# 7.4 Bond Schedules

## LEARNING OBJECTIVES

- Construct an amortization of the premium bond schedule.
- Construct an accumulation of the discount bond schedule.

Recall that the purchase price of a bond does not necessarily equal its face value. The bond may be purchased for more that its face value, resulting in a premium, or the bond may be purchased for less that its face value, resulting in a discount. When an investor purchases a bond, the investor receives the bond payments as well as the bond’s face value at maturity. What are the tax implications associated with owning a bond? For example, the bond payments are taxable income for the investor. But, what are the tax implications of the premium or discount that is realized when the bond is redeemed at maturity?

The **book value of a bond** is the face value of the bond plus or minus any unamortized premium or discount. The initial book value of a bond is its purchase price. A common practice to deal with the tax implications of the premium or discount is to make adjustments to the book value of the bond over the holding period of the bond. The adjustments to the book value involve amortizing small parts of the premium or discount during each payment interval until the book value of the bond equals the bond’s face value.

## Amortization of the Premium

When a bond is purchased for more than its face value, the result is a premium. For example, suppose an investor pays $1,050 for a $1,000 face value bond. If the investor holds onto the bond until maturity, only the redemption price of $1,000 is returned. The bond premium of $50 represents a capital loss for the investor because the premium is not recovered at maturity. A **capital loss** is the amount by which the current value of an asset falls short of the original purchase price. What does the investor do with this capital loss? One option is for the investor to apply the capital loss against their income at the time of redemption, which results in a lower taxable amount at the time of redemption. But, commonly accepted practices allow the investor to amortize the capital loss over the period of time that the bond is held, and not just in the period during which the capital loss actually occurs (at maturity). This allows the investor to spread the capital loss over each payment interval, claiming a little bit of the capital loss against each bond payment in order to lower the taxable amount for each payment interval. This process is called the **amortization of the premium**.

The amortization of the premium schedule records the bond payment, the amount of the premium that is claimed with each payment, the book value, and the remaining premium to be amortized. The amortization of the premium schedule shows how much premium is claimed against each payment and the resulting book value.

An amortization of the premium schedule has six columns:

**Payment Number.**There is a row for every bond payment during the holding period of the bond.**Bond Payment.**The periodic bond payment (PMT). All of the payments are the same, including the last payment.**Interest on Book Value at Yield.**For each row, the interest on book value entry is the interest portion of the current bond payment based on the book value from the previous row.**Amortized Premium.**For each row, the amortized premium entry is the amount of the premium that is claimed as a capital loss in that payment interval.**Book Value.**For each row, the book value is the face value of the bond plus the remaining unamortized premium.**Remaining Premium to be Amortized.**For each row, the remaining premium is the amount of premium that is left to claim against the payments.

To fill in an amortization of the premium schedule, you first need to have all of the details about the bond, including the face value (FV), the purchase price (PV), the bond payment (PMT), the number of payments (N), and the yield rate. If any of these quantities are missing, calculate out the missing value before completing the amortization of the premium schedule.

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Amortized Premium |
Book Value |
Remaining Premium to be Amortized |

0 | Purchase Price^{1} |
Premium^{1} |
|||

1 | PMT^{2} |
INT on BV^{3} |
AP^{4} |
BV^{5} |
RP^{6} |

2 | PMT^{2} |
INT on BV^{3} |
AP^{4} |
BV^{5} |
RP^{6} |

[latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] |

N-1 | PMT^{2} |
INT on BV^{3} |
AP^{4} |
BV^{5} |
RP^{6} |

N | PMT^{2} |
INT on BV^{3} |
AP^{4} |
Face Value^{8} |
0^{9} |

Totals |
Total Payments^{10} |
Total Interest^{12} |
Total Amortized Premium^{11} |
D |

Follow these steps to fill in the amortization of the premium schedule.

- In row 0, the only entries are in the book value and remaining premium to be amortized columns. The initial book value is the purchase price of the bond and the initial remaining premium is the premium.
- Each entry in the bond payment column is the bond payment. All of the payments in this column are the same, including the last payment.
- Calculate the interest on the book value at yield. The interest is the book value from the previous row times the periodic yield rate: [latex]\mbox{Interest on Book Value}=\mbox{Book Value from Previous Row} \times i[/latex].
**Note:**this calculation uses the periodic yield rate, not the periodic bond rate. - Calculate the amortized premium. The amortized premium is the difference between the payment and the interest on the book value: [latex]\mbox{Amortized Premium}=PMT-\mbox{Interest on Book Value}[/latex].
- Calculate the new book value. The book value is the difference between the book value in the previous row and the amortized premium: [latex]\mbox{Book Value}=\mbox{Book Value from Previous Row}-\mbox{Amortized Premium}[/latex]
- Calculate the remaining premium to be amortized. The remaining premium is the difference between the remaining premium from the previous row and the amortized premium: [latex]\mbox{Remaining Premium}=\mbox{Remaining Premium from Previous Row}-\mbox{Amortized Premium}[/latex].
- For each payment, repeat steps 2 through 6, including for the last row.
- The last book value is the face value of the bond. If the calculations are done correctly, this will automatically happen.
- The last remaining premium to be amortized is 0. If the calculations are done correctly, this will automatically happen.
- The total payments is the sum of the payment column: [latex]\mbox{Total Payments}=N \times PMT[/latex].
- The total amortized premium is the sum of the amortized premium column, and is just the premium: [latex]\mbox{Total Amortized Premium}=\mbox{Premium}[/latex].
- The total interest is the sum of the interest on book value column, and equals the difference between the other two column totals: [latex]\mbox{Total Interest}=\mbox{Total Payments}-\mbox{Total Amortized Premium}[/latex].

