CHAPTER 11PROJECT ANALYSIS AND EVALUATIONCopyright © 2019 Mc

CHAPTER 11
PROJECT ANALYSIS AND EVALUATION
Copyright © 2019 McGraw-Hill Education. All rights reserved.
No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
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Perform and interpret a sensitivity analysis for a proposed
investment
Perform and interpret a scenario analysis for a proposed
investment
Determine and interpret cash, accounting, and financial break-
even points
Explain how the degree of operating leverage can affect the
cash flows of a project
Discuss how capital rationing affects the ability of a company
to accept projects

Key Concepts and Skills
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Evaluating NPV Estimates
Scenario and Other What-If Analyses
Break-Even Analysis
Operating Cash Flow, Sales Volume, and Break-Even
Operating Leverage
Capital Rationing
Chapter Outline
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NPV estimates are just that – estimates.
A positive NPV is a good start – now we need to take a closer
look.

Forecasting risk – how sensitive is our NPV to changes in the
cash flow estimates; the more sensitive, the greater the
forecasting risk.
Sources of value – why does this project create value?
Evaluating NPV Estimates
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4
Section 11.1
There are two primary reasons for a positive NPV: (1) we have
constructed a good project or (2) we have done a bad job of
estimating NPV.
Lecture Tip: With the lower flat-tax for corporations, previously
unattractive projects may not have positive NPVs. So, there may
be a one-time exception to the two reasons for finding positive
NPV projects.
Lecture Tip: Perhaps the single largest source of positive NPVs
is the economic concept of monopoly rents – positive profits
that occur from being the only one able or allowed to do
something. Monopoly rents are often associated with patent
rights and technological edges and they quickly disappear in a
competitive market. Introducing this notion in class provides a
springboard for discussions of both business and financial
strategy, as well as for discussion of the application of
economic theory to the real world.

According to Alan Shapiro, the following are project
characteristics associated with positive NPVs.
1) Economies of scale
2) Product differentiation
3) Cost advantages
4) Access to distribution channels
5) Favorable government policy
What happens to the NPV under different cash flow scenarios?
At the very least, look at:
Best case – high revenues, low costs
Worst case – low revenues, high costs
Measures of the range of possible outcomes
Best case and worst case are not necessarily probable, but they
can still be possible.
Scenario Analysis
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5
Section 11.2 (B)
A good example of the worst case actually happening is the
sinking of the Titanic. There were a lot of little things that went
wrong, none of which were that important by themselves, but in
combination they were deadly.

A more recent example of the worst case scenario happening is
the 2004 hurricane season in Florida. During the months of
August and September, 4 hurricanes (Charley, Frances, Ivan,
Jeanne) hit the state of Florida (the most previously had been 3
in the state of Texas in the late 1880s). This is ignoring tropical
storm Bonnie that hit the panhandle a week before Charley came
through. The eyes of 3 of the 4 hurricanes (all but Ivan, who
tore through the panhandle) passed over Polk County in central
Florida. The probability of 3 hurricanes passing over the same
location in the span of 6 weeks is extremely low. The eyes of
two of the hurricanes (Frances and Jeanne) made landfall on the
east side of Florida within 10 miles of each other. Again, the
probability of this happening 3 weeks apart is very, very small.
To imagine anything more devastating would have been
difficult, making this truly a worst-case scenario…until Katrina
paid a visit to New Orleans and the levees failed!
Lecture Tip: A major misconception about a project’s estimated
NPV at this point is that it depends upon how the cash flows
actually turn out. This thinking misses the point that NPV is an
ex ante valuation of an uncertain future. The distinction
between the valuation of what is expected versus the ex post
value of what transpired is often difficult for students to
appreciate.
A useful analogy for getting this point across is the market
value of a new car. The potential to be a “lemon” is in every
car, as is the possibility of being a “cream puff.” The greater
the likelihood that a car will have problems, the lower the price
will be. The point, however, is that a new car doesn’t have
many different prices right now – one for each conceivable
repair record. Rather, there is one price embodying the different
potential outcomes and their expected value. So it is with NPV
– the potential for good and bad cash flows is reflected in a
single market value.

Consider the project discussed in the text in section 11.2.
The initial cost is $200,000, and the project has a 5-year life.
There is no salvage.
Depreciation is straight-line, the required return is 12%, and the
tax rate is 21%.
The base, lower, and upper values are given for unit sales, price
per unit, variable costs per unit, and fixed costs.
Click on the Excel icon to see base case, best case, and worst
case scenarios results.
New Project Example
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6
Section 11.2 (B)
Click on the Excel icon to go to a spreadsheet that includes both
the scenario analysis and the sensitivity analysis presented in
the book.
ScenarioNet IncomeCash FlowNPVIRRBase
case23,70063,70029,62417.8%Worst Case-18,56521,435-
122,732-17.7%Best Case71,495111,495201,91547.9%
Summary of Scenario Analysis

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7
Section 11.2 (B)
Lecture Tip: You may wish to integrate this discussion of risk
with some of the topics to be discussed in forthcoming chapters.
The variability between best- and worst-case scenarios is the
essence of forecasting risk. Similarly, we link the risk of a
security with the variability of its expected return. This point
provides another opportunity to link economic theory
(investor/manager rationality versus required returns) with real -
world decision-making. You might also want to point out that
the cases examined in this type of analysis typically aren’t
literally the best and worst cases possible. The true worst-case
scenario is something absurdly unlikely, such as an earthquake
that swallows our production plant. Instead, the worst-case used
in scenario analysis is simply a pessimistic (but possible)
forecast used to develop expected cash flows.
What happens to NPV when we change one variable at a time?
This is a subset of scenario analysis where we are looking at the
effect of specific variables on NPV.
The greater the volatility in NPV in relation to a specific
variable, the larger the forecasting risk associated with that
variable, and the more attention we want to pay to its
estimation.
Sensitivity Analysis

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8
Section 11.2 (C)
Click on the Excel icon to return to the new project spreadsheet.
If desired, it may be a good point at which to demonstrate the
Solver function in Excel, as you can identify how high/low an
input could go before NPV becomes negative.
ScenarioUnit SalesCash FlowNPVIRRBase
case6,00063,70029,62417.8%Worst case5,50055,800
1,14712.2%Best case6,50071,60058,10223.2%
Summary of Sensitivity Analysis for New Project
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Section 11.2 (C)
Using an older standard tax rate of 34%, the worst case scenario
gives a negative NPV. This illustrates that the reduction in
taxes will make some previously unattractive investments
favorable.
9

Simulation is really just an expanded sensitivity and scenario
analysis.
Monte Carlo simulation can estimate thousands of possible
outcomes based on conditional probability distributions and
constraints for each of the variables.
The output is a probability distribution for NPV with an
estimate of the probability of obtaining a positive net present
value.
The simulation only works as well as the information that is
entered, and very bad decisions can be made if care is not taken
to analyze the interaction between variables.
Simulation Analysis
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10
Section 11.2 (D)
Lecture Tip: A very useful software is Crystal Ball, which is a
simulation package that integrates with Excel. It is relatively
inexpensive, yet it is very useful for basic-to-moderate
simulation analysis. For example, the software allows you to
build models (such as NPV) in Excel, then define the
assumptions behind the inputs (such as distribution, possible
extreme values, etc.), as well as the interaction (i.e.,
correlation) between the inputs. Output is then generated based
on a simulation of 1,000 runs, providing distribution analysis

and numerical summary statistics.
Beware “Paralysis of Analysis”
At some point you have to make a decision.
If the majority of your scenarios have positive NPVs, then you
can feel reasonably comfortable about accepting the project.
If you have a crucial variable that leads to a negative NPV with
a small change in the estimates, then you may want to forego
the project.
Making a Decision
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Section 11.2 (D)
11
Common tool for analyzing the relationship between sales
volume and profitability
There are three common break-even measures:
Accounting break-even:
sales volume at which NI = 0
Cash break-even:
sales volume at which OCF = 0

Financial break-even:
sales volume at which NPV = 0
Break-Even Analysis
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Section 11.3
12
There are two types of costs that are important in breakeven
analysis: variable and fixed.
Total variable costs = quantity × cost per unit
Fixed costs are constant, regardless of output, over some time
period.
Total costs = fixed + variable = FC + vQ
Example:
Your firm pays $3,000 per month in fixed costs. You also pay
$15 per unit to produce your product.
What is your total cost if you produce 1,000 units?
What if you produce 5,000 units?
Example: Costs
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13
Section 11.3 (A)
Produce 1000 units: TC = 3000 + 15 × 1000 = 18,000
Produce 5000 units: TC = 3000 + 15 × 5000 = 78,000
Lecture Tip: You may wish to emphasize that, in computing
total variable costs, the only relevant costs are those that are
directly related to the manufacture and sale of the product.
Allocated (or indirect) costs should not enter the analysis.
Suggest to the students that when they are uncertain, they
should use the “with/without” criterion: will the costs be
different if the investment is made? If not, the cost is, by
definition, not directly related to the decision and should not be
included.
Average Cost
TC / # of units
Will decrease as # of units increases
Marginal Cost
The cost to produce one more unit
Same as variable cost per unit
Example: What is the average cost and marginal cost under each
situation in the previous example?
Produce 1,000 units: Average = 18,000 / 1000 = $18
Produce 5,000 units: Average = 78,000 / 5000 = $15.60
Average vs. Marginal Cost

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14
Section 11.3 (A)
Lecture Tip: Students should recognize that as quantity
increases, total fixed costs remain constant, but on a per unit
basis, they decrease with increasing volume. And, as quantity
increases, total cost per unit approaches variable cost per unit.
If a company expects a high unit sales volume, the company
may desire to exploit the possible economies of scale by
investing more in fixed costs in an effort to lower variable cost
per unit. However, this could create future financial problems if
sales expectations fail to materialize. You might mention that
this sensitivity to earnings declines will be examined later in
this chapter through the discussion of the degree of operating
leverage.
The quantity that leads to a zero net income
NI = (Sales – VC – FC – D)(1 – T) = 0
QP – vQ – FC – D = 0
Q(P – v) = FC + D
Q = (FC + D) / (P – v)
Accounting Break-Even

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Section 11.3 (B)
15
Accounting break-even is often used as an early stage screening
number.
If a project cannot break-even on an accounting basis, then it is
not going to be a worthwhile project.
Accounting break-even gives managers an indication of how a
project will impact accounting profit.
Using Accounting Break-Even
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Section 11.3 (C)
16
We are more interested in cash flow than we are in accounting
numbers.
As long as a firm has non-cash deductions, there will be a
positive cash flow.

If a firm just breaks even on an accounting basis, cash flow =
depreciation.
If a firm just breaks even on an accounting basis, NPV will
generally be < 0.
Accounting Break-Even and Cash Flow
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Section 11.4 (A)
17
Consider the following project:
A new product requires an initial investment of $5 million and
will be depreciated to an expected salvage of zero over 5 years.
The price of the new product is expected to be $25,000, and the
variable cost per unit is $15,000.
The fixed cost is $1 million.
What is the accounting break-even point each year?
Depreciation = 5,000,000 / 5 = 1,000,000
Q = (1,000,000 + 1,000,000)/(25,000 – 15,000) = 200 units
Example

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18
Section 11.4 (A)
What is the operating cash flow at the accounting break-even
point (ignoring taxes)?
OCF = (S – VC – FC - D) + D
OCF = (200 × 25,000 – 200 × 15,000 – 1,000,000 -1,000,000) +
1,000,000 = 1,000,000
What is the cash break-even quantity (ignoring taxes)?
OCF = [(P-v)Q – FC – D] + D = (P-v)Q – FC
Q = (OCF + FC) / (P – v)
Q = (0 + 1,000,000) / (25,000 – 15,000) = 100 units
Sales Volume and Operating Cash Flow
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19
Section 11.4 (B)
Cash break-even occurs where operating cash flow = 0.
Accounting Break-even

Where NI = 0
Q = (FC + D)/(P – v)
Cash Break-even
Where OCF = 0
Q = (FC + OCF)/(P – v); (ignoring taxes)
Financial Break-even
Where NPV = 0
Cash BE < Accounting BE < Financial BE
Three Types of Break-Even Analysis
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20
Section 11.4 (C)
Lecture Tip: Inquisitive students may ask how the computations
change when you include taxes. The equations change as
follows:
OCF = [(P − v)Q − FC − D](1 − T) + D
Use a tax rate = 21% and rework the Wettways example from
the book:
Need 1170 in OCF to break-even on a financial basis
OCF = [(40 − 20)(Q) − 500 − 700](1 − .21) + 700 = 1170
Q = 89.75

You end up with a new quantity of 90 units. The firm must sell
an additional 16 units to offset the effects of taxes. Although,
with the recent tax cuts, this difference is not as large as it
previously was.
Consider the previous example.
Assume a required return of 18%
Accounting break-even = 200
Cash break-even = 100 (ignoring taxes)
What is the financial break-even point (ignoring taxes)?
What OCF (or payment) makes NPV = 0?
N = 5; PV = 5,000,000; I/Y = 18;
CPT PMT = 1,598,889 = OCF
Q = (1,000,000 + 1,598,889) / (25,000 – 15,000)
= 260 units (ignoring taxes)
The question now becomes: Can we sell at least 260 units per
year?
Example: Break-Even Analysis
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21
Section 11.4 (C)
Assumptions: Cash flows are the same every year, no salvage

and no NWC. If there were salvage and NWC, you would net it
out to year 0 so that all you have in future years is OCF.
Operating leverage is the relationship between sales and
operating cash flow.
Degree of operating leverage measures this relationship.
The higher the DOL, the greater the variability in operating
cash flow.
The higher the fixed costs, the higher the DOL.
DOL depends on the sales level you are starting from.
DOL = 1 + (FC / OCF)
Operating Leverage
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Section 11.5
22
Consider the previous example.
Suppose sales are 300 units.
This meets all three break-even measures.
What is the DOL at this sales level?
OCF = (25,000 – 15,000) × 300 – 1,000,000 = 2,000,000
DOL = 1 + 1,000,000 / 2,000,000 = 1.5
What will happen to OCF if unit sales increases by 20%?
Percentage change in OCF = DOL × Percentage change in Q

Percentage change in OCF = 1.5(.2) = .3 or 30%
OCF would increase to 2,000,000(1.3) = 2,600,000
Example: DOL
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Section 11.5 (C)
23
Capital rationing occurs when a firm or division has limited
resources.
Soft rationing – the limited resources are temporary, often self-
imposed
Hard rationing – capital will never be available for this project
The profitability index is a useful tool when a manager is faced
with soft rationing.
Capital Rationing
11-‹#›

24
Section 11.6
If you face hard rationing, you need to reevaluate your analysis.
If you truly estimated the required return and expected cash
flows appropriately and computed a positive NPV, then capital
should be available.
Lecture Tip: In 2008, the economy was suffering from a real
estate and credit crisis. As a result, lenders essentially withdrew
from the market and credit dried up. This is a perfect example
of an issue that would create a situation very close to hard
rationing for many businesses.
Lecture Tip: If lower tax rates result in higher cash flows and
more attractive projects, then the issue of capital rationing will
become even more pronounced.
What is sensitivity analysis, scenario analysis and simulation?
Why are these analyses important, and how should they be
used?
What are the three types of break-even analysis, and how should
each be used?
What is the degree of operating leverage?
What is the difference between hard rationing and soft
rationing?
Quick Quiz

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Section 11.7
25
Is it ethical for a medical patient to pay for a portion of R&D
costs (since experimental procedures are not covered by
insurance) prior to the introduction of the final product?
Is it proper for physicians to recommend this procedure when
they have a vested interest in its usage?
Ethics Issues
11-‹#›
26
Case: Researchers associated with South Miami Hospital (SMH)
developed a new experimental laser treatment for heart patients.
Its development team and the physicians who use the laser
consider it to be a lifesaving advance. It should be noted that
the physicians who are touting the laser hold a significant stake
in the company that produces the laser. To offer a substitute for
a balloon angioplasty to treat heart blockages, the experimental
laser was developed at a cost of $250,000. SMH estimates that
it will cost $20,000 to install the laser. The procedure requires a
nurse at $50 per hour, a technician at $30 per hour, and a

physician who is paid $750 per hour. Patients are billed $3,000
for the procedure compared to $1,500 for the traditional balloon
treatment.
Now ask the students to determine the break-even quantity for
the new procedure:
Fixed cost = 250,000 + 20,000 = 270,000
Variable cost = 50 + 30 + 750 = 830 per hour
Cash Break-Even = 250,000 / (3,000 – 830) = 115.2 hours,
or approximately 116 patients (assuming a one-hour procedure
per patient).
A project requires an initial investment of $1,000,000 and is
depreciated straight-line to zero salvage over its 10-year life.
The project produces items that sell for $1,000 each, with
variable costs of $700 per unit. Fixed costs are $350,000 per
year.
What is the accounting break-even quantity, operating cash flow
at accounting break-even (ignoring taxes), and DOL at that
output level?
Comprehensive Problem
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27
Section 11.7
Accounting break-even:

Q = (FC + D) / (P – V) = ($350,000 + $100,000) / ($1,000 -
$700) = 1,500 units
OCF = ( S – VC – FC – D) + D = (1,500 × $1,000 – 1,500 ×
$700 - $350,000 - $100,000) + $100,000 = $100,000
DOL = 1 + (FC / OCF) = 1 + ($350,000 / 100,000) = 4.5
End of Chapter
CHAPTER 11
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11-‹#›
Microsoft Excel
97-2003 Worksheet
ScenarioBaseLowerUpperUnit
Sales600055006500Depreciation40000Price per unit807585VC
per unit605862No NWCFC per unit500004500055000Base Case
AnalysisBest CaseWorst CasePro Forma StatementPro Forma
StatementPro Forma
StatementSales480000Sales552500Sales412500VC360000VC37
7000VC341000FC50000FC45000FC55000Depreciation40000De
preciation40000Depreciation40000EBIT30000EBIT90500EBIT-
23500Taxes6300Taxes19005Taxes-4935NI23700NI71495NI-

