

How Do Companies Decide Which Projects to Undertake?
This discussion of net present value and internal rate of return examines how companies use NPV and IRR as decision tools to evaluate whether investments or projects are worth pursuing.
The perpetuities
topic introduced the dividend discount model as a way to value a
company's stock. Aside from buying stock back from the public, cash dividends are the common way that a company returns cash payments to its equity investors.
Where does the company get the cash to pay the dividends? The
company generates cash from the projects that it undertakes with the
cash that it acquires from investors (both debt and equity investors)
and any excess cash generated by its projects.
Investors base their required rate of return on the riskiness of the company's projects. The more risk
investors perceive in the cash flows from the projects, the higher the
rate of return investors will require from the company. The combined
return required by debt and equity capital is the weighted average cost of capital, or simply the cost of capital.
Investment versus financing
A key concept of business finance requires separating the investment
decision from the financing decision. The liabilities and net worth
comprise the source of funds—where the company gets its money. The
assets represent how those funds are invested—the investment decision.
The key point is that it does not matter what the source of funds is
when evaluating an investment.
For example, suppose an airline wanted to acquire an airplane for a
new route. The investment project is the new route. The airplane and
crew, ground personnel, fuel, ticketing, airport fees, and so on
represent the costs, or cash outflows, of this project. The revenues
from ticket sales are the revenues or cash inflows of the project.
These aspects of the project will remain the same regardless of how the
airline finances the project. The cost of the airplane is considered a
cash outflow. The firm could pay for the airplane with cash raised by
selling other assets, by using available cash, by borrowing from a
bank, issuing bonds, or issuing more stock. Even a lease represents a
form of financing. It does not matter what the source of funds is when
evaluating an investment.
So how does the firm take financing costs into account? Firms use the cost of capital.
Cost of capital
The cost of capital reflects the minimum amount that a firm must earn
on its assets in order for those assets to add value to the firm.
Expressed as a percent, the cost of capital is the rate at which assets
must provide cash inflows to justify their cost. Therefore, if the rate
of return of the net cash flows from a project, including the initial
investment and all future net cash flows, exceeds the cost of capital,
the project will add to the value of the firm. This was the case in
Challenge B. There you found that the assets generated a return of
21.31 percent, while the cost of capital was assumed to be 15 percent.
Understanding the derivation of the cost of capital requires a
review of how equity markets work, which goes beyond the scope of this
course. For the purpose of this topic, assume that the company's
financial manager has derived a value for the cost of capital to use in
evaluating projects.
Value additivity
Value additivity is the concept that the present value of a company
equals the sum of the present value of independent projects. For
projects to be evaluated independently, the cash flows from new
projects must include the effects that new projects have on existing
projects. This simple concept compels financial managers to go back and
reevaluate existing projects and helps managers focus on all of the
relevant cash flows attributable to the new project.
Relevant cash flows
Constructing the relevant cash flows for project evaluation is important and sometimes difficult. Some financial instruments,
such as bonds and mortgages, present a fairly welldefined set of cash
flows. Other financial instruments, such as options, futures, and
derivatives, can have complex cash flows dependent on several factors.
Most business projects require as much art as science in projecting the
cash inflows and cash outflows of a project, including the effects of
proposals on existing undertakings.
More advanced courses will deal with computing the cost of capital
projects and relevant cash flows of the firm's projects. The remainder
of this section will consider two methods analysts use to evaluate the
cash flows of different projects, regardless of when those cash flows
occur. These two methods are the net present value (NPV) and the
internal rate of return (IRR).

Does using returnable containers save a company money? Read about how to evaluate this option using net present value.
Can You Justify Returnables?

Projects with a Positive NPV Add Value to the Firm
The net present value, or NPV, is one of the most common methods
used to evaluate investments. At its simplest, NPV is the present value
computed by using the firm's cost of capital as the discount rate of
cash inflows, minus the present value of cash outflows, including the
initial investment.
Cash inflows and outflows can occur at any time during the project.
The NPV of the project is the sum of the present values of the net cash
flows for each time period t, where t takes on the values 0 (the beginning of the project) through N (the end of the project).


This can be expressed as
Sometimes, for convenience, this can be written with the initial cash flow listed separately as
C_{0} is negative if there is an initial cash outflow.


