The Mathematics of Money

(Darren Dugan) #1

Copyright © 2008, The McGraw-Hill Companies, Inc.


required rate of return; two different people, with different rate of return expectations, can
come up with very different results and draw very different conclusions as a result. Far
worse, these present values are based on projections of profits far into the future, and as we
well know even the most well-founded predictions often miss the mark by quite a bit.^2
In this section, we will consider an alternative method, also widely used, to evaluate the
payoff of an investment.

The Payback Period Method


Suppose that the town where Brant lives has just built a new recreation and fitness center,
and he’s just decided its time to get into better shape. He plans to work out at the fitness
center three times a week. To use the center, you either must pay $3.50 each visit, or buy an
annual pass for $200. Before working up a sweat at the fitness center, Brant needs to work
up a mental sweat to decide which way to pay.
On the one hand, he can work out the math fairly easily. Three visits a week times
52 weeks a year is 156 visits; 156 visits times $3.50 per visit works out to (156)($3.50) 
$546, far more than the $200 annual pass. It seems that the annual pass is a no-brainer ...
but then again, Brant knows that he has a bit of a history of deciding to get serious about
working out, going great guns at it for a few weeks, and then just losing steam. If he doesn’t
stick with it, or if he does keep working out but not quite as often as planned, he might not
really get the equivalent of $546 from his annual pass; in fact, he might not even get the
equivalent of $200 from his $200 annual pass.
The payback period of an investment is the amount of time it takes for the profits
from the investment to equal the amount invested. In a case like this one, calculating a
payback period may be helpful, since it will tell us how long Brant would need to keep
up his plans in order for the annual pass to be worthwhile. The $200 cost of the pass can
be thought of as an investment, and the profit is the $3.50 per visit that he avoids by hav-
ing the pass.
In Brant’s case, he would be paying 3($3.50)  $10.50 per week. The payback period of
the annual pass would be $200/$10.50  19.04 weeks, or just over 19 weeks. So, if Brant
sticks to his 3 visits a week plan for 19 weeks, the annual pass will have “paid for itself.”
Knowing this may help him decide whether or not the annual pass is a good idea.
Note that the “periods” of the payback period were weeks, because we expressed the
rate of return as $10.50 per week. We could have alternatively used a rate per month or
any other reasonable period of time, in which case the payback period would be expressed
in that same unit of time. Or, we could even use something other than a standard unit of
time. If we looked at the return on his investment as $3.50 per visit instead of $10.50
per week we could calculate the payback period as $200/$3.50  57.14 visits. This form
might be preferable if Brant is committed to keep working out, but not sure if he can keep
up the pace of 3 visits per week.
We can generalize this idea into a formula:

FORMULA 14.2.1


The Payback Period of an Investment

P = __RI

where
P represents the PAYBACK PERIOD
I represents the AMOUNT INVESTED
and
R represents the INVESTMENT RETURN PER PERIOD

(^2) For an illustration, levitate your fusion-powered hovercar to the space colony library and take a look back to predictions
from 50 years ago about what life would be like today. (You may need to ask the reference android for help.) The world in
which you live today is almost entirely unlike the world foreseen for today 50 years ago. There is no reason to think that
predictions made for 50 years in the future will be any more reliable than predictions made 50 years in the past.
14.2 The Payback Period Method 573

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