### NOTES

- The manual calculation of the interest on the book value entry above is based on the assumption that the payment frequency and the compounding frequency are equal. For bonds this is generally not an issue because most bonds have semi-annual payments and a yield rate that compounds semi-annually. If the payment frequency and the compounding frequency are not equal, an interest conversion would be required to convert the interest rate to the equivalent rate with the compounding frequency equal to the payment frequency. However, if you use the TI BAII Plus’s built-in amortization worksheet (described below), no interest conversion is required.
- As you fill in the schedule, round the entries to two decimal places.
- The amortized premium column represents how much of the premium is claimed as a capital loss against the payment for that payment interval.
- The amortization of the premium schedule presented here assumes the bond is purchased on an interest payment date. Bonds that are not purchased on an interest payment date have added complications that will not be addressed here.

## EXAMPLE

A $2,000 bond with a coupon rate of 8% is redeemable in two years. The bond was purchased when the yield rate was 5%. Construct the amortization of the premium schedule.

**Solution:**

**Step 1:** Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual. The given information is

[latex]\begin{eqnarray*} FV & = & \$2,000 \\ P/Y & = & 2 \\ I/Y & = & 5\% \\ C/Y & = & 2 \\ t & = & 2 \mbox{ years} \\ \mbox{Coupon Rate} & = & 8\% \end{eqnarray*}[/latex]

**Step 2:** Calculate the bond payment.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 2,000\times \frac{0.08}{2}\\ & = & \$80\end{eqnarray*}[/latex]

**Step 3:** Calculate the purchase price.

PMT Setting |
END |

N |
[latex]2 \times 2=4[/latex] |

PV |
? |

FV |
[latex]2,000[/latex] |

PMT |
[latex]80[/latex] |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$2,112.86[/latex]

The purchase price of the bond is $2,112.86.

**Step 4:** Calculate the premium.

[latex]\begin{eqnarray*} \mbox{Premium} & = & \mbox{Purchase Price}-\mbox{Face Value} \\ & = & 2,112.86-2,000 \\ & = & \$112.86 \end{eqnarray*}[/latex]

The premium is $112.86.

**Step 5:** Complete the amortization of premium schedule. Because the payment frequency and the compounding frequency are equal, no interest conversion is required. The calculations for each entry are shown in blue. The periodic yield rate is [latex]i=\frac{5\%}{2}=2.5\%[/latex].

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Amortized Premium |
Book Value |
Remaining Premium to be Amortized |

0 | [latex]\$2,112.86[/latex] | [latex]\$112.86[/latex] | |||

1 | [latex]\$80[/latex] | [latex]\$52.82[/latex] [latex]\textcolor{blue}{(2,112.86 \times 0.025)}[/latex] | [latex]\$27.18[/latex] [latex]\textcolor{blue}{(80-52.82)}[/latex] | [latex]\$2,085.68[/latex][latex]\textcolor{blue}{(2,112.86-27.18)}[/latex] | [latex]\$85.68[/latex][latex]\textcolor{blue}{(112.86-27.18)}[/latex] |

2 | [latex]\$80[/latex] | [latex]\$52.14[/latex][latex]\textcolor{blue}{(2,085.68 \times 0.025)}[/latex] | [latex]\$27.86[/latex][latex]\textcolor{blue}{(80-52.14)}[/latex] | [latex]\$2,057.82[/latex][latex]\textcolor{blue}{(2,085.68-27.86)}[/latex] | [latex]\$57.82[/latex][latex]\textcolor{blue}{(85.68-27.86)}[/latex] |

3 | [latex]\$80[/latex] | [latex]\$51.45[/latex][latex]\textcolor{blue}{(2,057.82 \times 0.025)}[/latex] | [latex]\$28.55[/latex][latex]\textcolor{blue}{(80-51.45)}[/latex] | [latex]\$2,029.27[/latex][latex]\textcolor{blue}{(2,057.82-28.55)}[/latex] | [latex]\$29.27[/latex][latex]\textcolor{blue}{(57.82-28.55)}[/latex] |

4 | [latex]\$80[/latex] | [latex]\$50.73[/latex] [latex]\textcolor{blue}{(2,029.27 \times 0.025)}[/latex] | [latex]\$29.27[/latex][latex]\textcolor{blue}{(80-50.73)}[/latex] | [latex]\$2,000[/latex] | [latex]\$0[/latex][latex]\textcolor{blue}{(29.27-29.27)}[/latex] |

Totals |
[latex]\$320[/latex][latex]\textcolor{blue}{(4 \times 80)}[/latex] | [latex]\$207.14[/latex][latex]\textcolor{blue}{(320-112.86)}[/latex] | [latex]\$112.86[/latex] |

### NOTES

- In the previous example, the result of the amortization of the premium schedule is that the investor ends up paying taxes only on $207.14 (the total of the interest on the book value at yield column) instead of $320 (the payments total). That is, by spreading the $112.86 capital loss over the holding period of the bond, the total taxable amount is reduced from $320 down to $207.14.
- Sometimes there is a minor difference in the face value and the remaining premium to be amortized in the last row, where the last entries in these columns are off by a cent or two. This is a normal occurrence that happens because of the rounding of the entries in the schedule.

