Business Mathematics (19)
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
PV = $20000 FV = 0 n =12* 20 = 240 PMT =
1.
12
-179.95 20 000
9 0
240
2. & 3.
(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?
FV of the Original Loan
FV of the Payments already made
Balance
This is now rearranged to isolate the “Balance” Balance
FV of the Original Loan
FV of the Payments already made
Consider that…
An Original Loan = The PV of ALL of the Payments (discounted at the contractual rate of interest on the loan)
Also, that…
A Balance = The PV of the remaining Payments
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.
(discounted at the contractual rate of interest on the loan)
Then…
…this can be expressed as …the Statement of Economic Equivalence
For a focal date of the original date of the loan, PV of first x Payments PV of the Balance just after the xth Payment
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period. (1) Calculate the monthly payment. (2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?
N == FV
-673.79 25.457
1. Calculate …the number of payments
0
2. Calculate …the balance after the 2nd to last payment
25
3.
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.
FV= 17,741.05 PV= 17,741.88
Difference ($0.83) is because the Prospective Method assumes that the final payment is the same as all the others.
The Retrospective Method is based on payments already made.
Balance = PV of remaining 180 payments
12
PV= 17,741.88
179.95 0 180 9
Comparison of Methods
Retrospective Method for Loan Balances Prospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for the effect of rounding the regular payment. Calculate the exact n for monthly payments of $179.95 to repay a $20,000 loanate…
Retrospective Retrospective
Retrospective Method for Loan Balances
Suppose we locate the Focal Date… of the xth payment, the Statement of Economic Equivalence becomes…
Balance = FV of $20,000 – FV of first 60 payments
12
FV= 17,741.05 179.95 20,000 60 9
Prospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. Total payments = 12* 20 Years = 240 - 60 made = 180 remaining
Balance after 10 payments = FV of $28,000 after 10 quarters – FV of 10 payments
FV= - 19,037.29 1500 28,000 10
Balance after 10 payments
10
B.
1. 2. 3.
Needed
Solve using… Retrospective Method Prospective Method
Then compare…
Retrospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. 12 * 5 Years
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the size of the final payment. Final Payment = (1+i) * (Balance after 2nd to last payment) Balance after 239 payments = FV of $20,000 after 239 months – FV of 239 payments
20 000
N=
239.982
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for the effect of rounding the regular payment.
Prospective Method for Loan Balances
… is based on PAYMENTS YET to be MADE!`
Retrospective Retrospective
Retrospective Method for Loan Balances
… is based on PAYMENTS ALREADY MADE!`
2.
PV = ?
FV = 0
n =12*20 = 240
PMT = 179.95
179.95 PV = 20,000.5345
179.95
3.
The difference of $0.5345 is due to rounding the monthly payment to the nearest cent!
N = = 17,741.05 P/V 179.9821 239.982 After 5 years, 239.982 – 60 = 179.982 payments remain. Therefore, balance (after 5 years) = PV of 179.982 payments of $179.95 60
Application
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years.
3. Calculate …the final payment
Final Payment = (1+0.10/4) * 673.79
= $690.63
LO 3.
A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term by equal monthly payments.
179.95
12
239 20,000
9
FV=
- 175.42
Final Payment = (1+0.09/12) * 175.42
= $176.74
A.
4
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.
1.
12
-179.95 20 000
9 0
240
2. & 3.
(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?
FV of the Original Loan
FV of the Payments already made
Balance
This is now rearranged to isolate the “Balance” Balance
FV of the Original Loan
FV of the Payments already made
Consider that…
An Original Loan = The PV of ALL of the Payments (discounted at the contractual rate of interest on the loan)
Also, that…
A Balance = The PV of the remaining Payments
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.
(discounted at the contractual rate of interest on the loan)
Then…
…this can be expressed as …the Statement of Economic Equivalence
For a focal date of the original date of the loan, PV of first x Payments PV of the Balance just after the xth Payment
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period. (1) Calculate the monthly payment. (2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?
N == FV
-673.79 25.457
1. Calculate …the number of payments
0
2. Calculate …the balance after the 2nd to last payment
25
3.
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.
FV= 17,741.05 PV= 17,741.88
Difference ($0.83) is because the Prospective Method assumes that the final payment is the same as all the others.
The Retrospective Method is based on payments already made.
Balance = PV of remaining 180 payments
12
PV= 17,741.88
179.95 0 180 9
Comparison of Methods
Retrospective Method for Loan Balances Prospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for the effect of rounding the regular payment. Calculate the exact n for monthly payments of $179.95 to repay a $20,000 loanate…
Retrospective Retrospective
Retrospective Method for Loan Balances
Suppose we locate the Focal Date… of the xth payment, the Statement of Economic Equivalence becomes…
Balance = FV of $20,000 – FV of first 60 payments
12
FV= 17,741.05 179.95 20,000 60 9
Prospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. Total payments = 12* 20 Years = 240 - 60 made = 180 remaining
Balance after 10 payments = FV of $28,000 after 10 quarters – FV of 10 payments
FV= - 19,037.29 1500 28,000 10
Balance after 10 payments
10
B.
1. 2. 3.
Needed
Solve using… Retrospective Method Prospective Method
Then compare…
Retrospective Method for Loan Balances
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years. 12 * 5 Years
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the size of the final payment. Final Payment = (1+i) * (Balance after 2nd to last payment) Balance after 239 payments = FV of $20,000 after 239 months – FV of 239 payments
20 000
N=
239.982
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its 20-year amortization period.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for the effect of rounding the regular payment.
Prospective Method for Loan Balances
… is based on PAYMENTS YET to be MADE!`
Retrospective Retrospective
Retrospective Method for Loan Balances
… is based on PAYMENTS ALREADY MADE!`
2.
PV = ?
FV = 0
n =12*20 = 240
PMT = 179.95
179.95 PV = 20,000.5345
179.95
3.
The difference of $0.5345 is due to rounding the monthly payment to the nearest cent!
N = = 17,741.05 P/V 179.9821 239.982 After 5 years, 239.982 – 60 = 179.982 payments remain. Therefore, balance (after 5 years) = PV of 179.982 payments of $179.95 60
Application
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95 during its 20-year amortization period. Calculate the exact balance after 5 years.
3. Calculate …the final payment
Final Payment = (1+0.10/4) * 673.79
= $690.63
LO 3.
A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term by equal monthly payments.
179.95
12
239 20,000
9
FV=
- 175.42
Final Payment = (1+0.09/12) * 175.42
= $176.74
A.
4
Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment. Payments of $1,500 are made every 3 months. A. Calculate the balance after the 10th payment. B. Calculate the final payment.