Loan Amortization
4.1 Loan Amortization: Calculator Approach
A. Amortization Definitions
Amortization of a loan is the process of the gradual reduction in the loan amount through periodic repayments, usually of equal size, over a predetermined length of time. The amortization period is the time period set to repay the loan. An amortization schedule is a detailed table, similar to what is shown in Table 4.1.1, that breaks down each loan payment amount into its interest (INT) and principal portions (PRN). It also shows the outstanding principal balance (BAL) after each payment is made.
Table 4.1.1 An Amortization Schedule
| Payment Number | Payment Amount PMT ($) | Interest Portion INT ($) | Principal Portion PRN ($) | Loan Balance BAL ($) |
| 0 | n/a | n/a | n/a | Loan Amount (PV) |
| 1 | [asciimath]PMT[/asciimath] | [asciimath]INT_1[/asciimath] | [asciimath]PRN_1[/asciimath] | [asciimath]BAL_1[/asciimath] |
| 2 | [asciimath]PMT[/asciimath] | [asciimath]INT_2[/asciimath] | [asciimath]PRN_2[/asciimath] | [asciimath]BAL_2[/asciimath] |
| [asciimath]vdots[/asciimath] | [asciimath]vdots[/asciimath] | [asciimath]vdots[/asciimath] | [asciimath]vdots[/asciimath] | [asciimath]vdots[/asciimath] |
| N-1 | [asciimath]PMT[/asciimath] | [asciimath]INT_(N-1)[/asciimath] | [asciimath]PRN_(N-1)[/asciimath] | [asciimath]BAL_(N-1)[/asciimath] |
| N | [asciimath]PMT_N[/asciimath] | [asciimath]INT_N[/asciimath] | [asciimath]PRN_N[/asciimath] | 0 |
| Totals | Total Amount Paid | Total Interest Portion | Total Principal Portion | n/a |
Note that the size of payment (PMT) is the same for every period except the last payment as it needs to be adjusted to fully repay the loan. In banking, all actual monetary values are rounded to the nearest cent (two decimal places). Because of this rounding, the final payment needs to be adjusted and thus is almost always different from other payments.
B. Amortization Schedules for Loans
For each payment (row) in the schedule, you can use the Amortization (AMORT) Worksheet in a financial calculator to calculate the principal portion (PRN), interest portion (INT), and the balance (BAL) after that payment. Given all the other information about the loan has already been entered, set P1 and P2 to that payment number to obtain the principal portion and interest portion of a particular payment number. For example, when P1 = 5 and P2 = 5, PRN displays the principal portion of the 5th payment and INT displays the interest portion of the 5th payment. BAL displays the principal balance after the 5th payment.
Figure 4.1.1 The keys on the AMORT worksheet on a financial calculator
How to Construct an Amortization Schedule: Calculator Approach
1. Enter the given information about the loan annuity in the TVM worksheet and compute the unknown variable, which is usually either the size of payment (PMT) or the number of payments in the term (N).
- Unknown PMT: compute PMT first using the TVM worksheet, round it off to the nearest cent (if needed), re-enter the rounded value (ensure to keep the same sign) to the PMT button in the calculator, and then use the Amortization Worksheet.
- Unknown N: compute N first using the TVM worksheet, round it up to the nearest whole payment, re-enter it to the N button in the calculator (optional), and then use the Amortization Worksheet.
2. In the first row of the schedule, enter zero for the payment number and the loan amount (PV) under the balance.
To construct the other rows except for the last two rows, complete steps 3-7:
3. Enter the size of payment (PMT) against its respective payment number. The periodic payment amount is usually a rounded amount (to the nearest cent).
4. In the AMORT worksheet, set both P1 and P2 to the payment number to obtain the principal portion and interest portion of that particular payment number. Use the down arrow key to move to BAL.
5. Enter the outstanding principal balance (BAL) in the 5th column of the schedule. Use the down arrow key to move to PRN.
6. Enter the principal portion (PRN) of the payment in the 4th column. Use the down arrow key to move to INT.
7. Enter the interest portion (INT) of the payment in the 3rd column.
To construct the row of last payment, complete steps 8-11:
8. For the final payment, we only take the interest portion from the AMORT worksheet, and we will calculate the BAL and the size of the payment as below.
9. In the row of the final payment number, the principal balance will be zero because the loan will be fully paid off.
10. For the final payment, the principal portion equals the remaining balance before the last payment ([asciimath]PRN_N=[/asciimath] [asciimath]BAL_(N-1)[/asciimath])
11. Since PMT or N might be rounded, the final payment may be different than PMT. The final payment amount has to be calculated using any of the two methods:
-
- [asciimath]PMT_N=PRN_(N)+INT_N[/asciimath] where [asciimath]PRN_N=BAL_(N-1)[/asciimath]
- [asciimath]PMT_N=PMT-"Overpayment"[/asciimath] where the overpayment is the absolute value of BAL on the last payment ([asciimath]BAL_(N)[/asciimath] ).