If the present value of the
cash flows, discounted at the cost of capital, exceeds the cost of the
investment, then the investment will add to the value of the company.
The positive NPVs calculated in Challenges B and C, where the discount
rates equaled the costs of capital, indicated that those projects would
add value to the firm.
Example: Single future cash flow
Consider two simple investments, each with only one future net cash flow.
Assume that each project costs $900. The net present value of each project using different costs of capital are:
The net present value indicates that project B is more valuable at a
cost of capital of 10 percent, that project A is slightly more valuable
at 15 percent, and that neither project would add to the value of the
company at a cost of capital of 20 percent.
Which project would you choose? If the projects are
independent, you would choose both if the cost of capital is 15 percent
or less. Neither project would be acceptable at a cost of capital of 20
percent, since this would lower the present value of the whole firm. If
these are mutually exclusive projects, take the one that adds more to
the value of the firm. In this case, that answer depends on the actual
cost of capital.
What happens if the projects have different levels of risk? Bond markets require higher rates of return for bonds from companies in poor financial condition (the socalled junk bond market) than the markets do for bonds issued by companies in excellent financial condition (investmentgrade
bonds). One way to account for different levels of risk between
competing projects is to increase the cost of capital by some amount to
reflect the higher risk. If project B is substantially more risky than
project A, such that the NPV of project A is computed at 15 percent
whereas the NPV of project B is computed at 20 percent, then the
investment decision takes on a new dimension. In this case, only
project A would be acceptable.
As with the cost of capital itself, the process for adjusting projects for risk goes beyond the scope of this topic.
Later, in the spreadsheet examples, the NPV of multiple net cash
flows is calculated. As the Challenge problems demonstrate, the
analysis of net present value is the same whether you have one future
cash flow or many.
The NPV calculation is one method analysts use to decide whether a
potential project, or investment, can add value to the firm. As you may
have seen in Challenge A and Challenge B, the NPV is often calculated
assuming a required rate of return on the investment, a rate given in
the assumptions of the factual situation. The NPV calculation provides
a dollar measure of how much a project is expected to add to a firm's
value. Analysts may also want to know what the rate of return on a
project is in order to compare it to the cost of capital. This rate is
called the internal rate of return, or IRR.
Projects with an IRR Greater than the Cost of Capital Add Value to the Firm
The IRR is the discount rate that makes the present value of the
cash inflows equal to the present value of the cash outflows. This is
the same as saying that the IRR is the discount rate that makes the net
present value equal to zero.
What is the IRR for the two projects above, each of which has an
initial investment of $900? Project A provides a net cash inflow of
$2,000 at the end of five years. Project B provides a net cash inflow
of $3,000 at the end of eight years.
Unlike projects with multiple future net cash flows, the IRR for a
single future net cash flow, and a single initial investment, can be
computed with a relatively simple formula.
In the "Time Value of Money" section, you learned how to calculate
the present value (PV) of a future net cash flow (FV) received N periods from now, discounted at a periodic interest rate of r.
Solving for r, the rate of return, produces
Let IRR_{x} be the internal rate of return for project x.
For project A,
 
FV = $2,000
N = 5
PV = $900

Therefore, the IRR is
For project B,
 
FV = $3,000
N = 8
PV = $900

Therefore, the IRR is
You will find businesses using both the NPV and IRR calculations to
aid in making investment decisions. IRR is a potentially flawed
decision tool because it can be easily misapplied. IRR problems include:
 Lending versus borrowing. For some projects that have cash inflows followed by cash outflows, the NPV rises as the discount rate is increased. In this case, projects in which the IRR is less than the cost of capital are acceptable. 
 Multiple
rates of return. If there is more than one change in the sign of the
cash flows, the project may have several IRRs, or no IRR. 
 Mutually exclusive projects. The IRR rule may not accurately rank mutually exclusive projects that vary in time or scale. 
 Shortterm
interest rates may be different from longterm rates. In a single
project, the cost of capital for oneyear cash flows can differ from
the cost of capital for twoyear cash flows, and so on. This does not
allow you to compare the project's IRR with the cost of capital. In
these cases, there is no straightforward method for calculating the
project's IRR. 
The following animation demonstrates how IRR can lead to faulty decisionmaking.
Evaluating multiple net cash flows
The basic approach used for projects with a single net cash inflow
after the initial investment also applies to projects with multiple
future net cash inflows. Annuity formulas aid computations when all
cash flows are equal in amount, but when cash inflows vary over time,
computations are more tedious.
Example Analyze a project with quarterly net cash flows using
a discount rate of 3 percent per quarter. The following table shows the
cash flows by quarter.
For this project, the investment is the initial cash flow. By
convention, outflows appear as negative numbers; net cash inflows are
positive.

Obtain the cost of capital, 3 percent per quarter, and generate
discount factors. From the present value of a single payment, the discount factor for period t at a periodic interest rate of r is


Compute present values of each net cash flow. Multiply the net cash
flow for each period by its discount factor to obtain its present value.


Sum the present values of each cash flow to calculate the NPV.
The NPV for this project is $9.32.

Find the IRR, the discount rate, that makes the NPV zero. See the spreadsheet example below.
Spreadsheet Examples


Net present value
Most modern spreadsheets have financial spreadsheet functions that can compute NPV and IRR.
Although the exact form may vary, the NPV spreadsheet function typically takes the form
NPV (discount rate, net cash flows for t = 1 to N)
The discount rate is a value or cell address for the periodic rate.
Since the cash flows occur at quarterly intervals, the discount rate
must be entered as a decimal quarterly rate.
Net cash flows for t = 1 to N means that the cell
range for only the future net cash flows must be entered. As a result,
the NPV is not NPV as defined here, but the PV of future cash flows.
The initial investment must be subtracted from the result of the NPV
function to get the actual net present value. (See the cell range for
IRR by comparison.)
Net Present Value = NPV function + initial investment (as negative number).
Using the quarterly cash flows in the example above, enter in a new spreadsheet, in cells A1 through A5, the values: 900, 250, 300, 400, and 20.
In cell B6, enter the quarterly discount rate 0.03.
In cell A6, enter the formula =NPV(B6,A2:A5)+A1.
The result should be $9.32.

