Chapter 11
Cash Flows and Capital Budgeting
Learning Objectives
1. Explain why incremental after-tax free cash flows are relevant in evaluating a project,
and be able to calculate them for a project.
2. Discuss the five general rules for incremental after-tax free cash flow calculations, and
explain why cash flows stated in nominal (real) dollars should be discounted using a
nominal (real) discount rate.
3. Describe how distinguishing between variable and fixed costs can be useful in
forecasting operating expenses.
4. Explain the concept of equivalent annual cost, and be able to use it to compare projects
with unequal lives, decide when to replace an existing asset, and calculate the
opportunity cost of using an existing asset.
5. Determine the appropriate time to harvest an asset.
I.
Chapter Outline
Calculating Project Cash Flows
•
In capital budgeting, we estimate the NPV of the cash flows that a project is expected to
produce in the future.
•
A.
All of the cash flow estimates are forward-looking.
Incremental After-Tax Free Cash Flows
•
The cash flows we discount in an NPV analysis are the incremental after-tax free
cash flows , which refers to the fact that these cash flows reflect the amount by
which the firm’s total after-tax free cash flows will change if the project is
adopted. See Equation 11.1:
o FCFProject = FCFFirm with project – FCFFirm without project
•
The term free cash flows (FCF) refers to the fact that the firm is free to distribute
these cash flows to creditors and stockholders because these are the cash flows that
are left over after a firm has made necessary investments in working capital and
long-term assets.
B.
The FCF Calculation
•
Referring to 11.2, which is a more detailed version of 11.1:
FCF = [(Revenue – Op Exp – D&A) x (1 – t)] + D&A – Cap Ex – Add WC
•
We first compute the incremental cash flow from operations (CF Opns), which is
the cash flow that the project is expected to generate after all operating expenses
and taxes have been paid.
•
We then subtract the incremental capital expenditures (Cap Ex) and incremental
additions to working capital (Add WC) required for the project to obtain FCF.
•
The FCF is therefore a measure of the after-tax cash flows from operations over
and above what is necessary to make any required investments.
•
The idea that we can evaluate the cash flows from a project independently of the
cash flows for the firm is known as the stand-alone principle. It is another way of
saying that we can treat the project as if it is a stand-alone firm that has its own
revenue, expenses, and investment requirements.
C. Cash Flows from Operations
•
Note that the incremental cash flow from operations, CF Opns, equals the
incremental net operating profits after tax (NOPAT) plus the incremental
depreciation and amortization (D&A) associated with the project.
•
We exclude interest expenses when calculating NOPAT because the cost of
financing a project is reflected in the discount rate that is used in the NPV
calculation.
•
We use the firm’s marginal tax rate (t) to calculate NOPAT because the profits
from a project are assumed to be incremental to the firm.
•
We add incremental depreciation and amortization (D&A) to NOPAT when
calculating CF Opns because, as in the accounting statement of cash flows, D&A
represents a noncash charge that reduces the firm’s tax obligation.
•
However, since D&A is a noncash charge, we have to add it back to
NOPAT in order to get the cash flow from operations right.
D. Cash Flows Associated with Investments
•
Once we have estimated CF Opns, we simply subtract cash flows associated with
required investment to obtain FCF for a project in a particular period.
•
Investments can be required to purchase long-term tangible assets and intangible
assets, or to fund current assets.
E. FCF versus Accounting Earnings
•
The impact of a project on a firm’s overall value or on its stock price
does not depend on how the project affects the company’s accounting earnings. It
depends only on how the project affects the company’s free cash flows.
•
Accounting earnings can differ from cash flows for a number of
reasons, making accounting earnings an unreliable measure of the costs and
benefits of a project.
•
Accounting earnings also reflect noncash charges, such as depreciation
and amortization, which are intended to account for the costs associated with
deterioration of the assets in a business as those assets are used.
11.2
Estimating Cash Flows in Practice
A.
Five General Rules for Incremental Cash Flow Calculations
•
Rule 1: Include cash flows and only cash flows in your calculations. Do not
include allocated costs or overhead unless they reflect cash flows.
•
Rule 2: Include the impact of the project on cash flows from other product lines. If
the product associated with a project is expected to cannibalize or boost sales of
another product, you must include the expected impact of the new project on the
cash flows from the other product in the analysis.
•
Rule 3: Include all opportunity costs. By opportunity costs, we mean the cost of
giving up another opportunity.
•
Rule 4: Forget sunk costs. Sunk costs are costs that have already been incurred,
but all that matters when you evaluate a project at a particular point in time is how
much you have to invest in the future and what you could expect to receive in
return for that investment; this means that past investments are irrelevant.
•
Rule 5: Include only after-tax cash flows in the cash flow calculations. The
incremental pretax earnings of a project only matter to the extent that they affect
the after-tax cash flows that the firm’s investors receive.
B.
Nominal versus Real Cash Flows
•
Nominal dollars are the dollars that we typically think of. They represent the actual
dollar amounts that we expect a project to generate in the future, without any
adjustments.
•
When prices are going up, a given nominal dollar amount will buy less and less
over time.
o Real dollars represent dollars stated in terms of constant purchasing power.
•
We can write the cost of capital , k, as
o 1 + k = (1 + ∆Pe) x (1 + r)
o r is the real cost of capital
o (∆Pe) is the expected rate of inflation
o (r) is the real rate of return
•
It is important to make sure that all cash flows are stated in either nominal dollars
or real dollars.
C.
Tax Rates and Depreciation
•
A progressive tax system, which we have in the United States, is one in which the
marginal tax rate at low levels of income is lower than the marginal tax rate at high
levels of income.
•
One especially important difference from a capital budgeting perspective is that the
depreciation methods allowed by GAAP differ from those allowed by the IRS.
o The straight-line depreciation method illustrated earlier in this chapter in
the NASCAR racetrack example is allowed by GAAP and is often used for
financial reporting.
o An “accelerated” method of depreciation, called the Modified Accelerated
Cost Recovery System (MACRS), has been in use for U.S. federal tax
calculations since the Tax Reform Act of 1986 went into effect.

MACRS thus enables a firm to deduct depreciation charges sooner,
thereby realizing the tax savings sooner and increasing the present
value of the tax savings.
D.
Computing the Terminal-Year FCF
•
The FCF in the last, or terminal, year of a project often includes cash flows that are
not typically included in the calculations for other years.
o For instance, in the final year of a project, the assets acquired during the
life of the project may be sold and the working capital that has been
invested may be recovered.
 Add WC = Change in cash and cash equivalents + Change in
accounts receivable + Change in inventories – Change
in accounts payable.
o When an asset is expected to have a salvage value, we must include the
salvage value realized from the sale (net of any tax consequences) of the
asset and the impact of the sale on the firm’s taxes in the terminal-year
FCF calculations.
E.
Expected Cash Flows
•
We are estimating when we forecast FCF in an NPV analysis.
o The expected FCF for a particular year equals the sum of the products of
the possible outcomes (FCFs) and the probabilities that those outcomes
will be realized.
11.3.1 Forecasting Free Cash Flows
A.
Cash Flows from Operations
•
To forecast incremental cash flows from operations we must forecast the
incremental net revenue, operating expenses, and depreciation and amortization
associated with the project, as well as the firm’s marginal tax rate
•
When forecasting operating expenses, analysts often distinguish between variable
costs and fixed costs
B.
Investment Cash Flows
•
We must consider two general classes of investments when calculating FCF:
incremental capital expenditures and incremental additions to working capital
1. Capital Expenditures
o Capital expenditure forecasts in an NPV analysis reflect the expected level
of investment during each year of the project’s life.
o Capital expenditures are typically required at the beginning of a project
2. Working Capital
o Cash flow forecasts in an NPV analysis include four working capital items:
1)cash and cash equivalents, 2) accounts receivable, 3) inventories, and 4)
accounts payable
11.4
Special Cases
A.