Although the calculations in an amortization of premium schedule are relatively straightforward, the manual calculations are time-consuming, especially when the schedule has a lot of rows. The amortization worksheet on a financial calculator, such as the TI BAII Plus, can be used to quickly calculate the entries for each row of the schedule.

## USING THE TI BAII PLUS CALCULATOR TO CONSTRUCT AN AMORTIZATION OF THE PREMIUM SCHEDULE

To use the amortization worksheet to complete an amortization of the premium schedule:

- Solve for any unknown quantities about the bond. You need to know all of the information about the bond first before you can use the amortization worksheet.
- Enter all the value of all seven time value of money variables into the calculator (N, PV, FV, PMT, I/Y, P/Y, C/Y). If you calculated the purchase price (PV) in the first step, you must
**re-enter**it rounded to two decimals and with the correct cash flow sign. Make sure the payment setting is set to END, and obey the cash flow sign convention. Because this is a bond, PV (the purchase price) is negative, FV (the face value) is positive, and PMT (the bond payment) is positive. - Go to the amortization worksheet by pressing 2nd AMORT (the PV button).
- To view the entries for a specific row of the schedule, set P1 and P2 to the row number. For example, to view the entries for row 5, set P1=5 and P2=5:
- At the P1 prompt, enter the row number and press ENTER.
- Press the down arrow.
- At the P2 prompt, enter the row number and press ENTER.
- Press the down arrow.
- The BAL entry is the book value entry for the corresponding row.
- Press the down arrow.
- The PRN entry is the amortized premium entry for the corresponding row.
- Press the down arrow.
- The INT entry is the interest on book value at yield entry for the corresponding row.
- Press the down arrow the return to the P1 screen.

- Repeat the previous step with a different row number to view the entries for a different row.

### NOTES

- On the amortization worksheet, BAL is the book value entry, PRN is the amortized premium entry, and INT is the interest on book value at yield entry.
- You cannot get the entries for the last column, remaining premium to be amortized, from the amortization worksheet on the calculator. This entry will still need to be calculated manually.
- Make sure to re-enter PV rounded to 2 decimal places before using the amortization worksheet. If you enter PV with all of the decimal places, you will not get the correct entries for the amortization of the premium schedule.
- As you read the entries off of the amortization worksheet on the calculator and put them in the schedule, round the entries to 2 decimal places.

## EXAMPLE

A $5,000 bond has a 4% coupon and is redeemable in two years. It was purchased to yield 3%. Construct the amortization of the premium schedule.

**Solution:**

**Step 1:** Calculate the purchase price.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 5,000\times \frac{0.04}{2}\\ & = & \$100\end{eqnarray*}[/latex]

PMT Setting |
END |

N |
[latex]2 \times 2=4[/latex] |

PV |
? |

FV |
[latex]5,000[/latex] |

PMT |
[latex]100[/latex] |

I/Y |
[latex]3[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$5,096.36[/latex]

**Step 2:** Calculate the premium.

[latex]\begin{eqnarray*} \mbox{Premium} & = & \mbox{Purchase Price}-\mbox{Face Value} \\ & = & 5,096.36-5,000 \\ & = & \$96.36 \end{eqnarray*}[/latex]

**Step 3:** Enter the information into the time value of money buttons on the calculator.

PMT Setting |
END |

N |
[latex]4[/latex] |

PV |
[latex]-5,096.36[/latex] |

FV |
[latex]5,000[/latex] |

PMT |
[latex]100[/latex] |

I/Y |
[latex]3[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

**Step 4:** Complete the amortization of the premium schedule using the amortization worksheet on the calculator. Remember, the calculator will not tell you the entries for the last column (remaining premium to be amortized), so you will need to complete this column manually.

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Amortized Premium |
Book Value |
Remaining Premium to be Amortized |

0 | [latex]\$5,096.36[/latex] | [latex]\$96.36[/latex] | |||

1 | [latex]\$100[/latex] | [latex]\$76.45[/latex] | [latex]\$23.55[/latex] | [latex]\$5,072.81[/latex] | [latex]\$72.81[/latex] |

2 | [latex]\$100[/latex] | [latex]\$76.09[/latex] | [latex]\$23.91[/latex] | [latex]\$5,048.90[/latex] | [latex]\$48.90[/latex] |

3 | [latex]\$100[/latex] | [latex]\$75.73[/latex] | [latex]\$24.27[/latex] | [latex]\$5,024.63[/latex] | [latex]\$24.63[/latex] |

4 | [latex]\$100[/latex] | [latex]\$75.37[/latex] | [latex]\$24.63[/latex] | [latex]\$5,000[/latex] | [latex]\$0[/latex] |

Totals |
[latex]\$400[/latex] | [latex]\$303.63[/latex] | [latex]\$96.36[/latex] |

**Row 1:**In the amortization worksheet, set P1=1 and P2=1. The entry for the last column (remaining premium to be amortized) is [latex]96.36-23.55[/latex].**Row 2:**In the amortization worksheet, set P1=2 and P2=2. The entry for the last column (remaining premium to be amortized) is [latex]72.81-23.91[/latex].**Row 3:**In the amortization worksheet, set P1=3 and P2=3. The entry for the last column (remaining premium to be amortized) is [latex]48.90-24.27[/latex].**Row 4:**In the amortization worksheet, set P1=4 and P2=4. The entry for the last column (remaining premium to be amortized) is [latex]24.63-24.63[/latex].**Totals Row:**- The amortized premium total is the premium ([latex]\$96.36[/latex]).
- The payments total is the sum of the payments: [latex]4 \times 100=400[/latex].
- The interest on book value total is the difference in the other two column totals: [latex]400-96.36=303.63[/latex].

