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Lecture for Section 3-3 of Barnett's "Finite Mathematics."
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Math 1300 Finite MathematicsSection 3.4 Present Value of an Annuity; Amortization
Jason Aubrey
Department of MathematicsUniversity of Missouri
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Present Value
Present value is the value on a given date of a futurepayment or series of future payments, discounted to reflectthe time value of money and other factors such asinvestment risk.
Present value calculations are widely used in business andeconomics to provide a means to compare cash flows atdifferent times on a meaningful "like to like" basis.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Present Value
Present value is the value on a given date of a futurepayment or series of future payments, discounted to reflectthe time value of money and other factors such asinvestment risk.Present value calculations are widely used in business andeconomics to provide a means to compare cash flows atdifferent times on a meaningful "like to like" basis.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all paymentsPMT = periodic paymenti = rate per periodn = number of periods
Note: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all payments
PMT = periodic paymenti = rate per periodn = number of periods
Note: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all paymentsPMT = periodic payment
i = rate per periodn = number of periods
Note: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all paymentsPMT = periodic paymenti = rate per period
n = number of periodsNote: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all paymentsPMT = periodic paymenti = rate per periodn = number of periods
Note: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Theorem (The present value of an ordinary annuity)
PV =
[1− (1 + i)−n
i
]PMT
wherePV = present value of all paymentsPMT = periodic paymenti = rate per periodn = number of periods
Note: Payments are made at the end of each period.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period?
Here m = 1; n = 10; i = rm = 0.0665
1 = 0.0665; PMT = $5, 000.So,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0665)−10
.0665
]($5, 000) = $35, 693.18
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period?
Here m = 1; n = 10; i = rm = 0.0665
1 = 0.0665; PMT = $5, 000.So,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0665)−10
.0665
]($5, 000) = $35, 693.18
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period?
Here m = 1; n = 10; i = rm = 0.0665
1 = 0.0665; PMT = $5, 000.So,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0665)−10
.0665
]($5, 000) = $35, 693.18
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: American General offers a 10-year ordinary annuitywith a guaranteed rate of 6.65% compounded annually. Howmuch should you pay for one of these annuities if you want toreceive payments of $5,000 annually over the 10-year period?
Here m = 1; n = 10; i = rm = 0.0665
1 = 0.0665; PMT = $5, 000.So,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0665)−10
.0665
]($5, 000) = $35, 693.18
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: Recently, Lincoln Benefit Life offered an ordinaryannuity that earned 6.5% compounded annually. A personplans to make equal annual deposits into this account for 25years in order to then make 20 equal annual withdrawals of$25,000, reducing the balance in the account to zero. Howmuch must be deposited annually to accumlate sufficient fundsto provide for these payments? How much total interest isearned during this entire 45-year process?
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
We first find the present value necessary to provide for thewithdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.065)−20
.065
]($25, 000) = $275, 462.68
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
We first find the present value necessary to provide for thewithdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.065)−20
.065
]($25, 000) = $275, 462.68
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
We first find the present value necessary to provide for thewithdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.065)−20
.065
]($25, 000) = $275, 462.68
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
We first find the present value necessary to provide for thewithdrawals.
In this calculation, PMT = $25,000, i = 0.065 and n = 20.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.065)−20
.065
]($25, 000) = $275, 462.68
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
FV =
[(1 + i)n − 1
i
]PMT
$275, 462.68 =
[(1.065)25 − 1
.065
]PMT
PMT =
[.065
(1.065)25 − 1
]($275, 462.68) = $4, 677.76
Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
FV =
[(1 + i)n − 1
i
]PMT
$275, 462.68 =
[(1.065)25 − 1
.065
]PMT
PMT =
[.065
(1.065)25 − 1
]($275, 462.68) = $4, 677.76
Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
FV =
[(1 + i)n − 1
i
]PMT
$275, 462.68 =
[(1.065)25 − 1
.065
]PMT
PMT =
[.065
(1.065)25 − 1
]($275, 462.68) = $4, 677.76
Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
FV =
[(1 + i)n − 1
i
]PMT
$275, 462.68 =
[(1.065)25 − 1
.065
]PMT
PMT =
[.065
(1.065)25 − 1
]($275, 462.68) = $4, 677.76
Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
FV =
[(1 + i)n − 1
i
]PMT
$275, 462.68 =
[(1.065)25 − 1
.065
]PMT
PMT =
[.065
(1.065)25 − 1
]($275, 462.68) = $4, 677.76
Thus, depositing $4,677.76 annually for 25 years will providefor 20 annual withdrawals of $25,000.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization
In business, amortization is the distribution of a singlelump-sum cash flow into many smaller cash flowinstallments, as determined by an amortization schedule.