18565Cash
FlowsYearOCFNCSCFFAYearOCFNCSCFFAYearOCFNCSCFF
A0-200000-2000000-200000-2000000-200000-
20000016370063700111149511149512143521435263700637002
11149511149522143521435363700637003111495111495321435
21435463700637004111495111495421435214355637006370051
1149511149552143521435NPV$29,624.24NPV$201,914.52NPV
-$122,731.62Sensitivity Analysis For Unit SalesPro Forma
StatementBaseLowerUpperSales480000440000520000VC36000
0330000390000FC500005000050000Depreciation400004000040
000EBIT300002000040000Taxes630042008 400NI23700158003
1600Cash FlowsYear0-200,000-200,000-
200,0001637005580071600263700558007160036370055800716
0046370055800716005637005580071600NPV$29,624.24$1,146
.51$58,101.98Numbers in blue were computed in Excel.
SensitivityBaseLowerUpperUnit
Sales600055006500Depreciation40000Price per unit807585VC
per unit605862No NWCFC per unit500004500055000Base Case
AnalysisBest CaseWorst CasePro Forma StatementPro Forma
StatementPro Forma
StatementSales480000Sales552500Sales412500VC360000VC37
7000VC341000FC50000FC45000FC55000Depreciation40000De
preciation40000Depreciation40000EBIT30000EBIT90500EBIT-
23500Taxes6300Taxes19005Taxes-4935NI23700NI71495NI-
18565Cash
FlowsYearOCFNCSCFFAYearOCFNCSCFFAYearOCFNCSCFF
A0-200000-2000000-200000-2000000-200000-
20000016370063700111149511149512143521435263700637002
11149511149522143521435363700637003111495111495321435
21435463700637004111495111495421435214355637006370051
1149511149552143521435NPV$29,624.24NPV$201,914.52NPV
-$122,731.62Sensitivity Analysis For Unit SalesPro Forma
StatementBaseLowerUpperSales480000440000520000VC36000
0330000390000FC500005000050000Depreciation400004000040
000EBIT300002000040000Taxes630042008400NI23700158003

1600Cash FlowsYear0-200,000-200,000-
200,0001637005580071600263700558007160036370055800716
0046370055800716005637005580071600NPV$29,624.24$1,146
.51$58,101.98Numbers in blue were computed in Excel.
CHAPTER 9
NET PRESENT VALUE AND
OTHER INVESTMENT CRITERIA
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Show the reasons why the net present value criterion is the best
way to evaluate proposed investments
Discuss the payback rule and some of its shortcomings
Discuss the discounted payback rule and some of its
shortcomings

Explain accounting rates of return and some of the problems
with them
Present the internal rate of return criterion and its strengths and
weaknesses
Calculate the modified internal rate of return
Illustrate the profitability index and its relation to net present
value
9-‹#›
Net Present Value
The Payback Rule
The Discounted Payback
The Average Accounting Return
The Internal Rate of Return
The Profitability Index
The Practice of Capital Budgeting
Chapter Outline

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8.3
Lecture Tip: A logical prerequisite to the analysis of investment
opportunities is the creation of investment opportunities. Unlike
the field of investments, where the analyst more or less takes
the investment opportunity set as a given, the field of capital
budgeting relies on the work of people in the areas of
engineering, research and development, information technology
and others for the creation of investment opportunities. As such,
it is important to remind students of the importance of creativity
in this area, as well as the importance of analytical techniques.
We need to ask ourselves the following questions when
evaluating capital budgeting decision rules:
Does the decision rule adjust for the time value of money?
Does the decision rule adjust for risk?
Does the decision rule provide information on whether we are
creating value for the firm?
Good Decision Criteria

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8.4
Section 9.1
Economics students will recognize that the practice of capital
budgeting defines the firm’s investment opportunity schedule.
The difference between the market value of a project and its
cost
How much value is created from undertaking an investment?
The first step is to estimate the expected future cash flows.
The second step is to estimate the required return for proj ects of
this risk level.
The third step is to find the present value of the cash flows and
subtract the initial investment.
Net Present Value
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8.5
Section 9.1 (A)

We learn how to estimate the cash flows and the required return
in subsequent chapters.
The NPV measures the increase in firm value, which is also the
increase in the value of what the shareholders own. Thus,
making decisions with the NPV rule facilitates the achievement
of our goal in Chapter 1 – making decisions that will maximize
shareholder wealth.
Lecture Tip: Although this point may seem obvious, it is often
helpful to stress the word “net” in net present value. It is not
uncommon for some students to carelessly calculate the PV of a
project’s future cash flows and fail to subtract out its cost (after
all, this is what the programmers of Lotus and Excel did when
they programmed the NPV function). The PV of future cash
flows is not NPV; rather, NPV is the amount remaining after
offsetting the PV of future cash flows with the initial cost.
Thus, the NPV amount determines the incremental value created
by undertaking the investment.
You are reviewing a new project and have estimated the
following cash flows:
Year 0: CF = -165,000
Year 1: CF = 63,120; NI = 13,620
Year 2: CF = 70,800; NI = 3,300
Year 3: CF = 91,080; NI = 29,100
Average Book Value = 72,000
Your required return for assets of this risk level is 12%.
Project Example Information

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8.6
Section 9.1 (B)
This example will be used for each of the decision rules so that
the students can compare the different rules and see that
conflicts can arise. This illustrates the importance of
recognizing which decision rules provide the best information
for making decisions that will increase owner wealth.
If the NPV is positive, accept the project.
A positive NPV means that the project is expected to add value
to the firm and will therefore increase the wealth of the owners.
Since our goal is to increase owner wealth, NPV is a direct
measure of how well this project will meet our goal.
NPV – Decision Rule
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8.7
Section 9.1 (B)
Lecture Tip: Here’s another perspective on the meaning of NPV.
If we accept a project with a negative NPV of -$2,422, this is

financially equivalent to investing $2,422 today and receiving
nothing in return. Therefore, the total value of the firm would
decrease by $2,422. This assumes that the various components
(cash flow estimates, discount rate, etc.) used in the
computation are correct.
Lecture Tip: In practice, financial managers are rarely presented
with zero NPV projects for at least two reasons. First, in an
abstract sense, zero is just another of the infinite number of
values the NPV can take; as such, the likelihood of obtaining
any particular number is small. Second, and more pragmatically,
in most large firms, capital investment proposals are submitted
to the finance group from other areas for analysis. Those
submitting proposals recognize the ambivalence associated with
zero NPVs and are less likely to send them to the finance group
in the first place.
Using the formulas:
NPV = -165,000 + 63,120/(1.12) + 70,800/(1.12)2 +
91,080/(1.12)3 = 12,627.41
Using the calculator:
CF0 = -165,000; C01 = 63,120; F01 = 1; C02 = 70,800; F02 = 1;
C03 = 91,080; F03 = 1; NPV; I = 12; CPT NPV = 12,627.41
Do we accept or reject the project?
Computing NPV for the Project
9-‹#›

8.8
Section 9.1 (B)
Again, the calculator used for the illustration is the TI BA-II
plus. The basic procedure is the same; you start with the year 0
cash flow and then enter the cash flows in order. F01, F02, etc.
are used to set the frequency of a cash flow occurrence. Many
calculators only require you to use this functio n if the frequency
is something other than 1.
Since we have a positive NPV, we should accept the project.
Does the NPV rule account for the time value of money?
Does the NPV rule account for the risk of the cash flows?
Does the NPV rule provide an indication about the increase in
value?
Should we consider the NPV rule for our primary decision rule?
Decision Criteria Test – NPV
9-‹#›
8.9
Section 9.1 (B)
The answer to all of these questions is yes.

The risk of the cash flows is accounted for through the choice
of the discount rate.
Lecture Tip: The new tax law contains a provision that allows
firms, in some cases, to take bonus depreciation in year one up
to 100 percent of the cost of the asset. This will, all else equal,
increase the NPV of proposed projects.
Spreadsheets are an excellent way to compute NPVs, especially
when you have to compute the cash flows as well.
Using the NPV function
The first component is the required return entered as a decimal.
The second component is the range of cash flows beginning
with year 1.
Subtract the initial investment after computing the NPV.
Calculating NPVs with a Spreadsheet
9-‹#›
8.10
Section 9.1 (B)
Click on the Excel icon to go to an embedded Excel worksheet
that has the cash flows along with the right and wrong way to

compute NPV. Click on the cell with the solution to show the
students the difference in the formulas.
How long does it take to get the initial cost back in a nominal
sense?
Computation
Estimate the cash flows.
Subtract the future cash flows from the initial cost until the
initial investment has been recovered.
Decision Rule – Accept if the payback period is less than some
preset limit.
Payback Period
9-‹#›
Section 9.2 (A)
8.11
Assume we will accept the project if it pays back within two
years.
Year 1: 165,000 – 63,120 = 101,880 still to recover
Year 2: 101,880 – 70,800 = 31,080 still to recover
Year 3: 31,080 – 91,080 = -60,000 project pays back in year 3
Computing Payback

9-‹#›
8.12
Section 9.2 (A)
The payback period is year 3 if you assume that the cash flows
occur at the end of the year, as we do with all of the other
decision rules.
If we assume that the cash flows occur evenly throughout the
year, then the project pays back in 2.34 years.
Either way, the payback rule would say to reject the project.
Does the payback rule account for the time value of money?
Does the payback rule account for the risk of the cash flows?
Does the payback rule provide an indication about the increase
in value?
Should we consider the payback rule for our primary decision
rule?
Decision Criteria Test – Payback

