Econ final test finance&econ interest

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Econ 325 — Review Questions Topic: Loans and Key Historical Consumer Lending Laws

INTRODUCTION: HISTORICAL LOAN TYPES AND IMPORTANT LENDING LAWS

Key Laws

Truth in Lending Act of 1968

• Total Charge on a Loan (or Total Interest Paid on a Loan) • APR of a Loan, where APR is based on the equivalent actuarial loan that yields the same total

interest • Truth in lending tables were created to help lenders who used non-actuarial loans to comply

with the law. Fair Credit Reporting Act 1970

• People have a right to know the information used against them in employment, insurance and credit

• Rumors or innuendos are prohibited from credit reports • Incorrect information in credit reports must be fixed or the consumer can provide a statement

that must be included in the credit report • Consumers can buy their own credit reports for a reasonable fee

Equal Credit Opportunity Act 1974 (and amendments)

• No bias in lending based on sex, marital status, age, national origin, ethnicity, race, color, religion and the receipt of public assistant benefits

• If denied credit a consumer must be given a reason why Fair Debt Collection Practices Act (1978)

• Debt collectors can’t threaten consumers about a debt; claim to represent gov’t; call at odd hours and generally inconvenience the consumer

Fair and Accurate Credit Transactions Act of 2003

• Free annual credit report; http://annualcreditreport.com

Readings

• See D2L for readings that were discussed in class or are useful to understand the period when the laws were passed.

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Lending Laws

Part A: Understanding Problems

1. What does “Total Interest Charge” for a loan mean? 2. What is prepayment on a loan? 3. What two things did the 1968 Truth-in-Lending Act require lenders to provide to borrowers? 4. Describe how the Truth-in-Lending act defines the Annual Percentage Rate on a loan. 5. Provide examples of what loan collectors did that caused the Fair Debt Collection Practices Act.

6. Why do you think the Fair Credit Reporting Act allowed for “reasonable fee” for someone to get their credit report?

7. What useful information is provided on Truth-in-Lending Tables? 8. What kind of loans would benefit from the Truth-in-Lending Tables?

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CALCULATING LOAN PAYMENTS AND LOAN INTEREST Key Terms Historical Types of Loans (with monthly payments)

• Add-On Loan • Discount per year Loan • Actuarial Loan

Loan Vocabulary

• Total payment for month • Interest payment for month • Principal payment for month • Loan balance • Face value of a loan • APR, number of periods to the loan and the periodic interest rate

LOANS

Part B: Understanding Problems

1. What is an actuarial loan? How does it work? 2. How does an add-on rate loan work? 3. How does a discount rate per year loan work? 4. Provide an example of a discount rate per year loan. 5. Provide an example of an add-on rate loan. 6. Which loan(s) allow for easy calculation of the total charge (or total interest on the life of the loan)

to the customer? 7. Which loan(s) allow for easy calculation of the APR for the loan? 8. Which loan(s) make it easy to determine the loan balance for a loan? Which make it difficult to do

so? 9. Which loan(s) require the borrower to pay the total interest for the life of the loan, even if loan is

paid early? 10. Which loans allow the borrower to avoid interest charges if the loan is paid off early? 11. Which loan(s) have fixed payments? Which loan(s) have payments that change during the life of

the loan? 12. Which loan(s) have balances that fall proportionally (or fall linearly) with the number of payments

made? 13. Which loans(s) have balances that do not fall proportionally (or fall linearly) with the number of

payments made? 14. Which loans require truth-in-lending tables in order to comply with federal laws?

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Add-on Rate Loans, Discount Rate per Year and Actuarial Loans

Check D2L for PowerPoint decks.

Part C: Add-On Rate Loans

1. Suppose you have a 5% Add-on loan to buy a $4,000 refrigerator for 24 months.

(a) What is your loan’s face value?

(b) What is your monthly payment?

(c) What is your total interest paid on the loan?

(d) How much total interest do you avoid if you pay off your loan early after having made 12 monthly payments.

2. Suppose you have a 6% Add-on loan to buy a $1,000 refrigerator for 12 months.

(a) What is your loan’s face value?

(b) What is your monthly payment?

(c) What is your total interest paid on the loan?