Unlike other repayment models, each repaymentinstallment consists of both principal and interest.Amortization is chiefly used in loan repayments (a commonexample being a mortgage loan) and in sinking funds.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization
In business, amortization is the distribution of a singlelump-sum cash flow into many smaller cash flowinstallments, as determined by an amortization schedule.Unlike other repayment models, each repaymentinstallment consists of both principal and interest.Amortization is chiefly used in loan repayments (a commonexample being a mortgage loan) and in sinking funds.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization
Payments are divided into equal amounts for the durationof the loan, making it the simplest repayment model.
A greater amount of the payment is applied to interest atthe beginning of the amortization schedule, while moremoney is applied to principal at the end.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization
Payments are divided into equal amounts for the durationof the loan, making it the simplest repayment model.A greater amount of the payment is applied to interest atthe beginning of the amortization schedule, while moremoney is applied to principal at the end.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment.
We note that m = 12; i = 0.07212 = 0.006; n = 20× 12 = 240;
PV = $50, 000
PV =
[1− (1 + i)−n
i
]PMT
$50, 000 =
[1− (1.006)−240
.006
]PMT
PMT =
[.006
1− (1.006)−240
]($50, 000) = $393.67
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment.
We note that m = 12; i = 0.07212 = 0.006; n = 20× 12 = 240;
PV = $50, 000
PV =
[1− (1 + i)−n
i
]PMT
$50, 000 =
[1− (1.006)−240
.006
]PMT
PMT =
[.006
1− (1.006)−240
]($50, 000) = $393.67
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment.
We note that m = 12; i = 0.07212 = 0.006; n = 20× 12 = 240;
PV = $50, 000
PV =
[1− (1 + i)−n
i
]PMT
$50, 000 =
[1− (1.006)−240
.006
]PMT
PMT =
[.006
1− (1.006)−240
]($50, 000) = $393.67
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment.
We note that m = 12; i = 0.07212 = 0.006; n = 20× 12 = 240;
PV = $50, 000
PV =
[1− (1 + i)−n
i
]PMT
$50, 000 =
[1− (1.006)−240
.006
]PMT
PMT =
[.006
1− (1.006)−240
]($50, 000) = $393.67
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family has a $50,000, 20-year mortgage at 7.2%compounded monthly. Find the monthly payment.
We note that m = 12; i = 0.07212 = 0.006; n = 20× 12 = 240;
PV = $50, 000
PV =
[1− (1 + i)−n
i
]PMT
$50, 000 =
[1− (1.006)−240
.006
]PMT
PMT =
[.006
1− (1.006)−240
]($50, 000) = $393.67
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaidbalance after 5 years.In these "unpaid balance after" problems, n represents thenumber of interest periods remaining.
Here n = 240− 60 = 180. Therefore,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−180
.006
](393.67) = $43, 258.22
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaidbalance after 5 years.In these "unpaid balance after" problems, n represents thenumber of interest periods remaining.
Here n = 240− 60 = 180. Therefore,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−180
.006
](393.67) = $43, 258.22
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaidbalance after 5 years.In these "unpaid balance after" problems, n represents thenumber of interest periods remaining.
Here n = 240− 60 = 180. Therefore,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−180
.006
](393.67) = $43, 258.22
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, find the unpaid balance after 5 years.
We use the value of PMT=$393.67 to find the unpaidbalance after 5 years.In these "unpaid balance after" problems, n represents thenumber of interest periods remaining.
Here n = 240− 60 = 180. Therefore,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−180
.006
](393.67) = $43, 258.22
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10years.