Projects with Different Lives
•
A problem that arises quite often in capital budgeting involves choosing
between two mutually exclusive investments where the investments have different
lives.
o
In a situation like this, we can effectively make the lives of the mowers
the same by assuming repeated investments over some identical period and
then comparing the NPVs of their costs.
o
A less cumbersome and more powerful method to handle the problem
is to compute the equivalent annual cost (EAC). The EAC can be calculated
as follows:
EACi = kNPVi [(1 + k)t / (1 + k)t –t 1]


( 1+ k )
EACi = k NPVi 

t
−1

 ( 1+ k )
 k is the opportunity cost of capital
 NPVi is the normal NPV of the investment i
 t is the life of the investment
(11.5)
 EAC simply reflects the annuity that has the same present value as the
∆FCFs of an investment over the investment period we are considering.
B.
When to Harvest an Asset
•
The optimal time to harvest is the point in time at which the rate of increase in
cash flows, from period to period, is no longer greater than the cost of capital.
 At this point in time, it becomes optimal to harvest the trees and invest the
proceeds in alternative investments that yield the opportunity cost of capital.
C.
When to Replace an Existing Asset
•
Two fundamental questions: Do the benefits of replacing the existing asset exceed
the costs, and, if they do not now, when will they?
•
Solving this problem is simply a matter of computing the EAC for the new asset
and comparing it with the annual cash inflows from the old asset.
D.
The Cost of Using an Existing Asset
•
The third rule of calculating incremental after-tax cash flows is to include all
opportunity costs that are not always directly observable.
o Sometimes they have to be computed by first figuring the EAC for a given set
of cash flows and then adjusting the EAC by the appropriate discount rate and
time, if the EAC is not in present value form.
II.
Suggested and Alternative Approaches to the Material
This is a very important chapter for all students. Finance majors will need the basic capital budgeting
concepts introduced here for their more advanced course work, whereas nonmajors will require a
basic foundation for the material in their professional lives as they will undoubtedly be involved in
project analysis at some point in their careers. The chapter begins with a general framework for
capital budgeting, so that the overall approach to capital budgeting can be seen before proceeding to
the detailed aspects of the subject matter. Free cash flow (FCF) is then formally introduced, along
with individual subcalculations that make it up. The chapter includes a number of examples to
accompany the individual aspects of the material and finishes by introducing cases where the net
present value of the projects FCFs will require further adjustments.
Thorough coverage of the material is recommended before proceeding to the next chapter
where the tools are then applied further. As such, it may be necessary to allocate additional time for
this chapter as the material is central to Finance while it also incorporates a number of tools that were
learned in earlier chapters.
III. Summary of Learning Objectives
1. Explain why incremental after-tax cash flows are relevant in evaluating a project, and be
able to calculate them for a project.
The incremental after-tax cash flows, also called the incremental free cash flows (FCFs), for a
project equal the expected change in the total after-tax cash flows of the firm if the project is
adopted. The impact of a project on the firm’s total cash flows is the appropriate measure of cash
flows because these are the cash flows that reflect all of the costs and benefits from the project
and only the costs and benefits from the project. The incremental after-tax cash flows are
calculated using Equation 11.2. The calculation is also illustrated in Exhibit 11.1.
2. List and explain the five general rules for incremental after-tax free cash flow calculations
and why cash flows stated in nominal (real) dollars should be discounted using a nominal
(real) discount rate.
The five general rules are:
Rule 1:
Include cash flows and only cash flows in your calculations. Stockholders only care
about the impact of a project on the firm’s cash flows.
Rule 2:
Include the impact of the project on cash flows from other product lines. If a project
affects the cash flows from other projects, we must take this fact into account in NPV
analysis in order to fully capture the impact of the project on the firm’s total cash
flows.
Rule 3:
Include all opportunity costs. If an asset is used for a project, the relevant cost for that
asset is the value that could be realized from its most valuable alternative use. By
including this cost in the NPV analysis, we capture the change in the firm’s cash flows
that is attributable to the use of this asset for the project.
Rule 4:
Forget sunk costs. The only costs that matter are those to be incurred from this point
on.
Rule 5:
Include only after-tax cash flows in the cash flow calculations. Since stockholders
receive cash flows after taxes have been paid, they are only concerned about after-tax
cash flows.
Since a nominal rate reflects both the expected rate of inflation and a real return, we would be
overadjusting for inflation if we discounted a real cash flow with a nominal rate. Similarly, if we
discounted a nominal cash flow using a real discount rate, we would be undercompensating for
expected inflation in the discounting process. This is why we discount nominal cash flows using
only a nominal discount rate and we discount real cash flows using only a real discount rate.
3. Describe how distinguishing between variable and fixed costs can be useful in forecasting
operating expenses.
Variable costs vary directly with the number of units sold, while fixed costs do not. When
forecasting operating expenses, it is often useful to treat variable and fixed costs separately. We
can forecast variable costs by multiplying unit variable costs by the number of units sold. Fixed
costs are more accurately based on the specific characteristics of those costs, rather than as a
function of sales. Separating fixed costs from the variable also makes it easier to identify the
factors that will cause them to change over time and therefore easier to forecast them.
4. Explain the concept of equivalent annual cost, and be able to use it to compare projects with
unequal lives, decide when to replace an existing asset, and calculate the opportunity cost of
using an existing asset.
The equivalent annual cost (EAC) is the annualized cost of an investment that is stated in nominal
dollars. In other words, it is the annual payment from an annuity that has the same NPV and the
same life as the project. Since it is a measure of the annual cost or cash inflow from a project, the
EAC for one project can be compared directly with the EAC from another project, regardless of
the lives of those two projects. Applications of the EAC concept are presented in Section 11.4.
5.
Determine the appropriate time to harvest an asset. The appropriate time to harvest an
asset is that point in time where harvesting the asset yields the largest present value, in today’s
dollars, of the project NPV.
IV. Summary of Key Equations
Equation
Description
Formula
11.1
Incremental free cash
flow definition
FCFProject = FCFFirm with project – FCFFirm without project
11.2
Incremental free dash
flow calculation
FCF = [(Revenue – Op Exp – D&A) x (1 – t)] + D&A – Cap Ex – Add WC
11.3
Inflation and real
components of cost of
capital
11.4
Incremental additions to
working capital
11.5
Equivalent annual cost
1 + k = (1 + ∆Pe) x (1 + r)
Add WC = Change in cash and cash equivalents + Change in
accounts receivable + Change in inventories –
Change in accounts payable
EACi = kNPVi [(1 + k)t / (1 + k)t – 1]
V.
Before You Go On Questions and Answers
Section 11.1
1. Why do we care about incremental cash flows at the firm level when we evaluate a project?
We care about incremental cash flows at the firm level because they reflect the impact of the
project on the total cash flows that the firm produces. This is what the stockholders care about.
The difference between the present value of the expected cash flows from the firm with the
project and the present value of the expected cash flows from the firm without the project is
precisely what the NPV of a project is. Our NPV estimate will be incorrect if we do not account
for all of the incremental cash flows at the firm level.
2. Why is D&A first subtracted and then added back in FCF calculations?
By subtracting D&A, calculating the tax obligation, and then adding back D&A, we are
accounting for the fact that D&A is a noncash charge that reduces the firm’s tax obligation by the
product of D&A and the tax rate (D&A x t). If we did not do this, we would overstate the tax
obligation and understate FCF.
3. What types of investments should be included in FCF calculations?
All investments directly associated with the project should be included in FCF calculations. These
can include both investments in tangible and intangible assets. They can also include investments
in additions to working capital, such as for the credit a firm extends to its customers and
inventories.
Section 11.2
1. What are the five general rules for calculating FCF?
(1) Include cash flows and only cash flows in your calculations.
(2) Include the impact of the project on cash flows from other product lines.
(3) Include all opportunity costs.
(4) Forget sunk costs.
(5) Include only after-tax cash flows in the cash flow calculations.
2. What is the difference between nominal and real dollars? Why is it important not to mix them
in an NPV analysis?
When most people talk about dollar amounts, they are referring to nominal dollars. Nominal
dollars do not take into account changes in purchasing power. Real dollars are dollar amounts that
are adjusted for changes in purchasing power. For example, 100 real dollars have the same
purchasing power whether they are received today or at some future date. It is important not to
mix nominal and real dollars in an NPV analysis because the discount rate is either a nominal
rate, which is used to discount nominal dollars, or a real rate, which is used to discount real
dollars. Since the discount rate must be either a nominal rate or a real rate, if real and nominal
dollars are mixed in an NPV analysis, the NPV will be calculated incorrectly.
3. What is a progressive tax system? What is the difference between a firm’s marginal and
average tax rates?
A progressive tax system is one in which the marginal tax rate at low levels of income is lower
than the marginal tax rate at high levels of income. A firm’s marginal tax rate is the rate that it
pays on the last dollar earned while the average tax rate is the average rate paid on the firm’s total
earnings (tax paid divided by taxable income).
4. How can FCF in the terminal year of a project’s life differ from FCF in the other years?
FCF in the terminal year can differ from FCF in other years in several ways. The terminal year
cash flows can include cash flows from the asset sales, including the actual proceeds from the
sales themselves and taxes due or received if there is a gain or loss on the sale. Terminal year
cash flows can also include cash flows associated with recovery of working capital.
5. Why is it important to understand that cash flow forecasts in an NPV analysis are expected
values?
It is important to recognize that we are forecasting expected cash flows in an NPV analysis
because uncertainties regarding project cash flows that are unique to the project should be
reflected in the cash flow forecasts.
Section 11.3
1. What is the difference between variable and fixed costs, and what are examples of each?
Variable costs vary directly with unit sales. Fixed costs do not vary with unit sales. For an
example of each, see the video game player scenario on page 379. Variable costs are those
associated with purchasing the components for the player, the labor required, and sales and
marketing. These costs will vary according to the number of units produced. Fixed costs are those
associated with assembly space, and administrative expenses.
2. How are working capital items forecast? Why are accounts receivable typically forecast as a
percentage of revenue and accounts payable, and inventories as percentages of the cost of good
sold?
Working capital items are forecast using 1) cash and cash equivalents, 2) accounts receivable, 3)
inventories, and 4) accounts payable
Inventories are forecast as a percentage of the cost of goods sold because the COGS represent a
measure of the amount of money invested in inventories. Accounts payable are forecast this way
because the COGS is a measure of the amount of money actually owed to suppliers.
Section 11.4
1. When can we not simply compare the NPVs of two mutually exclusive projects?
If we expect to replace at least one of the projects at the end of its life, we cannot simply compare
the NPVs. Doing so would ignore the subsequent investment(s). You can only directly compare
the NPVs of mutually exclusive projects under one condition—that is, if you expect to terminate
the project that is chosen (e.g., sell the lawn mower) on or before the end of the life of the shorterlived project.
2. How do we decide when to harvest an asset?
We choose the harvest date that maximizes the NPV of the asset. To identify this date, we
compare the NPVs expected from harvesting the asset for each of the feasible harvest dates. The
best date to harvest the asset is the date that produces the largest NPV, once the NPVs for all of
the alternative harvest dates have been discounted to the same point in time.
3. Under what circumstance would you replace an old machine that is still operating with a new
one?
You should replace the old machine when the EAC of the new machine is lower than the EAC of
the old machine (if revenues are the same for both machines) or when the annualized cash inflow
from the replacement is greater.
VI. Self Study Problems
11.1
Explain why the announcement of a new investment is usually accompanied by a change in
the firm’s stock price.
Solution:
A firm’s investments cause changes in its future after-tax cash flows and stockholders are the
residual claimants (owners) of those cash flows. Therefore, the stock price should increase
when stockholders expect an investment to have a positive NPV, and decrease when it is
expected to have a negative NPV.
11.2
In calculating the NPV of a project, should we use all of the cash flows associated with the
project, or incremental cash flows of the project? Why?
Solution:
We should use incremental cash flows of the project. Incremental cash flows reflect the
amount by which the firm’s total cash flows will change if the project is adopted. In other
words, incremental cash flows represent the net difference in cash revenues, costs, and
investment outlays at the firm level with and without the project, which is precisely what the
stockholders care about.
11.3
You are considering opening another restaurant in the food chain of TexasBurgers. The new
restaurant will have annual revenue of $300,000 and operating expenses of $150,000. The
annual depreciation and amortization for the assets used in the restaurant will equal $50,000.
An annual capital expenditure of $10,000 will be required to offset wear-and-tear on the
assets used in the restaurant, but no additions to working capital will be required. The
marginal tax rate will be 40 percent. Calculate the incremental annual free cash flow for the
project
Solution:
The incremental annual free cash flow is calculated as:
FCF = ($300,000-$150,000-$50,000) x (1-0.4) + $50,000-$10,000 = $100,000
11.4
Sunglass Heaven, Inc., is launching a new store in a shopping mall in Houston. The store’s
annual revenue depends on Houston’s weather conditions in the summer. The annual revenue
will be $240,000 in a sizzling summer, with probability of 0.3; $80,000 in a cool summer,
with probability of 0.2; and $150,000 in a normal summer, with probability of 0.5. What is the
expected annual revenue of the store?
Solution:
The expected annual revenue is:
(0.3 x $240,000) + (0.2 x $80,000) + (0.5 x $150,000) = $163,000
11.5
Sprigg Lane needs to purchase a new central air-conditioning system for a plant. There are
two choices. The first system costs $50,000 and is expected to last 10 years, and the second
system costs $72,000 and is expected to last 15 years. Assume that the opportunity cost of
capital is 10 percent. Which air-conditioning system should you purchase?
Solution:
The equivalent annual cost for each system is:
EAC1 = (0.1)($50,000)[((1.1)10)/((1.1)10-1)] = $8,137.27
EAC2 = (0.1)($72,000)[((1.1)15)/((1.1)15-1)] = $9,466.11
Therefore Sprigg Lane should purchase the first one.
VII. Critical Thinking Questions
11.1
Do you agree or disagree with the following statement: given the techniques discussed in this
chapter? We can estimate future cash flows precisely and obtain an exact value for the NPV
of an investment.
The statement is not true. Given the nature of the real business world, it is almost certain that
the cash flows generated by a project will differ from the forecasts used to decide whether to
proceed with the project. However, techniques discussed in this chapter provide an important
and useful framework that helps minimize errors and ensures that forecasts are internally
consistent.
11.2
What are the differences between forecasted cash flows used in capital budgeting calculations
and past accounting earnings?
Cash flows used in capital budgeting calculations are forward looking; they are incremental
after-tax cash flows based on forecast. Accounting earnings are backward looking; they
represent a record of past performance and may not accurately reflect cash flows.
11.3
Suppose that FRA Corporation already has divisions in both Dallas and Houston. FRA is now
considering setting up a third division in Austin. This expansion will require one senior
manager from Dallas and one from Houston to relocate to Austin. Ignore relocation expenses.
Is their annual compensation relevant to the decision to expand?
The annual compensations of existing senior managers are not incremental to the new
investment and therefore are not relevant for capital budgeting analysis. This is consistent
with our Rule 1 for incremental cash flow calculations: Include cash flows and only cash
flows; do not include allocated costs unless they reflect cash flows.
11.4
MusicHeaven, Inc., is a producer of MP3 players which currently have either 20 gigabytes or
30 gigabytes of storage. Now the company is considering launching a new production line
making mini MP3 players with 5 gigabytes of storage. Analysts forecast that your company
will be able to sell 1 million such mini MP3 players if the investment is taken. In making the
investment decision, discuss what the company should consider other than the sales of the
mini MP3 players.
