7.4 Bond Schedules

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 Price1	Premium1
1	PMT2	INT on BV3	AP4	BV5	RP6
2	PMT2	INT on BV3	AP4	BV5	RP6
[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]
N-1	PMT2	INT on BV3	AP4	BV5	RP6
N	PMT2	INT on BV3	AP4	Face Value8	09
Totals	Total Payments10	Total Interest12	Total Amortized Premium11		D

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

1. 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.
2. 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.
3. 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.
4. 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].
5. 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]
6. 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].
7. For each payment, repeat steps 2 through 6, including for the last row.
8. The last book value is the face value of the bond.  If the calculations are done correctly, this will automatically happen.
9. The last remaining premium to be amortized is 0.  If the calculations are done correctly, this will automatically happen.
10. The total payments is the sum of the payment column:  [latex]\mbox{Total Payments}=N \times PMT[/latex].
11. 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].
12. 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].

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.

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 Price1	Discount1
1	PMT2	INT on BV3	AD4	BV5	RD6
2	PMT2	INT on BV3	AD4	BV5	RD6
[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]	[latex]\vdots[/latex]
N-1	PMT2	INT on BV3	AD4	BV5	RD6
N	PMT2	INT on BV3	AD4	Face Value8	09
Totals	Total Payments10	Total Interest12	Total Accumulated Discount11

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

1. 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.
2. 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.
3. 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.
4. 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].
5. 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].
6. 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].
7. For each payment, repeat steps 2 through 6, including for the last row.
8. The last book value is the face value of the bond.  If the calculations are done correctly, this will automatically happen.
9. The last remaining discount to be accumulated is 0.  If the calculations are done correctly, this will automatically happen.
10. The total payments is the sum of the payment column:  [latex]\mbox{Total Payments}=N \times PMT[/latex].
11. 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].
12. 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].

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.

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Exercises

1. 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]

    

2. 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]

    

3. 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]

    

4. 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]