Corporate Finance
Lecture: Investment Decision Rules
SHEN Tao （沈涛） Tsinghua University
Outline
1 The NPV Decision Rule
2 Using the NPV Rule
3 Alternative Decision Rules
4 Choosing Between Projects
5 Evaluating Projects with Different Lives
6 Choosing Among Projects When Resources Are Limited
Investment Decision
We have spent the past few weeks using TVM
relationships to evaluate the NPV from cash flows over
time.
We will now use these methods to evaluate investment
projects.
The firm’s decision to pursue a project should be based
on whether or not the project meets the objective of
maximizing the value of the firm to the owners or
shareholders.
Investment Decision - Basics
Four steps in the investment decision making:
1. Estimate cash flows in the project. Essential and
most difficult step.
2. Select an appropriate opportunity cost of capital.
3. Apply an evaluation technique.
4. Determine the selected alternative and implement.
For now we will focus on the third step and evaluate
various techniques.
Project Evaluation Methods
Best Evaluation Method:
Net Present Value (NPV) Rule
Similar but Potentially Inferior Evaluation Method
Internal Rate of Return (IRR) Rule
Modified IRR Rule
Commonly Used Clearly Inferior Evaluation Methods
Payback Rule
The NPV Decision Rule
Logic of the decision rule:
When making an investment decision, take the alternative with the
highest NPV, which is equivalent to receiving its NPV in cash today
A simple example:
In exchange for $500 today, your firm will receive $550 in one
year. If the interest rate is 8% per year:
PV(Benefit)= ($550 in one year) ÷ ($1.08 $ in one year/$ today) =
$509.26 today
This is the amount you would need to put in the bank today to
generate $550 in one year
NPV= $509.26 - $500 = $9.26 today
The NPV Decision Rule
You should be able to borrow $509.26 and use the $550 in
one year to repay the loan
This transaction leaves you with $509.26 - $500 = $9.26
today
As long as NPV is positive, the decision increases the value of
the firm regardless of current cash needs or preferences
The NPV Decision Rule
The NPV decision rule implies that we should:
Accept positive-NPV projects; accepting them is equivalent to
receiving their NPV in cash today, and
Reject negative-NPV projects; accepting them would reduce
the value of the firm, whereas rejecting them has no cost (NPV
= 0)
Using the NPV Rule
A take-it-or-leave-it decision:
A fertilizer company can create a new environmentally friendly
fertilizer at a large savings over the company’s existing fertilizer
The fertilizer will require a new factory that can be built at a
cost of $81.6 million. Estimated return on the new fertilizer
will be $28 million after the first year, and last four years
Using the NPV Rule
Computing NPV
The following timeline shows the estimated return:
Using the NPV Rule
Given a discount rate r, the NPV is:
28 28 28 28
NPV 81.6
1 r (1 r )2 (1 r )3 (1 r ) 4
We can also use the annuity formula:
28 1
NPV 81.6 1 4
r (1 r )
If the company’s cost of capital is 10%, the NPV is $7.2 million and
they should undertake the investment
Using the NPV Rule
The NPV depends on cost of capital
NPV profile graphs the NPV over a range of discount rates
Based on this data the NPV is positive only when the discount
rates are less than 14%
IRR is defined as the discount rate(s) that set the NPV equal to
zero.
8.2 Using the NPV Rule
Payback Rule
1. Calculate the amount of time it takes to pay back the initial
investment, called the payback period.
2. Accept the project if the payback period is less than a pre-
specified length of time—usually a few years.
3. Reject the project if the payback period is greater than that
pre-specified length of time
Payback Example
Considering the following four projects:
Payback
Project CF0 CF1 CF2 CF3 CF4 Period NPV @10%
Q -5,000 0 5,000 0 0 2 ?
R -5,000 2,500 2,500 0 0 2 ?
S -5,000 2,500 2,500 2,500 0 2 ?
T -5,000 2,000 2,000 2,000 2,000 ? ?
The highest NPV is project ?
If we set the required payback period to be 2 years, we will turn
down project ?
