Installment Buying. MATH 100 Survey of Mathematical Ideas. J. Robert Buchanan. Summer Department of Mathematics

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1 Installment Buying MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018

2 Introduction Today we will focus on borrowing (to purchase something) and paying the loan back over time. This is called installment buying. There are two types: closed-end credit: there are a fixed number of equal payments that are made until the loan is repaid (e.g., mortgages, auto loans, etc.) open-end credit: the borrower pays until the loan is paid off, additional credit may be offered to the borrower before the initial loan is repaid (e.g., credit cards, store charge cards, home equity line of credit, etc.)

3 Add-On Interest Closed-end credit installment loans are sometimes set up so that the amount repaid is the principal plus simple interest. The future value of the loan is divided by the number of payments (usually made monthly) to determine the recurring payment.

4 Add-On Interest Closed-end credit installment loans are sometimes set up so that the amount repaid is the principal plus simple interest. The future value of the loan is divided by the number of payments (usually made monthly) to determine the recurring payment. Example Suppose you buy a dishwasher for $1600. You make a down payment of $500 and agree to make monthly payments for 12 months for which you are charged 24% add-on interest. What is the total interest paid? What is the total amount repaid? What is the monthly payment?

5 Open-End Credit For a typical credit card, the cardholder is authorized a credit limit, and then may make any number of purchases against that credit (up to the credit limit). The payment is usually made monthly and can be any amount from some preset minimum payment, up to the full balance of the card. At the end of each billing period (month) the cardholder receives an itemized billing listing purchases, cash advances, finance charges, total balance, and minimum payment.

6 Unpaid Balance Method Some credit cards use the unpaid balance method to calculate finance charges and total balance. Interest is charged on the unpaid balance from the end of the previous billing period (month). The interest is added to the unpaid balance and any new charges (that did not appear on the previous account statement) to determine the new total balance.

7 Example Complete the statement summary for the following credit account using the unpaid balance method and assuming a monthly interest rate of 2%. Unpaid Unpaid Balance at Purchases Balance Beginning Finance During at End Month of Month Charge Month Returns Payment of Month October $ $ $23.15 $ November $ $0.00 $ December $ $17.15 $ January $27.84 $ $200.00

8 Example Complete the statement summary for the following credit account using the unpaid balance method and assuming a monthly interest rate of 2%. Unpaid Unpaid Balance at Purchases Balance Beginning Finance During at End Month of Month Charge Month Returns Payment of Month October $ $16.57 $ $23.15 $ $ November $ $ $0.00 $ December $ $17.15 $ January $27.84 $ $200.00

9 Example Complete the statement summary for the following credit account using the unpaid balance method and assuming a monthly interest rate of 2%. Unpaid Unpaid Balance at Purchases Balance Beginning Finance During at End Month of Month Charge Month Returns Payment of Month October $ $16.57 $ $23.15 $ $ November $ $16.51 $ $0.00 $ $ December $ $ $17.15 $ January $27.84 $ $200.00

10 Example Complete the statement summary for the following credit account using the unpaid balance method and assuming a monthly interest rate of 2%. Unpaid Unpaid Balance at Purchases Balance Beginning Finance During at End Month of Month Charge Month Returns Payment of Month October $ $16.57 $ $23.15 $ $ November $ $16.51 $ $0.00 $ $ December $ $19.27 $ $17.15 $ $ January $ $27.84 $ $200.00

11 Example Complete the statement summary for the following credit account using the unpaid balance method and assuming a monthly interest rate of 2%. Unpaid Unpaid Balance at Purchases Balance Beginning Finance During at End Month of Month Charge Month Returns Payment of Month October $ $16.57 $ $23.15 $ $ November $ $16.51 $ $0.00 $ $ December $ $19.27 $ $17.15 $ $ January $ $19.26 $27.84 $ $ $670.16

12 Average Daily Balance Method An alternative to the unpaid balance method is the average daily balance method. With this method the card issuer tracks the balance on the card each day of the billing cycle and then calculates finance charges on the average daily balance during the billing cycle.

13 Example Suppose an account has a previous balance of $ and activity on the account during the billing period as follows. January 27 Billing date February 9 Candy $11.08 February 13 Returns $26.54 February 20 Payment $59.00 February 25 Repairs $71.19 Find the average daily balance on the account. Assume the billing date is the 27th of each month.

14 Solution Date Running Balance January 27 $ February 9 $ February 13 $ February 20 $ February 25 $ Number of Days Until Balance Changed Totals: ( ) ( ) Running Number Balance of Days

15 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ February 13 $ February 20 $ February 25 $ Totals:

16 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ February 20 $ February 25 $ Totals:

17 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ $ February 20 $ February 25 $ Totals:

18 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ $ February 20 $ $ February 25 $ Totals:

19 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ $ February 20 $ $ February 25 $ $ Totals:

20 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ $ February 20 $ $ February 25 $ $ Totals: 31 $

21 Solution Date ( ) ( ) Running Number of Days Running Number Balance Until Balance Changed Balance of Days January 27 $ $ February 9 $ $ February 13 $ $ February 20 $ $ February 25 $ $ Totals: 31 $ Thus the average daily balance is $15, = $