9-‹#›
8.13
Section 9.2 (B)
The answer to all of these questions is no.
Lecture Tip: The payback period can be interpreted as a naïve
form of discounting if we consider the class of investments with
level cash flows over arbitrarily long lives. Since the present
value of a perpetuity is the payment divided by the discount
rate, a payback period cutoff can be seen to imply a cer tain
discount rate. That is:
cost/annual cash flow = payback period cutoff
cost = annual cash flow times payback period cutoff
The PV of a perpetuity is: PV = annual cash flow / R. This
illustrates the inverse relationship between the payback period
cutoff and the discount rate.
Advantages
Easy to understand
Adjusts for uncertainty of later cash flows
Biased toward liquidity
Disadvantages
Ignores the time value of money
Requires an arbitrary cutoff point
Ignores cash flows beyond the cutoff date
Biased against long-term projects, such as research and
development, and new projects
Advantages and Disadvantages of Payback

9-‹#›
8.14
Section 9.2 (D)
Teaching the payback rule seems to put one in a delicate
situation – as the text indicates, the rule is flawed as an
indicator of project desirability. Yet, past surveys suggest that
practitioners often use it as a secondary decision measure. How
can we explain this apparent discrepancy between theory and
practice? While the payback period is widely used in practice, it
is rarely the primary decision criterion. As William Baumol
pointed out in the early 1960s, the payback rule serves as a
crude “risk screening” device – the longer cash is tied up, the
greater the likelihood that it will not be returned. The payback
period may be helpful when mutually exclusive projects are
compared. Given two similar projects with different paybacks,
the project with the shorter payback is often, but not always, the
better project. Similarly, the bias toward liquidity may be
justifiable in such industries as healthcare, where technology
changes rapidly, requiring quick payback to make machines
justifiable, or in international investments where the possibility
of government seizure of assets exists.
Compute the present value of each cash flow and then determine
how long it takes to pay back on a discounted basis.
Compare to a specified required period.

Decision Rule: Accept the project if it pays back on a
discounted basis within the specified time.
Discounted Payback Period
9-‹#›
Section 9.3
8.15
Assume we will accept the project if it pays back on a
discounted basis in 2 years.
Compute the PV for each cash flow and determine the payback
period using discounted cash flows.
Year 1: 165,000 – 63,120/1.121 = 108,643
Year 2: 108,643 – 70,800/1.122 = 52,202
Year 3: 52,202 – 91,080/1.123 = -12,627 project pays back in
year 3
Computing Discounted Payback

9-‹#›
8.16
Section 9.3
No – it doesn’t pay back on a discounted basis within the
required 2-year period.
Does the discounted payback rule account for the time value of
money?
Does the discounted payback rule account for the risk of the
cash flows?
Does the discounted payback rule provide an indication about
the increase in value?
Should we consider the discounted payback rule for our primary
decision rule?
Decision Criteria Test – Discounted Payback
9-‹#›
8.17
Section 9.3
The answer to the first two questions is yes.
The answer to the third question is no because of the arbitrary

cut-off date.
Since the rule does not indicate whether or not we are creating
value for the firm, it should not be the primary decision rule.
Advantages
Includes time value of money
Easy to understand
Does not accept negative estimated NPV investments when all
future cash flows are positive
Biased towards liquidity
Disadvantages
May reject positive NPV investments
Requires an arbitrary cutoff point
Ignores cash flows beyond the cutoff point
Biased against long-term projects, such as R&D and new
products
Advantages and Disadvantages of Discounted Payback
9-‹#›
Section 9.3
8.18
There are many different definitions for average accounting
return.
The one used in the book is:
Average net income / average book value

Note that the average book value depends on how the asset is
depreciated.
Need to have a target cutoff rate
Decision Rule: Accept the project if the AAR is greater than a
preset rate.
Average Accounting Return
9-‹#›
8.19
Section 9.4
The example in the book uses straight line depreciation to a
zero salvage; that is why you can take the initial investment and
divide by 2. If you use MACRS, you need to compute the BV in
each period and take the average in the standard way.
Assume we require an average accounting return of 25%.
Average Net Income:
(13,620 + 3,300 + 29,100) / 3 = 15,340
AAR = 15,340 / 72,000 = .213 = 21.3%

Computing AAR
9-‹#›
8.20
Section 9.4
Students may ask where you came up with the 25%. Point out
that this is one of the drawbacks of this rule. There is no good
theory for determining what the return should be. We generally
just use some rule of thumb.
This rule would indicate that we reject the project.
Does the AAR rule account for the time value of money?
Does the AAR rule account for the risk of the cash flows?
Does the AAR rule provide an indication about the increase in
value?
Should we consider the AAR rule for our primary decision rule?
Decision Criteria Test – AAR

9-‹#›
8.21
Section 9.4
The answer to all of these questions is no. In fact, this rule is
even worse than the payback rule in that it doesn’t even use
cash flows for the analysis. It uses net income and book value.
Thus, it is not surprising that most surveys indicate that few
large firms employ the payback and/or AAR methods
exclusively.
Advantages
Easy to calculate
Needed information will usually be available
Disadvantages
Not a true rate of return; time value of money is ignored
Uses an arbitrary benchmark cutoff rate
Based on accounting net income and book values, not cash
flows and market values
Advantages and Disadvantages of AAR
9-‹#›
8.22
Section 9.4

Lecture Tip: An alternative view of the AAR is that it is the
micro-level analogue to the ROA discussed in a previous
chapter. As you remember, firm ROA is normally computed as
Firm Net Income / Firm Total Assets. And, it is not uncommon
to employ values averaged over several quarters or years in
order to smooth out this measure. Some analysts ask, “If the
ROA is appropriate for the firm, why is it less appropriate for a
project?” Perhaps the best answer is that whether you compute
the measure for the firm or for a project, you need to recognize
the limitations – it doesn’t account for risk or the time value of
money and it is based on accounting, rather than market, data.
This is the most important alternative to NPV.
It is often used in practice and is intuitively appealing.
It is based entirely on the estimated cash flows and is
independent of interest rates found elsewhere.
Internal Rate of Return
9-‹#›
8.23
Section 9.5
The IRR rule is very important. Management, and individuals in
general, often have a much better feel for percentage returns,
and the value that is created, than they do for dollar increases.
A dollar increase doesn’t appear to provide as much information

if we don’t know what the initial expenditure was. Whether or
not the additional information is relevant is another issue.
Definition: IRR is the return that makes the NPV = 0
Decision Rule: Accept the project if the IRR is greater than the
required return.
IRR – Definition and Decision Rule
9-‹#›
Section 9.5
8.24
If you do not have a financial calculator, then this becomes a
trial and error process.
Calculator
Enter the cash flows as you did with NPV.
Press IRR and then CPT.
IRR = 16.13% > 12% required return
Computing IRR

9-‹#›
8.25
Section 9.5
Many of the financial calculators will compute the IRR as soon
as it is pressed; others require that you press compute.
NPV Profile for the Project
IRR = 16.13%
9-‹#›
8.26
Section 9.5
Note that the NPV profile is also a form of sensitivity analysis.
NPV00.020.040.060.080.10.120.140000000000000010.160.180.
20.22600005076042121340312644619324126276323381-5227-
10525-
1553600.020.040.060.080.10.120.140000000000000010.160.180
.20.2200.020.040.060.080.10.120.140000000000000010.160.18
0.20.2200.020.040.060.080.10.120.140000000000000010.160.1
80.20.22
Discount Rate

NPV
Does the IRR rule account for the time value of money?
Does the IRR rule account for the risk of the cash flows?
Does the IRR rule provide an indication about the increase in
value?
Should we consider the IRR rule for our primary decision
criteria?
Decision Criteria Test - IRR
9-‹#›
8.27
Section 9.5
The answer to all of these questions is yes, although it is not
always as obvious.
The IRR rule accounts for time value because it is finding the
rate of return that equates all of the cash flows on a time value
basis.
The IRR rule accounts for the risk of the cash flows because
you compare it to the required return, which is determined by

the risk of the project.
The IRR rule provides an indication of value because we will
always increase value if we can earn a return greater than our
required return.
We could consider the IRR rule as our primary decision criteria,
but as we will see, it has some problems that the NPV does not
have. That is why we end up choosing the NPV as our ultimate
decision rule.
Knowing a return is intuitively appealing
It is a simple way to communicate the value of a project to
someone who doesn’t know all the estimation details.
If the IRR is high enough, you may not need to estimate a
required return, which is often a difficult task.
Advantages of IRR
9-‹#›
8.28
Section 9.5
You should point out, however, that if you get a very large IRR
then you should go back and look at your cash flow estimates
again. In competitive markets, extremely high IRRs should be
rare. Also, since the IRR calculation assumes that you can
reinvest future cash flows at the IRR, a high IRR may be
unrealistic.

