and original effective yield, each value can be checked against the next. The direction of a valid sequence also helps identify whether the bond was issued at a premium or discount to its redemption value.

The key is to keep the quantities on the same time scale. If coupons are quarterly, use the effective yield for one quarter and the coupon paid each quarter. Treat each book value as measured immediately after a coupon has been paid, unless the problem states a different timing convention.

Definition: For a bond with redemption value $R$, level coupon $C$ per period, and effective yield $j$ per coupon period, let $B_k$ be the book value immediately after the $k$th coupon. Under the standard amortized-value method, the recurrence is $$ B_{k+1}=B_k(1+j)-C. $$ It assumes the same yield $j$ is used throughout, the coupon is paid at the end of each period, and there are no additional cash flows or transactions between the stated book values.

The interest earned on the opening book value for one period is $jB_k$. The coupon paid is $C$. The book-value change is therefore $$ B_{k+1}-B_k=jB_k-C. $$ If interest earned exceeds the coupon, book value increases; if it is less than the coupon, book value decreases. This comparison checks the change from one period to the next, but classifying a bond as a premium or discount bond also requires comparing its book value with $R$.

What Premium and Discount Patterns Mean

For a standard level-coupon bond with $m$ coupons remaining, its book value immediately after a coupon is the present value, at the original yield, of the remaining coupons and redemption payment: $$ B_k=C\sum_{r=1}^{m}(1+j)^{-r}+R(1+j)^{-m}. $$ The coupon rate per period, measured against redemption value, is $C/R$. The difference between book value and redemption value can be written as $$ B_k-R=(C-jR)\sum_{r=1}^{m}(1+j)^{-r}. $$ This follows by subtracting $R$ from the present-value expression and using $1-(1+j)^{-m}=j\sum_{r=1}^{m}(1+j)^{-r}$. The sum is positive, so the sign of $B_k-R$ is the sign of $C-jR$.

This result explains the usual patterns for a consistent bond schedule. When $C>jR$, the bond is at a premium: $B_k>R$, and its book value declines toward $R$ as coupons are paid. When $C<jR$, it is at a discount: $B_k<R$, and book value rises toward $R$. When $C=jR$, it is at par: book value equals redemption value throughout the schedule. These conclusions concern a schedule based on the bond’s original yield and specified cash flows; a later market yield does not replace that yield in the book-value recurrence.

Key check: First calculate the interest $jB_k$ and compare it with $C$. Then compare $B_k$ with $R$ and, for a standard consistent schedule, check that the direction of change agrees with premium, discount, or par status. A trend alone cannot validate every reported entry.

Worked Examples

Worked Example: Confirming Premium Amortization

A bond has redemption value $\$1{,}000$, pays a coupon of $\$45$ each period, and has an effective yield of $4\%$ per coupon period. Its book value immediately after a coupon is $\$1{,}040$. Determine its status, calculate the next two book values, and check whether a report of $\$1{,}033.064$ after the second period is consistent.

Method. Use the recurrence with $j=0.04$ and $C=45$, matching the yield and coupon to one period. The coupon rate against redemption value is $45/1{,}000=0.045=4.5\%$, which exceeds the yield of $4\%$. Thus this is a premium bond, and a consistent schedule should decline toward $\$1{,}000$.

For the first period, interest on the opening book value is $$ 1{,}040(0.04)=41.60. $$ The change is $41.60-45=-3.40$, so $$ B_1=1{,}040+41.60-45=1{,}036.60. $$ For the second period, interest is $$ 1{,}036.60(0.04)=41.464. $$ Therefore, $$ B_2=1{,}036.60+41.464-45=1{,}033.064. $$ The reported second-period value agrees exactly with the recurrence. It is below $B_1$, as premium amortization requires, and remains above redemption value. The common mistake to avoid is subtracting the coupon without first calculating interest on the opening book value.

Worked Example: Confirming Discount Accretion

A bond has redemption value $\$2{,}000$, a coupon of $\$70$ per period, and an effective yield of $4\%$ per period. Its opening book value is $\$1{,}900$. A record gives the next book value as $\$1{,}906$. Check the entry and calculate the following book value.

Method. The coupon rate is $70/2{,}000=3.5\%$, less than the $4\%$ yield. The bond is at a discount, so a valid book-value sequence should rise toward $\$2{,}000$.

Interest for the first period is $$ 1{,}900(0.04)=76. $$ The book-value change is $76-70=6$, giving $$ B_1=1{,}900+76-70=1{,}906. $$ The recorded entry is consistent. For the next period, interest is $$ 1{,}906(0.04)=76.24, $$ so $$ B_2=1{,}906+76.24-70=1{,}912.24. $$ Both computed values are below redemption value and increase from one period to the next. The common mistake is to call any increasing book value a discount without checking the stated cash flows and redemption value; here both the rate comparison and the recurrence support that classification.