3. Suppose you have a 6% Add-on loan to buy a $1,000 refrigerator for 24 months.

(a) What is your loan’s face value?

(b) What is your monthly payment?

(c) What is your total interest paid on the loan?

4. Suppose you have a 6.5% Add-on loan to buy a $1,000 refrigerator for 18 months.

(a) What is your loan’s face value?

(b) What is your monthly payment?

(c) What is your total interest paid on the loan?

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Part D: Discount Rate Per Year Loans

5. Suppose you have a 5% discount per year loan. You borrow to buy a refrigerator for $5,000 with for an 18-month loan.

(a) What is your loan balance before you make any payments (or gross loan amount)?

(b) What is your monthly payment?

(c) What is the total interest you pay during the life of the loan?

(d) Do you avoid future interest is you completely prepay your loan at 12-months?

6. Suppose you have a 4% discount per year loan. You wish to buy a $5,000 car with a 48-month loan.

(a) What is your loan balance before you make any payments (or gross loan amount)?

(b) What is your monthly payment?

(c) What is the total interest you pay during the life of the loan?

7. Suppose you have an 8% discount per year loan. You wish to buy a $1,000 car with a 6-month loan. (a) What is your monthly payment?

(b) What is the total interest you pay during the life of the loan?

8. Suppose you have a 5% discount per year loan. You wish to borrow $1,000 at a bank and have 12 monthly payments. (a) What is your monthly payment?

(b) What is the total interest you pay during the life of the loan?

9. Suppose you have a 8% discount per year loan. You wish to borrow $2,000 at a bank and have 15 monthly payments. (a) What is your monthly payment?

(b) What is the total interest you pay during the life of the loan?

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Part E: Actuarial Loans—Intro

10. Actuarial Loans--Intro. Suppose you have a monthly interest rate of 20% a month. If you can make 3 monthly payments of $300, starting next month, how much can you borrow today?

11. Actuarial Loans--Intro. Suppose you have a monthly interest rate of 20% a month. If you can make 5 monthly payments of $300, starting next month, how much can you borrow today?

12. Actuarial Loans--Intro. Suppose you have a monthly interest rate of 20% a month. If you can make a monthly payment of $300 in two months, a monthly payment of $300 in five months, and a monthly payment of $3000 in seven months, how much can you borrow today?

13. Actuarial Loans--Intro. Suppose you have a monthly interest rate of 2% a month. What would the APR of a monthly loan be?

Actuarial Loans – Basic Payments and Total Interest

14. Actuarial loan. Suppose a person buys a $20,000 car with an actuarial loan. The loan is for 48 months and the monthly interest rate is 0.4%. (a) What are the monthly payments? (b) How much total interest will be paid?

15. Actuarial loan. Suppose a person buys a $30,000 car with an actuarial loan. The loan is for 60 months and the monthly interest rate is 0.5%. (a) What are the monthly payments? (b) How much total interest will be paid?

16. Actuarial Loan. You took out a 360-month mortgage with 0.5% monthly interest rate that has $800 monthly payments. (a) How much did you borrow?

17. Actuarial Loan. You took out a 360-month mortgage with 0.5% monthly interest rate and borrowed $250,000. (a) What is your monthly payment? (b) How much interest will you pay if you make all the payments for 360 months?

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Actuarial Loans – Loan Balance After a Certain Number of Payments?

18. Actuarial Loan. You took out a 360-month mortgage with 0.5% monthly interest rate and borrowed $250,000. (a) What is your monthly payment? (b) How much interest will you pay if you make all the payments for 360 months? (c) What will your loan balance after you have made 60 monthly payments? (d) Would you avoid future interest if you prepay your loan after 60-months? (e) After 60-monthly payments, how much would you need to pay to completely prepay your loan?

19. Actuarial Loan. You took out a 360-month mortgage with 0.5% monthly interest rate and borrowed $250,000. (a) What is your monthly payment? (b) What will your loan balance after you have made 60 monthly payments? (c) What will your loan balance after you have made 120 monthly payments? (d) After you have made 180 monthly payments, do you expect your loan balance to be; (i) greater

than $125,000, (ii) or less than $125,000, (iii) or equal to $125,000? Explain your answer. (e) What actually will your loan balance after you have made 180 monthly payments?