Here n = 240− 120 = 120 and so,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−120
.006
]($393.67) = $33, 606.26
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10years.
Here n = 240− 120 = 120 and so,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−120
.006
]($393.67) = $33, 606.26
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10years.
Here n = 240− 120 = 120 and so,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−120
.006
]($393.67) = $33, 606.26
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
For the same mortgage, compute the unpaid balance after 10years.
Here n = 240− 120 = 120 and so,
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.006)−120
.006
]($393.67) = $33, 606.26
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000debt that is to be amortized in eight equal quarterly paymentsat 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we havem = 4; n = 8; i = r
m = 0.0284 = 0.007. Then
PV =
[1− (1 + i)−n
i
]PMT
$5, 000 =
[1− (1.007)−8
.007
]PMT
PMT =
[.007
1− (1.007)−8
]($5, 000) = $644.85
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000debt that is to be amortized in eight equal quarterly paymentsat 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we havem = 4; n = 8; i = r
m = 0.0284 = 0.007. Then
PV =
[1− (1 + i)−n
i
]PMT
$5, 000 =
[1− (1.007)−8
.007
]PMT
PMT =
[.007
1− (1.007)−8
]($5, 000) = $644.85
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000debt that is to be amortized in eight equal quarterly paymentsat 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we havem = 4; n = 8; i = r
m = 0.0284 = 0.007. Then
PV =
[1− (1 + i)−n
i
]PMT
$5, 000 =
[1− (1.007)−8
.007
]PMT
PMT =
[.007
1− (1.007)−8
]($5, 000) = $644.85
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000debt that is to be amortized in eight equal quarterly paymentsat 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we havem = 4; n = 8; i = r
m = 0.0284 = 0.007. Then
PV =
[1− (1 + i)−n
i
]PMT
$5, 000 =
[1− (1.007)−8
.007
]PMT
PMT =
[.007
1− (1.007)−8
]($5, 000) = $644.85
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Example: Construct the amortization schedule for a $5,000debt that is to be amortized in eight equal quarterly paymentsat 2.8% compounded quarterly.
First we calculate the quarterly payment. Here we havem = 4; n = 8; i = r
m = 0.0284 = 0.007. Then
PV =
[1− (1 + i)−n
i
]PMT
$5, 000 =
[1− (1.007)−8
.007
]PMT
PMT =
[.007
1− (1.007)−8
]($5, 000) = $644.85
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001
$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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $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 =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85
$8.93 $635.50 $640.35
8 $644.85
$4.48 $640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93
$635.50 $640.35
8 $644.85
$4.48 $640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93 $635.50
$640.35
8 $644.85
$4.48 $640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93 $635.50 $640.358 $644.85
$4.48 $640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93 $635.50 $640.358 $644.85 $4.48
$640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93 $635.50 $640.358 $644.85 $4.48 $640.37
$0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Amortization Schedules
Period Payment Payment Int. Bal. Reduction Unpaid Bal.0 $5,0001 $644.85 $35 $609.85 $4,390.152 $644.85 $30.73 $614.12 $3,776.033 $644.85 $26.43 $618.42 $3,157.614 $644.85 $22.10 $622.75 $2,534.875 $644.85 $17.74 $627.11 $1,907.766 $644.85 $13.35 $631.50 $1,276.267 $644.85 $8.93 $635.50 $640.358 $644.85 $4.48 $640.37 $0.00*
Interest owed during a period =(Balance during period)(Interest rate per period)
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A family purchased a home 10 years ago for$80,000. The home was financed by paying 20% down andsigning a 30-year mortgage at 9% on the unpaid balance. Thenet market value of the house (amount recieved aftersubtracting all costs involved in selling the house) is now$120,000, and the family wishes to sell the house. How muchequity (to the nearest dollar) does the family have in the housenow after making 120 monthly payments?
[Equity = (current net market value) - (unpaid loan balance)]
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 1. Find the monthly payment:
Here PV = (0.80)($80,000) = $64,000, i = rm = 0.09
12 = 0.0075and n = 360.
