Introduction to Finance

Figure 17.1
shows this relationship, called the 
NPV profi le
, between 
NPV and the cost of capital. As the cost of capital on a project rises, the NPV changes from pos-
itive, to zero, to negative. The cost of capital at which the NPV is zero deserves special attention.
NPV profi le 
the graphical 
relationship between a project’s 
NPV and cost of capital

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17.4 Capital Budgeting Techniques—Internal Rate of Return
531
While the NPV method tells us that projects A and B provide expected returns greater 
than 10 percent, we do not know the actual rates of return. The 
internal rate of return (IRR) 
method
fi nds the return that causes the NPV to be zero; namely, the point where the NPV 
profi le crosses the 0 axis in Figure 17.1. Net present value will equal zero when the PV of the 
cash fl ows equals the project’s initial investment, as seen in equation 17-2:
9
NPV =
∑
n
t
= 1
[
CF
t
∕(1 + IRR)
t
− Initial Investment
]
= 0 (17-2)
A trial-and-error process can be used to fi nd the internal rate of return (IRR), but fi nancial 
calculators and computer spreadsheets (such as Excel’s IRR function) provide a much quicker 
means of estimating internal rates of return.
Let’s illustrate the IRR process fi rst for project A. Because the cash infl ows form an annu-
ity, the IRR is easy to fi nd. From Chapter 9, we know the following:
Present value of an annuity = PVIFA × annuity cash fl ow
Rearranging, we divide the initial capital budgeting outlay (PV annuity) by the cash infl ow 
annuity amount to fi nd the present value interest factor (PVIF) for an ordinary annuity:
PVIFA = PV annuity∕Annual receipt
For project A, the PVIFA is 3.448 ($20,000∕5,800). We know this PVIFA of 3.448 is for fi ve 
years. By turning to Table 4 in the Appendix, “Present Value of a $1 Ordinary Annuity,” we 
can read across the fi ve-year row until we fi nd a PVIFA close to 3.448. It falls between 3.605 
(12 percent) and 3.433 (14 percent) but is closer to the PVIFA at 14 percent. Thus, the IRR for 
project A is slightly less than 14 percent.
Of course, with fi nancial calculators and computer spreadsheets, there are other, more 
precise, ways to fi nd the internal rate of return. As an example, we will use a fi nancial calcu-
lator and project A’s cash fl ows: