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Section 5.4 Amortization
Question 1 – How do you find the present value of an annuity?
Question 2 – How is a loan amortized?
Question 3 - How do you make an amortization table?
Question 1 – How do you find the present value of an annuity?
Key Terms
Present value
Summary
For an ordinary annuity whose present value is PV, the future value is
1 i n 1
FV PV 1 i
n
PMT
i
if the payments PMT are made into the annuity which earns interest per period i over n periods.
Since the payments are made into the annuity, the second term is added. The future value of the
annuity increases.
If the payments are made from the annuity, the second term is subtracted to give
1 i n 1
FV PV 1 i
n
PMT
i
In this case, the future value of the annuity decreases since money is removed from the annuity.
In some applications, we wish to find the present value (what must be in the account today) so
that the account ends up with some amount in the future. To find the percent value, we need to
substitute values for FV, i, PMT, and n and solve the resulting equation for PV.
Notes
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Guided Example
Find the present value of an ordinary annuity with payments of $10,000 paid semiannually for 15
years at 5% compounded semiannually.
Solution We’ll use the formula
1 i n 1
FV PV 1 i
n
PMT
i
to find the present value PV. Think of this as a decreasing annuity problem where we want the future
value to be zero. From the problem statement, we know that
PMT 10000
0.05
i 0.025
2
n 15 2 30
Put these values into the formula and solve for PV:
1 0.025 30 1
0 PV 1 0.025
30
10000
0.025
1 0.025 30 1
PV 1 0.025
30
10000 Move the second term to the left side
0.025
1 0.025 30 1
10000
0.025
PV Isolate PV using division.
1 0.025
30
If we evaluate this in a graphing calculator, we get approximately $209,302.93. This means that if we
deposit $209,302.93 today, we can make semiannual payments of $10,00 from the annuity for 15
years before there is nothing left in the annuity.
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Practice
1. Find the present value of an ordinary annuity with payments of $90,000 paid annually for 25 years
at 8% compounded annually.
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Question 2 – How is a loan amortized?
Key Terms
Payment Amortization
Summary
Decreasing annuities may be used in auto or home loans. In these types of loans, some amount of
money is borrowed. Fixed payments are made to pay off the loan as well as any accrued interest.
This process is called amortization.
In the language of finance, a loan is said to be amortized if the amount of the loan and interest
are paid using fixed regular payments. From the perspective of the lender, this type of loan is a
decreasing annuity. The amount of the loan is the present value of the annuity. The payments
from the annuity (to the lender) reduce the value of the annuity until the future value is zero.
Suppose a loan of PV dollars is amortized by periodic payments of PMT at the end of each
period. If the loan has an interest rate of i per period over n periods, the payment is
i PV
PMT=
1 1 i
n
Suppose you want to borrow $10,000 for an automobile. Navy Federal Credit Union offers a
loan at an annual rate of 1.79% amortized over 12 months. To find the payment, identify the key
quantities in the formula:
i 0.0179
12
PV 10, 000
n 12
Put these values into the payment formula to get
i PV 0.0179
10000
PMT= 12
841.44
1 1 i 1 1 0.0179
12
n 12
Pay careful attention to how problems must be rounded. Rounding up, down, or to the nearest
cent can change answers drastically.
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Notes
Guided Example Practice
Find the payment necessary to amortize a loan of 1. Find the payment necessary to amortize a loan
$7400 at an interest rate of 6.2% compounded of $25,000 at an interest rate of 8.4%
semiannually in 18 semiannual payments. compounded quarterly in 24 quarterly
payments.
Solution To find the payment, use the formula
i PV
PMT=
1 1 i
n
In this case,
PV 7400
0.062
i 0.031
2
n 18
Put the values in the formula to give
0.031 7400
PMT= 542.60
1 1 0.031
18
To find the total payments, multiply the amount
of each payment by 18 to get
542.60 18 9766.80
To find the total amount of interest paid, subtract
the original loan amount from the total payments,
9766.80 7400 2366.80
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Question 3 – How do you make an amortization table?
Key Terms
Amortization table
Summary
An amortization table (also called an amortization schedule) records the portion of the payment
that applies to the principal and the portion that applies to interest. Using this information, we
can determine exactly how much is owed on the loan at the end of any period.
The amortization table generally has 5 columns and rows corresponding to the initial loan
amount and the payments. The heading for each column are shown below.
Amount in Outstanding
Payment Amount of Interest in Payment Balance at the
Number Payment Payment Applied to End of the
Balance Period
To fill out the table, you need to carry out a sequence of steps to get each row of the table.
1. The first row of the table corresponds to the initial loan balance. Call this payment 0 and
place the amount loaned in the column title “Outstanding Balance at the End of the
Period”.
2. Go to the next line in the table and enter the payment calculated on the loan.
3. In the same row, use I PV rt to find the interest on the outstanding balance. Place this
under the column titled “Interest in Payment”.
4. To find the “Amount in Payment Applied to Balance”, subtract the “Interest in the
Payment” from the “Amount of Payment”.