20. Actuarial loan. Suppose an actuarial loan with monthly loan payments and a monthly interest rate that is used to computer loan interest has: (a) An APR = 18%, what is the monthly interest rate? (b) An APR = 12%, what is the monthly interest rate? (c) An APR = 6%, what is the monthly interest rate?

21. Actuarial loan. Suppose you want an actuarial loan with monthly loan payments and a monthly interest rate that is used to computer loan interest to borrow money to purchase a house. You wish to have a 30-year loan (i.e. have 360 monthly payments). How much money can you borrow if: (a) You can pay $1,500 a month and the loan has an APR=6.0%? (b) You can pay $1,500 a month and the loan has an APR=4.8%? (c) You can pay $1,500 a month and the loan has an APR=3.6%?

22. Actuarial Loan. You take a loan with 0.4% monthly interest with 120-monthly payments and

you borrow $50,000. (a) What are your payments? (b) How much will your loan balance be reduced by your first payment? (c) How much of your first loan payment go to paying interest?

23. Actuarial Loan. Sketch the loan balance for an actuarial loan over the life of the loan. How do

things change if the interest rate is increased?

24. Actuarial Loan. When is the loan balance for an actuarial loan over the life of the loan a straight line?

Math Review

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MATH REVIEW FOR ACTUARIAL LOANS

ANNUAL PERCENTAGE RATE: APR

Definitions

Definitions to know

a. Annual percentage rate (APR) b. Periodic rate c. Periods per year

Formulas

• APRperiodic rate r

=

• Note we will only be working with loans with monthly payments and monthly interest rates.

Examples

• 1% a month gives us an APR = 12% • An APR=9% gives us a monthly interest rate of 0.75% a month or r = 0.0075 • Remember

o 1% is equal to 0.01 o 0.1% is equal to 0.001 o 0.01% is equal to 0.0001 o 0.51% is equal to 0.0051

Simple Problems

Determine the following periodic interest rates if you know the APR for a loan.

Annual Percentage

Rate (r)

Payments Periods Per

Year

Monthly Interest Rate

4% Monthly 12 1/3% 6% Monthly 12 12% Monthly 12

ADD-ON RATE LOANS; DISCOUNT RATE PER YEAR LOANS; EQUAL PRINCIPAL PAYMENT LOANS

• Check your class notes for the mathematics associated with these three types of loans.

Math Review

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ACTUARIAL LOAN PAYMENTS AND LOAN BALANCES

Definitions

Definitions to know

• Payment (or cash flow) • Loan Balance or Loan Principal • Loan payment • Loan interest rate • APR • PVAIF (Present Value Annuity Interest Factor)

• ( )

11 1 nr

PVAIF r

 − + =       

• = ×Loan Balance $payment PVAIF

• = Loan Balancepayment

PVAIF

• 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑛𝑛 × 𝑟𝑟, where n is the number of periods in a year and r is the period interest rate.

• Loan balance after a certain number of payments is equal to the payment times the PVAIF using the loan interest rate and the number of payments remaining.

• Total interest paid on the loan: 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑛𝑛𝑡𝑡𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡 = (𝑛𝑛 × 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡) − 𝑖𝑖𝑛𝑛𝑖𝑖𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛 𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖

• The ith payment on a loan with n payments, reduces the loan balance by 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

(1+𝑟𝑟)𝑛𝑛+1−𝑖𝑖

Key Points • For an actuarial loan, the loan balance falls non-linearly with time (or number of payments made). • The loan balance falls slowly at the beginning and more rapidly towards the end.

Simple Calculation Problems

$X = monthly payment

Periodic interest rate (r)

Number of Loan Payments (n)

PVAIF How Much Can Borrow?