The company’s launch of the new mini MP3 players may reduce its current sales of MP3
players of bigger storage. This impact has to be considered. This is consistent with our Rule 2
for incremental cash flow calculations: Include the impact of the project on cash flows from
other product lines.
11.5
QualityLiving Trust is a real estate investment company that builds and remodels apartment
buildings in northern California. It is currently considering remodeling a few idle buildings in
its possession into luxury apartment buildings in San Jose. The company bought those
buildings eight months ago. How should the market value of the buildings be treated in
evaluating this project?
Although the buildings are not currently in use, the company can sell them at their market
value rather than remodel them into apartments. Therefore, the market value of the buildings
is the opportunity cost of the project and should be considered as cash outflow in the
investment decision. This is consistent with our Rule 3 for incremental cash flow calculations:
Include all opportunity costs.
11.6
High-End Fashions, Inc., bought a production line of ankle-length skirts last year at a cost of
$500,000. This year, however, miniskirts are hot in the market and ankle-length skirts are
completely out of fashion. High-End has the option to rebuild the production line and use it to
produce miniskirts, with a cost of $300,000 and expected revenue of $700,000. How should
the company treat the cost of $500,000 of the old production line in evaluating the rebuilding
plan?
The cost of the old production line occurred in the past. It cannot be changed whether or not
the company rebuilds it into the miniskirt production line. Therefore, High-End should not
consider the cost of $500,000. This is consistent with our Rule 4 for incremental cash flow
calculations: Forget sunk costs.
11.7
How is the MACRS depreciation method under IRS rules different from that under GAAP
rules? What is the implication of incremental after-tax cash flows from firms’ investments?
GAAP allows the straight-line depreciation method. In contrast, an “accelerated” method of
depreciation, Modified Accelerated Cost Recovery System (MACRS), has been used for U.S.
federal tax calculations. The advantage of MACRS, relative to straight-line depreciation, is
that it enables a firm to deduct depreciation changes sooner, thereby realizing the tax saving
sooner and increasing the present value of the tax savings.
11.8
Explain the difference between marginal and average tax rates, and identify which of these
rates is used in capital budgeting and why.
The marginal tax rate is the rate paid on the next dollar earned. The average tax rate is the
dollar value of total taxes paid divided by total income. The marginal tax rate is the
appropriate rate to use in capital budgeting analysis because this is the tax rate that will be
paid on the incremental income earned by the project.
11.9
When two mutually exclusive projects have different lives, how should we make the capital
budgeting decision? What is the underlying assumption in this method?
When we choose from mutually exclusive projects with different lives, instead of electing the
project with higher NPV or lower net present value of costs, we should choose the project
with higher Equivalent Annual Revenue or lower Equivalent Annual Cost. The underlying
assumption is that we will continue to operate with the same equivalent annual revenue or
equivalent annual cost in the future.
11.10 What is the opportunity cost of using an existing asset? Give an example of the opportunity
cost of using the excess capacity of a machine.
The opportunity cost of using an existing asset in a project is the present value of the change
in the firm’s cash flows that is attributed to the fact that this asset is being used in the project.
For example, by using the excess capacity of a machine, you may accelerate the wear-andtear of the machine and hence will need to replace it sooner. The present value of the added
annualized costs is the opportunity cost of using the excess capacity.
11.11 You are providing financial advice to a shrimp farmer who will be harvesting his last crop of
farm-raised shrimp. His current shrimp crop is very young and will therefore grow and
become more valuable as their weight increases. Describe how you would determine the
appropriate time to harvest the entire crop of shrimp.
Assuming that the price of shrimp is directly (and linearly) related to the weight of the
shrimp, then the optimal point in time to harvest the shrimp would be where the rate of
weight increase is no longer greater than the opportunity cost of capital for the shrimp farmer.
Alternatively, the appropriate time is when the value increase of the shrimp is no longer
greater than the opportunity cost of capital.
VIII.
BASIC
Questions and Problems
11.1 Calculating project cash flows: Why do we use forecasted cash flows instead of forecasted
accounting earnings in estimating the NPV of a project?
Solution:
Accounting earnings can differ from cash flows for a number of reasons, making accounting
earnings an unreliable measure of the costs and benefits of a project. For example, ease of
manipulating earnings components such as accounts receivable and depreciation may result in
distorted estimation of capital budgeting; using forecasted cash flows eliminates such
possibilities. In addition, because there is time value of money, cash flows better reflect the
actual available funds to be distributed to shareholders at each point in time.
11.2 The FCF calculation: How do we calculate incremental free cash flow from the forecasted
earnings of a project? What are the common adjustment items?
Solution:
We need to adjust for the depreciation and amortization tax shield, capital expenditures, and
changes in working capital (including receivables and payables).
11.3 The FCF calculation: How do we adjust for depreciation when we calculate incremental aftertax cash flow from EBITDA? What is the intuition for the adjustment?
Solution:
There are two ways to adjust for depreciation: (1) subtract depreciation from EBITDA,
multiply it by (1 – tax rate), and then add depreciation back; (2) add the tax shield from
depreciation (depreciation multiplied by tax rate) to revenue. These two methods yield the
same results. The intuition is that although depreciation itself is not a cash flow inflow or
outflow, increase in depreciation will result in a decrease in taxable income. This saving on
tax is treated as cash inflow in calculating incremental after-tax cash flows.
11.4
Nominal versus real cash flows: What is the difference between nominal and real dollars?
Which rate of return should we use to discount each type of these cash flows in the future?
Solution:
Nominal dollars are dollars stated as we usually think of them, without any adjustment for
changes in purchasing power over time. Real dollars are dollars stated so that their purchasing
power remains constant. We should use nominal rate of return to discount future nominal
dollars and real rate of return to discount future real dollars. By doing this, we will get
meaningful present values in today’s dollars and purchasing power.
11.5
Taxes and depreciation: What is the difference between the average tax rate and the
marginal tax rate? Which one should we use in calculating the incremental after-tax cash
flows?
Solution:
In a progressive tax system, the marginal tax rate is different from the average tax rate. The
average tax rate is the total amount of tax divided by total amount of money earned, while the
marginal tax rate is the rate paid on the last dollar earned. Since a firm already pays taxes, the
appropriate tax rate used for the firm’s new project is the tax rate that the firm will pay on any
additional profits that are earned because the project is adopted. Therefore, we use the
marginal tax rate in calculating incremental after-tax cash flows.
11.6
Computing terminal-year FCF: Five years ago, a pharmaceutical company bought a
machine that produces pain-reliever medicine at a cost of $2 million. The machine has been
depreciated over the past five years, and the current book value is $800,000. The company
decides to sell the machine now at its market price of $1 million. The marginal tax rate is 30
percent. What are the relevant cash flows? How do they change if the market price of the
machine is $600,000 instead?
Solution:
The relevant cash flows include the sale price of the machine, as well as the tax on the capital
gain:
1,000,000 – 0.3(1,000,000 – 800,000) = $940,000
When the market price of the machine is changed to $600,000, the relevant cash flows
include the sale price and tax saving on capital loss:
6,000,000 + 0.3(800,000 – 600,000) = $6,060,000
11.7
Cash flows from operations: What are variable costs and fixed costs? What are some
examples of each? How are these costs estimated in forecasting operating expenses?
Solution:
Variable costs vary directly with unit sales. Fixed costs do not vary with unit sales. For an
example of each, see the video game player scenario on page 379. Variable costs are those
associated with purchasing the components for the player, the labor required, and sales and
marketing. These costs will vary according to the number of units produced. Fixed costs are those
associated with assembly space, and administrative expenses.
11.8
Investment cash flows: Six Twelve is considering opening up a new convenience store in
downtown New York City. The expected annual revenue is $800,000. To estimate the increase
in working capital, analysts estimate the ratio of cash and cash equivalents to revenue to be
0.03, and the ratio of receivables to revenue to be 0.05 in the same industry. What are the
incremental cash flows related to working capital when the store is opened?