You start with the cash flows the same as you did for the NPV.
You use the IRR function.
You first enter your range of cash flows, beginning with the
initial cash flow.
You can enter a guess, but it is not necessary.
The default format is a whole percent – you will normally want
to increase the decimal places to at least two.
Calculating IRRs With A Spreadsheet
9-‹#›
8.29
Section 9.5
Click on the Excel icon to go to an embedded spreadsheet so
that you can illustrate how to compute IRR on the spreadsheet.
SummaryNet Present ValueAcceptPayback
PeriodRejectDiscounted Payback PeriodRejectAverage
Accounting ReturnRejectInternal Rate of ReturnAccept
Summary of Decisions for the Project

9-‹#›
8.30
Section 9.5
So, what should we do?
We have two rules that indicate to accept and three that indicate
to reject.
NPV and IRR will generally give us the same decision.
Exceptions:
Nonconventional cash flows – cash flow signs change more than
once
Mutually exclusive projects
Initial investments are substantially different (issue of scale).
Timing of cash flows is substantially different.
NPV vs. IRR
9-‹#›
Section 9.5 (A)
8.31

When the cash flows change sign more than once, there is more
than one IRR.
When you solve for IRR you are solving for the root of an
equation, and when you cross the x-axis more than once, there
will be more than one return that solves the equation.
If you have more than one IRR, which one do you use to make
your decision?
IRR and Nonconventional
Cash Flows
9-‹#›
8.32
Section 9.5 (A)
Lecture Tip: A good introduction to mutually exclusive projects
and non-conventional cash flows is to provide examples that
students can relate to. An excellent example of mutually
exclusive projects is the choice of which college or university
to attend. Many students apply and are accepted to more than
one college, yet they cannot attend more than one at a time.
Consequently, they have to decide between mutually exclusive
projects.
Nonconventional cash flows and multiple IRRs occur when
there is a net cost to shutting down a project. The most common
examples deal with collecting natural resources. After the

resource has been harvested, there is generally a cost associated
with restoring the environment.
Suppose an investment will cost $90,000 initially and will
generate the following cash flows:
Year 1: 132,000
Year 2: 100,000
Year 3: -150,000
The required return is 15%.
Should we accept or reject the project?
Another Example: Nonconventional Cash Flows
9-‹#›
8.33
Section 9.5 (A)
NPV = – 90,000 + 132,000 / 1.15 + 100,000 / (1.15)2 – 150,000
/ (1.15)3 = 1,769.54
Calculator: CF0 = -90,000; C01 = 132,000; F01 = 1; C02 =
100,000; F02 = 1; C03 = -150,000; F03 = 1; I = 15; CPT NPV =
1769.54
If you compute the IRR on the calculator, you get 10.11%
because it is the first one that you come to. So, if you just
blindly use the calculator without recognizing the uneven cash
flows, NPV would say to accept and IRR would say to reject.

Another type of nonconventional cash flow involves a
“financing” project, where there is a positive cash flow
followed by a series of negative cash flows. This is the opposite
of an “investing” project. In this case, our decision rul e
reverses, and we accept a project if the IRR is less than the cost
of capital, since we are borrowing at a lower rate.
NPV Profile
IRR = 10.11% and 42.66%
9-‹#›
8.34
Section 9.5 (A)
You should accept the project if the required return is between
10.11% and 42.66%.
NPV00.050.10.150.20.250.30.350.40.450.50.5500000000000000
4-8000-3158.41-
52.591769.542638.8928002435.141681.15641.4-605.6-2000-
3496.0200.050.10.150.20.250.30.350.40.450.50.5500000000000
000400.050.10.150.20.250.30.350.40.450.50.550000000000000
0400.050.10.150.20.250.30.350.40.450.50.55000000000000004
Discount Rate
NPV

The NPV is positive at a required return of 15%, so you should
Accept.
If you use the financial calculator, you would get an IRR of
10.11% which would tell you to Reject.
You need to recognize that there are non-conventional cash
flows and look at the NPV profile.
Summary of Decision Rules
9-‹#›
Section 9.5 (A)
8.35
If you choose one, you can’t choose the other.
Example: You can choose to attend graduate school at either
Harvard or Stanford, but not both.
Intuitively, you would use the following decision rules:
NPV – choose the project with the higher NPV
IRR – choose the project with the higher IRR
IRR and Mutually Exclusive Projects

9-‹#›
Section 9.5 (A)
8.36
PeriodProject AProject B0-500-
40013253252325200IRR19.43%22.17%NPV64.0560.74
Example With Mutually
Exclusive Projects
The required return for both projects is 10%.
Which project should you accept and why?
9-‹#›
8.37
Section 9.5 (A)
As long as we do not have limited capital, we should choose
project A. Students will often argue that you should choose B
because then you can invest the additional $100 in another good
project, say C. The point is that if we do not have limited
capital, we can invest in A and C and still be better off.
If we have limited capital, then we will need to examine what

combinations of projects with A provide the highest NPV and
what combinations of projects with B provide the highest NPV.
You then go with the set that will create the most value. If you
have limited capital and a large number of mutually exclusive
projects, then you will want to set up a computer program to
determine the best combination of projects within the budget
constraints. The important point is that we DO NOT use IRR to
choose between projects regardless of whether or not we have
limited capital.
Embedded in the analysis, we may want to calculate the NPV of
the incremental project, i.e., the additional CF represented by
project A above project B. The IRR of this CF stream is the
crossover point and provides the return on the incremental
investment.
NPV Profiles
IRR for A = 19.43%
IRR for B = 22.17%
Crossover Point = 11.8%
9-‹#›
8.38
Section 9.5 (A)
If the required return is less than the crossover point of 11.8%,
then you should choose A.

If the required return is greater than the crossover point of
11.8%, then you should choose B.
A00.020.040.060.080.10.120.140000000000000010.160.180.20.
220.24150131.01112.9895.8579.5664.0549.2735.159999999999
99721.78.83-3.47-15.25-
26.53B00.020.040.060.080.10.120.140000000000000010.160.18
0.20.220.24125110.8697.4184.672.3960.7449.6238.9799999999
9999728.819.0599999999999999.72000000000000060.77-7.83
Discount Rate
NPV
NPV directly measures the increase in value to the firm.
Whenever there is a conflict between NPV and another decision
rule, you should always use NPV.
IRR is unreliable in the following situations:
Nonconventional cash flows
Conflicts Between
NPV and IRR
9-‹#›
Section 9.5 (A)

8.39
Calculate the net present value of all cash outflows using the
borrowing rate.
Calculate the net future value of all cash inflows using the
investing rate.
Find the rate of return that equates these values.
Benefits: single answer and specific rates for borrowing and
reinvestment
Modified IRR
9-‹#›
8.40
Section 9.5 (C)
Measures the benefit per unit cost, based on the time value of
money.
A profitability index of 1.1 implies that for every $1 of
investment, we create an additional $0.10 in value.
This measure can be very useful in situations in which we have
limited capital.
Profitability Index

9-‹#›
Section 9.6
8.41
Advantages
Closely related to NPV, generally leading to identical decisions
Easy to understand and communicate
May be useful when available investment funds are limited
Disadvantages
May lead to incorrect decisions in comparisons of mutually
exclusive investments
Advantages and Disadvantages of Profitability Index
9-‹#›
Section 9.6
8.42
We should consider several investment criteria when making

decisions.
NPV and IRR are the most commonly used primary investment
criteria.
Payback is a commonly used secondary investment criteria.
Capital Budgeting In Practice
9-‹#›
8.43
Section 9.7
Even though payback and AAR should not be used to make the
final decision, we should consider the project very carefully if
they suggest rejection. There may be more risk than we have
considered or we may want to pay additional attention to our
cash flow estimations. Sensitivity and scenario analysis can be
used to help us evaluate our cash flows.
The fact that payback is commonly used as a secondary criterion
may be because short paybacks allow firms to have funds sooner
to invest in other projects without going to the capital markets.
Why are smaller firms more likely to use payback as a primary
decision criterion?
Small firms don’t have direct access to the capital markets and
therefore find it more difficult to estimate discount rates based
on funds cost;

the AAR is the project-level equivalent to the ROA measure
used for analyzing firm profitability; and
(3) some small firm decision-makers may be less aware of DCF
approaches than their large firm counterparts.
When managers are judged and rewarded primarily on the basis
of periodic accounting figures, there is an incentive to evaluate
projects with methods such as payback or average accounting
return. On the other hand, when compensation is tied to firm
value, it makes more sense to use NPV as the primary decision
tool.
Net present value
Difference between market value and cost
Take the project if the NPV is positive.
Has no serious problems
Preferred decision criterion
Internal rate of return
Discount rate that makes NPV = 0
Take the project if the IRR is greater than the required return.
Same decision as NPV with conventional cash flows
IRR is unreliable with nonconventional cash flows or mutually
exclusive projects.
Profitability Index
Benefit-cost ratio
Take investment if PI > 1
Cannot be used to rank mutually exclusive projects
May be used to rank projects in the presence of capital rationing
Summary – DCF Criteria

9-‹#›
8.44
Section 9.8
For IRR, we assume a conventional investment project. For a
financing project, we accept if the IRR is less than the
“required” rate.
Payback period
Length of time until initial investment is recovered
Take the project if it pays back within some specified period.
Doesn’t account for time value of money, and there is an
arbitrary cutoff period
Discounted payback period
Length of time until initial investment is recovered on a
discounted basis
Take the project if it pays back in some specified period.
There is an arbitrary cutoff period.
Summary – Payback Criteria
9-‹#›

Section 9.8
8.45
Average Accounting Return
Measure of accounting profit relative to book value
Similar to return on assets measure
Take the investment if the AAR exceeds some specified return
level.
Serious problems and should not be used
Summary – Accounting Criterion
9-‹#›
Section 9.8
8.46
Consider an investment that costs $100,000 and has a cash
inflow of $25,000 every year for 5 years. The required return is
9%, and required payback is 4 years.
What is the payback period?
What is the discounted payback period?
What is the NPV?
What is the IRR?
Should we accept the project?