12. The last row lists all the totals, which can be used to cross-check the calculations:
- Total Principal Portion = Original loan amount
- Total Amount Paid = Total Interest Portion + Total Principal Portion
Note that for clear presentation all values in the table should be entered as nonnegative values.
Pearline took out a loan of $10,000 from TD Bank to buy office supplies. The loan has an annual interest rate of 10%, compounded annually, and is to be repaid over four years. a) Determine the amount of her payments due at the end of each year. b) Create an amortization schedule for her loan.
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Given information:
- Interest is compounded annually so [asciimath]C//Y = 1[/asciimath]
- Deposits are made at the end of every year so [asciimath]P//Y = 1[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Loan amortization period: [asciimath]t = 4[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=1 (4) =4[/asciimath]
- Nominal interest rate:[asciimath]I//Y = 10%[/asciimath]
- Loan amount is cash inflow: [asciimath]PV = $10,000[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
a) First, we use the TVM worksheet to compute PMT.
Thus, the size of the payment is $3,154.71.
Note that the PMT value should be rounded to the nearest cent. After rounding, this adjusted value should then be re-entered into the PMT key with the same negative sign.
b) Next, we follow the steps given in the “How To – Calculator Approach” to fill in the amortization schedule. Steps 1 to 3 are already done. We now use the AMORT worksheet to obtain BAL, PRN, and INT for each payment.
It is important to note that when entering values into the amortization schedule, they should be rounded to two decimal places. Also, notice that the balance after payment number 4 is negative, which indicates overpayment.
Therefore, we need to compute the final PMT separately (Steps 8-11):
Method 1:
[asciimath]PMT_N=PRN_(N)+INT_N[/asciimath] where [asciimath]PRN_N=BAL_(N-1)[/asciimath]
[asciimath]PMT_4=BAL_3+INT_4[/asciimath] [asciimath]=2867.91+286.79=$3154.70[/asciimath]
Method 2:
[asciimath]PMT_N=PMT-"Overpayment"[/asciimath]
Here the overpayment is the absolute value of BAL on the last payment ([asciimath]BAL_(4)[/asciimath]).
[asciimath]PMT_4=3154.71-0.01=$3154.70[/asciimath]
Step 12: Finally, we check that all totals add up as we did before using the formula approach.
| Payment Number | Payment Amount PMT ($) | Interest Portion INT ($) | Principal Portion PRN ($) | Loan Balance BAL ($) |
| 0 | n/a | n/a | n/a | [asciimath]10,000[/asciimath] |
| 1 | [asciimath]3154.71[/asciimath] | [asciimath]1000[/asciimath] | [asciimath]2154.71[/asciimath] | [asciimath]7845.29[/asciimath] |
| 2 | [asciimath]3154.71[/asciimath] | [asciimath]784.53[/asciimath] | [asciimath]2370.18[/asciimath] | [asciimath]5475.11[/asciimath] |
| 3 | [asciimath]3154.71[/asciimath] | [asciimath]547.51[/asciimath] | [asciimath]2607.20[/asciimath] | [asciimath]2867.91[/asciimath] |
| 4 | [asciimath]3154.70[/asciimath] | [asciimath]286.79[/asciimath] | [asciimath]2867.91[/asciimath] | [asciimath]0[/asciimath] |
| Totals | [asciimath]$12,618.83[/asciimath] | [asciimath]$2618.83[/asciimath] | [asciimath]$10,000[/asciimath] | n/a |
The figure above features a bar chart that shows the interest and principal portions of each payment for this example. Additionally, it includes a line chart that tracks the remaining principal balance after each payment is made. An important observation is that as the outstanding balance reduces over time, the interest portion of each payment also decreases. Since the payment amount (PMT) is constant for each period (except for the last payment), the diminishing interest portion results in an increased proportion of each payment being allocated to reduce the loan principal.
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A $15,000 loan is settled by end-of-quarter payments of $4500. The interest is 6.8% compounded semi-annually. a) Find the number of payments needed to settle the loan. b) construct an amortization schedule for the loan.
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Given information:
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- Payments are made at the end of every quarter so [asciimath]P//Y = 4[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Nominal interest rate: [asciimath]I//Y = 6.8%[/asciimath]
- Loan principal is cash inflow: [asciimath]PV = $15,000[/asciimath]
- Periodic payments are cash outflow: [asciimath]PMT = -$4500[/asciimath]
- The loan is fully paid: [asciimath]FV=0[/asciimath]
a) [asciimath]N=?[/asciimath]
First, we use the TVM worksheet to compute N.
