Why Construct a Book Value Table?
A bond’s price at the yield used to value its cash flows is its initial book value. At each coupon date, the book value changes as interest is recognized and the coupon is paid. A period-by-period table makes those changes explicit: it shows the opening book value, effective interest for the period, coupon payment, premium amortization or discount accretion, and closing book value.
The central calculation is not to apply the coupon rate to the book value. Instead, calculate interest using the yield rate and the opening book value. The coupon is the contractual cash payment; the difference between that payment and effective interest explains how the book value changes.
Assume throughout that coupons are level, paid at the end of each period, and that \(i\) is the effective yield for one such period. The bond is valued using that same yield in every period, with no default, fees, or changes in contractual cash flows. For annual coupons, \(i\) is an effective annual yield. The table concerns coupon dates; it does not calculate book value between coupon dates.
How the Table Is Built
Each row starts with the previous row’s closing book value. Multiply that opening amount by the yield to find effective interest. Compare effective interest with the coupon to identify the period’s premium amortization or discount accretion, then calculate the closing value.
For the first period this is the bond’s purchase price at the stated yield. For later periods it is the preceding closing book value.
Multiply the opening book value by the effective yield for one coupon period: \(iB_{k-1}\).
If the coupon is larger, premium amortization is \(C-iB_{k-1}\). If interest is larger, discount accretion is \(iB_{k-1}-C\).
Use \(B_k=B_{k-1}+iB_{k-1}-C\), then carry that value forward as the next period’s opening value.
A table can use one column labelled “Premium amortization / discount accretion.” In a premium period, record the positive amount of amortization and subtract it from the opening value. In a discount period, record the positive amount of accretion and add it. The coupon itself is still shown as a cash payment in either case.
Why the Recurrence Holds
The recurrence follows from the value of the bond at the next coupon date. Immediately after the current coupon date, the bond is worth \(B_{k-1}\). Over one period, that value accumulates at the yield to \(B_{k-1}(1+i)\). At the next coupon date, the holder receives cash \(C\); the value remaining in the bond after that payment is the accumulated value less the coupon.
Rearranging gives \(B_k-B_{k-1}=iB_{k-1}-C\). This signed change is positive when effective interest exceeds the coupon, so a discount book value rises. It is negative when the coupon exceeds effective interest, so a premium book value falls. This also explains why a coupon payment is not itself the amount by which book value changes.
Worked Example: A Premium Bond with Annual Coupons
A hypothetical bond has a redemption value of $1,000, pays an annual coupon of $80, and has three years remaining. Its effective annual yield is 5%. Construct its book value table from the purchase date through maturity. Use the yield to calculate interest, carry full precision through the calculations, and report dollar amounts to the nearest cent.
Method. First determine the initial book value as the present value of the three $80 coupons and the $1,000 redemption payment at 5%. Then apply the recurrence in each year.
The initial book value is
So the purchase price is $1,081.70 to the nearest cent. This is a premium because the coupon of $80 exceeds the first year’s effective interest at this book value.
Period 1. Effective interest is \(0.05(1081.6974409\ldots)=54.0848720\ldots\), or $54.08. Premium amortization is \(80-54.0848720\ldots=25.9151280\ldots\), or $25.92. The closing value is \(1081.6974409\ldots+54.0848720\ldots-80=1055.7823129\ldots\), or $1,055.78.
Period 2. Using the unrounded opening value, interest is \(0.05(1055.7823129\ldots)=52.7891156\ldots\), or $52.79. Premium amortization is \(80-52.7891156\ldots=27.2108844\ldots\), or $27.21. The closing value is \(1055.7823129\ldots+52.7891156\ldots-80=1028.5714286\ldots\), or $1,028.57.
Period 3. Interest is \(0.05(1028.5714286\ldots)=51.4285714\ldots\), or $51.43. Premium amortization is \(80-51.4285714\ldots=28.5714286\ldots\), or $28.57. The closing value is \(1028.5714286\ldots+51.4285714\ldots-80=1000\).
| Year | Opening book value | Effective interest | Coupon paid | Premium amortization | Closing book value |
|---|---|---|---|---|---|
| 1 | $1,081.70 | $54.08 | $80.00 | $25.92 | $1,055.78 |
| 2 | $1,055.78 | $52.79 | $80.00 | $27.21 | $1,028.57 |
| 3 | $1,028.57 | $51.43 | $80.00 | $28.57 | $1,000.00 |
Check. In each row, opening value plus effective interest less coupon equals closing value when calculations use unrounded amounts. The last closing value is the $1,000 redemption amount. Displayed cents can produce a one-cent difference if the rounded columns are recomputed; the table carries unrounded values between periods.
Discount Accretion in a Table
A discount bond follows the same recurrence and table procedure. Its effective interest exceeds its coupon, so the book value increases by the difference. Do not change the interest calculation merely because the bond is at a discount: interest remains the yield multiplied by the opening book value.
Worked Example: A Discount Bond with Annual Coupons
A hypothetical bond has a redemption value of $1,000, pays an annual coupon of $40, and has three years remaining. Its effective annual yield is 6%. Construct the table, retaining full precision internally and reporting dollars to the nearest cent.
Method. Value the remaining payments at 6%, then calculate each period’s effective interest and discount accretion.
The initial value is $946.54, below redemption value. In year 1, interest is \(0.06(946.539761\ldots)=56.792386\ldots\), or $56.79. Accretion is \(56.792386\ldots-40=16.792386\ldots\), or $16.79, and the closing value is \(946.539761\ldots+56.792386\ldots-40=963.332147\ldots\), or $963.33.