What decision rule should be the primary decision method?
When is the IRR rule unreliable?
Quick Quiz
9-‹#›
8.47
Section 9.8
Payback period = 4 years
The project does not pay back on a discounted basis.
NPV = -2,758.72
IRR = 7.93%
An ABC poll in the spring of 2004 found that one-third of
students age 12 – 17 admitted to cheating and the percentage
increased as the students got older and felt more grade pressure.
If a book entitled “How to Cheat: A User’s Guide” would
generate a positive NPV, would it be proper for a publishing
company to offer the new book?
Should a firm exceed the minimum legal limits of government
imposed environmental regulations and be responsible for the
environment, even if this responsibility leads to a wealth
reduction for the firm? Is environmental damage merely a cost
of doing business?

Should municipalities offer monetary incentives to induce firms
to relocate to their areas?
Ethics Issues
9-‹#›
8.48
Case 1:
Assume the publishing company has a cost of capital of 8% and
estimates it could sell 10,000 volumes by the end of year one
and 5,000 volumes in each of the following two years. The
immediate printing costs for the 20,000 volumes would be
$20,000. The book would sell for $7.50 per copy and net the
company a profit of $6 per copy after royalties, marketing costs
and taxes. Year one net would be $60,000. From a capital
budgeting standpoint, is it financially wise to buy the
publication rights? What is the NPV of this investment? The
year 0 cash flow is -20,000, year 1 is 60,000, and years 2 and 3
are 30,000 each. Given a cost of capital of 8%, the NPV is just
over $85,000. It looks good, right? Now ask the class if the
publishing of this book would encourage cheating and if the
publishing company would want to be associated with this text
and its message. Some students may feel that one should accept
these profitable investment opportunities, while others might
prefer that the publication of this profitable text be rejected due
to the behavior it could encourage. Although the example is
simplistic, this type of issue is not uncommon and serves as a
starting point for a discussion of the value of “reputational
capital.”

Case 2:
Assume that to comply with the Air Quality Control Act of
1989, a company must install three smoke stack scrubber units
to its ventilation stacks at an installed cost of $355,000 per unit.
An estimated $100,000 per unit in fines could be saved each
year over the five-year life of the ventilation stacks. The cost of
capital is 14% for the firm. The analysis of the investment
results in a NPV of -$35,076. Could investment in a healthier
working environment result in lower long-term costs in the form
of lower future health costs? If so, might this decision result in
an increase in shareholder wealth? Notice that if the answer to
this second question is yes, it suggests that our original analysis
omitted some side benefits to the project.
An investment project has the following cash flows: CF0 = -
1,000,000; C01 – C08 = 200,000 each
If the required rate of return is 12%, what decision should be
made using NPV?
How would the IRR decision rule be used for this project, and
what decision would be reached?
How are the above two decisions related?
Comprehensive Problem
9-‹#›

8.49
Section 9.8
NPV = -$6,472; reject the project since it would lower the value
of the firm.
IRR = 11.81%, so reject the project since it would tie up
investable funds in a project that will provide insufficient
return.
The NPV and IRR decision rules will provide the same decision
for all independent projects with conventional/normal cash flow
patterns. If a project adds value to the firm (i.e., has a positive
NPV), then it must be expected to provide a return above that
which is required. Both of those justifications are good for
shareholders.
End of Chapter
CHAPTER 9
9-‹#›
9-‹#›

Sheet1Year0123Cash Flows-165000631207080091080Required
Return0.12NPV - WRONG$11,274.48NPV - RIGHT$12,627.41
Sheet2
Sheet3
Sheet1Year0123Cash Flows-165000631207080091080Required
Return0.12NPV - WRONG$11,274.48NPV -
RIGHT$12,627.41IRR16%16.13%Default Format
Sheet2
Sheet3
CHAPTER 6
DISCOUNTED CASH FLOW VALUATION (CALCULATOR)
6C-‹#›
5.1
This version relies primarily on the financial calculator with a
brief presentation of formulas. The calculator discussed is the
TI-BA-II+. The slides are easy to modify for whatever

calculator you prefer.
Determine the future and present value of investments with
multiple cash flows
Explain how loan payments are calculated and how to find the
interest rate on a loan
Describe how loans are amortized or paid off
Show how interest rates are quoted (and misquoted)
6C-‹#›
Future and Present Values of Multiple Cash Flows
Valuing Level Cash Flows: Annuities and Perpetuities
Comparing Rates: The Effect of Compounding
Loan Types and Loan Amortization
Chapter Outline

6C-‹#›
You think you will be able to deposit $4,000 at the end of each
of the next three years in a bank account paying 8 percent
interest.
You currently have $7,000 in the account.
How much will you have in three years?
How much will you have in four years?
Multiple Cash Flows – FV (Example 6.1)
6C-‹#›
Section 6.1 (A)
5.4
Find the value at year 3 of each cash flow and add them
together.
Today’s (year 0) CF: 3 N; 8 I/Y; -7,000 PV; CPT FV = 8817.98
Year 1 CF: 2 N; 8 I/Y; -4,000 PV; CPT FV = 4,665.60
Year 2 CF: 1 N; 8 I/Y; -4,000 PV; CPT FV = 4,320
Year 3 CF: value = 4,000
Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 =
21,803.58
Value at year 4: 1 N; 8 I/Y; -21,803.58 PV; CPT FV =
23,547.87
Multiple Cash Flows – FV (Example 6.1, CTD.)

6C-‹#›
5.5
Section 6.1 (A)
The students can read the example in the book. It is also
provided here.
You think you will be able to deposit $4,000 at the end of each
of the next three years in a bank account paying 8 percent
interest. You currently have $7,000 in the account. How much
will you have in three years? In four years?
Point out that there are several ways that this can be worked.
The book works this example by rolling the value forward each
year. The presentation will show the second way to work the
problem, finding the future value at the end for each cash flow
and then adding. Point out that you can find the value of a set of
cash flows at any point in time, all you have to do is get the
value of each cash flow at that point in time and then add them
together.
I entered the PV as negative for two reasons. (1) It is a cash
outflow since it is an investment. (2) The FV is computed as
positive, and the students can then just store each calculation
and then add from the memory registers, instead of writing
down all of the numbers and taking the risk of keying something
back into the calculator incorrectly.
Formula:
Today (year 0): FV = 7000(1.08)3 = 8,817.98

Year 1: FV = 4,000(1.08)2 = 4,665.60
Year 2: FV = 4,000(1.08) = 4,320
Year 3: value = 4,000
Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 =
21,803.58
Value at year 4 = 21,803.58(1.08) = 23,547.87
Suppose you invest $500 in a mutual fund today and $600 in
one year.
If the fund pays 9% annually, how much will you have in two
years?
Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = 594.05
Total FV = 594.05 + 654.00 = 1,248.05
Multiple Cash Flows – FV Example 2
6C-‹#›
5.6
Section 6.1 (A)
Formula: FV = 500(1.09)2 + 600(1.09) = 1,248.05
How much will you have in 5 years if you make no further
deposits?

First way:
Total FV = 769.31 + 846.95 = 1,616.26
Second way – use value at year 2:
3 N; -1,248.05 PV; 9 I/Y; CPT FV = 1,616.26
Multiple Cash Flows – FV Example 2 (ctd.)
6C-‹#›
5.7
Section 6.1 (A)
Formula:
First way: FV = 500(1.09)5 + 600(1.09)4 = 1,616.26
Second way: FV = 1248.05(1.09)3 = 1,616.26
Suppose you plan to deposit $100 into an account in one year
and $300 into the account in three years.
How much will be in the account in five years if the interest
rate is 8%?
Total FV = 136.05 + 349.92 = 485.97
Multiple Cash Flows – FV Example 3

6C-‹#›
5.8
Section 6.1 (A)
Formula:
FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97
Find the PV of each cash flow and add them
Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178.57
Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93
Multiple Cash Flows – pv (Example 6.3)
6C-‹#›

5.9
Section 6.1 (B)
The students can read the example in the book.
You are offered an investment that will pay you $200 in one
year, $400 the next year, $600 the next year and $800 at the end
of the fourth year. You can earn 12 percent on very similar
investments. What is the most you should pay for this one?
Point out that the question could also be phrased as “How much
is this investment worth?”
Remember the sign convention. The negative numbers imply
that we would have to pay 1,432.93 today to receive the cash
flows in the future.
Formula:
Year 1 CF: 200 / (1.12)1 = 178.57
Year 2 CF: 400 / (1.12)2 = 318.88
Year 3 CF: 600 / (1.12)3 = 427.07
Year 4 CF: 800 / (1.12)4 = 508.41
Example 6.3 Timeline
0
1
2
3
4

200
400
600
800
178.57
318.88
427.07
508.41
1,432.93
6C-‹#›
Section 6.1 (B)
5.10
You can use the PV or FV functions in Excel to find the present
value or future value of a set of cash flows.
Setting the data up is half the battle – if it is set up properly,
then you can just copy the formulas.

