CP

alternative to end up with $127.63 after five years. Conversely, if the security cost

more than $100, you should not buy it, because you would have to invest only $100

in a similar-risk alternative to end up with $127.63 after five years. If the price were

exactly $100, then you should be indifferent—you could either buy the security or

turn it down. Therefore, $100 is defined as the security’s fair,or equilibrium,

value.

In general, the present value of a cash flow due n years in the future is the amount which,

if it were on hand today, would grow to equal the future amount.Since $100 would grow to

$127.63 in five years at a 5 percent interest rate, $100 is the present value of $127.63

due in five years when the opportunity cost rate is 5 percent.

Finding present values is calleddiscounting,and it is the reverse of compound-

ing—ifyouknowthePV,youcancompoundtofindtheFV,whileifyouknowtheFV,

youcandiscounttofindthePV.Whendiscounting,youwouldfollowthesesteps:

Time Line:

(^0) 5% 12345
PV ? 127.63
Equation:
To develop the discounting equation, we begin with the future value equation, Equa-
tion 2-1:
(2-1)
Next, we solve it for PV in several equivalent forms:
(2-3)
The last form of Equation 2-3 recognizes that the Present Value Interest Factor for
i and n, PVIFi,n, is shorthand for the formula in parentheses in the second version of
the equation.

1. NUMERICAL SOLUTION

05%1 2 3 4 5


 100  ←—-105.00←—-110.25←—-115.76←—-121.55←—-127.63

1.05 1.05 1.05 1.05 1.05

Divide $127.63 by 1.05 five times, or by (1.05)^5 , to find PV $100.

2. FINANCIAL CALCULATOR SOLUTION

Inputs: 5 5 0 127.63

Output: 
 100

Enter N 5, I 5, PMT 0, and FV 127.63, and then press PV to get PV 
100.

PV

FVn

(1i)n

FVna

1

1 i

b

n

FVn(PVIFi,n).

FVnPV(1i)nPV(FVIFi,n).

Present Value 65

Time Value of Money 63