Solution:
Cash flow related to working capital in year0 = $(800,000)*(0.03 + 0.05) = $(64,000)
11.9
Investment cash flows: Keswick Supply Company wants to set up a division providing copy
and fax business. Customers will be given 20 days to pay for such services. The annual
revenue of the division is estimated to be $25,000. What is the incremental cash flow
associated with accounts receivable?
Solution:
The customers are expected to take 20 days to pay, and the average accounts receivable
balance will be (20days/365days/year)*100%*25,000 = 5.48%*25,000 = $1,370.
Alternatively, the average daily credit sale is $25,000 / 365 = $68.49, and it takes 20 days, on
average, to collect the sale. Therefore, 20 x $68.49 = $1,369.86, or about $1,370.
11.10 Expected cash flows: Define expected cash flows and explain why this concept is important
in evaluating projects.
Solution:
Expected cash flows are probability-weighted averages of the future cash flows generated by
a project under alternative scenarios. In the real business world there are a lot of uncertainties.
Future cash flows may vary across different states of the world. It is not possible to estimate a
unique number of cash flow for all states. We can estimate the expected cash flows across
different states and use that as an estimation of future cash flows. The cash flows that are
discounted in an NPV analysis are the expected incremental cash flows the project will
produce.
11.11 Projects with different lives: Explain the concept of equivalent annual cost and how it is
used to compare projects with different lives.
Solution:
The equivalent annual cost (EAC) is the annualized cost of an investment stated in nominal
dollars. In other words, it is the annual payment from an annuity with a life equal to that of a
project that has the same NPV as the project. Since it is a measure of the annual cost or cash
inflow from a project, the EAC for one project can be compared directly with the EAC from
another project, regardless of the lives of those two projects.
11.12 Replace an existing asset: Explain how we decide the optimal time to replace an existing
asset with a new one.
Solution:
The optimal time to replace an existing asset with a new one is if the benefits of replacing the
machine exceed the costs.
INTERMEDIATE
11.13 Nominal versus real cash flows: You are buying a sofa. You will pay $200 today and make
three consecutive annual payments of $300 in the future. The real rate of return is 10 percent,
and the expected inflation rate is 4 percent. What is the actual price of the sofa?
Solution:
We can calculate it in two different ways:
(1)
Use nominal dollars and nominal rate of return:
Nominal rate of return = (1 + 10%)*(1 + 4%)-1 = 14.4%
Price = 200 + 300 / (1 + 14.4%) + 300/(1 + 14.4%)2 + 300 / (1 + 14.4%)3 = 891.84
(2)
Use real dollars and a real rate of return:
Real annual payments are: 300 / (1 + 4%) = 288.46, 300 / (1 + 4%)2 = 277.37, and
300 / (1 + 4%)3 = 266.70
Price = 200 + 288.46/(1 + 10%) + 277.37/(1 + 10.4%)2+266.7 / (1+10.4%)3 =891.84
Note that we get identical results as long as we are consistent in using nominal or real cash
flows and corresponding discount rates.
11.14 Nominal versus real cash flows: You are graduating in two years. You want to invest your
current savings of $5,000 in bonds for the down payment on a new car when you graduate and
start to work. You can invest the money in either Bond A, a two-year bond with a 3 percent
annual interest rate, or Bond B, an inflation-indexed two-year bond paying 1 percent real
interest above the inflation rate (assume this bond makes annual interest payments). The
inflation rate over the next two years is expected to be 1.5 percent. Assume that both bonds
are default free and have the same market price. Which one should you invest in?
Solution:
The nominal interest rate is 3 percent for bond A, and (1 + 1%)*(1 +1.5%) –1 = 2.52% for the
inflation-indexed bond B. You should invest in bond A.
11.15 Marginal and average tax rates: Given the U.S. Corporate Tax Rate Schedule in Exhibit
11.5, what is the marginal tax rate and average tax rate of a corporation that generates a
taxable income of $12 million in 2007?
Solution:
The marginal tax rate is 35 percent.
The total tax payable is 3,400,000 + (12,000,000-10,000,000)*35% = $4,100,000 Therefore
the average tax rate = 4,100,000 / 12,000,000 = 34.2%
11.16 Investment cash flows: Healthy Potions, Inc., is considering investing in a new production
line of eye drops. Other than investing in the equipment, the company needs to increase its cash
and cash equivalents by $10,000, increase the level of inventory by $30,000, increase accounts
receivable by $25,000, and increase accounts payable by $5,000 at the beginning of the
investment. Healthy Potions will recover these changes in working capital at the end of the
project 10 years later. Assume the appropriate discount rate to be 12 percent. What are the
relevant cash flows given the above information?
Solution:
The relevant cash flow related to working capital at the beginning of the project is:
$(10,000)-$30,000-$25,000 + $5,000 = $(60,000)
The present value of relevant cash flow related to working capital at the end of the project is:
60,000 / (1 + 12%)10 = $19,318.39
11.17 Cash flows from operations: Given the soaring price of gasoline, Ford is considering
introducing a new production line of gas-electric hybrid sedans. The expected annual sales
number of such hybrid cars is 30,000; the price is $22,000 per car. Variable costs of
production amount to $10,000 per car. The fixed overhead including salary of top executives
is $80 million per year. However, the introduction of the hybrid sedan will decrease Ford’s
sales of regular sedans by 10,000 cars per year; the regular sedans have a unit price of $20,000
and unit variable cost of $12,000, and fixed costs of $250,000 per year. Depreciation costs of
the production plant are $50,000 per year. The marginal tax rate is 40 percent. What is the
incremental annual cash flow from operations?
Solution:
Step One: Revenue: $22,000 x 30,000 machines =$660,000,000
Step Two: Op Exp: $10,000 x 30,000 machines = $300,000,000, plus lost net revenue from
regular sedans = ($20,000 – $12,000) x 10,000 = $80,000,000; total Op Exp =
$380,000,000
Step Three: D&A: $50,000
Step Four: Plug information into the text book template as below.
=
=
x
=
+
=
=
ΔNR
ΔOpEx
ΔEBITDA
ΔD&A
ΔEBIT
(1-t)
ΔNOPAT
ΔD&A
ΔCFO
ΔCapEx
ΔAWC
ΔFCF
660,000,000
-380,000,000
280,000,000
-50,000
279,950,000
0.60
167,970,000
50,000
168,020,000
0
0
168,020,000
Alternatively, the incremental annual cash flow from operations is:
((22,000-10,000)*30,000-(20,000-12,000)*10,000)*(1-0.4) + 50,000*0.4 = 168,020,000
Note that the fixed costs are not included in the incremental cash flows calculations, since
they exist regardless of the hybrid sedan investment.
11.18 FCF and NPV for a project: Archers Daniels Midland Company is considering buying a
new farm that it plans to operate for 10 years. The farm will require an initial investment of
$12 million. The investment will consist of $2 million for land and $10 million for trucks and
other equipments. The land, all trucks, and all other equipment are expected to be sold at the
end of 10 years at a price of $5 million, $2 million above book value. The farm is expected to
produce revenue of $2 million each year, and annual cash flow from operates equals $1.8
million. The marginal tax rate is 35 percent, and the appropriate discount rate is 10 percent.
Calculate the NPV of this investment.
Solution:
Cash flow of investment in year 0 is: $(12,000,000)
PV of annual investment year 1 to 10 is found using the present value factor for an annuity:
Payment * ( 1 – {1 / (1 + i)n}) / i)
$(1,000,000)( 1 – {1 / (1.1)10}) / 0.1) = $(6,144,567.11)
PV of sales and tax on capital gain in year 10 is:
(5,000,000-[2,000,000*0.35]) / (1.1)10 = 1,657,836.15
Therefore, PV of investment cash flows = $(16,486,730.97)
The company should not buy the farm.
