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College of Science and Engineering
Mathworks

Cost of Credit

car_loan_application

What happens if instead of depositing money in the bank, we borrow money from the bank. For example, if we use a credit card. Do you think the interest rate will be the same, smaller, or larger? Now A will be the amount we owe. How can we show that we owe the bank instead of the bank owing us?

Usually a traditional loan uses the compound interest formula. As with the simple interest formula, the principal P is the amount of the loan , and the amount A is the total amount to be repaid. Here is an example that uses the compound interest formula to understand loans.

EXAMPLE 1

You use a credit card to purchase a $100 MP3 player. The credit card company charges 24% per year compounded monthly. How much do you owe after t months if you don’t pay the credit card company any money? How much do you owe at the end of the year?

Solution You used the card to purchase a $100 MP3 player so the initial value is $100. To indicate that you owe the money, write P = −100. The annual rate is 24%, or i = 2% per month. So the amount after t months is:
A = −100(1 + .02)^t = −100 · 1.02^t
At the end of the year, the amount is
A = −100 · 1.02^12 = −126.8242
Hence you owe $126.82 to the credit card company.

EXAMPLE 2

Janice has a credit card bill of $2500 that she needs to pay off. There are several options she is considering below. What will her monthly payment be, and what will be the total cost of the loan be for each option?

  • Repay the amount over a period of two years at an interest rate of 6% compounded monthly.
  • Repay the amount over a period of four years at an interest rate of 6% compounded monthly.
  • Repay the amount over a period of six years at an interest rate of 6% compounded monthly.
  • Repay the amount over a period of two years at an interest rate of 18% compounded monthly.
  • Repay the amount over a period of four years at an interest rate of 18% compounded monthly.
  • Repay the amount over a period of six years at an interest rate of 18% compounded monthly.
Monthly Payment And Total Cost Of The Loan

Amt. borrowed

Months

Interest Rate

Monthly Pmt.

Total Cost

Interest Paid
$2500 24 6% $110.80 $2,659.20 $159.20
$2500 48 6% $58.71 $2,818,08 $318.08
$2500 72 6% $41.43 $2,982,96 $482.96
$2500 24 18% $134.81 $3,235.44 $735.44
$2500 48 18% $73.44 $3,525.12 $1,025.12
$2500 72 18% $57.02 $4,105.44 $1,605.44

PROBLEM 1

John has a yearly income of $45,000. He is able to pay all of his bills on time each month, and has $425 after regular expenses. John is considering buying a new car. Assuming that John qualifies for a loan at 3% for 6 years, how expensive a car can he afford? John has three cars he is considering:

  • Car 1: Costs $18,000 with other monthly expenses for insurance, gas and repairs of $120 per month.
  • Car 2: Costs $24,000 with other monthly expenses for insurance, gas and repairs of $150 per month.
  • Car 3: Costs $30,000 with other monthly expenses for insurance, gas and repairs of $180 per month.

Which of these three options can John afford to buy? Would it be financially wise to spend his entire $425 for the car? Explain.

EXPLORATION 1

Many people have money in savings accounts that pay interest and at the same time owe money to credit card companies. Suppose we put $1000 dollars in the bank and the bank pays 12% compounded monthly. At the same time we use the credit card to purchase $1000 of clothes. The credit card company charges 24% compounded monthly. How much do we have in the bank after t months? How much do we owe to the credit  card company after t months? Does this make sense to you? What function could we graph to investigate what happens? Graph the function and see what happens at the end of the year.

When you borrow money to purchase a very expensive item like a car or a house, the lender (usually a bank) asks you to pay something each month. After each payment is made, that much less is owed the bank. In this way, you slowly pay off your loan. The formulas in this situation are complicated, but we can compute a few steps to explore how this works. Then we can use an online calculator to analyze a more realistic situation.

EXPLORATION 2

Imagine you borrow $10000 to purchase a car. The bank says it will charge 10 % annual interest rate.

1. Suppose you pay the bank $500 every 6 months.

  • a. How long do you think it will take to repay the loan?
  • b. Since you are paying each six months, the bank recomputes the interest and the money owed each six months. The money owed is called the principal of the loan. How much interest do you owe for the first 6 months? Adding the initial principal, $10000, and the interest, how much do you owe right before your first payment? How much do you owe after your first payment? What do you notice?
  • c. Repeat the calculations above for the second 6 month period: How much interest do you owe for the second 6 months? Adding the amount owed after the first payment, and the interest, how much do you owe right before your second payment? How much do you owe after your second payment?
  • d. At this rate, how long will it take to pay off the loan?

