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 lumpsum. Instead, they usually pay it back back gradually over time, in what are called installments. Installment Loans with addon 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 addon 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 = ADDON 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 addon 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 addon 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 addon installments, it is given by: APR Formula for AddOn Installments: APR = Nr ( ) 1 (exact) 2 3 N APR 2Nr N + 1 (approximate, when N > 5) Notice the APR is always larger than the addon 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 addon rate! Example 3: Suppose you re ready to buy the car of your dreams, and you are faced with 2 financing options (both monthly addon installment plans): (a) pay the stickerprice of $60,000 at 0% APR over 60 months, (b) pay the stickerprice 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 agreedupon final payment date (the maturity date). For that reason, we say credit cards are a form of revolving credit, while installment loans are closedend. 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
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