Weakness of Payback Rule
1. Ignores the time value of money.
2. Ignores cash-flows after the payback period.
3. Lacks a decision criterion grounded in economics
(what is the right number of years to require for a
payback period?).
IRR Rule
IRR Investment Rule: Take any investment
opportunity where IRR exceeds the opportunity cost of
capital. Turn down any opportunity whose IRR is less
than the opportunity cost of capital.
In most cases IRR rule agrees with NPV for stand alone
project’s if all of the project’s negative cash flows precede
the positive cash flows. In other cases the IRR may
disagree with NPV.
Pitfall: Delayed Investment
Suppose by taking a project, you have the following cash
flows:
Question:
What is the IRR for this project?
Assume that the cost of capital is 10%, should you take
the project or not?
What is NPV when r=10%?
Pitfall: Delayed Investment
500,000 500, 000 500, 000
NPV 1, 000,000 2
3
$243,426
1.1 1.1 1.1
Multiple IRRs
When the cash flows flip signs more than once, there are,
in general, multiple IRRs.
For example, a ten-year project has following cash flows:
CF0 = +1,000,000
CF1 = CF2 = CF3 = -500,000
CF10 = +600,000
CFn = 0 for other n
500,000 500,000 500,000 600,000
NPV ( r ) 1,000,000
1 r (1 r ) 2
(1 r ) 3
(1 r )10
Modified IRR
Used to overcome problem of multiple IRRs
Assume positive CF is reinvested at the Cost of
Capital and calculate the resulting IRR which we call
MIRR.
Bring all negative cash flows to the present and
compound all of the positive cash flows to the end.
Modified IRR
Example:
Assume a project creating CF0=-$1000, CF1=$2500,
CF3=-1540. Assume discount rate is 15%.
So NPV profile is:
2500 1540
NPV(r) -1000 -
1 r (1 r) 2
Note there are multiple IRRs for these cash flows.
Modified IRR
Modified IRR
C0 C1 C2
Before -1000 2500 -1540
After
So Modified IRR is calculated by solving:
2875
2164.46 0
(1 r ) 2
=> IRR=15.25%
Modified IRR
There is now only a single IRR, at 15.25%. Because our cost
of capital is 15%, we would properly accept the project using
the IRR rule
Modified IRR
It solves the problem of multiple IRRs.
But there is considerable debate about whether it is
appropriate to move cash flows of projects.
It does not solve other pitfalls:
Delayed Investment
Different Project Scale
Different Cash Flow Timing
IRR - Different Project Scale
Project C0 C1 IRR NPV at 10%
E -10,000 +20,000 100% +8,182
F -20,000 +35,000 75% +11,818
If the projects are mutually exclusive, which project should be accepted
according to IRR?
If the projects are mutually exclusive, which project should be accepted
according to the NPV rule?
Which project should we choose? What’s going on?
IRR - Different Timing of Cash Flows
Cash Flow, Dollars
Project C0 C1 C2 C3 C4 Etc. IRR NPV@10%
G -9000 +6000 +5000 +4000 0 … 33% $3592
H -9000 +1800 +1800 +1800 +1800 … 20% $9000
I -6000 +1200 +1200 +1200 +1200 … 20% $6000
If the projects are mutually exclusive, which project should be
accepted according to IRR?
If the projects are mutually exclusive, which project should be
accepted according to the NPV rule?
Which project should we choose? What’s going on?
Computing the Crossover Point
The crossover point is the discount rate that makes the
NPV of the two alternatives equal.
We can find the discount rate by setting the equations for
the NPV of each project equal to each other and solving
for the discount rate.
In general, we can always compute the effect of choosing
Project A over Project B as the difference of the NPVs.
At the crossover point the difference is 0.
Computing the Crossover Point
Problem:
Solve for the crossover point for the following two projects.
