The Mathematics of Money

576 Chapter 14 Evaluating Projected Cash Flows

Payback Period Where Payments Vary

Formula 14.2.1 and all of the examples above assume that the “profits” continue at a con-

stant rate. This is often realistic. Even though Jennifer does not keep her light on for exactly

the same amount of time (and hence the savings from the more efficient lightbulb will not

be exactly the same) each month, and even though Harold and Olivia’s interest savings will

change as the balance on their loan declines, the differences are small enough in the near

term to be ignored.

When differences are not too small to be ignored, the calculation of the payback period

can require a bit more effort. In those cases, we need to make a chronological list of the

savings to figure out when payback is reached.

Example 14.2.6 Melina is considering joining her local YMCA at a cost of $840 per

year. Right now, she pays $65 a month for a membership at another gym, which she

would quit if she joined the Y. Three months from now, her daughter will be going to

a summer camp, and as a Y member she will get a discount of $280, and 6 months

from now her daughter will be starting an after-school program, for which Y members

get a $45 per month discount. Find the payback period for Melina’s membership.

To do this, we list the savings in each month, and also keep a running total of the cumulative

savings from the start of the membership.

Month Savings in Month Cumulative Savings

1 $65.00 $65.00

2 $65.00 $130.00

3 $345.00 $475.00

4 $65.00 $540.00

5 $65.00 $605.00

6 $110.00 $715.00

7 $110.00 $825.00

8 $110.00 $935.00

We carry the table only as far as month 8 because we can see that the savings exceed the

$840 investment between the seventh and eighth months.

The following exercises will provide additional examples of where payback periods may

be useful.

A. Payback Periods

1. A $3,500 investment is expected to return $500 per year. Calculate its payback period in years.


2. An investment of $525,000 is expected to return $4,000 per month. Calculate its payback period in months.


3. A tae kwon do program charges $125 a month. Students are given the opportunity to instead pay for a full year in
   advance, in which case the cost is $1,000. What is the payback period, in months, for paying for the year in advance?


4. Cattarauqua Ginseng Enterprises is considering switching to new packaging equipment that would allow the company
   to package its products more quickly and with less-expensive packaging materials. The new equipment would

EXERCISES 14.2