Click on the Excel icon for an example.
Multiple Cash Flows Using a Spreadsheet
6C-‹#›
5.11
Section 6.1 (B)
Click on the tabs at the bottom of the worksheet to move from a
future value example to a present value example.
Lecture Tip: The present value of a series of cash flows depends
heavily on the choice of discount rate. You can easily illustrate
this dependence in the spreadsheet on Slide 6.10 by changing
the cell that contains the discount rate. A separate worksheet on
the slide provides a graph of the relationship between PV and
the discount rate.
You are considering an investment that will pay you $1,000 in
one year, $2,000 in two years, and $3,000 in three years.
If you want to earn 10% on your money, how much would you
be willing to pay?
N = 1; I/Y = 10; FV = 1,000; CPT PV = -909.09
N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.89
N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.94
PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.93

Multiple Cash Flows – PV Another Example
6C-‹#›
5.12
Section 6.1 (B)
Formula:
PV = 1000 / (1.1)1 = 909.09
PV = 2000 / (1.1)2 = 1,652.89
PV = 3000 / (1.1)3 = 2,253.94
PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.92
Another way to use the financial calculator for uneven cash
flows is to use the cash flow keys.
Press CF and enter the cash flows beginning with year 0.
You have to press the “Enter” key for each cash flow.
Use the down arrow key to move to the next cash flow.
The “F” is the number of times a given cash flow occurs in
consecutive periods.
Use the NPV key to compute the present value by entering the
interest rate for I, pressing the down arrow, and then computing
the answer.
Clear the cash flow worksheet by pressing CF and then 2nd
CLR Work.
Multiple Uneven Cash Flows – Using the Calculator

6C-‹#›
5.13
Section 6.1 (B)
The next example will be worked using the cash flow keys.
Note that with the BA-II Plus, the students can double check the
numbers they have entered by pressing the up and down arrows.
It is similar to entering the cash flows into spreadsheet cells.
Other calculators also have cash flow keys. You enter the
information by putting in the cash flow and then pressing CF.
You have to always start with the year 0 cash flow, even if it is
zero.
Remind the students that the cash flows have to occur at even
intervals, so if you skip a year, you still have to enter a 0 cash
flow for that year.
Your broker calls you and tells you that he has this great
investment opportunity.
If you invest $100 today, you will receive $40 in one year and
$75 in two years.
If you require a 15% return on investments of this risk, should
you take the investment?
Use the CF keys to compute the value of the investment.
CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1

NPV; I = 15; CPT NPV = 91.49
No – the broker is charging more than you would be willing to
pay.
Decisions, Decisions
6C-‹#›
5.14
Section 6.1 (B)
You can also use this as an introduction to NPV by having the
students put –100 in for CF0. When they compute the NPV, they
will get –8.51. You can then discuss the NPV rule and point out
that a negative NPV means that you do not earn your required
return. You should also remind them that the sign convention on
the regular TVM keys is NOT the same as getting a negative
NPV.
You are offered the opportunity to put some money away for
retirement.
You will receive five annual payments of $25,000 each
beginning in 40 years.
How much would you be willing to invest today if you desire an
interest rate of 12%?
Use cash flow keys:
CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5; NPV; I
= 12; CPT NPV = 1,084.71
Saving For Retirement

6C-‹#›
Section 6.1 (B)
5.15
Saving For Retirement Timeline
0 1 2 … 39 40 41 42 43 44
0 0 0 … 0 25K 25K 25K 25K 25K
Notice that the year 0 cash flow = 0 (CF0 = 0)
The cash flows in years 1 – 39 are 0 (C01 = 0; F01 = 39)
The cash flows in years 40 – 44 are 25,000 (C02 = 25,000; F02
= 5)

6C-‹#›
Section 6.1 (B)
5.16
Suppose you are looking at the following possible cash flows:
Year 1 CF = $100;
Years 2 and 3 CFs = $200;
Years 4 and 5 CFs = $300.
The required discount rate is 7%.
What is the value of the cash flows at year 5?
What is the value of the cash flows today?
What is the value of the cash flows at year 3?
Quick Quiz – Part I
6C-‹#›
5.17
Section 6.1
The easiest way to work this problem is to use the uneven cash
flow keys and find the present value first and then compute the
others based on that.
CF0 = 0; C01 = 100; F01 = 1; C02 = 200; F02 = 2; C03 = 300;
F03 = 2; I = 7; CPT NPV = 874.17

Value in year 5: PV = 874.17; N = 5; I/Y = 7; CPT FV =
1,226.07
Value in year 3: PV = 874.17; N = 3; I/Y = 7; CPT FV =
1,070.90
Using formulas and one CF at a time:
Year 1 CF: FV5 = 100(1.07)4 = 131.08; PV0 = 100 / 1.07 =
93.46; FV3 = 100(1.07)2 = 114.49
Year 2 CF: FV5 = 200(1.07)3 = 245.01; PV0 = 200 / (1.07)2 =
174.69; FV3 = 200(1.07) = 214
Year 3 CF: FV5 = 200(1.07)2 = 228.98; PV0 = 200 / (1.07)3 =
163.26; FV3 = 200
Year 4 CF: FV5 = 300(1.07) = 321; PV0 = 300 / (1.07)4 =
228.87; PV3 = 300 / 1.07 = 280.37
Year 5 CF: FV5 = 300; PV0 = 300 / (1.07)5 = 213.90; PV3 =
300 / (1.07)2 = 262.03
Value at year 5 = 131.08 + 245.01 + 228.98 + 321 + 300 =
1,226.07
Present value today = 93.46 + 174.69 + 163.26 + 228.87 +
213.90 = 874.18 (difference due to rounding)
Value at year 3 = 114.49 + 214 + 200 + 280.37 + 262.03 =
1,070.89 (difference due to rounding)
Annuity – finite series of equal payments that occur at regular
intervals
If the first payment occurs at the end of the period, it is called
an ordinary annuity.
If the first payment occurs at the beginning of the period, it is
called an annuity due.
Perpetuity – infinite series of equal payments
Annuities and Perpetuities Defined

6C-‹#›
Section 6.2
5.18
Perpetuity: PV = C / r
Annuities:
Annuities and Perpetuities – Basic Formulas
6C-‹#›
5.19
Section 6.2
Lecture Tip: The annuity factor approach is a short-cut
approach in the process of calculating the present value of
multiple cash flows and it is only applicable to a finite series of
level cash flows. Financial calculators have reduced the need
for annuity factors, but it may still be useful from a conceptual
standpoint to show that the PVIFA is just the sum of the PVIFs
across the same time period.

You can use the PMT key on the calculator for the equal
payment.
The sign convention still holds.
Ordinary annuity versus annuity due
You can switch your calculator between the two types by using
the 2nd BGN 2nd Set on the TI BA-II Plus.
If you see “BGN” or “Begin” in the display of your calculator,
you have it set for an annuity due.
Most problems are ordinary annuities.
Annuities and the Calculator
6C-‹#›
5.20
Section 6.2
Other calculators also have a key that allows you to switch
between Beg/End.
After carefully going over your budget, you have determined
you can afford to pay $632 per month toward a new sports car.
You call up your local bank and find out that the going rate is 1
percent per month for 48 months.
How much can you borrow?
To determine how much you can borrow, we need to calculate
the present value of $632 per month for 48 months at 1 percent
per month.

Annuity – Example 6.5
6C-‹#›
Section 6.2 (A)
5.21
You borrow money TODAY so you need to compute the present
value.
48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54 ($24,000)
Formula:
Annuity – Example 6.5 (ctd.)
6C-‹#›
5.22
Section 6.2 (A)
The students can read the example in the book.

After carefully going over your budget, you have determined
you can afford to pay $632 per month towards a new sports car.
You call up your local bank and find out that the going rate is 1
percent per month for 48 months. How much can you borrow?
Note that the difference between the answer here and the one in
the book is due to the rounding of the Annuity PV factor in the
book.
Suppose you win the Publishers Clearinghouse $10 million
sweepstakes.
The money is paid in equal annual end-of-year installments of
$333,333.33 over 30 years.
If the appropriate discount rate is 5%, how much is the
sweepstakes actually worth today?
30 N; 5 I/Y; 333,333.33 PMT;
CPT PV = 5,124,150.29
Annuity – Sweepstakes Example
6C-‹#›
5.23
Section 6.2 (A)
Formula:
PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
You are ready to buy a house, and you have $20,000 for a down

payment and closing costs.
Closing costs are estimated to be 4% of the loan value.
You have an annual salary of $36,000, and the bank is willing
to allow your monthly mortgage payment to be equal to 28% of
your monthly income.
The interest rate on the loan is 6% per year with monthly
compounding (.5% per month) for a 30-year fixed rate loan.
How much money will the bank loan you?
How much can you offer for the house?
Buying a House
6C-‹#›
5.24
Section 6.2 (A)
It might be good to note that the outstanding balance on the
loan at any point in time is simply the present value of the
remaining payments.
Bank loan
Monthly income = 36,000 / 12 = 3,000
Maximum payment = .28(3,000) = 840
30×12 = 360 N
.5 I/Y
-840 PMT
CPT PV = 140,105
Total Price
Closing costs = .04(140,105) = 5,604

Down payment = 20,000 – 5,604 = 14,396
Total Price = 140,105 + 14,396 = 154,501
Buying a House (ctd.)
6C-‹#›
5.25
Section 6.2 (A)
You might point out that you would probably not offer 154,501.
The more likely scenario would be 154,500 , or less if you
assumed negotiations would occur.
Formula
PV = 840[1 – 1/1.005360] / .005 = 140,105
The present value and future value formulas in a spreadsheet
include a place for annuity payments.
Click on the Excel icon to see an example.
Annuities on the Spreadsheet – Example

6C-‹#›
Section 6.2 (A)
5.26
You know the payment amount for a loan, and you want to know
how much was borrowed. Do you compute a present value or a
future value?
You want to receive 5,000 per month in retirement.
If you can earn 0.75% per month and you expect to need the
income for 25 years, how much do you need to have in your
account at retirement?
Quick Quiz – Part II
6C-‹#›
5.27
Section 6.2 (A)
Calculator
PMT = 5,000; N = 25×12 = 300; I/Y = .75; CPT PV = 595,808
Formula
PV = 5000[1 – 1 / 1.0075300] / .0075 = 595,808
Suppose you want to borrow $20,000 for a new car.
You can borrow at 8% per year, compounded monthly (8/12 =
.66667% per month).

