Loan Amortization
4.3 Loan Amortization: Formula 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 4.3.1, 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 4.3.1 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
How to Construct an Amortization Schedule: Algebraic (Formula) Approach
1. Calculate the periodic interest rate per payment period.
- For simple annuities, it will be [asciimath]i=(I//Y)/(C//Y)[/asciimath]
- For general Annuities, it will be [asciimath]i_2=(1+i)^c-1[/asciimath] where [asciimath]c=(C//Y)/(P//Y)[/asciimath] .
2. In the first row, enter zero for the payment number and the loan amount (PV) under the balance.
3. In the second row, 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. Enter the interest portion (INT) of the payment in the 3rd column. This is the previous principal balance multiplied by the periodic interest rate:
- [asciimath]INT_1=BAL_0xxi[/asciimath] for simple annuities or
- [asciimath]INT_1=BAL_0xxi_2[/asciimath] for general annuities.
5. Enter the principal portion (PRN) of the payment in the 4th column. This is the difference between the periodic payment amount and the interest portion: [asciimath]PRN_1=PMT-INT_1[/asciimath]
6. Enter the outstanding principal balance (BAL) in the 5th column. This is the difference between the previous principal balance and the principal portion: [asciimath]BAL_1=BAL_0-PRN_1[/asciimath]
7. Construct the remaining rows of the schedule by doing the same calculations in Steps 3 – 6 until the final payment number, where the principal balance will be zero.
8. The final payment amount is calculated by adding the previous principal balance and the interest charged on the previous principal balance. [asciimath]PMT_N=PRN_(N)+INT_N[/asciimath] where [asciimath]PRN_N=BAL_(N-1)[/asciimath] and
- [asciimath]INT_N=PRN_(N-1)xxi[/asciimath] for simple annuities
- [asciimath]INT_N=PRN_(N-1)xxi_2[/asciimath] for general annuities
9. 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
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]
- Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (10%)/1 =10%[/asciimath]
- Loan amount: [asciimath]PV = $10,000[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
a) First, we need to find the size of the payment (PMT). Substituting the values into Formula 3.9b gives
[asciimath]PMT=(PV*i)/(1-(1+i)^(-N))[/asciimath]
[asciimath]PMT=(10000(10%))/(1-(1+10%)^(-4))[/asciimath]
[asciimath]=3154.708...[/asciimath]
[asciimath]~~$3154.71[/asciimath]
b) Next, we follow the steps given in the “How To – Formula Approach” to fill in the amortization schedule. For simple annuities, the interest per payment period is equal to the interest per compounding period, so we use [asciimath]i=10%[/asciimath] for the payment interest portion (INT) calculation.
| Payment Number | Payment Amount PMT ($) | 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]10,000(10%)[/asciimath]
[asciimath]=1000[/asciimath] |
[asciimath]3154.71-1000[/asciimath]
[asciimath]=2154.71[/asciimath] |
[asciimath]10,000-2154.71[/asciimath]
[asciimath]=7845.29[/asciimath] |
| 2 | [asciimath]3154.71[/asciimath] | [asciimath]7845.29(10%)[/asciimath]
[asciimath]=784.53[/asciimath] |
[asciimath]3154.71-784.53[/asciimath]
[asciimath]=2370.18[/asciimath] |
[asciimath]7845.29-2370.18[/asciimath]
[asciimath]=5475.11[/asciimath] |
| 3 | [asciimath]3154.71[/asciimath] | [asciimath]5475.11(10%)[/asciimath]
[asciimath]=547.51[/asciimath] |
[asciimath]3154.71-547.51[/asciimath]
[asciimath]=2607.20[/asciimath] |
[asciimath]5475.11-2607.20[/asciimath]
[asciimath]=2867.91[/asciimath] |
| 4 | [asciimath]286.79+2867.91[/asciimath]
[asciimath]=3154.70[/asciimath] |
[asciimath]2867.91(10%)[/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 |
Step 9: Once the schedule is filled, we compute the totals and check that
- Total Principal Portion = Original loan amount
- Total Amount Paid = Total Interest Portion + Total Principal Portion
which are correct for this schedule.
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]
- Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (6.8%)/2 =3.4%[/asciimath]
- Loan principal: [asciimath]PV = $15,000[/asciimath]
- Size of payments: [asciimath]PMT = $4500[/asciimath]
- The loan is fully paid: [asciimath]FV=0[/asciimath]
a) [asciimath]N=?[/asciimath]
The term duration and thus the number of payments in the term are unknown. Since the present value of the annuity is known, we can apply Formula 3.10c to determine the number of payments, N, for an ordinary general annuity. The first step in this process is to calculate the interest rate per payment period, [asciimath]i_2[/asciimath].
[asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]
[asciimath]i_2=(1+0.034)^(2/4)-1[/asciimath]
[asciimath]=0.016857...[/asciimath]
Applying Formula 3.10c yields
[asciimath]N=-(ln[1-(PV*i_2)/(PMT)])/(ln(1+i_2))[/asciimath]
[asciimath]N=-(ln[1-(15,000(0.016857... ))/(4500)])/(ln(1+0.016857... ))[/asciimath]
[asciimath]=3.4...[/asciimath]
[asciimath]=4[/asciimath] (Rounded up)
The number of payments in the term, N, is rounded up to 4 payments.
b)
Next, we follow the steps given in the “How To – Formula Approach” to fill in the amortization schedule. For general annuities, the interest per payment period, [asciimath]i_2=0.016857...[/asciimath], should be used for the payment interest portion (INT) calculation.
| Payment Number | Payment Amount PMT ($) | 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]15,000(0.016857... )[/asciimath] [asciimath]=252.87[/asciimath] | [asciimath]4500-252.87[/asciimath] [asciimath]=4247.13[/asciimath] | [asciimath]15,000-4247.13[/asciimath] [asciimath]=10,752.87[/asciimath] |
| 2 | [asciimath]4500[/asciimath] | [asciimath]10,752.87(0.016857... )[/asciimath] [asciimath]=181.27[/asciimath] | [asciimath]4500-181.27[/asciimath] [asciimath]=4318.73[/asciimath] | [asciimath]10,752.87-4318.73[/asciimath] [asciimath]=6434.14[/asciimath] |
| 3 | [asciimath]4500[/asciimath] | [asciimath]6434.14(0.016857... )[/asciimath] [asciimath]=108.47[/asciimath] | [asciimath]4500-108.47[/asciimath] [asciimath]=4391.53[/asciimath] | [asciimath]6434.14-4391.53[/asciimath] [asciimath]=2042.61[/asciimath] |
| 4 | [asciimath]34.43+2042.61[/asciimath] [asciimath]=2077.04[/asciimath] | [asciimath]2042.61(0.016857...)[/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 formula approach outlined below.
The retrospective method can be used to calculate the principal balance at a specific focal point in time. This method involves the calculation of the future values of both the payments made and the loan principal up to that focal point. The principal balance at the focal date is then found by taking the difference between these future values.
Figure 4.3.2 Timeline for Retrospective method of calculating the balance at a specific focal date
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("Payments up to focal date")[/asciimath]Formula 4.1
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]
- Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (4%)/4 =1%[/asciimath]
- Loan principal: [asciimath]PV = $20,000[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
i) [asciimath]PMT=?[/asciimath]
First, we need to find the size of the payment (PMT). Substituting the values into Formula 3.9b gives
[asciimath]PMT=(PV*i)/(1-(1+i)^(-N))[/asciimath]
[asciimath]PMT=(20,000(1%))/(1-(1+1%)^(-32))[/asciimath]
[asciimath]=733.417...[/asciimath]
[asciimath]~~$733.42[/asciimath]
Thus, the size of the payment is $733.42.
ii) Next, we use Formula 4.1 to calculate the balance at the end of the first year (focal date). By the end of the first year, four payments will have been made.
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("PMTs made up to focal date")[/asciimath]
To find the future value of the original loan, we use the future value formula of compound interest (Formula 2.4a), and to calculate the future value of the payments, we apply the future value formula of the ordinary simple annuity (Formula 3.5a).
The number of payments up to the focal date of [asciimath]t=1[/asciimath] year is [asciimath]N= 4(1)=4[/asciimath].
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("PMTs made up to focal date")[/asciimath]
[asciimath]BAL_("Focal date")=PV(1+i)^N-[/asciimath] [asciimath]PMT[((1+i)^N-1)/i][/asciimath]
[asciimath]=20,000(1+0.01)^4-[/asciimath] [asciimath]733.42[((1+0.01)^4-1)/0.01][/asciimath]
[asciimath]=20,812.08-2977.98[/asciimath]
[asciimath]=$17,834.10[/asciimath]
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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 within the loan term. The interest portion for any payment is calculated by multiplying the principal balance outstanding after the previous payment and the periodic interest rate per payment period. After calculating the interest portion, the principal portion of the payment can be found by deducting the interest portion from the payment amount.
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]
- Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (4.62%)/12 =0.385%[/asciimath]
- Loan principal: [asciimath]PV = $308,000[/asciimath]
- Periodic payments: [asciimath]PMT=$2,375.11[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
To find the interest portion of the 21st payment, we need to find the principal balance outstanding after the previous payment (20th payment) is made.
i) Finding the previous outstanding balance
We use Formula 4.1 to calculate the balance after 20 payments (focal date).