1000 4% 20 1000 6% 20 1000 10% 30 1000 15% 10 1000 20% 5

Solution Key

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APPENDIX—ANSWER KEY

Part A: Lending Laws—Understanding Problems Question 1 The “Total Interest Charge” is the amount of interest paid if all loan payments are made as specified by the loan agreement. That is, no early payments or loan prepayments are made. Question 2 A loan prepayment is when loan payments are made earlier than required by the loan agreement. A total loan prepayment will reduce the loan balance to zero. Question 3 The Truth-in-Lending Act requires that lenders provide total interest charge on a loan and also provide an Annual Percentage Rate. Question 4 For installment loans (where interest is computed each month), the APR = r x 12, where r is the monthly interest rate. The APR for actuarial loans with monthly payments and a monthly interest rate is also APR = r x 12. The APR for Add-On Rate Per Year or Discount Rate Per Year Loans is harder to computer. Based on a loan’s terms, you can compute the total interest for the loan. Based on the Add-on Rate of Discount Rate Loan’s total interest charge, then you create an actuarial loan that comes up with the same total interest charge. The interest rate for the equivalent actuarial loan is then used to compute the APR. Question 5 Loan Collectors might make phone calls at odd hours, that is, very early in the morning, or very late at night. Also, some loan collectors would pretend to work for the Federal, State, or Local government. Question 6 Creating and mailing credit reports would involve costs to the credit bureau. Allowing the credit bureau to charge a “reasonable fee” would mean that the credit bureaus would not lose money when providing consumers with their credit reports. Question 7 The Truth-in-Lending tables provides “total interest charge” and “APR” for various types of loans, interest rates, and months of repayment. Question 8 The Truth-in-Lending tables make it easy for Add-On Rate Loans and Discount Rate Per Year loans to provide an APR to the consumer. The Actuarial Loans made it easy for lenders to provide a total interest charge.

Solution Key

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Part B: Loans—Understanding Problems

No answers provided.

Part C: Add-On Rate Loans

Question 1. (a) You have two years of 5% interest, or 5% x 2 = 10% is added on. 10% of 4000 = 400. Loan

amount = $4000+$400 (b) $4,400/24 = $183.33 rounded to nearest $0.01 (c) As calculated in part (a), the total interest is $400

Question 2.

(a) 12 months = 1 year. Interest = 6% of $1,000 = $60. Loan balance = $1,000 + $60 = $1060. (b) Payment = $1,060/12 = 88.33 rounded to the nearest $0.01 (c) Total interest paid is $60.

Question 3.

(b) 24 months = 2 years; Interest = 6% x 2 = 12%; 12% of $1,000 = $120; Loan balance = $1,000 + $120 = $1,120.

(c) Payment = $1,120/24 = $46.67 rounded to the nearest $0.01 (d) Total interest paid is $120

Question 4.

(a) 18 months = 18/12 years = 1.5 years; Interest = 6.5% x 1.5 = 9.75%; 9.75% of $1,000 = $97.50; Loan balance = $1,000 + $97.50 = $1,097.50

(b) Payment = $1,097.50/18 = $60.97 rounded to the nearest $0.01 (c) Total interest paid is $97.50

Part D: Discount Rate Per Year Loans

Question 5.

18 months = 1.5 years; 1.5 x 5% = 7.5%.

(a) Loan Balance = $X. $𝑋𝑋(1 − 7.5%) = $5,000

$𝑋𝑋 = $5,000

(1 − 0.075)

$𝑋𝑋 = $5,000 0.925

$𝑋𝑋 = $5,405.41 (b) Monthly payment = $5,405.41

18 = $300.30 rounded to nearest $0.01

(c) Total interest = $5,405.41 - $5,000 = $405.41 (d) You don’t avoid interest with loan prepayment

Solution Key

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Question 6.

48 months = 4 years; 4 x 4% = 16%.

(a) Loan Balance = $X. $𝑋𝑋(1 − 16%) = $5,000

$𝑋𝑋 = $5,000

(1 − 0.19)

$𝑋𝑋 = $5,000

0.84

$𝑋𝑋 = $5,952.38 (b) Monthly payment = $5,952.38

48 = $124.01 rounded to nearest $0.01

(c) Total interest = $5,952.38- $5,000 = $952.38

Question 7.

6 months = 1/2 year; 8% x 1/2 = 4%.

(a) Loan Balance = $X. $𝑋𝑋(1 − 4%) = $1,000

$𝑋𝑋 = $1,000

(1 − 0.04)

$𝑋𝑋 = $1,000

0.96

$𝑋𝑋 = $1,041.67 (b) Monthly payment = $1,041.67

6 = $173.61 rounded to nearest $0.01

(c) Total interest = $1,041.67 - $1,000 = $41.67

Question 8.