## TRY IT

A $10,000 bond has a coupon rate of 6.5%. The bond was purchased when there was three years to maturity and the yield rate was 5%. Construct the amortization of the premium schedule.

**Click to see Solution**

PMT Setting |
END |

N |
[latex]2 \times 3=6[/latex] |

PV |
? |

FV |
[latex]10,000[/latex] |

PMT |
[latex]325[/latex] |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$10,413.11[/latex]

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Amortized Premium |
Book Value |
Remaining Premium to be Amortized |

0 | [latex]\$10,413.11[/latex] | [latex]\$413.11[/latex] | |||

1 | [latex]\$325[/latex] | [latex]\$260.33[/latex] | [latex]\$64.67[/latex] | [latex]\$10,348.44[/latex] | [latex]\$348.44[/latex] |

2 | [latex]\$325[/latex] | [latex]\$258.71[/latex] | [latex]\$66.29[/latex] | [latex]\$10,282.15[/latex] | [latex]\$282.15[/latex] |

3 | [latex]\$325[/latex] | [latex]\$257.05[/latex] | [latex]\$67.95[/latex] | [latex]\$10,214.20[/latex] | [latex]\$214.20[/latex] |

4 | [latex]\$325[/latex] | [latex]\$255.36[/latex] | [latex]\$69.64[/latex] | [latex]\$10,144.56[/latex] | [latex]\$144.56[/latex] |

5 | [latex]\$325[/latex] | [latex]\$253.61[/latex] | [latex]\$71.39[/latex] | [latex]\$10,073.17[/latex] | [latex]\$73.17[/latex] |

6 | [latex]\$325[/latex] | [latex]\$251.83[/latex] | [latex]\$73.17[/latex] | [latex]\$10,000[/latex] | [latex]\$0[/latex] |

Totals |
[latex]\$1,950[/latex] | [latex]\$303.63[/latex] | [latex]\$413.11[/latex] |

## Accumulation of the Discount

When a bond is purchased for less than its face value, the result is a discount. For example, suppose an investor pays $950 for a $1,000 face value bond. If the investor holds onto the bond until maturity, the investor receives the redemption price of $1,000. The bond discount of $50 represents a capital gain for the investor. A **capital gain** is the amount by which the current value of an asset exceeds the original purchase price. What does the investor do with this capital gain? One option is for the investor to claim all of the capital gain as taxable income at the time of redemption, which results in a higher taxable amount at the time of redemption. But, commonly accepted practices allow the investor to distribute the capital gain over the period of time that the bond is held, and not just in the period during which the capital gain actually occurs (at maturity). This allows the investor to spread the capital gain over each payment interval, claiming a little bit of the capital gain along with each bond payment. This results in a higher taxable amount for each payment interval, but results in lower taxes overall because the capital gain is not claimed as a single lump-sum taxable expense on the maturity date. This process is called the **accumulation of the discount**.

The accumulation of the discount schedule records the bond payment, the amount of the discount that is claimed with each payment, the book value, and the remaining discount to be accumulated. The accumulation of the discount schedule shows how much discount is claimed with each payment and the resulting book value.

An accumulation of the discount schedule has six columns:

**Payment Number.**There is a row for every bond payment during the holding period of the bond.**Bond Payment.**The periodic bond payment (PMT). All of the payments are the same, including the last payment.**Interest on Book Value at Yield.**For each row, the interest on book value entry is the interest portion of the current bond payment based on the book value from the previous row.**Accumulated Discount.**For each row, the accumulated discount entry is the amount of the discount that is claimed as a capital gain in that payment interval.**Book Value.**For each row, the book value is the face value of the bond minus the remaining unaccumulated discount.**Remaining Discount to be Accumulated.**For each row, the remaining discount is the amount of discount that is left to claim with the payments.

To fill in an accumulation of the discount schedule, you first need to have all of the details about the bond, including the face value (FV), the purchase price (PV), the bond payment (PMT), the number of payments (N), and the yield rate. If any of these quantities are missing, calculate out the missing value before completing the accumulation of the discount schedule.

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Accumulated Discount |
Book Value |
Remaining Discount to be Accumulated |

0 | Purchase Price^{1} |
Discount^{1} |
|||

1 | PMT^{2} |
INT on BV^{3} |
AD^{4} |
BV^{5} |
RD^{6} |

2 | PMT^{2} |
INT on BV^{3} |
AD^{4} |
BV^{5} |
RD^{6} |

[latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] | [latex]\vdots[/latex] |

N-1 | PMT^{2} |
INT on BV^{3} |
AD^{4} |
BV^{5} |
RD^{6} |

N | PMT^{2} |
INT on BV^{3} |
AD^{4} |
Face Value^{8} |
0^{9} |

Totals |
Total Payments^{10} |
Total Interest^{12} |
Total Accumulated Discount^{11} |

Follow these steps to fill in the accumulation of the discount schedule.

