Ch. 11.2: Installment Buying

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1 Ch. 11.2: Installment Buying When people take out a loan to make a big purchase, they don t often pay it back all at once in one lump-sum. Instead, they usually pay it back back gradually over time, in what are called installments. Installment Loans with add-on Interest The traditional installment loan is a loan agreement in which the borrower agrees to repay in a fixed number of equal installments at regular time intervals. The simplest method for calculating interest is called add-on interest: the total amount repaid in interest is calculated by a simple interest formula applied to the initial amount borrowed (the principle, or amount financed). We ll assume that installment payments are made monthly: Installment Loan Formulas: TOTAL INTEREST REPAID: TOTAL AMOUNT REPAID: NUMBER OF PAYMENTS: AMOUNT OF EACH PAYMENT: I = P rt A = P + I N = 12t m = A N P = AMOUNT TO BE FINANCED r = ADD-ON INTEREST RATE (per year) t = TIME TO REPAY THE LOAN (in years) Car financing is often done this way. Example 1: Say you re interested in buying a new car, and need to come up with $21,000 in financing. The dealership offers you an installment plan with an add-on interest rate of 3.5% over 5 years. What, then, will be the size of your monthly payments? 1

2 ANSWER: Comparing the quantities given in the question to our list of formulas, we see we are given: P = $21, 000 r = year 1 t = 5 years The total number of payment periods is: N = (12 per year) (5 years) = 60 The total amount to be paid back to the dealer will be: A = P + I = P (1 + rt) = ($21, 000)( ) = $24, 675 Thus, the amount of each monthly payment is: m = A N = $24, = $ Annual Percentage Rate (APR) You don t often here the term add-on interest unless you ask for it. You re more likely to be quoted the APR when buying a loan. Legally (in the US), this is the true interest rate, and lenders are required by law to give this to you. It is a very general term, and it s calculation depends on how payments are made and how interest is calculated. For add-on installments, it is given by: APR Formula for Add-On Installments: APR = Nr ( ) 1 (exact) 2 3 N APR 2Nr N + 1 (approximate, when N > 5) Notice the APR is always larger than the add-on interest rate, r. 2

3 Example 2: What is the APR of the installment plan offered in Example 1? ANSWER: We still have r = year 1 and N = 60. Since N is large enough, we can use the approximate formula... APR = 6.89% APR. This is almost twice as much as the add-on rate! Example 3: Suppose you re ready to buy the car of your dreams, and you are faced with 2 financing options (both monthly add-on installment plans): (a) pay the sticker-price of $60,000 at 0% APR over 60 months, (b) pay the sticker-price of $60,000 minus a rebate of $5,000, for 3% APR over 60 months (perhaps offered by your local credit union). What would be the better choice, and how much money would you save? ANSWER: Lets look at each option separately, and calculate how much we ll pay in total for each (the future value, A). But first, notice that we have N = 60 payment periods in either case, so t = N/12 = 5 years for the loan to mature. (a) We want to finance P = $60, 000. By the APR formula, we see that 0% APR necessarily means an that r = 0, so we will ultimately pay back... A = P (1 + rt) = P (1 + 0) = P = $60, 000. In other words, we pay no interest. 3

4 (b) Here, we want to finance P = $60, 000 $5, 000 = $55, 000. We use the APR formula to determine r... APR 2 60 r = ( ) 120 r = r = ( ) = So, we will pay back... A = P (1 + rt) = ($55, 000)( ) = $59, Thus, we ll save over $800 if we choose to to pay the APR interest we should choose the second option, (b). 4

5 Credit Cards Credit Cards are another example of installment buying: the account balance at any given time is the amount owed, and the monthly interest is called a finance charge. Unlike installment loans, however, there is no fixed monthly payment (only a minimum), and no agreed-upon final payment date (the maturity date). For that reason, we say credit cards are a form of revolving credit, while installment loans are closed-end. Credit Card rates are often given as a % monthly or % daily. These are just fractions of the APR: Credit Card APR Formulas: % DAILY = APR 365 % MONTHLY = APR 12 (Note that we use the convention of 365 days/year here.) The finance charge at the end of each month is usually calculated using the simple interest formula. However, the details vary widely between cards, so we won t worry about such questions. 5

$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months?

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