12 months = 1 year; 5% x 1 = 5%.

(a) Loan Balance = $X. $𝑋𝑋(1 − 5%) = $1,000

$𝑋𝑋 = $1,000

(1 − 0.05)

$𝑋𝑋 = $1,000

0.95

$𝑋𝑋 = $1,052.63 (b) Monthly payment = $1,052.63

12 = $87.72 rounded to nearest $0.01

(c) Total interest = $1,052.63 - $1,000 = $52.63

Solution Key

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Question 9.

15 months = 1.25 years; 1.25 x 8% = 10%.

(a) Loan Balance = $X. $𝑋𝑋(1 − 10%) = $2,000

$𝑋𝑋 = $2,000

(1 − 0.10)

$𝑋𝑋 = $2,000

0.90

$𝑋𝑋 = $2,222.22 (b) Monthly payment = $2,222.22

15 = $148.15 rounded to nearest $0.01

(c) Total interest = $2,222.22 - $2,000 = $222.22

Part E: Actuarial Loans—Intro

Question 10.

You can borrow the value of the three-monthly payments.

𝐵𝐵𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝐵𝐵 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = $300 1.20

+ $300

(1.20)2 +

$300 (1.20)3

= 250.00 + 208.3333 + 173.6111

= 631.9444 or $631.94

Question 11.

You can borrow the value of the five-monthly payments.

𝐵𝐵𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝐵𝐵 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = $300 1.20

+ $300

(1.20)2 +

$300 (1.20)3

+ $300

(1.20)4 +

$300 (1.20)5

= 250.00 + 208.3333 + 173.6111 + 144.6759 + 120.5633

= 897.18

Question 12.

𝐵𝐵𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝐵𝐵 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = $300

(1.20)2 +

$300 (1.20)5

+ $300

(1.20)7

= 208.3333 + 120.5633 + 83.7245

= 412.62

Solution Key

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Question 13.

The loan APR = 2% x 12 = 24%.

Part E: Actuarial Loans –Basic Payments and Total Interest

Question 14.

Loan amount = $20,000. n = 48. r = 0.004

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.004,𝑛𝑛 = 48) = 1 − 1(1.004)48

0.004

= 43.59424893

(a) 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 = $20,000 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

= $458.78 rounded to nearest $0.01 (b) Total Interest = Total Payments - $20,000

= (𝑛𝑛 × 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡) − $20,000 = $22,021.25− $20,000 = $2,021.25

Question 15.

Loan amount = $30,000. n = 60. r = 0.005

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 60) = 1 − 1(1.005)60

0.005

= 51.72556075

(a) 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 = $20,000 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

= $458.78 rounded to nearest $0.01 (b) Total Interest = Total Payments - $20,000

= (𝑛𝑛 × 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡) − $20,000 = $22,021.25− $20,000 = $2,021.25

Question 16.

Payment = $800. n = 360. r = 0.005

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 360) = 1 − 1(1.005)360

0.005

Solution Key

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= 166.7916144

(a) 𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃 = $800 × 166.7916144 = $133,433.29

Question 17.

Loan Amount = $250,000. n = 360. r = 0.005

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 360) = 1 − 1(1.005)360

0.005

= 166.7916144

(a) 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 = 𝐿𝐿𝐿𝐿𝑝𝑝𝑝𝑝 𝑃𝑃𝑝𝑝𝐿𝐿𝐴𝐴𝑝𝑝𝑝𝑝 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

= $250,000

166.7916144

= $1,498.88 (b) 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑃𝑃𝑛𝑛𝑡𝑡𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡 = 𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡𝑖𝑖 − $250,000

= (360 × $1,498.88)− 250,000 = 539,595.47− 250,000 = $289,595.47 when rounded to nearest $0.01

Solution Key

Copyright © 2012–2013, 2016-2020 by A. B. Sugiyama, PhD. All Rights Reserved. Page 16 of 18

Part F: Actuarial Loans—Loan Balance After a Certain Number of Payments

Question 18.

Loan amount = $250,000; n=360; r=0.5% or r=0.005. The numbers are similar to Question 20.