- In row 0, the only entries are in the book value and remaining discount to be accumulated columns. The initial book value is the purchase price of the bond and the initial remaining discount is the discount.
- Each entry in the bond payment column is the bond payment. All of the payments in this column are the same, including the last payment.
- Calculate the interest on the book value at yield. The interest is the book value from the previous row times the periodic yield rate: [latex]\mbox{Interest on Book Value}=\mbox{Book Value from Previous Row} \times i[/latex].
**Note:**this calculation uses the periodic yield rate, not the periodic bond rate. - Calculate the accumulated discount. The accumulated discount is the difference between the interest on the book value and the payment: [latex]\mbox{Accumulated Discount}=\mbox{Interest on Book Value}-PMT[/latex].
- Calculate the new book value. The book value is sum of the book value in the previous row and the accumulated discount: [latex]\mbox{Book Value}=\mbox{Book Value from Previous Row}+\mbox{Accumulated Discount}[/latex].
- Calculate the remaining discount to be accumulated. The remaining discount is the difference between the remaining discount from the previous row and the accumulated discount: [latex]\mbox{Remaining Discount}=\mbox{Remaining Discount from Previous Row}-\mbox{Accumulated Discount}[/latex].
- For each payment, repeat steps 2 through 6, including for the last row.
- The last book value is the face value of the bond. If the calculations are done correctly, this will automatically happen.
- The last remaining discount to be accumulated is 0. If the calculations are done correctly, this will automatically happen.
- The total payments is the sum of the payment column: [latex]\mbox{Total Payments}=N \times PMT[/latex].
- The total accumulated discount is the sum of the accumulated discount column, and is just the discount: [latex]\mbox{Total Accumulated Discount}=\mbox{Discount}[/latex].
- The total interest is the sum of the interest on book value column, and equals the sum of the other two column totals: [latex]\mbox{Total Interest}=\mbox{Total Payments}+\mbox{Total Accumulated Discount}[/latex].

### NOTES

- The manual calculation of the interest on the book value entry above is based on the assumption that the payment frequency and the compounding frequency are equal. For bonds this is generally not an issue because most bonds have semi-annual payments and a yield rate that compounds semi-annually. If the payment frequency and the compounding frequency are not equal, an interest conversion would be required to convert the interest rate to the equivalent rate with the compounding frequency equal to the payment frequency. However, if you use the TI BAII Plus’s built-in amortization worksheet (described below), no interest conversion is required.
- As you fill in the schedule, round the entries to two decimal places.
- The accumulated discount column represents how much of the discount is claimed as a capital gain with the payment for that payment interval.
- The accumulation of the discount schedule presented here assumes the bond is purchased on an interest payment date. Bonds that are not purchased on an interest payment date have added complications that will not be addressed here.

## EXAMPLE

A $3,000 bond with a coupon rate of 5% is redeemable in two years. The bond was purchased when the yield rate was 8%. Construct the accumulation of the discount schedule.

**Solution:**

**Step 1:** Because no other information is given, the frequency of the payments and the compounding frequencies (for the coupon rate and the yield rate) are assumed to be semi-annual. The given information is

[latex]\begin{eqnarray*} FV & = & \$3,000 \\ P/Y & = & 2 \\ I/Y & = & 8\% \\ C/Y & = & 2 \\ t & = & 2 \mbox{ years} \\ \mbox{Coupon Rate} & = & 5\% \end{eqnarray*}[/latex]

**Step 2:** Calculate the bond payment.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 3,000\times \frac{0.05}{2}\\ & = & \$75\end{eqnarray*}[/latex]

**Step 3:** Calculate the purchase price.

PMT Setting |
END |

N |
[latex]2 \times 2=4[/latex] |

PV |
? |

FV |
[latex]3,000[/latex] |

PMT |
[latex]75[/latex] |

I/Y |
[latex]8[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$2,836.65[/latex]

The purchase price of the bond is $2,836.65.

**Step 4:** Calculate the discount.

[latex]\begin{eqnarray*} \mbox{Discount} & = & \mbox{Face Value}-\mbox{Discount} \\ & = & 3,000-2,836.65 \\ & = & \$163.35 \end{eqnarray*}[/latex]

The discount is $163.35.

**Step 5:** Complete the accumulation of discount schedule. Because the payment frequency and the compounding frequency are equal, no interest conversion is required. The calculations for each entry are shown in blue. The periodic yield rate is [latex]i=\frac{8\%}{2}=4\%[/latex].

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Accumulated Discount |
Book Value |
Remaining Premium to be Amortized |

0 | [latex]\$2,836.65[/latex] | [latex]\$163.35[/latex] | |||

1 | [latex]\$75[/latex] | [latex]\$113.47[/latex][latex]\textcolor{blue}{(2,836.65 \times 0.04)}[/latex] | [latex]\$38.47[/latex][latex]\textcolor{blue}{(113.47-75)}[/latex] | [latex]\$2,875.12[/latex][latex]\textcolor{blue}{(2,836.65+38.47)}[/latex] | [latex]\$124.88[/latex][latex]\textcolor{blue}{(163.35-38.47)}[/latex] |

2 | [latex]\$75[/latex] | [latex]\$115[/latex][latex]\textcolor{blue}{(2,875.12 \times 0.04)}[/latex] | [latex]\$40[/latex] [latex]\textcolor{blue}{(115-75)}[/latex] | [latex]\$2,915.12[/latex][latex]\textcolor{blue}{(2,915.12+40)}[/latex] | [latex]\$84.88[/latex][latex]\textcolor{blue}{(124.88-40)}[/latex] |

