CHAPTER 5 Time Value of Money
Do time lines deal only with years, or can other periods be used?
Set up a time line to illustrate the following situation: You currently have
$2,000 in a 3-year certi! cate of deposit (CD) that pays a guaranteed 4%
annually.SELF^ TEST (^)
5-2 FUTURE VALUES
A dollar in hand today is worth more than a dollar to be received in the future
because if you had it now, you could invest it, earn interest, and end up with more
than a dollar in the future. The process of going to future values (FVs) from present
values (PVs) is called compounding. For an illustration, refer back to our 3-year
time line and assume that you plan to deposit $100 in a bank that pays a guaran-
teed 5% interest each year. How much would you have at the end of Year 3? We
! rst de! ne some terms, after which we set up a time line and show how the future
value is calculated.
Future Value (FV)
The amount to which a
cash flow or series of cash
flows will grow over a
given period of time when
compounded at a given
interest rate.
Present Value (PV)
The value today of a future
cash flow or series of cash
flows.
Compounding
The arithmetic process
of determining the final
value of a cash flow or
series of cash flows when
compound interest is
applied.
Future Value (FV)
The amount to which a
cash flow or series of cash
flows will grow over a
given period of time when
compounded at a given
interest rate.
Present Value (PV)
The value today of a future
cash flow or series of cash
flows.
Compounding
The arithmetic process
of determining the final
value of a cash flow or
series of cash flows when
compound interest is
applied.
PV! Present value, or beginning amount. In our example, PV! $100.
FVN! Future value, or ending amount, of your account after N periods.
Whereas PV is the value now, or the present value, FVN is the value N
periods into the future, after the interest earned has been added to the
account.
CFt! Cash " ow. Cash " ows can be positive or negative. The cash " ow for a par-
ticular period is often given as a subscript, CFt , where t is the period. Thus,
CF 0! PV! the cash " ow at Time 0, whereas CF 3 is the cash " ow at the
end of Period 3.
I! Interest rate earned per year. Sometimes a lowercase i is used. Interest
earned is based on the balance at the beginning of each year, and we assume
that it is paid at the end of the year. Here I! 5% or, expressed as a decimal,
Although the periods are often years, periods can also be quarters or months or
even days. Note that each tick mark corresponds to both the end of one period and
the beginning of the next one. Thus, if the periods are years, the tick mark at Time
2 represents the end of Year 2 and the beginning of Year 3.
Cash " ows are shown directly below the tick marks, and the relevant interest
rate is shown just above the time line. Unknown cash " ows, which you are trying
to! nd, are indicated by question marks. Here the interest rate is 5%; a single cash
out" ow, $100, is invested at Time 0; and the Time 3 value is an unknown in" ow. In
this example, cash " ows occur only at Times 0 and 3, with no " ows at Times 1 or
- Note that in our example, the interest rate is constant for all 3 years. That condi-
tion is generally true; but if it were not, we would show different interest rates for
the different periods.
Time lines are essential when you are! rst learning time value concepts, but
even experts use them to analyze complex! nance problems; and we use them
throughout the book. We begin each problem by setting up a time line to show
what’s happening, after which we provide an equation that must be solved to! nd
the answer. Then we explain how to use a regular calculator, a! nancial calculator,
and a spreadsheet to! nd the answer.