4.7: Short-Term Loans

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INTRODUCTION

Let’s start this collaboration by reviewing linear and exponential functions. As a group, try to come up with at least one example of each type:

Exponential growth
Exponential decay
Linear growth
Linear decay

To review, the following table summarizes exponential models. In this table, we assume that r > 0.

	Exponential Growth	Exponential Decay
Formula	y = A(1 + r)^x	y = A(1 – r)^x
Exponential growth/decay rate	r (percent of the previous value)	r (percent of the previous value)
Vertical intercept (starting value)	A	A
Applications	Any quantity (y) that increases by the same percent per unit increase in x	Any quantity (y) that decreases by the same percent per unit increase in x

To review, the following table summarizes linear models. In this table, we assume that m > 0.

	Linear Growth	Linear Decay
Formula	y = A + mx	y = A – mx
Linear growth/decay rate	m (constant)	m (constant)
Vertical intercept (starting value)	A	A
Applications	Any quantity (y) that increases by the same amount (m) per unit increase in x	Any quantity (y) that decreases by the same amount (m) per unit increase in x

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

compound interest can be applied to short-term loan scenarios.
linear and exponential equations for growth differ significantly in the long run.

By the end of this collaboration, you should be able to

build both exponential and linear models.
compare an exponential and linear model, and make a decision based on the comparison.

PROBLEM SITUATION: WHICH OPTION?

Suppose you have an unexpected expense of $100 and do not have the money to pay for it. Usually banks do not make loans for this small of an amount of money. In this collaboration, you will consider two options for short-term loans: 1. Payday loan, and 2. Borrow from a friend.

You will investigate the two options by considering how the unpaid balance accumulates over time. You should find specific values of the balance due for each option at 1, 4, 12, and 52 weeks; that is, 1 week, 1 month, 3 months, and 1 year. Interest is compounded weekly.

In your group, you will be discussing which option is best, assuming these are your only two options.

Option 1: Payday Loan

(1) A payday loan is a small, short-term loan that is intended to cover a borrower’s expenses until his or her next payday. Payday loan centers are often located in convenient locations such as shopping centers. Usually, they only require the borrower to have a checking account and a job. It is also assumed that the loan is short term, meaning it will be paid back in a few weeks. In many states, payday loans are somewhat regulated, with a maximum Annual Percentage Rate (APR) of 390%. For this option, assume the APR is 390%. Payday loan centers justify this high interest rate because the loan is short term and high risk.¹⁷

Although these loans are not exactly exponential (because of loan fees and payment schedules), exponential equations provide a good model to approximate what this loan costs the borrower. There are restrictions (usually around 45 days) for the amount of time you can hold the loan. However, many people just move the balance from center to center, letting the unpaid amount add up for quite a long time. That is, they use a new loan to pay off the old loan, and the balance continues to accumulate.

Step 1: Develop a formula for this scenario. Do this individually before sharing your ideas in your
group.

Step 2. Use the formula to complete the table in part (a).

Step 3: Use the information in the table to create a graph.

Step 4: Discuss your findings and record your observations.

(a) Work in your group and use your formula to complete the table below. Round your figures to
two decimal places.

Time in Weeks	Amount Owed
1
4
12
52

(b) Create a graph of the model from (a). Note: If completing this problem online, follow the instructions given online to create your graph.

Option 2: Borrow from a Friend

(2) You have a friend who offers to lend you the money. You agree to repay the original amount plus $10 for each week you owe her money.

Step 1: Develop a formula for this scenario.

Step 2. Use the formula to complete the table below.

Step 3: Use the information in the table to create a graph.

Step 4: Discuss your findings and record your observations.

(a) Work in your group using your formula to complete the table below.

Time in Weeks	Amount Owed
1
4
12
52

(b) Plot the data from the table above on the graph you created in Question 1(b). Note: If completing this problem online, follow the instructions given online to create your graph.

FURTHER APPLICATIONS

(3) Finance charges on credit cards are calculated by several different methods depending on the card and the company. None of these methods are exactly exponential, but like payday loans, an exponential model can be used to approximate the charges. According to Wikipedia, finance charges based on a credit card annual percentage rate (APR) of 29.99%, compounded daily, is about equivalent to an effective annual rate of 34.96%.¹⁸

For this problem, you will explore the finance charges for an average balance of $2,000 of debt on a credit card. Note that in reality, credit card balances go up and down each month. The customer is required to make a minimum payment each month, which lowers the balance, but the customer often makes new charges, which increases the balance.

Calculate the interest for one year on $2,000 with an APR of 29.99% compounded daily. (Hint: Remember your answer should be the interest, not the total balance.)

(4) Investopedia illustrates credit card interest with the following example.

Let’s say John and Jane both have $2,000 of debt on their credit cards, which require a minimum monthly payment of 3%, or $10, whichever is higher. Both are strapped for cash, but Jane manages to pay an extra $10 on top of her minimum monthly payments. John pays only the minimum.

Each month, John and Jane are charged a 20% annual interest on their cards’ outstanding balances. So, when John and Jane make payments, part of the payment goes to paying interest and part goes to the principal.¹⁹

(a) Assume the interest is compounded monthly. Calculate the interest charged for one month for John and Jane (the interest rate is the same for both John and Jane). Since 20% is the annual interest, the rate for one month is 20% ÷ 12 = 1.67%. Round your answer to the nearest cent.

(b) The minimum payment is $60. John pays the minimum and Jane pays an extra $10 for a payment of $70. Part of this pays the interest you calculated in (a), and the rest pays off the balance. Complete the table below. Round to the nearest cent.

	John	Jane
Original balance	$2,000	$2,000
Amount paid on the balance
New balance after the payment is subtracted

(c) Calculate the interest that John and Jane each pay in the second month. Remember to use the new balance for each. Round to the nearest cent.

John’s interest:

Jane’s interest:

(d) You might wonder if the small difference in the interest charges for John and Jane really matters. According to Investopedia, if John continues to make the minimum payment and Jane continues to pay an extra $10 each month, it will take John 15 years to pay off the original debt of $2,000, and he will pay a total of $4,240. Jane will pay off her balance in seven years and pay a total of $3,276.

(i) How much interest did John pay?

(ii) What percentage of John’s total payment is for interest?

(iii) How much interest did Jane pay?

(iv) What percentage of Jane’s total payment is for interest?

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

_________________________________

¹⁷ www.paydayloaninfo.org/state-information

¹⁸ http://en.Wikipedia.org/wiki/Credit_card_interest

¹⁹ www.investopedia.com/articles/01/061301.asp#axzz1bo1CMcfb

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