The Mathematics of Money

574 Chapter 14 Evaluating Projected Cash Flows

Example 14.2.1 Jeanette owns a pizzeria. Right now, she spends $350 each month on

advertisements in the local newspaper. The paper offers her the option of paying $2,500

all at once for the same advertising. What would the payback period be for this option?

Use the formula

P = __RI 

$2,500

_______

$350

= 7.14 months

This result tells Jeanette that after a bit more than 7 months, the annual payment option will

have “paid for itself.”

In Jeanette’s example, the payback period gives a sense of how good the deal really is; we

could think of the offer as getting a full year’s advertising for a bit more than the price of

7 months’ worth. The payback period can also be used to evaluate whether or not an offer

is worth taking, as the following example will illustrate.

Example 14.2.2 To dd has a no-contract cell phone. He pays no monthly or annual

fee, but pays a fl at 20 cents per minute used. He is considering signing up for a contract

that offers 800 minutes per month and costs $47.99 per month. He knows that he will

never actually use the full 800 minutes, but wonders how many minutes he would need

to use for it to be worth signing up for the new plan.

Todd’s investment would be $47.99 each month. The return would be avoiding paying the

20 cents per minute that he currently must pay. Thus, his payback period is $47.99/$0.20 


239.95, or about 240 minutes per month.

In the example of Todd’s cell phone, knowing the payback period allows Todd to make a

reasoned decision about whether the new plan is worth it. If he believes that he will use at least

240 minutes per month on average, the new plan is worth the $47.99 per month commitment.

If he has doubts that he will use that many minutes, he may decide to stick with his current

plan. Whichever he decides, knowing the payback period enables him to draw a conclusion.

Notice that in this second example, the “time” of the payback period was not a period

of time going by, but the minutes represented the amount of use. When we say that Todd’s

payback period is 240 minutes, we don’t mean that he recovers his investment 240 minutes

after signing up for the new plan; we mean that he recovers his investment if he uses

240 minutes per month. In our original discussion of Brant’s gym pass, we saw that we

could have measured his payback period in weeks (comparable to how we measured

Jeanette’s payback on her advertising costs) or in visits (comparable to how we measured

Todd’s payback on the new cell phone plan). Whichever type of units we use, though, the

payback period can be evaluated in much the same way.

There is some alternative terminology used for this sort of analysis, which can be helpful

if we want to emphasize that the payback period is based on something other than elapsed

time on the calendar or clock. The payback period is also sometimes called the break-even

point. Using that terminology, we would say that Jeanette breaks even after 7.14 months,

and that Todd breaks even at 240 minutes of monthly use. We will use the terms payback

and break-even interchangeably in this section.

More Involved Payback Calculations

Sometimes the calculation of the cost, and of the payback, can be a bit more involved.

Example 14.2.3 Jenny bought a compact fl uorescent (CF) lightbulb for $4.95.

A comparable standard incandescent bulb would have cost $0.24. The CF bulb will

last as long as eight incandescent bulbs, and also uses only about one-quarter the

electricity. Her electric rate is $0.12 per kilowatt-hour (kWh), and the fi xture in which

she is placing the new bulb uses about 14 kilowatt-hours per month with a standard

bulb. How long will it take for her investment in the CF bulb to pay for itself?

Incandescent bulbs that would last as long would have cost 8($0.24) 
 $1.92. So she has

invested an extra $4.95  $1.92 
 $3.03 in the CF bulb.