In year 2, interest is \(0.06(963.332147\ldots)=57.799929\ldots\), or $57.80. Accretion is \(57.799929\ldots-40=17.799929\ldots\), or $17.80, giving a closing value of \(981.132075\ldots\), or $981.13. In year 3, interest is \(0.06(981.132075\ldots)=58.867925\ldots\), or $58.87. Accretion is \(58.867925\ldots-40=18.867925\ldots\), or $18.87; after the coupon, the closing book value is $1,000.
| Year | Opening book value | Effective interest | Coupon paid | Discount accretion | Closing book value |
|---|---|---|---|---|---|
| 1 | $946.54 | $56.79 | $40.00 | $16.79 | $963.33 |
| 2 | $963.33 | $57.80 | $40.00 | $17.80 | $981.13 |
| 3 | $981.13 | $58.87 | $40.00 | $18.87 | $1,000.00 |
Check. The book value rises each year because the effective interest is greater than the coupon. The closing value after the final coupon equals the $1,000 redemption value.
Using Redemption Values Other Than $1,000
The coupon amount and redemption value are separate contractual inputs. The table recurrence does not require redemption to equal $1,000, nor does it require the coupon to be a percentage of the redemption value stated in the problem. Use the actual coupon cash flow and the actual redemption payment when finding the initial value.
Worked Example: A Premium Bond with a Different Redemption Value
A hypothetical two-year bond will redeem for $1,200 and pays $50 at the end of each year, including the final year when redemption is also paid. Its effective annual yield is 4%. Construct the table, reporting amounts to the nearest cent and carrying full precision between periods.
Method. The final cash flow is $1,250: the $50 coupon plus $1,200 redemption. Work backward from the final post-coupon book value of $1,200 to determine the earlier values, or value the cash flows at 4%.
In year 1, interest is \(0.04(1203.772189\ldots)=48.150888\ldots\), or $48.15. The coupon exceeds interest, so premium amortization is \(50-48.150888\ldots=1.849112\ldots\), or $1.85. The closing value is \(1203.772189\ldots+48.150888\ldots-50=1201.923077\ldots\), or $1,201.92.
In year 2, interest is \(0.04(1201.923077\ldots)=48.076923\ldots\), or $48.08. Premium amortization is \(50-48.076923\ldots=1.923077\ldots\), or $1.92. Subtracting the $50 coupon from the value after interest leaves $1,200, the redemption value.
| Year | Opening book value | Effective interest | Coupon paid | Premium amortization | Closing book value |
|---|---|---|---|---|---|
| 1 | $1,203.77 | $48.15 | $50.00 | $1.85 | $1,201.92 |
| 2 | $1,201.92 | $48.08 | $50.00 | $1.92 | $1,200.00 |
Check. The initial price is only slightly above $1,200 because the annual coupon exceeds 4% of the redemption value by $2. The resulting premium decreases to zero over the two periods.
Common Mistakes and Checks
The table is most reliable when every row is checked using the recurrence, not just by looking at whether the balance appears to move in the expected direction.
- Using the coupon rate to calculate interest. Effective interest is the yield multiplied by opening book value. The coupon rate determines the contractual coupon amount, not the effective interest entry.
- Reversing amortization and accretion. For a premium, subtract \(C-iB_{k-1}\). For a discount, add \(iB_{k-1}-C\). Both differences are positive when named as amortization or accretion.
- Using the previous row’s interest or opening value. Each period’s interest is calculated from that period’s own opening book value.
- Ignoring the final redemption payment when finding the initial book value. The maturity cash flow includes both the final coupon and redemption; the final coupon is still shown separately in the table.
For instance, in the premium example, the incorrect calculation \(54.08-80=-25.92\) treats premium amortization as interest less coupon. Subtracting that negative number would make the book value rise to $1,107.62, contrary to the fact that the coupon exceeds effective interest. The correct premium amortization is \(80-54.08=25.92\), and the closing value is approximately \(1081.70-25.92=$1,055.78\).
Check Your Understanding
For each problem, use the stated effective yield per annual coupon period. Report dollar amounts to the nearest cent, carrying unrounded values between periods.
- A two-year bond redeems for $5,000, pays $300 annually, and yields 5% effective annually. Find the initial book value and the year 1 effective interest, premium amortization, and closing book value. (Answer: Initial book value $5,092.97; year 1 interest $254.65; amortization $45.35; closing book value $5,047.62.)
- A two-year bond redeems for $2,000, pays $70 annually, and yields 4% effective annually. Find the initial book value and the book value after each coupon. (Answer: Initial book value $1,981.14; closing values $1,990.38 and $2,000.00; the respective annual discount accretions are $9.25 and $9.62.)
- A two-year bond redeems for $10,000, pays $350 annually, and yields 3% effective annually. Construct the first-period entries and find the second-period closing value. (Answer: Initial book value $10,095.67; year 1 interest $302.87; premium amortization $47.13; year 1 closing value $10,048.54; year 2 closing value $10,000.00.)
- A two-year bond redeems for $1,000, pays $30 annually, and yields 5% effective annually. Calculate the first-period effective interest and discount accretion, then explain why the book value increases. (Answer: Initial book value $962.81; first-period interest $48.14; accretion $18.14; first closing book value $980.95. The book value increases because interest exceeds the coupon.)