3 | [latex]\$75[/latex] | [latex]\$116.60[/latex][latex]\textcolor{blue}{(2,915.12 \times 0.04)}[/latex] | [latex]\$41.60[/latex][latex]\textcolor{blue}{(116.60-75)}[/latex] | [latex]\$2,956.72[/latex][latex]\textcolor{blue}{(2,915.12+41.60)}[/latex] | [latex]\$43.28[/latex][latex]\textcolor{blue}{(84.88-41.60)}[/latex] |

4 | [latex]\$75[/latex] | [latex]\$118.27[/latex] [latex]\textcolor{blue}{(2,029.27 \times 0.025)}[/latex] | [latex]\$43.27[/latex][latex]\textcolor{blue}{(80-50.73)}[/latex] | [latex]\$2999.99[/latex][latex]\textcolor{blue}{(2,956.72+43.27)}[/latex] | [latex]\$0.01[/latex][latex]\textcolor{blue}{(43.28-43.27)}[/latex] |

Totals |
[latex]\$300[/latex][latex]\textcolor{blue}{(4 \times 75)}[/latex] | [latex]\$463.34[/latex][latex]\textcolor{blue}{(300+163.34)}[/latex] | [latex]\$163.34[/latex] |

### NOTES

- In the previous example, there is a minor difference of $0.01 in the final book value and the final remaining discount to be accumulated in row 4 and the totals in the interest on the book value and accumulated discount totals. This sometimes happens, and is caused by rounding the entries at each step.
- In the previous example, the result of the accumulation of discount schedule is that the investor ends up paying taxes on $463.34 (the total of the interest on the book value at yield column) instead of $300 (the payments total). That is, by spreading the $163.35 capital gain over the holding period of the bond, the total taxable amount is increased from $300 to $463.34. Because the capital gain is spread out over the payments, this results in a lower tax amount overall.

Although the calculations in an accumulation of the discount schedule are relatively straightforward, the manual calculations are time-consuming, especially when the schedule has a lot of rows. The amortization worksheet on a financial calculator, such as the TI BAII Plus, can be used to quickly calculate the entries for each row of the schedule.

## USING THE TI BAII PLUS CALCULATOR TO CONSTRUCT AN ACCUMULATION OF THE DISCOUNT SCHEDULE

To use the amortization worksheet to complete an accumulation of the discount schedule:

- Solve for any unknown quantities about the bond. You need to know all of the information about the bond first before you can use the amortization worksheet.
- Enter all the value of all seven time value of money variables into the calculator (N, PV, FV, PMT, I/Y, P/Y, C/Y). If you calculated the purchase price (PV) in the first step, you must
**re-enter**it rounded to two decimals and with the correct cash flow sign. Make sure the payment setting is set to END, and obey the cash flow sign convention. Because this is a bond, PV (the purchase price) is negative, FV (the face value) is positive, and PMT (the bond payment) is positive. - Go to the amortization worksheet by pressing 2nd AMORT (the PV button).
- To view the entries for a specific row of the schedule, set P1 and P2 to the row number. For example, to view the entries for row 5, set P1=5 and P2=5:
- At the P1 prompt, enter the row number and press ENTER.
- Press the down arrow.
- At the P2 prompt, enter the row number and press ENTER.
- Press the down arrow.
- The BAL entry is the book value entry for the corresponding row.
- Press the down arrow.
- The PRN entry is the accumulated discount entry for the corresponding row.
- Press the down arrow.
- The INT entry is the interest on book value at yield entry for the corresponding row.
- Press the down arrow the return to the P1 screen.

- Repeat the previous step with a different row number to view the entries for a different row.

### NOTES

- On the amortization worksheet, BAL is the book value entry, PRN is the accumulation of discount entry, and INT is the interest on book value at yield entry.
- You cannot get the entries for the last column, remaining discount to be accumulated, from the amortization worksheet on the calculator. This entry will still need to be calculated manually.
- Make sure to re-enter PV rounded to 2 decimal places before using the amortization worksheet. If you enter PV with all of the decimal places, you will not get the correct entries for the accumulation of the discount schedule.
- As you read the entries off of the amortization worksheet on the calculator and put them in the schedule, round the entries to 2 decimal places.

## EXAMPLE

A $7,000 bond has a 3% coupon and is redeemable in two years. It was purchased to yield 5%. Construct the accumulation of the discount schedule.

**Solution:**

**Step 1:** Calculate the purchase price.

[latex]\begin{eqnarray*} PMT & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 7,000\times \frac{0.03}{2}\\ & = & \$105\end{eqnarray*}[/latex]

PMT Setting |
END |

N |
[latex]2 \times 2=4[/latex] |

PV |
? |

FV |
[latex]7,000[/latex] |

PMT |
[latex]105[/latex] |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$6,736.66[/latex]

**Step 2:** Calculate the discount.

[latex]\begin{eqnarray*} \mbox{Premium} & = & \mbox{Face Value}-\mbox{Purchase Price} \\ & = & 7,000-6,736.66 \\ & = & \$263.34 \end{eqnarray*}[/latex]

**Step 3:** Enter the information into the time value of money buttons on the calculator.

PMT Setting |
END |

N |
[latex]4[/latex] |

PV |
[latex]-6,736.66[/latex] |

FV |
[latex]7,000[/latex] |

PMT |
[latex]105[/latex] |

I/Y |
[latex]5[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

**Step 4:** Complete the accumulation of the discount schedule using the amortization worksheet on the calculator. Remember, the calculator will not tell you the entries for the last column (remaining discount to be accumulated), so you will need to complete this column manually.