You can use Excel to solve NPV
NPV Excel Tutorial

Internal rate of return
Unlike the spreadsheet version of NPV, the IRR function actually uses the values for time periods 0 through N:
IRR (net cash flows for t = 0 to N, {optional guess discount rate}).
The net cash flows for t = 0 to N are the range of cells with all net cash flows, including the initial investment as a negative number.
The optional "guess discount rate" allows the user to enter a value
or cell that contains a "guess" as to the value of the IRR in its
periodic form, such as .012 annually, .03 for quarterly, and .01 for
monthly.
Using the same information in the NPV example above, enter in cell A7 the formula =IRR(A1:A5).
The answer should be 3.4897 percent. (You may need to increase the
number of decimals displayed, or the result may appear to be 3
percent). This is the quarterly IRR.
To compute the effective annual rate
(EAR) needed to evaluate the annual cost of capital, use the formula
presented in Future Value in the "Time Value of Money" section:
The effective annual rate is
Here m is the compounding frequency.
Notice that this is not the same as multiplying the quarterly rate by
4, which is 13.96 percent. The annual nominal rate is 13.96 percent.
Hint: Do not compare the annualized nominal rate to the cost of
capital stated at an annual rate. Instead, compare the EAR with the
annual cost of capital.
Summary
 The
cost of capital is the discount rate companies use to evaluate
projects. It will vary from project to project depending on the
assessed risk of each project. 
 The investment decision is separate from the financing decision. 
 Value
additivity is the theory that the present value of a company is equal
to the sum of the present values of all of the company's independent
projects. 
 Relevant
cash flows include the initial investment, cash inflows, and cash
outflows for a new project, plus the changes in cash flow on existing
projects. 
 Net
present value is the net dollar benefit of a new project discounted at
the cost of capital. NPV must be positive to add value to the firm. 
 The
internal rate of return is the discount rate that makes the NPV equal
to zero. IRR must be greater than the cost of capital for a new project
to add value to the firm. 
How do spreadsheets find the IRR?
How many decimal places should you use?
1. Calculate the internal rate of return for the following set of
cash flows by first using trial and error. The initial cash outflow is
$8,145, followed by seven years of semiannual cash inflows of $890. The
associated discount rate is 5.6 percent.
Hint: There is a concise way to solve by trial and error.
Solution 1
2. Your company will invest $5 million to receive payments of $2
million for the next 10 years. Calculate the NPV if the required rate
of return is 14 percent per year.
Solution 2
Alternate Solution 2
3. After graduation, you landed a job at a large, multinational
media corporation. Your firm has been negotiating a license agreement
to use a certain documentary film for a term of 2.5 years. You expect
that the film will return cash flows of $12.5 million at the end of
each sixmonth period. The company licensing the rights to use the film
is asking $50 million. Your company's required rate of return is 17.5
percent. Should you purchase the license to show the film?
Solution 3
Alternate Solution 3
4. Consider the following information pertaining to a project that
your company is currently evaluating. The project calls for your
factory to add a second canning machine that will result in endofyear
cash flows of $3,200, $3,700, $4,100, $4,500, and $4,900 over the next
five years. The canning machine will cost $15,000, and your company
uses a 13 percent discount rate when evaluating projects. What is the
net present value of these cash flows?
Solution 4
Alternate Solution 4
5. Take a set of four annual cash flows starting at the end of year
0: $1,000, +$400, +$600, and +$800. Compute the IRR. Then compute the
FV of each cash flow using the IRR as the compounding interest rate.
Sum these FVs. What is the net future value? Using the sum of the
future values for the cash inflows in years 1, 2, and 3, what is the
IRR of this single future value against the initial investment of
$1,000? (Use the formula to compute the IRR of a single future cash
flow.)
Solution 5
6. Take the same cash flows in the question above. The IRR was
fairly high at 31.69 percent. What if the cash flows from the project
cannot be reinvested at the IRR? (No other project at that level
exists.) Compute the future value of the cash flows at the end of year
3 using a lower interest rate, such as 12 percent. Add the future cash
inflows to derive a single sum cash equivalent inflow as of the end of
year 3. Now compute the IRR using the formula for a single future cash
inflow at the end of year 3. What happens to the IRR?
Solution 6
7. A wine lover has decided to start a winery. The initial
investment will be $5 million. The winery will require additional
investments of $1 million per year at the end of the next five years
while the vines mature. Beginning at the end of year 6, the winery is
expected to produce net cash inflows of $2 million at the end of each
year, growing at 20 percent per year. How long will it take the project
to reach a positive net present value, assuming an annually compounded
discount rate of 15 percent?
Solution 7
8. Using the information about the winery in the previous question,
when would the IRR exceed the discount rate of 15 percent if there is
an additional $4 million expenditure in year 10, with no change in
revenues?
Solution 8

You can use Excel to solve IRR
IRR Excel Tutorial

