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140 PART 2 Important Financial Concepts
as the discount rate, required return, cost of capital, and opportunity cost. These
terms will be used interchangeably in this text.
EXAMPLE Paul Shorter has an opportunity to receive $300 one year from now. If he can
earn 6% on his investments in the normal course of events, what is the most he
should pay now for this opportunity? To answer this question, Paul must determine
how many dollars he would have to invest at 6% today to have $300 one
year from now. Letting PV equal this unknown amount and using the same notation
as in the future value discussion, we have
PV (1 0.06) $300 (4.7)
Solving Equation 4.7 for PV gives us Equation 4.8:
$300
PV
(4.8)
(1 0.06)
$283.02
The value today (“present value”) of $300 received one year from today,
given an opportunity cost of 6%, is $283.02. That is, investing $283.02 today at
the 6% opportunity cost would result in $300 at the end of one year.
The Equation for Present Value
The present value of a future amount can be found mathematically by solving
Equation 4.4 for PV. In other words, the present value, PV, of some future
amount, FVn, to be received n periods from now, assuming an opportunity cost of
i, is calculated as follows:
FVn
(1 i) n
1
(1 i) n
PV FV n (4.9)
Note the similarity between this general equation for present value and the
equation in the preceding example (Equation 4.8). Let’s use this equation in an
example.
EXAMPLE Pam Valenti wishes to find the present value of $1,700 that will be received 8
years from now. Pam’s opportunity cost is 8%. Substituting FV8$1,700, n8,
and i0.08 into Equation 4.9 yields Equation 4.10:
$1,700 $1,700
PV $918.42 (4.10)
(1 0.08) 8 1.851
The following time line shows this analysis.
Time line for present
value of a single
amount ($1,700 future
amount, discounted at
8%, from the end of
8 years)
0 1 2 3 4 5 6 7 8
PV = $918.42
End of Year
FV 8 = $1,700