Worked Example: Finding an Inconsistent Entry

A bond has a coupon of $\$30$ per period and an effective yield of $4\%$ per period. A schedule lists consecutive book values of $\$980$, $\$989.20$, and $\$998.00$, each measured immediately after a coupon. Check both transitions and identify any inconsistent entry.

Method. Apply the recurrence to each opening value. For the first transition, interest is $$ 980(0.04)=39.20. $$ After paying the $\$30$ coupon, the expected book value is $$ 980+39.20-30=989.20. $$ The first transition is consistent.

For the second transition, interest on $\$989.20$ is $$ 989.20(0.04)=39.568. $$ The expected next value is $$ 989.20+39.568-30=998.768. $$ The reported value, $\$998.00$, is $\$0.768$ below this result, so the second transition is inconsistent with the stated yield and coupon.

The same issue can be checked by inferring the yield implied by each pair. From the first transition, $$ \frac{989.20+30}{980}-1=0.04=4\%. $$ From the reported second transition, $$ \frac{998.00+30}{989.20}-1 =\frac{1{,}028}{989.20}-1 \approx 0.039224=3.9224\%. $$ The two implied yields differ. The common mistake is to check only whether the sequence rises; an increasing sequence can still contain an entry that fails the recurrence.

A Systematic Consistency Check

For each reported transition, the book value and coupon determine the yield that would make that transition work. Rearranging the recurrence gives $$ j=\frac{B_{k+1}+C}{B_k}-1. $$ This is the implied effective yield for that period, provided $B_k$ is nonzero. If a problem specifies a yield, compare each implied value with it. If the yield is not specified but the same yield is supposed to apply throughout, compare the implied yields across all transitions.

Set the period.
Identify the coupon interval and express the effective yield for exactly that interval. Do not mix an annual rate with a quarterly or semiannual coupon.

Calculate the expected next value.
For each transition, compute $B_k(1+j)-C$, using the opening value for that period.

Compare with the record.
State whether the reported value agrees with the expected value. If it does not, give the discrepancy and its direction.

Interpret the pattern.
Compare book value with redemption value and compare $C/R$ with $j$. Check whether the observed direction fits the stated bond terms.

Common Traps

Using the wrong rate period. A nominal annual quote convertible $m$ times per year must first be converted to the effective rate per coupon period. For example, a nominal rate $i^{(2)}$ convertible semiannually corresponds to a periodic rate $i^{(2)}/2$, not the full nominal annual quote.
Reversing the cash-flow order. Under the stated convention, interest is earned on the opening book value and the coupon is then paid: $B_{k+1}=B_k(1+j)-C$. Adding the coupon to book value reverses the investor’s cash-flow perspective.
Treating a trend as proof of consistency. A rising sequence may suggest discount accretion, but every consecutive pair must still satisfy the recurrence.
Confusing market yield with the carrying yield. The recurrence uses the yield underlying the book-value schedule. A change in market yield does not by itself recalculate previously established book values.
Classifying from the first change alone. The sign of $jB_k-C$ gives that period’s change. Use redemption value and the specified bond terms to identify premium or discount status.

Key takeaway: A reliable diagnosis combines three checks: the period-matched recurrence, the relation of book value to redemption value, and the comparison of coupon rate $C/R$ with effective yield $j$. If reported values fail the recurrence, flag the entry even when the overall pattern looks plausible.

Check Your Understanding

For each problem, use the stated timing convention that book values are immediately after coupons.

A bond has opening book value $\$1{,}500$, coupon $\$35$, and effective yield $2\%$ per period. Its next reported value is $\$1{,}495$. Check the transition and classify the pattern relative to redemption value $\$1{,}400$. (Answer: Interest is $\$30$; the change is $-\$5$, so $\$1{,}495$ is consistent. The bond is at a premium and the book value declines.)
A bond has opening book value $\$2{,}500$, coupon $\$28$, and effective yield $1.2\%$ per period. A reported next value is $\$2{,}501$. Is it consistent? (Answer: No. Interest is $\$30$, so the expected value is $\$2{,}502$; the reported value is $\$1$ too low.)
A schedule gives $B_0=\$1{,}200$, $B_1=\$1{,}186$, and $B_2=\$1{,}171.18$, with a coupon of $\$50$ each period. Find the yield implied by each transition and check whether one constant yield fits. (Answer: The first transition implies $3\%$. The second implies $1{,}221.18/1{,}186-1\approx2.9663\%$, so the entries do not fit one constant yield.)
A bond pays $\$30$ every six months. Its opening book value is $\$700$, and its nominal annual yield is $8\%$ convertible semiannually. Find the next book value and state the direction of change. (Answer: The periodic yield is $4\%$; interest is $\$28$, so the next book value is $\$698$, a decrease of $\$2$.)
A bond has redemption value $\$1{,}000$, coupon $\$40$ per period, effective yield $3.5\%$ per period, and opening book value $\$1{,}020$. Find the next book value and classify the pattern. (Answer: Interest is $\$35.70$; the next book value is $\$1{,}015.70$. Since $C/R=4\%>3.5\%$, it is a premium bond and the value declines toward redemption.)

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