PV =
[1− (1 + i)−n
i
]PMT
$64, 000 =
[1− (1.0075)−360
.0075
]PMT
PMT =
[.0075
1− (1.0075)−360
]($64, 000) = $514.96
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 1. Find the monthly payment:
Here PV = (0.80)($80,000) = $64,000, i = rm = 0.09
12 = 0.0075and n = 360.
PV =
[1− (1 + i)−n
i
]PMT
$64, 000 =
[1− (1.0075)−360
.0075
]PMT
PMT =
[.0075
1− (1.0075)−360
]($64, 000) = $514.96
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 1. Find the monthly payment:
Here PV = (0.80)($80,000) = $64,000, i = rm = 0.09
12 = 0.0075and n = 360.
PV =
[1− (1 + i)−n
i
]PMT
$64, 000 =
[1− (1.0075)−360
.0075
]PMT
PMT =
[.0075
1− (1.0075)−360
]($64, 000) = $514.96
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 1. Find the monthly payment:
Here PV = (0.80)($80,000) = $64,000, i = rm = 0.09
12 = 0.0075and n = 360.
PV =
[1− (1 + i)−n
i
]PMT
$64, 000 =
[1− (1.0075)−360
.0075
]PMT
PMT =
[.0075
1− (1.0075)−360
]($64, 000) = $514.96
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 1. Find the monthly payment:
Here PV = (0.80)($80,000) = $64,000, i = rm = 0.09
12 = 0.0075and n = 360.
PV =
[1− (1 + i)−n
i
]PMT
$64, 000 =
[1− (1.0075)−360
.0075
]PMT
PMT =
[.0075
1− (1.0075)−360
]($64, 000) = $514.96
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity):
Here PMT = $514.96, n = 12(20) = 240, i = 0.0912 = 0.0075.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0075)−240
.0075
]($514.96) = $57, 235
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity):
Here PMT = $514.96, n = 12(20) = 240, i = 0.0912 = 0.0075.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0075)−240
.0075
]($514.96) = $57, 235
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity):
Here PMT = $514.96, n = 12(20) = 240, i = 0.0912 = 0.0075.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0075)−240
.0075
]($514.96) = $57, 235
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2. Find unpaid balance after 10 years (the PV of a$514.96 per month, 20-year annuity):
Here PMT = $514.96, n = 12(20) = 240, i = 0.0912 = 0.0075.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0075)−240
.0075
]($514.96) = $57, 235
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
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 familywill have $62,765 after paying off the unpaid loan balance of$57,235.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 familywill have $62,765 after paying off the unpaid loan balance of$57,235.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 familywill have $62,765 after paying off the unpaid loan balance of$57,235.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 familywill have $62,765 after paying off the unpaid loan balance of$57,235.
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: A person purchased a house 10 years ago for$120,000 by paying 20% down and signing a 30-year mortgageat 10.2% compounded monthly. Interest rates have droppedand the owner wants to refinance the unpaid balance bysigning a new 20-year mortgage at 7.5% compounded monthly.How much interest will the refinancing save?
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
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 thatm = 12; n = 30× 12 = 360; i = r
m = 0.10212 = 0.0085.
$96, 000 =
[1− (1.0085)−360
0.0085
]PMT
PMT = $856.69
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
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 thatm = 12; n = 30× 12 = 360; i = r
m = 0.10212 = 0.0085.
$96, 000 =
[1− (1.0085)−360
0.0085
]PMT
PMT = $856.69
Jason Aubrey Math 1300 Finite Mathematics
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Present Value of an Ordinary AnnuityAmortization
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 thatm = 12; n = 30× 12 = 360; i = r
m = 0.10212 = 0.0085.
$96, 000 =
[1− (1.0085)−360
0.0085
]PMT
PMT = $856.69
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 thatm = 12; n = 30× 12 = 360; i = r
m = 0.10212 = 0.0085.
$96, 000 =
[1− (1.0085)−360
0.0085
]PMT
PMT = $856.69
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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 thatm = 12; n = 30× 12 = 360; i = r
m = 0.10212 = 0.0085.
$96, 000 =
[1− (1.0085)−360
0.0085
]PMT
PMT = $856.69
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time ofrefinancing).
