Chapter 5 Time Value of Money 1435-12 UNEVEN CASH FLOWS
The de! nition of an annuity includes the words constant payment—in other words,
annuities involve payments that are equal in every period. Although many
! nancial decisions involve constant payments, many others involve uneven, or
nonconstant, cash! ows. For example, the dividends on common stocks typically
increase over time, and investments in capital equipment almost always generate
uneven cash " ows. Throughout the book, we reserve the term payment (PMT) for
annuities with their equal payments in each period and use the term cash! ow
(CFt) to denote uneven cash " ows, where t designates the period in which the cash
" ow occurs.
There are two important classes of uneven cash " ows: (1) a stream that con-
sists of a series of annuity payments plus an additional! nal lump sum and (2) all
other uneven streams. Bonds represent the best example of the! rst type, while
stocks and capital investments illustrate the second type. Here are numerical ex-
amples of the two types of " ows:
Periods (^0) I = 12% 1 2 4
Cash "ows
3 5
$0 $100 $100 $100 $100 $ 100
$ 1,000
$1,100
- Annuity plus additional "nal payment:
Periods (^0) I = 12% 1 2 4
Cash "ows
3 5
$0 $100 $300 $300 $300 $500
- Irregular cash #ows:
We can! nd the PV of either stream by using Equation 5-7 and following the step-
by-step procedure, where we discount each cash " ow and then sum them to! nd
the PV of the stream:
PV!
CF 1
______(1 # I) 1 #
CF 2
(1 ______# I)^2 #^...^ #^CFN
_______
(1 # I)N! ∑
t! 1N
CFt__(1 # I) (^) t 5-7
If we did this, we would! nd the PV of Stream 1 to be $927.90 and the PV of Stream
2 to be $1,016.35.
The step-by-step procedure is straightforward; but if we have a large number
of cash " ows, it is time-consuming. However,! nancial calculators speed up the
process considerably. First, consider Stream 1; notice that we have a 5-year, 12% or-
dinary annuity plus a! nal payment of $1,000. We could! nd the PV of the annuity,
then! nd the PV of the! nal payment and sum them to obtain the PV of the stream.
Financial calculators do this in one simple step—use the! ve TVM keys, enter the
data as shown below, and press the PV key to obtain the answer, $927.90.
N I/YR PV PMT FV
5 12 100 1000
–927.90
The solution procedure is different for the second uneven stream. Here we
must use the step-by-step approach as shown in Figure 5-4. Even calculators and
spreadsheets solve the problem using the step-by-step procedure, but they do it
Payment (PMT)
This term designates equal
cash flows coming at
regular intervals.
Payment (PMT)
This term designates equal
cash flows coming at
regular intervals.
Uneven (Nonconstant)
Cash Flows
A series of cash flows
where the amount varies
from one period to the
next.
Uneven (Nonconstant)
Cash Flows
A series of cash flows
where the amount varies
from one period to the
next.
Cash Flow (CFt)
This term designates a
cash flow that’s not part of
an annuity.
Cash Flow (CFt)
This term designates a
cash flow that’s not part of
an annuity.