82 CHAPTER 2 Time Value of MoneyThe future value of our illustrative uneven cash flow stream is $2,124.92:01234567 6%
100 200 200 200 200 0 1,000
0
224.72
238.20
252.50
267.65
141.85
2,124.92Some financial calculators have a net future value (NFV) key which, after the cash
flows and interest rate have been entered, can be used to obtain the future value of an
uneven cash flow stream. Even if your calculator doesn’t have the NFV feature, you
can use the cash flow stream’s net present value to find its net future value: NFV
NPV (1 i)n. Thus, in our example, you could find the PV of the stream, then find
the FV of that PV, compounded for n periods at i percent. In the illustrative problem,
find PV 1,413.19 using the cash flow register and I 6. Then enter N 7, I 6,
PV 1413.19, and PMT 0, and then press FV to find FV 2,124.92, which
equals the NFV shown on the time line above.Solving for i with Uneven Cash Flow Streams
It is relatively easy to solve for i numerically when the cash flows are lump sums or an-
nuities. However, it is extremely difficultto solve for i if the cash flows are uneven, be-
cause then you would have to go through many tedious trial-and-error calculations.
With a spreadsheet program or a financial calculator, though, it is easy to find the
value of i. Simply input the CF values into the cash flow register and then press the
IRR key. IRR stands for “internal rate of return,” which is the percentage return on an
investment. We will defer further discussion of this calculation for now, but we will
take it up later, in our discussion of capital budgeting methods in Chapter 7.^8Give two examples of financial decisions that would typically involve uneven cash
flows. (Hint: Think about a bond or a stock that you plan to hold for five years.)
What is meant by the term “terminal value”?Growing Annuities
Normally, an annuity is defined as a series of constantpayments to be received over a
specified number of periods. However, the term growing annuityis used to describe
a series of payments that is growing at a constant rate for a specified number of
periods. The most common application of growing annuities is in the area of fi-
nancial planning, where someone wants to maintain a constant real, or inflation-
adjusted, income over some specified number of years. For example, suppose a
65-year-old person is contemplating retirement, expects to live for another 20 years,↑
↑
↑
↑
↑
↑(^8) To obtain an IRR solution, at least one of the cash flows must have a negative sign, indicating that it is an
investment. Since none of the CFs in our example were negative, the cash flow stream has no IRR. How-
ever, had we input a cost for CF 0 , say, $1,000, we could have obtained an IRR, which would be the rate of
return earned on the $1,000 investment. Here IRR would be 13.96 percent.