11.19 Projects with different lives: You are starting a family pizza restaurant and need to buy a
motorcycle for delivery orders. You have two models in mind. Model A costs $9,000 and is
expected to run for 6 years; model B is more expensive, with a price of $14,000 and an
expected life of 10 years. The annual maintenance costs are $800 for model A and $700 for
model B. Assume that the opportunity cost of capital is 10 percent. Which one should you
buy?
Solution:
You need first calculate the NPV of costs for each of the motorcycles:
NPVA = 9,000 + 800*(((1.1)6-1) / ((1.1)6*0.1)) = 12,484.21
NPVB = 14,000 + 700*(((1.1)10-1) / ((1.1)10*0.1)) = 18,301.20
Then you need to calculate the EAC of each model:
EACA = 12,484.21*10%*((1.1)6) / ((1.1)6-1) = 2,866.47
EACB = 18,301.20*10%*((1.1)10) / ((1.1)10-1) = 2,978.44
Therefore you should buy model A.
11.20 When to harvest an asset: Predator LLC, a leveraged buyout specialist, recently bought a
company and want to decide the optimal time to sell it. The partner in charge of this
investment has estimated the after-tax cash flows at different times as follows: $700,000 if
sold one year later; $1,000,000 if sold two years later; $1,200,000 if sold three years later; and
$1,300,000 if sold four years later. The opportunity cost of capital is 12 percent. When should
Predator sell the company? Why?
Solution:
The NPV of each choice is:
NPV1 = 700,000 / (1.12)1 = 625,000
NPV2 = 1,000,000 / (1.12)2 = 797,194
NPV3 = 1,200,000 / (1.12)3 = 854,136
NPV4 = 1,300,000 / (1.12)4 =826,174
Therefore you should sell the company three years later.
11.21 Replace an existing asset: Bell Mountain Vineyards is considering updating its current
accounting system with a high-end electronic system. While the new accounting system
would save the company money, the cost of the system continues to decline. Bell Mountain’s
opportunity cost of capital is 10 percent, and the costs and values of future savings choices
are as follows:
Year
Cost
Value of Future Savings (at time of purchase)
0
$5,000
$7,000
1
4,500
7,000
2
4,000
7,000
3
3,600
7,000
4
3,300
7,000
5
3,100
7,000
When should Bell Mountain buy the new accounting system?
Solution:
The NPV of each choice is:
NPV0 = 2,000
NPV1 = 2,500 / (1.1)1 = 2,275
NPV2 = 3,000 / (1.1)2 = 2,479
NPV3 = 3,400 /(1.1)3 =2,554
NPV4 = 3,700 / (1.1)4 = 2,527
NPV5 = 3,900 / (1.1)5 = 2,422
Therefore the company should buy it in year 3.
11.22 Replace an existing asset: You have a 1993 Nissan that is expected to run for another three
years, but you are considering buying a new Hyundai in a few years. You will donate the old
Nissan to Goodwill when you buy the new car. The annual maintenance cost is $1,500 per
year for the old Nissan and $200 for the new Hyundai. The price of your favorite Hyundai
model is $18,000; it is expected to run for 15 years. Your opportunity cost of capital is 3
percent. Ignore taxes. When should you buy the new Hyundai?
Solution:
NPV of cost of the new car is:
NPVHyundai= 18,000 + 200*(((1.03)15-1) / ((1.03)15*0.03)) = 20,387.59
EAC of the new car is:
EACHyundai = 20,387.59*0.03*((1.03)15) / ((1.03)15-1) = 1,707.8 > 1,500
Therefore, you should drive the 1993 Nissan for three more years and then buy a new
Hyundai.
11.23 Replace an existing asset: Assume that you are considering replacing your old Nissan with a
new Hyundai, as in the previous problem. However, the annual maintenance cost of the old
Nissan increases as time goes by. It is $1,200 in the first year, $1,500 in the second year, and
$1,800 in the third year. When should you replace it with the new Hyundai in this case?
Solution:
The EAC of the Hyundai remains at $1,707.8, as calculated in the previous problem.
Compare this amount with the annual maintenance costs of the Nissan and you will see that in
year 2 it is cheaper to drive the Nissan, but in year 3 it is cheaper to drive the Hyundai.
Therefore, the optimal time to replace the old car is at the end of year 2.
11.24 When to harvest an existing asset: Anaconda Manufacturing Company currently own a mine
that is known to contain a known amount of gold. Since Anaconda does not have any goldmining expertise, the company plans to sell the entire mine and base the selling price on a
fixed multiple of the spot price for gold at the time of the sale. Analysts at Anaconda have
forecast the price spot for gold and have determined that the price will increase by 14 percent,
12 percent, 9 percent, and 6 percent during the next one, two, three, and four years,
respectively. If Anaconda’s opportunity cost of capital is 10 percent, what is the optimal time
for Anaconda to sell the mine?
Solution:
The rate of gold price appreciation is greater than the opportunity cost of capital for the next
two years and then it drops below the opportunity cost of capital. Therefore, Anaconda should
sell the gold at the beginning of the third year (or at the end of the second year).
11.25 Replace an existing asset: You are thinking about delivering pizzas in your spare time. Since
you must use your own car to deliver the pizzas, you will wear out your current car one year
earlier, which is one year from today, than if you did not take on the delivery job. You
estimate that when you purchase a new car, regardless of when that occurs, you will pay
$20,000 for the car and it will last you five years. If your opportunity cost of capital is 7
percent, what is the opportunity cost of using your car to deliver pizzas?
Solution:
 (1 + 0.07)5 
EACNew CAr = .07 *$20,000 * 
 = $4,877.81
5
(1 + 0.07) − 1
Therefore, the opportunity cost of wearing out your car a year earlier is
NPVU sin g your car
ADVANCED
= −$4,877.81/(1.07)1 = $4,558,70
11.26 You are the CFO of SlimBody, Inc., a retailer of the exercise machine Slimbody6® and related
accessories. Your firm is considering opening up a new store in Los Angeles. The store will
have a life of 20 years. It will generate annual sales of 5,000 exercise machines, and the price
of each machine is $2,500. The annual sales of accessories will be $600,000, and the
operating expenses of running the store, including labor and rent, will amount to 50 percent of
the revenues concerning the exercise machines. The initial investment in the store will equal
$30 million and will be fully depreciated on a straight-line basis over the 20-year life of the
store. Your firm will need to invest $2 million in additional working capital immediately, and
recover it at the end of the investment. Your firm’s marginal tax rate is 30 percent. The
opportunity cost of opening up the store is 10 percent. What are the incremental cash flows
from this project at the beginning of the project as well as in years 1-19 and 20? Should you
approve it?
Solution:
Step One: Initial outlay = $30,000,000 + $2,000,000 (WC requirement) = $32,000,000
Step Two: ΔNR for years 1- 20: $2,500 x 5,000 machines = $12,500,000 plus $600,000 =
$13,100,000
Step Three: ΔOpExp for years 1- 20: $1,250 x 5,000 machines = $6,250,000
Step Four: ΔD&A for years 1- 20: $30,000,000 / 20 years = $1,500,000 / year
Step Five: Plug information into the text book template as below.
Step Six: Yr 20 recapture of WC requirements that were funded in year 0.
Yrs 1-19
ΔNR
ΔOpEx
ΔEBITDA
ΔD&A
ΔEBIT
(1-t)
ΔNOPAT
ΔD&A
ΔCFO
ΔCapEx
ΔAWC
ΔFCF
-30,000,000
-2,000,000
-32,000,000
13,100,000
-6250000
6,850,000
-1500000
5,350,000
0.7
3,745,000
1,500,000
5,245,000
0
0
5,245,000
Yr 20
13,100,000
-6250000
6,850,000
-1500000
5,350,000
0.7
3,745,000
1,500,000
5,245,000
0
2,000,000
7,245,000
Therefore the NPV of the project is:
NPV = -32,000,000+5,245,000*(((1.1)20-1)/ ((1.1)20*0.1))+2,000,000/(1.1)20
=14,483,370
You should approve the project since it has a positive NPV.