Expected Net Cash Flow
Year
Project A Project B
0 -$12,000 -$10,000
1 $5,000 $4,100
2 $5,000 $4,100
3 $5,000 $4,100
Computing the Crossover Point
Execute:
Setting the difference equal to 0:
$5,000 $5,000 $5,000 $4,100 $4,100 $4,100
NPV $12,000 0
(1 r) 2 (1 r)3 (1 r) 2 (1 r) 3
$10,000
1 r 1 r
$900 $900 $900
NPV $2,000 0
1 r (1 r) 2 (1 r)3
As you can see, solving for the crossover point is just like
solving for the IRR, so we will need to use a financial
calculator or spreadsheet
And we find that the crossover occurs at a discount rate of
16.65%.
Computing the Crossover Point
Just as the NPV of a project tells us the value impact of
taking the project, so the difference of the NPVs of two
alternatives tells us the incremental impact of choosing
one project over another.
The crossover point is the discount rate at which we
would be indifferent between the two projects because
the incremental value of choosing one over the other
would be zero.
NPV - Always the Preferred Choice
Payback
Time Value of Money is Ignored
Ignores Cash flows Beyond Payback Period
IRR Potential Pitfalls
Delayed investment
Sometimes there are multiple IRRs
Can be misleading when scale or timing of cashflows differ
Modified IRR
Solves the problem of multiple IRRs
Does not adjust for other pitfalls.
NPV is Always the Best Method
Projects with Different Lives
Often firms are faced with comparing to similar projects with
different lives.
Equivalent Annual Annuity provides a framework to
compare costs over different time horizons.
Steps:
Identify the yearly costs for both alternatives.
Calculate the PV of the costs using the cost of capital.
Convert the costs into an annuity to equally spread costs over
time horizon.
Compare equivalent annual costs and select lowest alternative
Projects with Different Lives
Choose from Network Server A or B?
Projects with Different Lives
Since the equivalent annual cost for project A is lower, so we
should choose Network Server A.
Question: Why not choose the one with lower PV of cost
(Server B)?
Answer:
In many real investment decision, we usually do not know, ex
ante, how long we will use these servers.
So under this uncertainty, it is reasonable to compare their
annual equivalent cost.
However…..
Projects with Different Lives
Some Considerations:
Required Life:
If we know, ex ante, that we will use the server for only three
years. This will potentially change our initial decision.
Replacement cost:
When we compare A and B, we assume the cost of servers will
not change over time. If we believe a dramatic drop in the cost
of servers by the third year, maybe project B is a better choice.
Constraints in Resources
Theoretically firms should purse all positive NPV projects
available to them.
In practice many firms face resource constrain the capital
projects they pursue.
If there is resource constraint, the Profitability Index (PI) is a
useful decision rule.
Definition: The profitability index is the net present
value of future cash flows generated by a project divided by
the resource consumed. (see next slide)
The higher the profitability index the more profitable
shareholders receive from the investment.
Profitability Index
Value Created NPV
PI
Resource Consumed Resource Consumed
Steps:
(1) Forecast cash flows (amount and timing of each).
(2) Estimate the opportunity cost of capital.
(3) Calculate the profitability index for each project.
(4) Rank projects by the profitability index and go down
the list until you run out of money.
Profitability Index Example
McKesson Inc. has $15 million to invest in the following projects:
Project Investment NPV PI (NPV/Investment)
L 4,000,000 1,000,000
M 7,000,000 5,000,000
N 4,000,000 2,000,000
O 3,000,000 1,000,000
P 4,000,000 3,000,000
Which projects should McKesson pursue?
Profitability Index Example
McKesson Inc. has $15 million to invest in the following projects:
Project Investment NPV PI (NPV/Investment)
L 4,000,000 1,000,000 .25
M 7,000,000 5,000,000 .71
N 4,000,000 2,000,000 .50
O 3,000,000 1,000,000 .33
P 4,000,000 3,000,000 .75
Which projects should McKesson pursue?
Profitability Index (PI)
In this case the constraint is cash, but the constraint could be
any limited resource.
In summary, if there is no constraints, you should always take
all positive-NPV projects when these projects are not
mutually exclusive. If the projects are mutually exclusive,
take the one with highest positive NPV.
When there is some constraints, use profitability index
procedure to choose projects.
Putting It All Together
Putting It All Together
