80 CHAPTER 2 Time Value of Money
We could find the PV of each individual cash flow using the numerical, financial cal-
culator, or spreadsheet methods, and then sum these values to find the present value of
the stream. Here is what the process would look like:
01234567 6%
100 200 200 200 200 0 1,000
94.34
178.00
167.92
158.42
149.45
0.00
665.06
1,413.19
All we did was to apply Equation 2-8, show the individual PVs in the left column
of the diagram, and then sum these individual PVs to find the PV of the entire
stream.
The present value of a cash flow stream can always be found by summing the pres-
ent values of the individual cash flows as shown above. However, cash flow regularities
within the stream may allow the use of shortcuts. For example, notice that the cash
flows in periods 2 through 5 represent an annuity. We can use that fact to solve the
problem in a slightly different manner:
01234567 6%
100 200 200 200 200 0 1,000
94.34
693.02
653.79
0.00
665.06
1,413.19
Cash flows during Years 2 to 5 represent an ordinary annuity, and we find its PV at
Year 1 (one period before the first payment). This PV ($693.02) must then be dis-
counted back one more period to get its Year 0 value, $653.79.
Problems involving uneven cash flows can be solved in one step with most finan-
cial calculators. First, you input the individual cash flows, in chronological order, into
the cash flow register. Cash flows are usually designated CF 0 , CF 1 , CF 2 , CF 3 , and so
on. Next, you enter the interest rate, I. At this point, you have substituted in all the
known values of Equation 2-8, so you only need to press the NPV key to find the pres-
ent value of the stream. The calculator has been programmed to find the PV of each
cash flow and then to sum these values to find the PV of the entire stream. To input
the cash flows for this problem, enter 0 (because CF 0 0), 100, 200, 200, 200, 200, 0,
1000 in that order into the cash flow register, enter I 6, and then press NPV to ob-
tain the answer, $1,413.19.
Two points should be noted. First, when dealing with the cash flow register, the
calculator uses the term “NPV” rather than “PV.” The N stands for “net,” so NPV is
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78 Time Value of Money