Alternative Solution:
Incremental cash flows in year 0 is:
FCF0 = -30,000,000-2,000,000= -32,000,000
Annual incremental cash flows through the life of the investment are:
FCFt = (2,500*2,500+600,000)*(1-0.3)+0.3*1,500,000 = 5 ,245,000
Additional incremental cash flows at the end of the project are:
2,000,000
Therefore the NPV of the project is:
NPV = -32,000,000+5,245,000*(((1.1)20-1)/ ((1.1)20*0.1))+2,000,000/(1.1)20
=14,483,370
You should approve the project since it has a positive NPV.
11.27 Rocky Mountain Lumber, Inc., is considering purchasing a new wood saw that costs $50,000.
The saw will generate revenues of $100,000 per year for five years. The cost of materials and
labor needed to generate these revenues will total $60,000 per year, and other cash expenses
will be $10,000 per year. The machine is expected to sell for $1,000 at the end of its five-year
life and will be depreciated on a straight-line basis over five years to zero. Rocky Mountain’s
tax rate is 34 percent, and its opportunity cost of capital is 10 percent. Should the company
purchase the saw? Explain why or why not.
Solution:
Step One: Initial outlay = $50,000
Step Two: ΔNR for years 1- 5: $100,000
Step Three: ΔOpExp for years 1- 5: $60,000 + $10,000 = $70,000
Step Four: ΔD&A for years 1- 5: $50,000 / 5 years = $10,000 / year
Step Five: Plug into the text book template as below.
Step Six: Yr 5: Capital recovery = $1,000 – (.34 x 1,000 gain on sale) = $660.
Yrs 1-4
ΔNR
ΔOpEx
ΔEBITDA
ΔD&A
ΔEBIT
(1-t)
ΔNOPAT
ΔD&A
ΔCFO
ΔCapEx
ΔAWC
ΔFCF
-50,000
0
-50,000
Yr 5
100,000
-70000
30,000
-10000
20,000
0.66
13,200
10,000
23,200
0
0
23,200
100,000
-70000
30,000
-10000
20,000
0.66
13,200
10,000
23,200
660
0
23,860
Therefore, NPV of investment is:
-50,000+23,200/(1.1)1+23,200/(1.1)2+23,200/(1.1)3+23,200/(1.1)4
+(23,200+660)/(1.1)5=$38,356
Therefore the company should buy the machine.
Alternatively:
The annual operating cash flows from year 1 to 5 are:
(100,000-60,000-10,000)*(1-0.34)+0.34*10,000=23,200
The after-tax terminal value in year 5 is:
1,000 -(.34)(1,000-0) = 660
Therefore, NPV of investment is:
-50,000+23,200/(1.1)1+23,200/(1.1)2+23,200/(1.1)3+23,200/(1.1)4
+(23,200+660)/(1.1)5=$38,356
Therefore the company should buy the machine.
11.28 A beauty product company is developing a new fragrance named Happy Forever. There is a
probability of 0.5 that consumers will love Happy Forever, and in this case, annual sales will
be 1 million bottles; a probability of 0.4 that consumers will find the smell acceptable and
annual sales will be 200,000 bottles; and a probability of 0.1 that consumers will find the
smell weird and annual sales will be only 50,000 bottles. The selling price is $38, and the
variable cost is $8 per bottle. There is a fixed production cost of $1 million per year, and
depreciation costs are $1.2 million. Assume that the marginal tax rate is 40 percent. What are
the expected annual incremental cash flows from the new fragrance?
Solution:
Step One: Expected sales units: (.5)1,000,000 + (.4)200,000 + (.1)50,000 = 585,000 units
Step Two: ΔNR: 585,000 units x $38 = $22,230,000
Step Three: ΔOpExp: 585,000 units x $8 + $1,000,000 = $5,680,000
Step Four: ΔD&A: $1,200,000
Step Five: Plug into the text book template as below.
=
=
x
=
+
=
=
ΔNR
ΔOpEx
ΔEBITDA
ΔD&A
ΔEBIT
(1-t)
ΔNOPAT
ΔD&A
ΔCFO
ΔCapEx
ΔAWC
ΔFCF
22,230,000
-5,680,000
16,550,000
-1,200,000
15,350,000
0.60
9,210,000
1,200,000
10,410,000
0
0
10,410,000
Alternatively, the expected annual incremental cash flows are:
(((0.5*1,000,000+0.4*200,000+0.1*50,000)*(38-8))-1,000,000)*(1-0.4)+1,200,000*0.4 =
10,410,000
11.29 Great Fit, Inc., is a company that makes clothing. The company has a product line that
produces women’s tops of regular sizes. The same machine could be used to produce petite
sizes as well. However, the life of the machines will be reduced from four more years to two
more years if the petite size production is added. The cost of identical machines with a life of
eight years is $2 million. Assume the opportunity cost of capital is 8 percent. What is the
opportunity cost of adding petite sizes?
Solution:
The opportunity cost is the incremental costs of the machine in year 3 and year 4 if petite
sizes are in production. The EAC of the machine is:
EAC=2,000,000*0.08*((1.08)8)/((1.08)8-1)=348,029.52
The present value of such cost in year 3 and year 4 is:
NPV=348,029.52/(1.08)3+348,029.52/(1.08)4=532,089.14
11.30 Biotech Partners LLC has been farming a new strain of radioactive-material-eating bacteria
that the electrical utility industry can use to help dispose of its nuclear waste. Two opposing
effects are affecting Biotech’s decision of when to harvest the bacteria. The bacteria are
currently growing at a 22 percent annual rate, but due to known competition from other top
firms, Biotech analysts estimate that the price for the bacteria will fluctuate according to the
scale below. If the opportunity cost of capital is 10 percent, then when should Biotech harvest
the entire bacteria colony at one time?
Year
Change in Price Due to Competition (5)
1
5%
2
-2
3
-8
4
-10
5
-15
6
-25
Solution:
Change in revenue:
Yr 1
(1.05)(1.22) = 1.2810 or 28.1%
Yr 2
(0.98)(1.22) = 1.1956 or 19.56%
Yr 3
(0.92)(1.22) = 1.1224 or 12.24%
Yr 4
(0.9)(1.22) = 1.0980 or 9.80%
Yr 5
(0.85)(1.22) = 1.037 or 3.70%
Yr 6
(0.75)(1.22) =- 0.9150 or -8.50%
Since the change in revenue is higher for the first two years, Biotech should sell its bacteria
colony at the beginning of the third year or at the end of the second year.
11.31 ACME manufacturing is considering replacing an existing production line with a new line
that has a greater output capacity and operates with less labor than the existing line. The new
line would cost $1 million, have a five-year life, and would be depreciated using MACRS
over three years. At the end of five years, the new line could be sold as scrap for $200,000 (in
year 5 dollars). Because the new line is more automated, it would require fewer operators,
resulting in a saving of $40,000 per year before tax and unadjusted for inflation (in today’s
dollars). Additional sales with the new machine are expected to result in additional net cash
inflows, before tax, of $60,000 per year (in today’s dollars). If ACME invests in the new line,
a one-time investment of $10,000 in additional working capital will be required. The tax rate
is 35 percent, the opportunity cost of capital is 10 percent, and the annual rate of inflation is 3
percent. What is the NPV of the new production line?
