Chapter 5 Time Value of Money 1315-3 PRESENT VALUES
Finding a present value is the reverse of! nding a future value. Indeed, we simply
solve Equation 5-1, the formula for the future value, for the PV to produce the
basic present value equation, 5-2:
Future value = FVN = PV(1 + I)N 5-1
Present value = PV =
FV N__ (1 + I) (^) N 5-2
We illustrate PVs with the following example. A broker offers to sell you a
Treasury bond that will pay $115.76 3 years from now. Banks are currently offer-
ing a guaranteed 5% interest on 3-year certi! cates of deposit (CDs); and if you
don’t buy the bond, you will buy a CD. The 5% rate paid on the CDs is de! ned as
your opportunity cost, or the rate of return you could earn on an alternative invest-
ment of similar risk. Given these conditions, what’s the most you should pay for
the bond? We answer this question using the four methods discussed in the last
section—step-by-step, formula, calculator, and spreadsheet. Table 5-2 summarizes
the results.
First, recall from the future value example in the last section that if you in-
vested $100 at 5%, it would grow to $115.76 in 3 years. You would also have $115.76
after 3 years if you bought the T-bond. Therefore, the most you should pay for the
bond is $100—this is its “fair price.” If you could buy the bond for less than $100,
you should buy it rather than invest in the CD. Conversely, if its price was more
than $100, you should buy the CD. If the bond’s price was exactly $100, you should
be indifferent between the T-bond and the CD.
The $100 is de! ned as the present value, or PV, of $115.76 due in 3 years when
the appropriate interest rate is 5%. In general, the present value of a cash! ow due N
years in the future is the amount which, if it were on hand today, would grow to equal the
given future amount. Because $100 would grow to $115.76 in 3 years at a 5% interest
rate, $100 is the present value of $115.76 due in 3 years at a 5% rate. Finding pres-
ent values is called discounting; and as noted above, it is the reverse of compound-
ing—if you know the PV, you can compound to! nd the FV, while if you know the
FV, you can discount to! nd the PV.
Opportunity Cost
The rate of return you
could earn on an
alternative investment
of similar risk.
Opportunity Cost
The rate of return you
could earn on an
alternative investment
of similar risk.
Discounting
The process of finding the
present value of a cash
flow or a series of cash
flows; discounting is the
reverse of compounding.
Discounting
The process of finding the
present value of a cash
flow or a series of cash
flows; discounting is the
reverse of compounding.
$115.76
A B C D E F G
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
Future payment
Interest rate
No. of periods
=
=
=
= $115.76
5.00%
3
CFN
I
N
FV
Periods:
Cash Flow Time Line:
0
PV =?
Step-by-Step Approach: $100.00 $105.00 $110.25
$115.76
1 2 3
Formula Approach: PV = FVN/(1 " I)N $100.00
Calculator Approach:
In the Excel formula, 0 indicates that there are no intermediate cash "ows.
Excel Approach: Fixed inputs:
Cell references:
N I/YR PV PMT
3 5
!$100.00
$0
FV
$115.76
=PV(0.05,3, 0 ,115.76) = !$100.00
=PV(C65,C66, 0 ,C64) = !$100.00
PV = $115.76/(1.05)^3
PV =
PV =
Tabl e 5 - 2 Summary of Present Value Calculations