Note that N is rounded up to 4 payments. The rounded value is then re-entered in the N key.
b) Next, we follow the steps given in the “How To – Calculator Approach” to fill in the amortization schedule. We now use the AMORT worksheet to obtain BAL, PRN, and INT for each payment.
Note that when entering values into the amortization schedule, they should be rounded to two decimal places. Additionally, observe that the balance after payment number 4 is negative, which indicates overpayment. Therefore, we need to adjust the final payment using the methods discussed in Steps 8-11:
Method 1:
[asciimath]PMT_N=PRN_(N)+INT_N[/asciimath] where [asciimath]PRN_N=BAL_(N-1)[/asciimath]
[asciimath]PMT_4=BAL_3+INT_4[/asciimath] [asciimath]=2042.61+34.43=$2077.04[/asciimath]
Method 2:
[asciimath]PMT_N=PMT-"Overpayment"[/asciimath]
Here the overpayment is the absolute value of BAL on the last payment ([asciimath]BAL_(4)[/asciimath]).
[asciimath]PMT_4=4500-2422.96=$2077.04[/asciimath]
Step 12: Finally, we check that all totals add up:
- Total Principal Portion = Original loan amount
- Total Amount Paid = Total Interest Portion + Total Principal Portion
| Payment Number | Payment Amount PMT ($) | Interest Portion INT ($) | Principal Portion PRN ($) | Loan Balance BAL ($) |
| 0 | n/a | n/a | n/a | [asciimath]15,000[/asciimath] |
| 1 | [asciimath]4500[/asciimath] | [asciimath]252.87[/asciimath] | [asciimath]4247.13[/asciimath] | [asciimath]10,752.87[/asciimath] |
| 2 | [asciimath]4500[/asciimath] | [asciimath]181.27[/asciimath] | [asciimath]4318.73[/asciimath] | [asciimath]6434.14[/asciimath] |
| 3 | [asciimath]4500[/asciimath] | [asciimath]108.47[/asciimath] | [asciimath]4391.53[/asciimath] | [asciimath]2042.61[/asciimath] |
| 4 | [asciimath]2077.04[/asciimath] | [asciimath]34.43[/asciimath] | [asciimath]2042.61[/asciimath] | [asciimath]0[/asciimath] |
| Totals | [asciimath]$15,577.04[/asciimath] | [asciimath]$577.04[/asciimath] | [asciimath]$15,000[/asciimath] | n/a |
The below figure features a bar chart that shows the interest and principal portions of each payment for this example. It also includes a line chart that depicts the remaining principal balance after each payment is made. It is important to note that as the outstanding balance reduces over time, the interest portion of each payment also decreases. Since the payment amount (PMT) is constant for each period (except for the last payment), the decreasing interest portion leads to an increased proportion of each payment being allocated to reduce the loan principal.
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C. Calculating The Principal Balance
At times, we may need to determine the principal balance of a loan before the loan term ends, such as when considering an early settlement or making a partial repayment. This can be achieved by constructing an amortization schedule to find the outstanding principal at the desired time, or by utilizing the calculator approach outlined below.
Calculator Approach:
- Enter the loan details into the Time Value of Money (TVM) worksheet of the financial calculator.
- Calculate any variable (if any) that is unknown, for example, the payment size (PMT) or the number of payments (N).
- Access the Amortization (AMORT) worksheet on the calculator. Input the range of payment periods for which the amortization needs to be calculated. For instance, to determine the balance at the end of the first year of a loan with monthly payments, set P2 to 12. P1 can be any value less than or equal to 12 since the calculator primarily considers P2 to calculate the remaining balance.
- The calculator will provide the balance at the end of the specified range, which represents the principal balance at that date.
Megan took out a $20,000 loan at an interest rate of 4% compounded quarterly. The loan is scheduled to be fully repaid over 8 years, with payments due at the end of each quarter. Calculate the remaining balance on the loan after the first year.
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Given information:
- Interest is compounded annually so [asciimath]C//Y = 4[/asciimath]
- Deposits are made at the end of every year so [asciimath]P//Y = 4[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Loan amortization period: [asciimath]t = 8[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=4 (8) =32[/asciimath]
- Nominal interest rate:[asciimath]I//Y = 4%[/asciimath]
- Loan principal is cash inflow: [asciimath]PV = $20,000[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
a) [asciimath]PMT=?[/asciimath]
First, we need to compute the size of the payment (PMT) using the TVM worksheet.
Thus, the size of the payment is $733.42. Note that PMT is rounded to the nearest cents, and then the rounded value is re-entered in the PMT key.
b) The number of payments up to the focal date of [asciimath]t=1[/asciimath] year is [asciimath]N= 4(1)=4[/asciimath].
To find the outstanding balance at the end of the first year, we use the AMORT worksheet and set P1 = P2 = 4 (P1 can be any value less than or equal to 4 but make sure P2 = 4).