5. To find the new “Outstanding Balance at the End of the Period”, subtract the “Amount in
Payment Applied to Balance” from the “Outstanding Balance at the End of the Period” in
the previous payment.
Fill out these quantities for all payments until the past payment. In the last payment, start by
paying off the loan by making “Amount in Payment Applied to Balance” equal to the
“Outstanding Balance at the End of the Period” in the second to last payment. This means the
loan will be paid off resulting in the “Outstanding Balance at the End of the Period” for the final
payment being 0. Finally, calculate the interest in the final payment and add it to the “Amount in
Payment Applied to Balance” to give the final payment. Because of rounding in the payment,
this may be slightly higher of lower than the other payments.
Let’s look at an example of a $10,000 for an automobile. Navy Federal Credit Union offers a
loan at an annual rate of 1.79% amortized over 12 months. The amortization table below shows
the calculation of the quantities for payment 1 and the last payment. Other payments follow a
similar process.
3. 38
. 0179
𝐼 = 𝑃𝑉 𝑟𝑡 = 10000 ∙ ∙1
12
Outstanding
Amount in
Payment Amount of Interest in Balance at the
Payment Applied
Number Payment the Payment End of the
to Balance
Period
2. 1. Starting
0 10000
Calculated balance
Payment 841.44 14.92 826.52 9173.48
1
2 841.44 13.68 827.76 8345.72 5.
3 841.44 12.45 828.99 7516.73 10000 – 826.52
4 841.44 11.21 830.23 6686.50
4.
841.44 – 14.92 5 841.44 9.97 831.47 5855.03
6 841.44 8.73 832.71 5022.32
7 841.44 7.49 833.95 4188.37
8 841.44 6.25 835.19 3353.18
9 841.44 5.00 836.44 2516.74
10 841.44 3.75 837.69 1679.05
11 841.44 2.50 838.94 840.11
12 841.36 1.25 840.11 0.00
Need to pay off the
loan in the last
payment
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Suppose a loan of $2500 is made to an individual at 6% interest compounded quarterly. The loan is
repaid in 6 quarterly payments.
a. Find the payment necessary to amortize the loan.
Solution To find the payment on the loan, use the formula
i PV
PMT
1 (1 i ) n
4 . The present value is PV 2500 and the
For this problem, the interest rate per period is i 0.06
number of periods is n 6 . Using these values gives
0.06
2500
PMT 4
438.813
1 1 0.06
4
6
Depending on how the rounding is done, this gives a payment of $438.81 or 438.82. For a calculated
payment, the payment is often rounded to the nearest penny. However, many finance companies will
round up to insure the last payment is no more than the other payments.
b. Find the total payments and the total amount of interest paid based on the calculated monthly
payments.
Solution The total payments (assuming the payment is rounded to the nearest penny) is
438.81 6 2632.86
The total amount of interest is
2632.86 2500 132.86
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c. Find the total payments and the total amount of interest paid based on an amortization table.
Solution Making the amortization table takes several steps. Let me take it in pieces using the payment
from above.
Amount in Outstanding
Payment Amount of Interest in Payment Balance at the
Number Payment Payment Applied to End of the
Balance Period
0 2500
1 438.81 37.50 401.31 2098.69
The next row is filled out in a similar manner.
Amount in Outstanding
Payment Amount of Interest in Payment Balance at the
Number Payment Payment Applied to End of the
Balance Period
0 2500
1 438.81 37.50 401.31 2098.69
2 438.81 31.48 407.33 1691.36
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Continue this process until the last row
Amount in Outstanding
Payment Amount of Interest in Payment Balance at the
Number Payment Payment Applied to End of the
Balance Period
0 2500
1 438.81 37.50 401.31 2098.69
2 438.81 31.48 407.33 1691.36
3 438.81 25.37 413.44 1277.92
4 438.81 19.17 419.64 850.28
5 438.81 12.87 425.94 432.34
6
After the fifth payment, we have $432.34 of principal left to pay in the final payment. So, this is the
principal in the sixth payment. The interest is found by paying interest on the outstanding balance,
0.06
4 432.34 6.49
This gives a final payment of
432.34 6.49 438.83
Now put these numbers into the amortization table.
Amount in Outstanding
Payment Amount of Interest in Payment Balance at the
Number Payment Payment Applied to End of the
Balance Period
0 2500
1 438.81 37.50 401.31 2098.69
2 438.81 31.48 407.33 1691.36
3 438.81 25.37 413.44 1277.92
4 438.81 19.17 419.64 850.28
5 438.81 12.87 425.94 432.34
6 438.83 6.49 432.34 0
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Since the payments had been rounded to the nearest penny (rounded down), the final payment
is slightly higher than the previous payments. Adding all of the payments we get a total of
$2632.88. Adding the interest amounts gives total interest of $132.88.
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Practice
1. Suppose a loan of $5000 is made to an individual at 4% interest compounded semiannually. The
loan is repaid in 6 semiannual payments.
a. Find the payment necessary to amortize the loan. Round the payment to the nearest penny.
b. Find the total payments and the total amount of interest paid based on the calculated monthly
payments.
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c. Find the total payments and the total amount of interest paid based on an amortization table.
