Copyright © 2008, The McGraw-Hill Companies, Inc.
Calculating Lease Payments
Suppose that the total price (including all fees, taxes, and other expenses) of her car comes
out to be $19,875. We’ve already (in Chapter 4) found how to calculate the monthly pay-
ments on a car loan; the following example will serve as a reminder.Example 10.4.1 Suppose that Jessie decides to fi nance the entire cost of the car
with a 5-year loan at an 8.4% interest rate. Calculate her monthly payment.Recall that her monthly payments form an annuity, and the amount borrowed is the present
value. Therefore:PV PMT a _n (^) | (^) i
$19,875 PMT a __ 60 | (^) .007
$19,875 PMT(48.85587164)
PMT $406.81
For comparison, now, let’s consider what would happen if she decided to lease the car. When
she borrows the money to buy the car, she will actually be borrowing $19,875, which she then
uses to buy the car. When she signs the lease, she doesn’t directly borrow any money, and so
a lease is not exactly the same as a loan. However, she does borrow something: the car itself.
Looking at the mathematics of the situation, we can think of it as a kind of loan. Since the car
is worth $19,875, she is, in effect, borrowing $19,875 after all, albeit in the form of the car.
There is a big difference though, in how this “loan” is repaid. With the actual loan, her
monthly payments repay all of the principal borrowed together with all of the interest. The
“loan” she takes out with the lease is only partly repaid with her monthly lease payments.
A large part of the “loan” will be repaid at the end of the lease by returning the car to the
leasing company.
Suppose that the leasing company has determined that, after 2 years of normal use
and proper maintenance, the value of this car should be $14,055. This is referred to as its
residual value. Of the $19,875 that she borrowed in the form of the car, then, she will repay
$14,055 by returning the car. So her lease payments need to cover the difference between
the original and residual values of the car, or $19,875 $14,055 $5,820, together with
the interest on this debt. In addition, even though $14,055 worth of principal will be repaid
by returning the car, she still must pay interest on the $14,055 as well. The situation can be
summed up in the following diagram:
Amount Interest Principal
$14,055 (residual value) Paid by monthly payments Paid by return of car
$5,820 (loss in value) Paid by monthly payments
Thus, her monthly lease payments are built of two parts: the principal and interest for the
loss in value, plus the interest (but not the principal) on the residual value. The first part
can be found in the same way as any other loan payment, using the present value annuity
formula. Since the interest on the residual value will be covered in each month’s payment,
it won’t ever compound, and so it can be found by using the simple interest formula. We
can sum this up with the following “formula”:
“FORMULA” 10.4.1
The (theoretical) monthly payment on a lease can be found by adding the following
two parts:- Payment on Loss: Subtract the residual value from the original price, and calculate
the annuity payment with this difference as the present value
PLUS- Interest on Residual: Use the residual value as principal and calculate the monthly
interest, using I PRT
10.4 Leasing 459