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("PMTs made up to focal date")[/asciimath]
To find the future value of the original loan, we use the future value formula of compound interest (Formula 2.4a), and to calculate the future value of the payments, we apply the future value formula of the ordinary simple annuity (Formula 3.5a). Note that since the annuity is simple, the number of compounding periods and payments are equal for any time period, so in both formulas [asciimath]N= 20[/asciimath].
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("PMTs made up to focal date")[/asciimath]
[asciimath]BAL_("Focal date")=PV(1+i)^N-[/asciimath] [asciimath]PMT[((1+i)^N-1)/i][/asciimath]
[asciimath]=308,000(1+0.00385)^20-[/asciimath] [asciimath]2375.11[((1+0.00385 )^20-1)/0.00385 ][/asciimath]
[asciimath]=332,603.78-49,280.39[/asciimath]
[asciimath]=$283,323.39[/asciimath]
ii) Calculating the interest portion of the 21st payment
a) The interest portion on a payment is the product of the outstanding balance and the interest rate per payment period (Step 4 of the “How to Construct an Amortization Schedule”). For simple annuities, the interest per payment period ([asciimath]i_2[/asciimath]) equals the interest rate per compounding period ([asciimath]i[/asciimath]). Thus the interest portion on the 21st payment is obtained by
[asciimath]INT_21=BAL_20*i[/asciimath]
[asciimath]=283,323.39(0.00385)[/asciimath]
[asciimath]~~$1090.80[/asciimath]
iii) Calculating the principal portion of the 21st payment
b) The principal portion of a payment is the difference between the amount of payment and the interest portion of that payment.
[asciimath]PRN_21=PMT-INT_21[/asciimath]
[asciimath]=2375.11-1090.80[/asciimath]
[asciimath]=$1284.31[/asciimath]
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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]
- Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (4.62%)/12 =0.385%[/asciimath]
- Loan principal: [asciimath]PV = $308,000[/asciimath]
- Periodic payments: [asciimath]PMT=$2,375.11[/asciimath]
- Loan is fully repaid, so [asciimath]FV=0[/asciimath]
The total amount of principal repaid over a specific time period is calculated by subtracting the principal balance at the end of the period from the principal balance at the beginning of that period.
[asciimath]"Principal repaid in year 8"=[/asciimath] [asciimath]BAL_("7th year")-BAL_("8th year")[/asciimath]
i) Finding the principal balance at the end of the 7th year (same as the beginning of the 8th year):
The number of payments up to the focal date of [asciimath]t=7[/asciimath] year is [asciimath]N= 12(7)=84[/asciimath].
[asciimath]BAL_("7th year")=FV_("Original loan")-[/asciimath] [asciimath]FV_("payments made")[/asciimath]
[asciimath]BAL_("7th year")=PV(1+i)^N-[/asciimath] [asciimath]PMT[((1+i)^N-1)/i][/asciimath]
[asciimath]=308,000(1+0.00385)^84-[/asciimath] [asciimath]2375.11[((1+0.00385 )^84-1)/0.00385 ][/asciimath]
[asciimath]=425,335.739-235,018.795[/asciimath]
[asciimath]~~$190,316.94[/asciimath]
ii) Finding the principal balance at the end of the 8th year:
The number of payments up to the focal date of [asciimath]t=8[/asciimath] year is [asciimath]N= 12(8)=96[/asciimath].
[asciimath]BAL_("8th year")=FV_("Original loan")-[/asciimath] [asciimath]FV_("payments made")[/asciimath]
[asciimath]BAL_("8th year")=PV(1+i)^N-[/asciimath] [asciimath]PMT[((1+i)^N-1)/i][/asciimath]
[asciimath]=308,000(1+0.00385)^96-[/asciimath] [asciimath]2375.11[((1+0.00385 )^96-1)/0.00385 ][/asciimath]
[asciimath]=445,407.736-275,222.203[/asciimath]
[asciimath]~~$170,185.53[/asciimath]
iii) principal repaid in the 8th year:
a)
[asciimath]"Principal repaid in year 8"=[/asciimath] [asciimath]BAL_("7th year")-BAL_("8th year")[/asciimath]
[asciimath]=190,316.94-170,185.53[/asciimath]
[asciimath]=$20,131.41[/asciimath]
b) The total interest paid in the 8th year of a loan is calculated by subtracting the amount of principal repaid during that year from the total payments made in the 8th year.
Interest paid in the 8th year = Total paid in the 8th year – Principal repaid in the 8th year
[asciimath]=(12 \ "payments"xx $2375.11)-$20,131.41[/asciimath]
[asciimath]=28,501.32-20,131.41[/asciimath]
[asciimath]=$8369.91[/asciimath]
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Section 4.3 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.
Show/Hide Answer
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.
Show/Hide Answer
a) PMT = $372.80
b) INT = $1,048.37
c) PRN = $3,425.23
- 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.
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a) PMT = $241.83
b) INT = $24.91
c) PRN = $216.92