(a) Payment = $1,498.88 (b) Total interest after 360 payments = $289,595.47 (c) 60 payments have been made; 300 payments remain

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 300)

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 300) = 1 − 1(1.005)300

0.005

= 155.206864 𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = $1,498.88 × 155.206864

= $232,636.46 (d) Yes, you could avoid future interest. (e) If you wish to pay off your loan at the 60th payment, you need to pay the monthly payment +

loan balance after the 60th payment. Or $232,636.46 = $1,498.88 + $232,636.46 = $234,135.34 If you wish to pay off your loan at the 61st payment, then you would need to pay $232,636.46 + one month’s interest on that amount. Or $232,636.46 + (0.005 x $232,636.46). Prepayment amount = $232,636.46 + 1163.18 = $233,799.64.

Question 19.

Loan amount = $250,000; n=360; r=0.5% or r=0.005. The question is similar to Question 20 and Question 21.

(a) Payment = $1,498.88 (b) Loan balance = $232,636.46 see question 21. (c) After 120 payments are made, there are 240 payments left.

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 240)

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 240) = 1 − 1(1.005)240

0.005

= 139.5807717

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = $1,498.88 × 139.5807717

= $209,214.83 (d) Because the loan balance is non-linear as payments are made, the loan balance is larger than

$125,000 after 180 (or ½) the payments are made. (e) After 180 payments are made, there are 180 payments left.

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = 𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 180)

Solution Key

Copyright © 2012–2013, 2016-2020 by A. B. Sugiyama, PhD. All Rights Reserved. Page 17 of 18

𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.005,𝑛𝑛 = 180) = 1 − 1(1.005)180

0.005

= 118.5035147

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑏𝑏𝑖𝑖 = $1,498.88 × 118.5035147

= $177,622.55

Question 20.

(a) The monthly interest rate is 18%/12 = 1.5% a month (b) The monthly interest rate is 12%/12 = 1% a month (c) The monthly interest rate is 6%/12 = 0.5% a month

Question 21.

(a) If APR = 6.0%, then r = 0.5% or r = 0.005. Then n = 360 and payment = $1,500. 𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑛𝑛 = 360, 𝑟𝑟 = 0.005)

(b) If APR = 4.8%, then r = 0.4% or r = 0.004. Then n = 360 and payment = $1,500.

𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑛𝑛 = 360, 𝑟𝑟 = 0.004)

(c) If APR = 3.6%, then r = 0.3% or r = 0.003. Then n = 360 and payment = $1,500. 𝐿𝐿𝑡𝑡𝑡𝑡𝑛𝑛 𝐴𝐴𝑝𝑝𝑡𝑡𝐴𝐴𝑛𝑛𝑡𝑡 = 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 × 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑛𝑛 = 360, 𝑟𝑟 = 0.003)

Question 22.

Loan amount = $50,000; r = 0.004; n = 120.

(a) 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃(𝑟𝑟 = 0.004,𝑛𝑛 = 120) = 1− 1(1.004)120

0.004 = 95.15596794

𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 = $50,000 𝐴𝐴𝑃𝑃𝐴𝐴𝑃𝑃𝑃𝑃

= $50,000

95.15596794 = $525.45

(b) The first payment reduces the loan balance by

𝐴𝐴𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 (1.004)120

= $525.45

1.614527836

= 325.45

(c) The amount of the first payment that goes to interest is 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑛𝑛𝑡𝑡𝑖𝑖𝑟𝑟𝑖𝑖𝑡𝑡 = 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 − 𝑝𝑝𝑟𝑟𝑖𝑖𝑛𝑛𝑏𝑏𝑖𝑖𝑝𝑝𝑡𝑡𝑡𝑡 𝑝𝑝𝑡𝑡𝑝𝑝𝑝𝑝𝑖𝑖𝑛𝑛𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑛𝑛𝑡𝑡𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑡𝑡 = $525.45 − $325.45

= $200.00 Note that the interest on $50,000 is $50,000 x 0.004 = $200.

Solution Key

Copyright © 2012–2013, 2016-2020 by A. B. Sugiyama, PhD. All Rights Reserved. Page 18 of 18

Question 22.

Number of Payments Made

Loan Balance

Question 24.

When the interest rate is 0%.

Number of Payments Made

Loan Balance