If you take a 4-year loan, what is your monthly payment?
4(12) = 48 N; 20,000 PV; .66667 I/Y; CPT PMT = 488.26
Finding the Payment
6C-‹#›
5.28
Section 6.2 (A)
Formula
20,000 = PMT[1 – 1 / 1.006666748] / .0066667
PMT = 488.26
Another TVM formula that can be found in a spreadsheet is the
payment formula.
PMT(rate, nper, pv, fv)
The same sign convention holds as for the PV and FV formulas.
Click on the Excel icon for an example.
Finding the Payment on a Spreadsheet

6C-‹#›
Section 6.2 (A)
5.29
You ran a little short on your spring break vacation, so you put
$1,000 on your credit card.
You can afford only the minimum payment of $20 per month.
The interest rate on the credit card is 1.5 percent per month.
How long will you need to pay off the $1,000?
Finding the Number of Payments – Example 6.6
6C-‹#›
Section 6.2 (A)
5.30
The sign convention matters!
1.5 I/Y
1,000 PV
-20 PMT
CPT N = 93.111 months = 7.75 years
And this is only if you don’t charge anything more on the card!
Finding the Number of Payments – Example 6.6 (ctd.)

6C-‹#›
5.31
Section 6.2 (A)
You ran a little short on your spring break vacation, so you put
$1,000 on your credit card. You can only afford to make the
minimum payment of $20 per month. The interest rate on the
credit card is 1.5 percent per month. How long will you need to
pay off the $1,000?
This is an excellent opportunity to talk about credit card debt
and the problems that can develop if it is not handled properly.
Many students don’t understand how it works, and it is rarely
discussed. This is something that students can take away from
the class, even if they aren’t finance majors.
1000 = 20(1 – 1/1.015t) / .015
.75 = 1 – 1 / 1.015t
1 / 1.015t = .25
1 / .25 = 1.015t
t = ln(1/.25) / ln(1.015) = 93.111 months = 7.75 years
Suppose you borrow $2,000 at 5%, and you are going to make
annual payments of $734.42.
How long before you pay off the loan?
Sign convention matters!!!
5 I/Y
2,000 PV
-734.42 PMT
CPT N = 3 years

Finding the Number of Payments – Another Example
6C-‹#›
5.32
Section 6.2 (A)
2000 = 734.42(1 – 1/1.05t) / .05
.136161869 = 1 – 1/1.05t
1/1.05t = .863838131
1.157624287 = 1.05t
t = ln(1.157624287) / ln(1.05) = 3 years
Suppose you borrow $10,000 from your parents to buy a car.
You agree to pay $207.58 per month for 60 months.
What is the monthly interest rate?
Sign convention matters!!!
60 N
10,000 PV
-207.58 PMT
CPT I/Y = .75%
Finding the Rate

6C-‹#›
Section 6.2 (A)
5.33
Trial and Error Process
Choose an interest rate and compute the PV of the payments
based on this rate.
Compare the computed PV with the actual loan amount.
If the computed PV > loan amount, then the interest rate is too
low.
If the computed PV < loan amount, then the interest rate is too
high.
Adjust the rate and repeat the process until the computed PV
and the loan amount are equal.
Annuity – Finding the Rate Without a Financial Calculator
6C-‹#›
Section 6.2 (A)
5.34
You want to receive $5,000 per month for the next 5 years.
How much would you need to deposit today if you can earn
0.75% per month?
What monthly rate would you need to earn if you only have
$200,000 to deposit?

Suppose you have $200,000 to deposit and can earn 0.75% per
month.
How many months could you receive the $5,000 payment?
How much could you receive every month for 5 years?
Quick Quiz – Part III
6C-‹#›
5.35
Section 6.2 (A)
Q1: 5(12) = 60 N; .75 I/Y; 5000 PMT; CPT PV = -240,867
PV = 5000(1 – 1 / 1.007560) / .0075 = 240,867
Q2: -200,000 PV; 60 N; 5000 PMT; CPT I/Y = 1.439%
Trial and error without calculator
Q3: -200,000 PV; .75 I/Y; 5000 PMT; CPT N = 47.73 (47
months plus partial payment in month 48)
200,000 = 5000(1 – 1 / 1.0075t) / .0075
.3 = 1 – 1/1.0075t
1.0075t = 1.428571429
t = ln(1.428571429) / ln(1.0075) = 47.73 months
Q4: -200,000 PV; 60 N; .75 I/Y; CPT PMT = 4,151.67
200,000 = C(1 – 1/1.007560) / .0075
C = 4,151.67
Suppose you begin saving for your retirement by depositing
$2,000 per year in an IRA.

If the interest rate is 7.5%, how much will you have in 40
years?
Remember the sign convention!
40 N
7.5 I/Y
-2,000 PMT
CPT FV = 454,513.04
Future Values for Annuities
6C-‹#›
5.36
Section 6.2 (B)
FV = 2000(1.07540 – 1)/.075 = 454,513.04
Lecture Tip: It should be emphasized that annuity factor tables
(and the annuity factors in the formulas) assumes that the first
payment occurs one period from the present, with the final
payment at the end of the annuity’s life. If the first payment
occurs at the beginning of the period, then FV’s have one
additional period for compounding and PV’s have one less
period to be discounted. Consequently, you can multiply both
the future value and the present value by (1 + r) to account for
the change in timing.
You are saving for a new house and you put $10,000 per year in
an account paying 8%. The first payment is made today.
How much will you have at the end of 3 years?

2nd BGN 2nd Set (you should see BGN in the display)
3 N
-10,000 PMT
8 I/Y
CPT FV = 35,061.12
2nd BGN 2nd Set (be sure to change it back to an ordinary
annuity)
Annuity Due
6C-‹#›
5.37
Section 6.2 (C)
Note that the procedure for changing the calculator to an
annuity due is similar on other calculators.
Formula:
FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12
What if it were an ordinary annuity? FV = 32,464 (so you
receive an additional 2,597.12 by starting to save today.)
Annuity Due Timeline

0 1 2 3
10000 10000 10000
32,464
35,016.12
6C-‹#›
5.38
Section 6.2 (C)
If you use the regular annuity formula, the FV will occur at the
same time as the last payment. To get the value at the end of the
third period, you have to take it forward one more period.
Suppose the Fellini Co. wants to sell preferred stock at $100 per
share.
A similar issue of preferred stock already outstanding has a
price of $40 per share and offers a dividend of $1 every quarter.
What dividend will Fellini have to offer if the preferred stock is
going to sell?
Perpetuity – Example 6.7

6C-‹#›
Section 6.2 (D)
5.39
Perpetuity formula: PV = C / r
Current required return:
40 = 1 / r
r = .025 or 2.5% per quarter
Dividend for new preferred:
100 = C / .025
C = 2.50 per quarter
Perpetuity – Example 6.7 (ctd.)
6C-‹#›
5.40
Section 6.2 (D)
This is a good preview to the valuation issues discussed in
future chapters. The price of an investment is just the present
value of expected future cash flows.
Example statement:

Suppose the Fellini Co. wants to sell preferred stock at $100 per
share. A very similar issue of preferred stock already
outstanding has a price of $40 per share and offers a dividend of
$1 every quarter. What dividend will Fellini have to offer if the
preferred stock is going to sell?
You want to have $1 million to use for retirement in 35 years.
If you can earn 1% per month, how much do you need to deposit
on a monthly basis if the first payment is made in one month?
What if the first payment is made today?
You are considering preferred stock that pays a quarterly
dividend of $1.50.
If your desired return is 3% per quarter, how much would you
be willing to pay?
Quick Quiz – Part IV
6C-‹#›
5.41
Section 6.2 (D)
Q1: 35(12) = 420 N; 1,000,000 FV; 1 I/Y; CPT PMT = 155.50
1,000,000 = C (1.01420 – 1) / .01
C = 155.50
Q2:Set calculator to annuity due and use the same inputs as
above. CPT PMT = 153.96

The payments would be smaller by one period’s interest. Divide
the above result by 1.01.
1,000,000 = C[(1.01420 – 1) / .01] ( 1.01)
C = 153.96
Q3: PV = 1.50 / .03 = $50
Another online financial calculator can be found at
MoneyChimp.
Go to the website and work the following example.
Choose calculator and then annuity
You just inherited $5 million. If you can earn 6% on your
money, how much can you withdraw each year for the next 40
years?
MoneyChimp assumes annuity due!
Payment = $313,497.81
Work the Web Example
6C-‹#›
Section 6.2 (D)
5.42
Table 6.2

6C-‹#›
Section 6.2 (D)
5.43
A growing stream of cash flows with a fixed maturity
Growing Annuity
6C-‹#›
Section 6.2 (E)
5.44
A defined-benefit retirement plan offers to pay $20,000 per year
for 40 years and increase the annual payment by three-percent
each year. What is the present value at retirement if the
discount rate is 10 percent?
Growing Annuity: Example

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CHAPTER 11PROJECT ANALYSIS AND EVALUATIONCopyright © 2019 Mc