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Accumulated Discount |
Book Value |
Remaining Discount to be Accumulated |

0 | [latex]\$6,736.66[/latex] | [latex]\$263.34[/latex] | |||

1 | [latex]\$105[/latex] | [latex]\$168.42[/latex] | [latex]\$63.42[/latex] | [latex]\$6,800.08[/latex] | [latex]\$199.92[/latex] |

2 | [latex]\$105[/latex] | [latex]\$170[/latex] | [latex]\$65[/latex] | [latex]\$6,865.08[/latex] | [latex]\$134.92[/latex] |

3 | [latex]\$105[/latex] | [latex]\$171.63[/latex] | [latex]\$66.63[/latex] | [latex]\$6,931.71[/latex] | [latex]\$68.29[/latex] |

4 | [latex]\$105[/latex] | [latex]\$173.29[/latex] | [latex]\$68.29[/latex] | [latex]\$7,000[/latex] | [latex]\$0[/latex] |

Totals |
[latex]\$420[/latex] | [latex]\$683.34[/latex] | [latex]\$263.34[/latex] |

**Row 1:**In the amortization worksheet, set P1=1 and P2=1. The entry for the last column (remaining discount to be accumulated) is [latex]263.34-63.42[/latex].**Row 2:**In the amortization worksheet, set P1=2 and P2=2. The entry for the last column (remaining discount to be accumulated) is [latex]199.92-65[/latex].**Row 3:**In the amortization worksheet, set P1=3 and P2=3. The entry for the last column (remaining discount to be accumulated) is [latex]134.92-66.63[/latex].**Row 4:**In the amortization worksheet, set P1=4 and P2=4. The entry for the last column (remaining discount to be accumulated) is [latex]68.29-68.29[/latex].**Totals Row:**- The accumulated discount total is the discount ([latex]\$263.34[/latex]).
- The payments total is the sum of the payments: [latex]4 \times 105=420[/latex].
- The interest on book value total is the sum of other two column totals: [latex]420+263.34=683.34[/latex].

## TRY IT

A $5,000 bond has a coupon rate of 3.6%. The bond was purchased when there was 2.5 years to maturity and the yield rate was 6%. Construct the accumulation of the discount schedule.

**Click to see Solution**

PMT Setting |
END |

N |
[latex]2 \times 2.5=5[/latex] |

PV |
? |

FV |
[latex]5,000[/latex] |

PMT |
[latex]90[/latex] |

I/Y |
[latex]6[/latex] |

P/Y |
[latex]2[/latex] |

C/Y |
[latex]2[/latex] |

[latex]PV=\$4,725.22[/latex]

Payment Number |
Bond Payment |
Interest on Book Value at Yield |
Accumulated Discount |
Book Value |
Remaining Discount to be Accumulated |

0 | [latex]\$4,725.22[/latex] | [latex]\$274.78[/latex] | |||

1 | [latex]\$90[/latex] | [latex]\$141.76[/latex] | [latex]\$51.76[/latex] | [latex]\$4,776.98[/latex] | [latex]\$223.02[/latex] |

2 | [latex]\$90[/latex] | [latex]\$143.31[/latex] | [latex]\$53.31[/latex] | [latex]\$4,830.29[/latex] | [latex]\$169.71[/latex] |

3 | [latex]\$90[/latex] | [latex]\$144.91[/latex] | [latex]\$54.91[/latex] | [latex]\$4,885.19[/latex] | [latex]\$114.80[/latex] |

4 | [latex]\$90[/latex] | [latex]\$146.56[/latex] | [latex]\$56.56[/latex] | [latex]\$4,941.75[/latex] | [latex]\$58.25[/latex] |

5 | [latex]\$90[/latex] | [latex]\$148.25[/latex] | [latex]\$58.25[/latex] | [latex]\$5,000[/latex] | [latex]\$0[/latex] |

Totals |
[latex]\$450[/latex] | [latex]\$703.33[/latex] | [latex]\$253.33[/latex] |

**Exercises**

- A $20,000 bond has a 5% coupon rate. The bond was purchased when there are three years to maturity and the yield rate was 6.75%. Construct the appropriate bond schedule for the bond.

**Click to see Answer****Payment Number****Bond Payment****Interest on Book Value at Yield****Accumulated Discount****Book Value****Remaining Discount to be Accumulated**0 [latex]\$19,063.66[/latex] [latex]\$936.34[/latex] 1 [latex]\$500[/latex] [latex]\$643.40[/latex] [latex]\$143.40[/latex] [latex]\$19,207.06[/latex] [latex]\$792.94[/latex] 2 [latex]\$500[/latex] [latex]\$648.24[/latex] [latex]\$148.24[/latex] [latex]\$19,355.30[/latex] [latex]\$644.70[/latex] 3 [latex]\$500[/latex] [latex]\$653.24[/latex] [latex]\$153.24[/latex] [latex]\$19,508.54[/latex] [latex]\$491.46[/latex] 4 [latex]\$500[/latex] [latex]\$658.41[/latex] [latex]\$158.41[/latex] [latex]\$19,666.95[/latex] [latex]\$333.05[/latex] 5 [latex]\$500[/latex] [latex]\$663.76[/latex] [latex]\$163.76[/latex] [latex]\$19,830.71[/latex] [latex]\$169.29[/latex] 6 [latex]\$500[/latex] [latex]\$669.29[/latex] [latex]\$169.29[/latex] [latex]\$20,000[/latex] [latex]\$0[/latex] **Totals**[latex]\$3,000[/latex] [latex]\$3,936.34[/latex] [latex]\$936.34[/latex] - A $50,000 bond with an 8% coupon is redeemable in two years. The bond was purchased when the yield to maturity was 5%. Construct the appropriate bond scheduled for the bond.