Solution:
(Revenue Op Exp)(1-t)
plus Tax x Deprec
minus Cap Exp
$1,000,000
plus tax on Salvage
minus Add WC
$66,950
$68,959
$71,027
$73,158
$116,655
$155,575
$51,835
$25,935
$75,353
$(200,000)
$(70,000)
$10,000
$(10,000)
Net Cash Flows
$(1,010,000)
$183,605
$224,534
$122,862
$99,093
$215,353
PV of Net Cash Flows
$(1,010,000)
$166,914
$185,565
$92,308
$67,682
$133,717
Net Present Value
$(363,814)
11.32 The alternative to investing in the new production line in Problem 11.31 is to overhaul the
existing line, which currently has both a book value and a salvage value of $0. It would cost
$300,000 to overhaul the existing line, but this expenditure would extend its useful life to five
years. The line would have a $0 salvage value at the end of five years. The overhaul outlay
would be capitalized and depreciated using MACRS over three years. Should ACME replace
or renovate the existing line?
(Revenue Op Ex)(1-t)
plus Tax x Deprec
minus Cap Exp
$34,997
$46,673
$15,551
$7,781
$300,000
plus tax on Salvage
minus Add WC
Net Cash Flows
$(300,000)
$34,997
$46,673
$15,551
$7,781
$0
PV of Net Cash Flows
$(300,000)
$31,815
$38,572
$11,683
$5,314
$0
Net Present Value
$(212,615)
NPVnew - NVPold
$(151,199)
Renovating the old line is less costly.
CFA Problems
11.33. FITCO is considering the purchase of new equipment. The equipment costs $350,000, and an
additional $110,000 is needed to install it. The equipment will be depreciated straight-line to
zero over a five-year life. The equipment will generate additional annual revenues of
$265,000, and it will have annual cash operating expenses of $83,000. The equipment will be
sold for $85,000 after five years. An inventory investment of $73,000 is required during the
life of the investment. FITCO is in the 40 percent tax bracket and its cost of capital is 10
percent. What is the project NPV?
A.
$47,818
B.
$63,658
C.
$80,189
D.
$97,449
Solution:
D is correct.
5
NPV = −533, 000 + 
146, 000 124, 000
Outlay = FCInv + NWCInv – Sal0 + T(Sal0 –
B0)
Outlay = (350,000 + 110,000) + 73,000 – 0 + 0
= $533,000
The installed cost is $350,000 +
$110,000 = $460,000, so the
annual depreciation is $460,000/5
= $92,000. The annual after-tax
operating cash flow for Years 1–
5 is
CF = (S – C – D)(1 – T) + D = (265,000 –
83,000 – 92,000)(1 – 0.40) + 92,000 CF =
$146,000
The terminal year after-tax nonoperating cash flow in Year 5 is TNOCF =
Sal5 + NWCInv – T(Sal5 – B5) = 85,000
+ 73,000 – 0.40(85,000 – 0)
TNOCF = $124,000
The NPV is
t =1
11.34. After estimating a project’s NPV, the analyst is advised that the fixed capital outlay will be
revised upward by $100,000. The fixed capital outlay is depreciated straight-line over an
eight-year life. The tax rate is 40 percent and the required rate of return is 10 percent. No
changes in cash operating revenues, cash operating expenses, or salvage value are expected.
What is the effect on the project NPV?
A.
$100,000 decrease
B.
$73,325 decrease
C.
$59,988 decrease
D.
No change
Solution:
B is correct. The additional annual depreciation is $100,000/8 = $12,500. The depreciation
tax savings is 0.40 ($12,500) = $5,000. The change in project NPV is
−100, 000 + 
8
t=1
5, 000
= −100, 000+ 26, 675= − $73, 325
t
(1.10)
11.35. When assembling the cash flows to calculate an NPV or IRR, the project’s after-tax interest
expenses should be subtracted from the cash flows for
A.
the NPV calculation, but not the IRR calculation.
B.
the IRR calculation, but not the NPV calculation.
C.
both the NPV calculation and the IRR calculation.
D.
neither the NPV calculation nor the IRR calculation.
Solution:
D is correct. Financing costs are not subtracted from the cash flows for either the NPV or the
IRR. The effects of financing costs are captured in the discount rate used.
Sample Test Problems
11.1
You purchased 100 shares of stocks of an oil company, TexasEnergy, Inc., at the price of
$50/share. The company has 1 million shares outstanding. Ten days later TexasEnergy
announced an investment in an oil field in east Texas. The probability of the investment being
successful and generating NPV of $10 million is 0.2; the probability of the investment will be
a failure and generate a negative NPV of negative $1 million is 0.8. How would you expect
the stock price to change upon the company’s announcement of the investment?
Solution:
The expected change in the stock price should be equal to the expected NPV of the project
divided by the number of shares outstanding. The expected NPV of the project is
0.2*10,000,000 + 0.8*(–1,000,000), such that:
Change in stock price = (0.2*10,000,000 + 0.8*(-1,000,000)) / 1,000,000 shares =
$1.2/share
Stock price of TexasEnergy, Inc., will increase by $1.20 upon the announcement of the
investment.
11.2
A chemical company is considering buying a magic fan for its plant. The magic fan is
expected to work forever and to help cool the machines in the plant and hence reduce their
maintenance costs by $4,000 per year. The cost of the fan is $30,000. The appropriate
discount rate is 10 percent, and the marginal tax rate is 40 percent. Should the company buy
the magic fan?
Solution:
The after-tax saving of maintenance costs is: $4,000*(1 – 40%) / 10% = $24,000, which is
less than the cost. Therefore the company should not buy the fan. If one fails to take into
consideration the tax effect on maintenance costs, the opposite conclusion will be made.
Therefore it is important to remember our Rule 5 for incremental cash flow calculations:
Include only after-tax cash flows.
11.3
Hogvertz Elvin Catering (HEC) is considering switching to a new Wonder Food Maker. Both
food makers will remain useful for the next ten years, but the new option will generate a
depreciation expense of $5,000 per year while the old food maker will generate a depreciation
expense of $4,000 per year. What is the after-tax cash flow effect from deprecation of
switching to the new food maker for HEC if the firm’s marginal tax rate is 40 percent and the
correct discount rate is 12 percent?
Solution:
Without the benefit of other information required in the cash flow calculation table, we must
isolate the cash flow effect of depreciation for a firm. We therefore find that we have a
deduction of D&A above the tax calculation line and an addition of D&A below the tax
calculation line. This means that the net yearly after-tax effect of depreciation and
amortization can be simplified to:
-(D&A)(1 – t) + (D&A) = – (D&A) + (D&A) t + (D&A) = (D&A) t
Using that result, we find that the net yearly effect of depreciation and amortization of cash
flow is:
(D&A) t = ( $5,000 – $4,000) x .4 = $400
and so the present value of the total after-tax cash flow effect from depreciation can be found
as follows:
$400 x (((1.12)10-1) / ((1.12)10*0.12)) = $400 x 5.650223 = 2,260.09
11.4
The Long-Term Financing Company has identified a new alternative project that is similar in
all respect to its current project except one. That is, the new project will reduce the need for
working capital by $10,000 during the 30-year life of the project. The firm’s cost of capital is
18 percent, and the marginal corporate tax rate for the firm is 34 percent. What is the after-tax
present value of this new alternative project?
Solution:
Because working capital has no effect on the income statement of the firm, there are no tax
effects from the two cash flows associated with the working capital change. Therefore, the
after-tax present value of the alternative is:
$10,000 – $10,000 x (1.18)-30 = $10,000 – ($10,000) x 0.006975 = $9,930.25
11.5
Choice Masters must choose between two projects of unequal lives. Project 1 has a NPV of
$50,000 and will be viable for five years. The discount rate for project 1 and project 2 is 10
percent. Project 2 will be viable for seven years. In order for Choice Master to be indifferent
between the two projects, what must the NPV of project 2 be?
Solution:
The EAC for project 1 is:
 (1 + .0.1) 5 
0.1*$50,000* 
 =$13,189.87
5
(
1
+
0.1
)
−1


which means that project 2 must also generate cash flow of $13,189.87 per year for seven
years. Therefore, the NPV of project 2 must be

1
 1 
$13,189.87*
−
=$64,213.83
 0.1 1 (1 + 0.1) 7 

