The document discusses present value calculations for annuities and amortization schedules. It provides the formula to calculate the present value of an ordinary annuity. It then works through two examples, one calculating the present value needed to fund annual withdrawals over 20 years, and the other calculating the monthly payment for a $50,000, 20-year mortgage at 7.2% interest compounded monthly.

1. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Math 1300 Finite Mathematics
Section 3.4 Present Value of an Annuity; Amortization
Jason Aubrey
Department of Mathematics
University of Missouri
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Jason Aubrey Math 1300 Finite Mathematics

2. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Present Value
Present value is the value on a given date of a future
payment or series of future payments, discounted to reflect
the time value of money and other factors such as
investment risk.
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Jason Aubrey Math 1300 Finite Mathematics

3. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Present Value
Present value is the value on a given date of a future
payment or series of future payments, discounted to reflect
the time value of money and other factors such as
investment risk.
Present value calculations are widely used in business and
economics to provide a means to compare cash flows at
different times on a meaningful "like to like" basis.
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Jason Aubrey Math 1300 Finite Mathematics

4. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

5. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
where
PV = present value of all payments
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Jason Aubrey Math 1300 Finite Mathematics

6. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
where
PV = present value of all payments
PMT = periodic payment
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Jason Aubrey Math 1300 Finite Mathematics

7. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
where
PV = present value of all payments
PMT = periodic payment
i = rate per period
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Jason Aubrey Math 1300 Finite Mathematics

8. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
where
PV = present value of all payments
PMT = periodic payment
i = rate per period
n = number of periods
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Jason Aubrey Math 1300 Finite Mathematics

9. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
1 − (1 + i)−n
PV = PMT
i
where
PV = present value of all payments
PMT = periodic payment
i = rate per period
n = number of periods
Note: Payments are made at the end of each period.
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Jason Aubrey Math 1300 Finite Mathematics

10. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuity
with a guaranteed rate of 6.65% compounded annually. How
much should you pay for one of these annuities if you want to
receive payments of $5,000 annually over the 10-year period?
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Jason Aubrey Math 1300 Finite Mathematics

11. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuity
with a guaranteed rate of 6.65% compounded annually. How
much should you pay for one of these annuities if you want to
receive payments of $5,000 annually over the 10-year period?
r 0.0665
Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.
So,
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Jason Aubrey Math 1300 Finite Mathematics

12. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuity
with a guaranteed rate of 6.65% compounded annually. How
much should you pay for one of these annuities if you want to
receive payments of $5,000 annually over the 10-year period?
r 0.0665
Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.
So,
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

13. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuity
with a guaranteed rate of 6.65% compounded annually. How
much should you pay for one of these annuities if you want to
receive payments of $5,000 annually over the 10-year period?
r 0.0665
Here m = 1; n = 10; i = m = 1 = 0.0665; PMT = $5, 000.
So,
1 − (1 + i)−n
PV = PMT
i
1 − (1.0665)−10
PV = ($5, 000) = $35, 693.18
.0665
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Jason Aubrey Math 1300 Finite Mathematics

14. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: Recently, Lincoln Benefit Life offered an ordinary
annuity that earned 6.5% compounded annually. A person
plans to make equal annual deposits into this account for 25
years in order to then make 20 equal annual withdrawals of
$25,000, reducing the balance in the account to zero. How
much must be deposited annually to accumlate sufficient funds
to provide for these payments? How much total interest is
earned during this entire 45-year process?
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Jason Aubrey Math 1300 Finite Mathematics

15. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
We first find the present value necessary to provide for the
withdrawals.
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Jason Aubrey Math 1300 Finite Mathematics

16. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
We first find the present value necessary to provide for the
withdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
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Jason Aubrey Math 1300 Finite Mathematics

17. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
We first find the present value necessary to provide for the
withdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

18. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
We first find the present value necessary to provide for the
withdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
1 − (1 + i)−n
PV = PMT
i
1 − (1.065)−20
PV = ($25, 000) = $275, 462.68
.065
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Jason Aubrey Math 1300 Finite Mathematics

19. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Now we find the deposits that will produce a future value of
$275,462.68 in 25 years.
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Jason Aubrey Math 1300 Finite Mathematics

20. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Now we find the deposits that will produce a future value of
$275,462.68 in 25 years.
Here we use FV = $275,462.68, i = 0.065 and n = 25.
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Jason Aubrey Math 1300 Finite Mathematics

21. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Now we find the deposits that will produce a future value of
$275,462.68 in 25 years.
Here we use FV = $275,462.68, i = 0.065 and n = 25.
(1 + i)n − 1
FV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

22. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Now we find the deposits that will produce a future value of
$275,462.68 in 25 years.
Here we use FV = $275,462.68, i = 0.065 and n = 25.
(1 + i)n − 1
FV = PMT
i
(1.065)25 − 1
$275, 462.68 = PMT
.065
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Jason Aubrey Math 1300 Finite Mathematics

23. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Now we find the deposits that will produce a future value of
$275,462.68 in 25 years.
Here we use FV = $275,462.68, i = 0.065 and n = 25.
(1 + i)n − 1
FV = PMT
i
(1.065)25 − 1
$275, 462.68 = PMT
.065
.065
PMT = ($275, 462.68) = $4, 677.76
(1.065)25 − 1
Thus, depositing $4,677.76 annually for 25 years will provide
for 20 annual withdrawals of $25,000.
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Jason Aubrey Math 1300 Finite Mathematics

24. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
The interest earned during the entire 45-year process is
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Jason Aubrey Math 1300 Finite Mathematics

25. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
The interest earned during the entire 45-year process is
interest = (total withdrawals) − (total deposits)
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Jason Aubrey Math 1300 Finite Mathematics

26. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
The interest earned during the entire 45-year process is
interest = (total withdrawals) − (total deposits)
= 20($25, 000) − 25($4, 677.76)
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Jason Aubrey Math 1300 Finite Mathematics

27. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
The interest earned during the entire 45-year process is
interest = (total withdrawals) − (total deposits)
= 20($25, 000) − 25($4, 677.76)
= $383, 056
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Jason Aubrey Math 1300 Finite Mathematics

28. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization
In business, amortization is the distribution of a single
lump-sum cash flow into many smaller cash flow
installments, as determined by an amortization schedule.
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Jason Aubrey Math 1300 Finite Mathematics

29. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization
In business, amortization is the distribution of a single
lump-sum cash flow into many smaller cash flow
installments, as determined by an amortization schedule.
Unlike other repayment models, each repayment
installment consists of both principal and interest.
Amortization is chiefly used in loan repayments (a common
example being a mortgage loan) and in sinking funds.
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Jason Aubrey Math 1300 Finite Mathematics

30. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization
Payments are divided into equal amounts for the duration
of the loan, making it the simplest repayment model.
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Jason Aubrey Math 1300 Finite Mathematics

31. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization
Payments are divided into equal amounts for the duration
of the loan, making it the simplest repayment model.
A greater amount of the payment is applied to interest at
the beginning of the amortization schedule, while more
money is applied to principal at the end.
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Jason Aubrey Math 1300 Finite Mathematics

32. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%
compounded monthly. Find the monthly payment.
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Jason Aubrey Math 1300 Finite Mathematics

33. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%
compounded monthly. Find the monthly payment.
0.072
We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;
PV = $50, 000
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Jason Aubrey Math 1300 Finite Mathematics

34. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%
compounded monthly. Find the monthly payment.
0.072
We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;
PV = $50, 000
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

35. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%
compounded monthly. Find the monthly payment.
0.072
We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;
PV = $50, 000
1 − (1 + i)−n
PV = PMT
i
1 − (1.006)−240
$50, 000 = PMT
.006
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Jason Aubrey Math 1300 Finite Mathematics

36. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%
compounded monthly. Find the monthly payment.
0.072
We note that m = 12; i = 12 = 0.006; n = 20 × 12 = 240;
PV = $50, 000
1 − (1 + i)−n
PV = PMT
i
1 − (1.006)−240
$50, 000 = PMT
.006
.006
PMT = ($50, 000) = $393.67
1 − (1.006)−240
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Jason Aubrey Math 1300 Finite Mathematics

37. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
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Jason Aubrey Math 1300 Finite Mathematics

38. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaid
balance after 5 years.
In these "unpaid balance after" problems, n represents the
number of interest periods remaining.
Here n = 240 − 60 = 180. Therefore,
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Jason Aubrey Math 1300 Finite Mathematics

39. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaid
balance after 5 years.
In these "unpaid balance after" problems, n represents the
number of interest periods remaining.
Here n = 240 − 60 = 180. Therefore,
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

40. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaid
balance after 5 years.
In these "unpaid balance after" problems, n represents the
number of interest periods remaining.
Here n = 240 − 60 = 180. Therefore,
1 − (1 + i)−n
PV = PMT
i
1 − (1.006)−180
PV = (393.67) = $43, 258.22
.006
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41. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10
years.
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Jason Aubrey Math 1300 Finite Mathematics

42. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10
years.
Here n = 240 − 120 = 120 and so,
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Jason Aubrey Math 1300 Finite Mathematics

43. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10
years.
Here n = 240 − 120 = 120 and so,
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

44. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10
years.
Here n = 240 − 120 = 120 and so,
1 − (1 + i)−n
PV = PMT
i
1 − (1.006)−120
PV = ($393.67) = $33, 606.26
.006
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45. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000
debt that is to be amortized in eight equal quarterly payments
at 2.8% compounded quarterly.
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Jason Aubrey Math 1300 Finite Mathematics

46. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000
debt that is to be amortized in eight equal quarterly payments
at 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we have
m = 4; n = 8; i = m = 0.028 = 0.007. Then
r
4
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Jason Aubrey Math 1300 Finite Mathematics

47. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000
debt that is to be amortized in eight equal quarterly payments
at 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we have
m = 4; n = 8; i = m = 0.028 = 0.007. Then
r
4
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

48. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000
debt that is to be amortized in eight equal quarterly payments
at 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we have
m = 4; n = 8; i = m = 0.028 = 0.007. Then
r
4
1 − (1 + i)−n
PV = PMT
i
1 − (1.007)−8
$5, 000 = PMT
.007
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Jason Aubrey Math 1300 Finite Mathematics

49. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000
debt that is to be amortized in eight equal quarterly payments
at 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we have
m = 4; n = 8; i = m = 0.028 = 0.007. Then
r
4
1 − (1 + i)−n
PV = PMT
i
1 − (1.007)−8
$5, 000 = PMT
.007
.007
PMT = ($5, 000) = $644.85
1 − (1.007)−8
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Jason Aubrey Math 1300 Finite Mathematics

50. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0
1
2
3
4
5
6
7
8
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

51. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1
2
3
4
5
6
7
8
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

52. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85
2 $644.85
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

53. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35
2 $644.85
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

54. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85
2 $644.85
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

55. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

56. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

57. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

58. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

59. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

60. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

61. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

62. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

63. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

64. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

65. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

66. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

67. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

68. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

69. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

70. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

71. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

72. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93 $635.50
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

73. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93 $635.50 $640.35
8 $644.85
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

74. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93 $635.50 $640.35
8 $644.85 $4.48
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

75. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93 $635.50 $640.35
8 $644.85 $4.48 $640.37
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

76. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.
0 $5,000
1 $644.85 $35 $609.85 $4,390.15
2 $644.85 $30.73 $614.12 $3,776.03
3 $644.85 $26.43 $618.42 $3,157.61
4 $644.85 $22.10 $622.75 $2,534.87
5 $644.85 $17.74 $627.11 $1,907.76
6 $644.85 $13.35 $631.50 $1,276.26
7 $644.85 $8.93 $635.50 $640.35
8 $644.85 $4.48 $640.37 $0.00*
Interest owed during a period = university-logo
(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics

77. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A family purchased a home 10 years ago for
$80,000. The home was financed by paying 20% down and
signing a 30-year mortgage at 9% on the unpaid balance. The
net market value of the house (amount recieved after
subtracting all costs involved in selling the house) is now
$120,000, and the family wishes to sell the house. How much
equity (to the nearest dollar) does the family have in the house
now after making 120 monthly payments?
[Equity = (current net market value) - (unpaid loan balance)]
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Jason Aubrey Math 1300 Finite Mathematics

78. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1. Find the monthly payment:
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Jason Aubrey Math 1300 Finite Mathematics

79. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1. Find the monthly payment:
r 0.09
Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075
and n = 360.
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Jason Aubrey Math 1300 Finite Mathematics

80. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1. Find the monthly payment:
r 0.09
Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075
and n = 360.
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

81. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1. Find the monthly payment:
r 0.09
Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075
and n = 360.
1 − (1 + i)−n
PV = PMT
i
1 − (1.0075)−360
$64, 000 = PMT
.0075
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Jason Aubrey Math 1300 Finite Mathematics

82. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1. Find the monthly payment:
r 0.09
Here PV = (0.80)($80,000) = $64,000, i = m = 12 = 0.0075
and n = 360.
1 − (1 + i)−n
PV = PMT
i
1 − (1.0075)−360
$64, 000 = PMT
.0075
.0075
PMT = ($64, 000) = $514.96
1 − (1.0075)−360
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Jason Aubrey Math 1300 Finite Mathematics

83. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a
$514.96 per month, 20-year annuity):
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Jason Aubrey Math 1300 Finite Mathematics

84. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a
$514.96 per month, 20-year annuity):
0.09
Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075.
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Jason Aubrey Math 1300 Finite Mathematics

85. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a
$514.96 per month, 20-year annuity):
0.09
Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075.
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

86. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a
$514.96 per month, 20-year annuity):
0.09
Here PMT = $514.96, n = 12(20) = 240, i = 12 = 0.0075.
1 − (1 + i)−n
PV = PMT
i
1 − (1.0075)−240
PV = ($514.96) = $57, 235
.0075
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Jason Aubrey Math 1300 Finite Mathematics

87. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3. Find the equity:
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Jason Aubrey Math 1300 Finite Mathematics

88. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3. Find the equity:
Equity = (current net market value) − (unpaid loan balance)
= $120, 000 − $57, 235
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Jason Aubrey Math 1300 Finite Mathematics

89. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3. Find the equity:
Equity = (current net market value) − (unpaid loan balance)
= $120, 000 − $57, 235
= $62, 765
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Jason Aubrey Math 1300 Finite Mathematics

90. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3. Find the equity:
Equity = (current net market value) − (unpaid loan balance)
= $120, 000 − $57, 235
= $62, 765
Thus, if the family sells the house for $120,000 net, the family
will have $62,765 after paying off the unpaid loan balance of
$57,235.
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Jason Aubrey Math 1300 Finite Mathematics

91. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: A person purchased a house 10 years ago for
$120,000 by paying 20% down and signing a 30-year mortgage
at 10.2% compounded monthly. Interest rates have dropped
and the owner wants to refinance the unpaid balance by
signing a new 20-year mortgage at 7.5% compounded monthly.
How much interest will the refinancing save?
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Jason Aubrey Math 1300 Finite Mathematics

92. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1: Find monthly payments.
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Jason Aubrey Math 1300 Finite Mathematics

93. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1: Find monthly payments.
The owner put 20% down at the time of purchase.
Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000.
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Jason Aubrey Math 1300 Finite Mathematics

94. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1: Find monthly payments.
The owner put 20% down at the time of purchase.
Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000.
We also have that
r 0.102
m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085.
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Jason Aubrey Math 1300 Finite Mathematics

95. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1: Find monthly payments.
The owner put 20% down at the time of purchase.
Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000.
We also have that
r 0.102
m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085.
1 − (1.0085)−360
$96, 000 = PMT
0.0085
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Jason Aubrey Math 1300 Finite Mathematics

96. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 1: Find monthly payments.
The owner put 20% down at the time of purchase.
Therefore, PV = $120, 000 − (0.2)($120, 000) = $96, 000.
We also have that
r 0.102
m = 12; n = 30 × 12 = 360; i = m = 12 = 0.0085.
1 − (1.0085)−360
$96, 000 = PMT
0.0085
PMT = $856.69
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Jason Aubrey Math 1300 Finite Mathematics

97. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time of
refinancing).
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Jason Aubrey Math 1300 Finite Mathematics

98. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time of
refinancing).
Here we apply the formula with i = 0.0085 and n = 240.
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Jason Aubrey Math 1300 Finite Mathematics

99. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time of
refinancing).
Here we apply the formula with i = 0.0085 and n = 240.
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

100. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time of
refinancing).
Here we apply the formula with i = 0.0085 and n = 240.
1 − (1 + i)−n
PV = PMT
i
1 − (1.0085)−240
PV = ($856.69) = $87, 568.38
.0085
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Jason Aubrey Math 1300 Finite Mathematics

101. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment after
refinancing.
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Jason Aubrey Math 1300 Finite Mathematics

102. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment after
refinancing.
0.075
Here we apply the formula with i = 12 = 0.00625 and
n = 240.
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Jason Aubrey Math 1300 Finite Mathematics

103. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment after
refinancing.
0.075
Here we apply the formula with i = 12 = 0.00625 and
n = 240.
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

104. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment after
refinancing.
0.075
Here we apply the formula with i = 12 = 0.00625 and
n = 240.
1 − (1 + i)−n
PV = PMT
i
1 − (1.00625)−240
$87, 568.38 = PMT
.00625
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Jason Aubrey Math 1300 Finite Mathematics

105. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment after
refinancing.
0.075
Here we apply the formula with i = 12 = 0.00625 and
n = 240.
1 − (1 + i)−n
PV = PMT
i
1 − (1.00625)−240
$87, 568.38 = PMT
.00625
.00625
PMT = ($87, 568.38) = $705.44
1 − (1.00625)−240
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Jason Aubrey Math 1300 Finite Mathematics

106. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 4: We now compare the amount he would have spent
without refinancing to the amount he spends after refinancing.
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Jason Aubrey Math 1300 Finite Mathematics

107. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 4: We now compare the amount he would have spent
without refinancing to the amount he spends after refinancing.
If the owner did not refinance, he would pay a total of
856.69 × 240 = $205, 605.60 in principal and interest
during the last 20 years of the loan.
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Jason Aubrey Math 1300 Finite Mathematics

108. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Step 4: We now compare the amount he would have spent
without refinancing to the amount he spends after refinancing.
If the owner did not refinance, he would pay a total of
856.69 × 240 = $205, 605.60 in principal and interest
during the last 20 years of the loan.
This would amount to a total of
$205, 605.60 − $87, 568.38 = $118, 037.22 in interest.
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Jason Aubrey Math 1300 Finite Mathematics

109. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
After refinancing, the owner pays a total of
$705.44x240 = $169, 305.60 in principal and interest.
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Jason Aubrey Math 1300 Finite Mathematics

110. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
After refinancing, the owner pays a total of
$705.44x240 = $169, 305.60 in principal and interest.
This would amount to a total of
$169, 305.60 − $87, 568.38 = $81, 737.22 in interest.
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Jason Aubrey Math 1300 Finite Mathematics

111. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
After refinancing, the owner pays a total of
$705.44x240 = $169, 305.60 in principal and interest.
This would amount to a total of
$169, 305.60 − $87, 568.38 = $81, 737.22 in interest.
Therefore refinancing results in a total interest savings of
$118, 037.22 − $81, 737.22 = $36, 299.84.
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Jason Aubrey Math 1300 Finite Mathematics

112. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. The
dealer offers you 0% financing for 60 months or a $5,000
rebate. You can obtain 6.3% financing for 60 months at the
local bank. Which option should you choose?
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Jason Aubrey Math 1300 Finite Mathematics

113. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. The
dealer offers you 0% financing for 60 months or a $5,000
rebate. You can obtain 6.3% financing for 60 months at the
local bank. Which option should you choose?
To answer this question, we determine which option gives the
lowest monthly payment.
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Jason Aubrey Math 1300 Finite Mathematics

114. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. The
dealer offers you 0% financing for 60 months or a $5,000
rebate. You can obtain 6.3% financing for 60 months at the
local bank. Which option should you choose?
To answer this question, we determine which option gives the
lowest monthly payment.
Option 1: If you choose 0% financing, your monthly payment
will be
$27, 300
PMT1 = = $455
60
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Jason Aubrey Math 1300 Finite Mathematics

115. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
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Jason Aubrey Math 1300 Finite Mathematics

116. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,
i = 0.063 = 0.00525 and n = 60.
12
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Jason Aubrey Math 1300 Finite Mathematics

117. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,
i = 0.063 = 0.00525 and n = 60.
12
1 − (1 + i)−n
PV = PMT
i
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Jason Aubrey Math 1300 Finite Mathematics

118. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,
i = 0.063 = 0.00525 and n = 60.
12
1 − (1 + i)−n
PV = PMT
i
1 − (1.00525)−60
$22, 300 = PMT
0.00525
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Jason Aubrey Math 1300 Finite Mathematics

119. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,
i = 0.063 = 0.00525 and n = 60.
12
1 − (1 + i)−n
PV = PMT
i
1 − (1.00525)−60
$22, 300 = PMT
0.00525
PMT = $434.24
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Jason Aubrey Math 1300 Finite Mathematics

120. Present Value of an Ordinary Annuity
Amortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate and
borrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,
i = 0.063 = 0.00525 and n = 60.
12
1 − (1 + i)−n
PV = PMT
i
1 − (1.00525)−60
$22, 300 = PMT
0.00525
PMT = $434.24
You should choose the rebate. You will save $455 - $434.24 =
$20.76 monthly, or ($20.76)(60) = $1,245.60 over the life of the
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