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 :
P =
r=
t=
I = P rt
A=P +I
N = 12t
A
m= N
AMOUNT TO BE FINANCED
ADD - ON INTEREST RATE
(per year)
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
ANSWER: Comparing the quantities given in the question to our list of formulas, we see
we are given:
P = $21, 000
r = 0.035 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)(1 + 0.035 × 5) = $24, 675
Thus, the amount of each monthly payment is:
m=
A
$24, 675
=
= $411.25
N
60
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
=
APR
≈
Nr
1 + + 13 + · · · +
1
2
2N r
N +1
1
N
(exact)
(approximate, when N > 5)
Notice the APR is always larger than the add-on interest rate, r.
2
Example 2: What is the APR of the installment plan offered in Example 1?
ANSWER: We still have r = 0.035 year−1 and N = 60. Since N is large enough, we
can use the approximate formula...
APR
≈
2 × 60 × 0.035
≈ 0.0689 = 6.89% APR.
60 + 1
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
(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
≈
=
60 + 1
r=
61
120
120
61
r = 0.03
0.03
= 0.01525
So, we will pay back...
A = P (1 + rt) = ($55, 000)(1 + 0.01525 × 5) = $59, 193.75.
Thus, we’ll save over $800 if we choose to to pay the APR interest–we should choose
the second option, (b).
4
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
=
MONTHLY
=
%
%
APR
365
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