2. Now suppose you pay the bank $1,000 every 6 months.

  • a. How long do you think it will take to repay the loan?
  • b. Since you are paying each six months, the bank recomputes the interest and the money owed each six months. The money owed is called the principal of the loan. How much interest do you owe for the first 6 months? Adding the initial principal, $10000, and the interest, how much do you owe right before your first payment? How much do you owe after your first payment? What do you notice?
  • c. Repeat the calculations above for the second 6 month period: How much interest do you owe for the second 6 months? Adding the amount owed after the first payment, and the interest, how much do you owe right before your second payment? How much do you owe after your second payment?
  • d. At this rate, how long will it take to pay off the loan?

EXPLORATION 3

Use an online calculator to investigate how car loans work. Typically, when you buy a car you negotiate with the car dealer or bank the terms of the loan. Two important features are the annual interest rate and total length of the loan. For a one year loan, you must pay back the whole loan in one year (12 monthly payments). For a 6 year loan, you pay the loan back over a longer period (72 monthly payments). As you saw in Exploration 2, the amount you still owe on the loan and the interest on the loan are calculated after each payment.

Different payment methods

Notice the significant difference that the higher interest rate makes both to monthly payments as well as the interest paid over the life of the loan. Also, the amount of interest increases with the  time it takes to repay the loan. If you borrow at a high interest and pay the loan off over a long period of time, the amount of interest paid can be very large. In Example 2 above, when the interest rate was 18% and the loan was over 6 years, the interest paid was $1605. This is 64.2% the original price of the car. However, some consumers cannot afford the larger monthly payment necessary to repay quickly, so they choose a longer term for the loan.

The benefits and costs of financial responsibility

Credit Bureaus are organizations that collect information about individuals’ borrowing and bill-paying habits. The credit bureau then uses this information to assign a credit score to everyone who wants to borrow money. Banks use this credit score to determine what interest rate they should charge a particular borrower. The lower the score, the higher the interest rate you will be charged. Hence, it is important to have a good credit history to qualify for a lower interest rate so borrowing is not so costly. In fact, with a good credit history, consumers may qualify for higher loan amounts than usual. Factors that influence a credit history include the timely payment of all bills are, a stable and adequate source of income, and the absence of bankruptcy history. Bankruptcy is a legal term used when a person cannot repay his debts.

It is financially beneficial to pay all credit card bills on time. It is absolutely financially necessary to make the minimum required payment. Not meeting the required payment usually results in a very high interest rate. The table above demonstrated that a higher interest rate results in much higher monthly payments, and costs much more money over the life of the loan.

Credit cards: The real story

Credit cards are a very convenient method of payment. It is important to recognize some key facts about credit cards:

  • Your credit card bill must always eventually be paid in full.
  • Interest rates on credit cards tend to be higher than other types of credit or loans.
  • Typically, if you pay the entire balance of the credit card bill at the end of each month, you are not charged any interest.
  • If you don’t pay the entire bill at the end of the month, the amount you owe can increase rapidly because of the interest charged.

Easy-access loans

One type of loan is called an easy-access loan. Many banks assess a penalty if a customer overdraws his account or charges more than his credit limit. To protect against these accidents, some people take out an ”easy access” loan, or line of credit. In this case, when the bank account is overdrawn or the credit card is charge too much, the bank will not charge an overdraft penalty. Instead, the bank will loan the money through as an easy-access loan. This loan is like a regular loan and will need to be repaid.

Typically, easy-access loans are for much shorter duration than a regular loan. Usually these will be repaid almost immediately, but nonetheless the customer is charged interest. In Problem 2 below, you will calculate the cost of an easy access loan with different interest rates and for different time periods. Again, use a calculator to find the monthly payment.

EXPLORATION 4

A credit card account charges no interest on purchases made during a month, if you pay the entire balance at the end of the month. If you don’t pay the entire bill at the end of the month, you are required to pay a minimum payment that is equal to 2% of the balance and you are charged 15% annual interest, compounded monthly on the unpaid charges. Make a table that identifies the advantages and disadvantages of paying the balance on a credit card monthly versus paying the minimum required. Use examples and calculations to support your claims.

PROBLEM 2

Sarah took out an easy-access loan of $750. The bank has two rates for easy-access loans. For customer with good credit, the rate is 6%. However, if a customer’s credit is not so good, the rate is 12%. And if any payments are missed, the rate is 18%. There are different options for the length of time to repay the easy-access loan. The options for payment periods are one month, three months, six months, or one year. Compare the monthly costs and the total cost of the loan between the three different interest rates.

Exercises

Use an online loan calculator when necessary.

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