**Click to see Answer****Payment Number****Bond Payment****Interest on Book Value at Yield****Amortized Premium****Book Value****Remaining Premium to be Amortized**0 [latex]\$52,821.48[/latex] [latex]\$2,821.48[/latex] 1 [latex]\$2,000[/latex] [latex]\$1,320.54[/latex] [latex]\$679.46[/latex] [latex]\$52,142.02[/latex] [latex]\$2,142.02[/latex] 2 [latex]\$2,000[/latex] [latex]\$1,303.55[/latex] [latex]\$696.45[/latex] [latex]\$51,445.57[/latex] [latex]\$1,445.57[/latex] 3 [latex]\$2,000[/latex] [latex]\$1,286.14[/latex] [latex]\$713.86[/latex] [latex]\$50,731.71[/latex] [latex]\$731.71[/latex] 4 [latex]\$2,000[/latex] [latex]\$1,268.29[/latex] [latex]\$731.71[/latex] [latex]\$50,000[/latex] [latex]\$0[/latex] **Totals**[latex]\$8,000[/latex] [latex]\$5,178.52[/latex] [latex]\$2,821.48[/latex] - Three years before maturity, a $55,000 face value bond carrying a 5.5% coupon is acquired when posted market rates are 4.77% compounded semi-annually. Construct the appropriate bond schedule for the bond.

**Click to see Answer****Payment Number****Bond Payment****Interest on Book Value at Yield****Amortized Premium****Book Value****Remaining Premium to be Amortized**0 [latex]\$56,110.02[/latex] [latex]\$1,110.02[/latex] 1 [latex]\$1,512.50[/latex] [latex]\$1,338.22[/latex] [latex]\$174.28[/latex] [latex]\$55,935.74[/latex] [latex]\$935.74[/latex] 2 [latex]\$1,512.50[/latex] [latex]\$1,334.07[/latex] [latex]\$178.43[/latex] [latex]\$55,757.31[/latex] [latex]\$757.31[/latex] 3 [latex]\$1,512.50[/latex] [latex]\$1,329.81[/latex] [latex]\$182.69[/latex] [latex]\$55,574.62[/latex] [latex]\$574.62[/latex] 4 [latex]\$1,512.50[/latex] [latex]\$1,325.45[/latex] [latex]\$187.05[/latex] [latex]\$55,387.58[/latex] [latex]\$387.58[/latex] 5 [latex]\$1,512.50[/latex] [latex]\$1,320.99[/latex] [latex]\$191.51[/latex] [latex]\$55,196.07[/latex] [latex]\$196.07[/latex] 6 [latex]\$1,512.50[/latex] [latex]\$1,316.43[/latex] [latex]\$196.07[/latex] [latex]\$55,000[/latex] [latex]\$0[/latex] **Totals**[latex]\$9,075[/latex] [latex]\$57,964.98[/latex] [latex]\$1,110.02[/latex] - When the current bond market is yielding 5.89% compounded semi-annually, Jennifer purchases a $10,000 face value bond carrying a 4.2% coupon with three years until maturity for her RRSP. Construct the appropriate bond schedule for the bond.

**Click to see Answer****Payment Number****Bond Payment****Interest on Book Value at Yield****Accumulated Discount****Book Value****Remaining Discount to be Accumulated**0 [latex]\$9,541.41[/latex] [latex]\$458.59[/latex] 1 [latex]\$210[/latex] [latex]\$280.99[/latex] [latex]\$70.99[/latex] [latex]\$9,612.40[/latex] [latex]\$387.60[/latex] 2 [latex]\$210[/latex] [latex]\$283.09[/latex] [latex]\$73.09[/latex] [latex]\$9,685.49[/latex] [latex]\$314.52[/latex] 3 [latex]\$210[/latex] [latex]\$285.24[/latex] [latex]\$75.24[/latex] [latex]\$9,760.73[/latex] [latex]\$239.27[/latex] 4 [latex]\$210[/latex] [latex]\$287.45[/latex] [latex]\$77.45[/latex] [latex]\$9,838.19[/latex] [latex]\$161.82[/latex] 5 [latex]\$210[/latex] [latex]\$289.73[/latex] [latex]\$79.73[/latex] [latex]\$9,917.92[/latex] [latex]\$82.08[/latex] 6 [latex]\$210[/latex] [latex]\$292.08[/latex] [latex]\$82.08[/latex] [latex]\$10,000[/latex] [latex]\$0[/latex] **Totals**[latex]\$1,260[/latex] [latex]\$1,718.59[/latex] [latex]\$458.59[/latex]

#### Attribution

“14.4: Debt Retirement & Amortization” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.