Here we apply the formula with i = 0.0085 and n = 240.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0085)−240
.0085
]($856.69) = $87, 568.38
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time ofrefinancing).
Here we apply the formula with i = 0.0085 and n = 240.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0085)−240
.0085
]($856.69) = $87, 568.38
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time ofrefinancing).
Here we apply the formula with i = 0.0085 and n = 240.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0085)−240
.0085
]($856.69) = $87, 568.38
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 2: Find amount owed after 10 years (at the time ofrefinancing).
Here we apply the formula with i = 0.0085 and n = 240.
PV =
[1− (1 + i)−n
i
]PMT
PV =
[1− (1.0085)−240
.0085
]($856.69) = $87, 568.38
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment afterrefinancing.
Here we apply the formula with i = 0.07512 = 0.00625 and
n = 240.
PV =
[1− (1 + i)−n
i
]PMT
$87, 568.38 =
[1− (1.00625)−240
.00625
]PMT
PMT =
[.00625
1− (1.00625)−240
]($87, 568.38) = $705.44
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment afterrefinancing.
Here we apply the formula with i = 0.07512 = 0.00625 and
n = 240.
PV =
[1− (1 + i)−n
i
]PMT
$87, 568.38 =
[1− (1.00625)−240
.00625
]PMT
PMT =
[.00625
1− (1.00625)−240
]($87, 568.38) = $705.44
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment afterrefinancing.
Here we apply the formula with i = 0.07512 = 0.00625 and
n = 240.
PV =
[1− (1 + i)−n
i
]PMT
$87, 568.38 =
[1− (1.00625)−240
.00625
]PMT
PMT =
[.00625
1− (1.00625)−240
]($87, 568.38) = $705.44
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment afterrefinancing.
Here we apply the formula with i = 0.07512 = 0.00625 and
n = 240.
PV =
[1− (1 + i)−n
i
]PMT
$87, 568.38 =
[1− (1.00625)−240
.00625
]PMT
PMT =
[.00625
1− (1.00625)−240
]($87, 568.38) = $705.44
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 3: We now calculate the owner’s monthly payment afterrefinancing.
Here we apply the formula with i = 0.07512 = 0.00625 and
n = 240.
PV =
[1− (1 + i)−n
i
]PMT
$87, 568.38 =
[1− (1.00625)−240
.00625
]PMT
PMT =
[.00625
1− (1.00625)−240
]($87, 568.38) = $705.44
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing.
If the owner did not refinance, he would pay a total of856.69× 240 = $205, 605.60 in principal and interestduring 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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing.
If the owner did not refinance, he would pay a total of856.69× 240 = $205, 605.60 in principal and interestduring 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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Step 4: We now compare the amount he would have spentwithout refinancing to the amount he spends after refinancing.
If the owner did not refinance, he would pay a total of856.69× 240 = $205, 605.60 in principal and interestduring 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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
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.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose?
To answer this question, we determine which option gives thelowest monthly payment.
Option 1: If you choose 0% financing, your monthly paymentwill be
PMT1 =$27, 300
60= $455
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose?
To answer this question, we determine which option gives thelowest monthly payment.
Option 1: If you choose 0% financing, your monthly paymentwill be
PMT1 =$27, 300
60= $455
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Example: You want to purchase a new car for $27,300. Thedealer offers you 0% financing for 60 months or a $5,000rebate. You can obtain 6.3% financing for 60 months at thelocal bank. Which option should you choose?
To answer this question, we determine which option gives thelowest monthly payment.
Option 1: If you choose 0% financing, your monthly paymentwill be
PMT1 =$27, 300
60= $455
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Present Value of an Ordinary AnnuityAmortization
Amortization Schedules
Option 2: Suppose that you choose the $5,000 rebate andborrow $22,300 for 60 months at 6.3% compounded monthly.
We compute the PMT for a loan with PV = $22,300,i = 0.063
12 = 0.00525 and n = 60.
PV =
[1− (1 + i)−n
i
]PMT
$22, 300 =
[1− (1.00525)−60
0.00525
]PMT
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 theloan.
Jason Aubrey Math 1300 Finite Mathematics