Therefore, the outstanding principal balance at the end of the first year is $17,834.10.
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D. Calculating the Interest and Principal Portions
Sometimes, it may be necessary to determine the interest or principal portion of a specific payment or a specific period of time within the loan term.
For a Specific Payment: To find the interest and principal portion of a specific payment, in the AMORT worksheet, set both P1 and P2 to that specific payment number. Then, review the details provided to see the breakdown of interest and principal for that specific payment.
For a Specific Period: To determine the total principal or interest paid during a certain period of a loan, such as the 5th year, first, identify the payment numbers that correspond to the start and end of this period. In the AMORT worksheet, set P1 to the payment number marking the start of the period and P2 to the payment number at the end of the period. The calculator will then display the cumulative interest and principal portions paid between these two points.
Samuel took out a $308,000 mortgage to buy an apartment. The mortgage is structured to be repaid with monthly payments of $2,375.11 at the end of each month. The interest rate on the mortgage is 4.62%, compounded monthly, and the loan is amortized over 15 years. a) Calculate the interest portion of the 21st payment. b) Calculate the principal portion of the 21st payment.
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Given information:
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Deposits are made at the end of every year so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Loan amortization period: [asciimath]t = 15[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (15) =180[/asciimath]
- Nominal interest rate:[asciimath]I//Y = 4.62%[/asciimath]
- Loan principal is cash inflow: [asciimath]PV = $308,000[/asciimath]
- Periodic payments are cash outflow: [asciimath]PMT=-$2,375.11[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
To find the interest and principal portions of the 21st payment, start by entering the loan information into the TVM worksheet. Next, access the AMORT worksheet and set both P1 and P2 to 21, which represents the 21st payment. Then, review the details provided to see the breakdown of interest and principal for that specific payment.
a) Therefore, the interest portion of the 21st payment is $1090.80.
b) The principal portion of the 21st payment is $1284.31.
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Samuel took out a $308,000 mortgage to buy an apartment. The mortgage is structured to be repaid with monthly payments of $2,375.11 at the end of each month. The interest rate on the mortgage is 4.62%, compounded monthly, and the loan is amortized over 15 years. a) Calculate the total principal amount repaid in the 8th year. b) Calculate the total amount of interest paid in the 8th year.
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Given information:
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Deposits are made at the end of every year so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Loan amortization period: [asciimath]t = 15[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (15) =180[/asciimath]
- Nominal interest rate:[asciimath]I//Y = 4.62%[/asciimath]
- Loan principal is cash inflow: [asciimath]PV = $308,000[/asciimath]
- Periodic payments are cash outflow: [asciimath]PMT=-$2,375.11[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
To determine the total principal or interest paid during a certain year of a loan, such as the 8th year, we need to identify the specific payment numbers that mark the beginning and end of that year. We then use the AMORT worksheet on a financial calculator to calculate those amounts:
i) To find the first payment of the 8th year, we calculate the number of payments made by the end of the 7th year, which is 12 payments per year times 7 years, equaling 84 payments. Therefore, the first payment of the 8th year is payment number 85 (84 from the previous years plus 1).
ii) The last payment of the 8th year is determined by the total number of payments made in 8 years, which is 12 payments per year times 8 years, equaling 96 payments.
iii) To calculate the interest and principal portions for the 8th year, you input the loan details into the TVM worksheet. Then, in the AMORT worksheet, set P1 to 85 and P2 to 96. The calculator will then display the cumulative interest and principal portions paid between these two points in time.
a) Therefore, the total principal amount repaid in the 8th year is $20,131.41.
b) The total interest amount paid in the 8th year is $8369.91.
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Section 4.1 Exercises
- Martina took out a $84,000 loan at an interest rate of 5.88% compounded semi-annually. The loan is scheduled to be fully repaid over 12 years, with payments due at the end of each month. a) Calculate the size of month-end payments. b) Calculate the principal balance at the end of year 2.
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a) PMT = $811.45
b) BAL = $73,724.15
- Erika took out a $32,600 loan at an interest rate of 4.83% compounded monthly. The loan is scheduled to be fully repaid over 9 years, with payments due at the end of each month. a) Calculate the size of month-end payments. b) Calculate the total interest amount paid in the 4th year. b) Calculate the total principal amount repaid in the 4th year.
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a) PMT = $372.80
b) INT = $1,048.36
c) PRN = $3,425.24
- Johnetta took out a $20,200 loan at an interest rate of 3.53% compounded monthly. The loan is scheduled to be fully repaid over 8 years, with payments due at the end of each month. a) Calculate the size of month-end payments. b) Calculate the interest portion of the 60th payment. c) Calculate the principal portion of the 60th payment.
Show/Hide Answer
a) PMT = $241.83
b) INT = $24.91
c) PRN = $216.92
