The Mathematics of Money

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

14.1 The Present Value Method 567

To see why, let’s go back to one of the alternative formulas for the present value annuity

factor from Chapter 4. There, we saw that:

a n (^) | (^) i

1  (1  i)n
__i
If you plug in very large values for n, the (1  i)n in the numerator gets very close to zero.
You can verify this for yourself on a calculator; the larger the value of n is, the closer the
value of (1  i)n gets to zero. The table below illustrates this, using an 8% rate:
n (1
i)n
10 0.46319349
30 0.09937733
50 0.02132123
75 0.00311328
100 0.00045459
250 0.0000000044061688
500 0.000000000000000019
So for very large values of n, this part of the formula becomes irrelevant, leaving us for all
practical purposes with just 1/i. This leads us to:

FORMULA 14.1.1

Present Value of a Perpetuity

The present value of a stream of payments carrying on indefi nitely into the future is

PV  _____PMTi

where

PV represents the PRESENT VALUE

PMT represents the amount of the PAYMENT per period

and

i represents the INTEREST RATE per period

Example 14.1.2 Find the present value of a $7,500 per year perpetuity assuming

an interest rate of 8%.

PV
_____PMTi

$7,500

___0.08
$93,750

Note that the present value assuming payments never stop is not that much larger than the

present value assuming payments stop after 20 or 50 years. This may seem surprising at

first, but actually it agrees with common sense. How much are payments that won’t be

received until 50 or more years in the future worth today? The present value of those far-

future profits is so small that they make very little difference in the overall present value.

Of course, assuming that the investment will keep churning out $7,500 per year forever

is not realistic. Sooner or later the business will fail, the mountains will crumble, the sun

will burn out. But using a perpetuity does not mean that we literally expect everything to

keep churning on forever. Since the far-future payments contribute so little to the pres-

ent value, in cases where there is no clear ending point, we can use a perpetuity as a very

reasonable approach to determining a present value. The fact that the perpetuity formula is

surprisingly simple is another attraction.

This is not to say that a perpetuity is always the appropriate choice, however. Care

should be taken to use this tool only in cases where the stream of payments can be expected

to continue on into the indefinite future. If there is reason to believe that there will be a

foreseeable ending point, a standard annuity is the more appropriate tool.

Example 14.1.3 A laundromat owner is considering investing in more effi cient

washing machines. From her knowledge of her business and research on the new

machines, she believes that each machine will save her $245 per year in lower utility