# Corporate Finance 370 Answer Key - PowerPoint

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```					Corporate Finance
180.366
Spring 2011
Prof. Duffee
VI. Investment decision rules

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Subjects in this chapter

• Introduce concept “opportunity cost of capital”

• Review some alternative rules for deciding whether to
take on a project

• Modify NPV rule for projects other than stand-alone
projects

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PVs and discount rates

• PV of a project’s risk-free cash flow is price of a risk-
free bond with same future payoff (no-arbitrage)
– How much do you have to spend today on Treasury
securities to replicate a fixed payoff of \$7MM in six months?

• Equivalent: discount cash flow by rate inferred from
bond
– \$7MM/(1.0016)1/2 = \$6.994MM
– Identical calculation, because source of 1.0016 is square root
of (\$1/price of Treasury bond paying \$1 in six months)

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Discount rates for uncertain cash flows

• PV price-based rule is same for uncertain cash
flows…
– What is price of instrument today that pays off in one year
expected \$15MM, standard deviation \$4MM, perfectly

• … but harder to apply in practice
– Cannot easily find expected prices of financial instruments

• Easier to use discount rates directly, calculated from a
mathematical model

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Opportunity cost of capital

• Model we use is outlined after first exam (and is focus
of Investments course)

• Intuition for now: discount rate = opportunity cost of
capital
– Highest expected return available in financial marketplace for
investment with economically equivalent cash flow

– Shorthand: “cost of capital,” “opportunity cost”

• Logic – could commit resources (capital) to project, or
to an alternative investment

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Review: Stand-alone projects

• Definition: taking on project does not preclude taking on others

• NPV rule: Take on project if NPV>0
– Use expected cash flows
– Use discount rate appropriate for cash-flow characteristics (risk)
• Later we will infer discount rates from the Capital Asset Pricing Model

• Logic: Taking on project equivalent to putting NPV in pocket
– Can sell off all cash flows today at their PVs; NPV is left over

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Example 1

• (Done in previous class)
• Annual expected cash flows are C0=-100, C1=50, C2=55,
C3=10, all others zero
• Assume discount rate for all expected cash flows is 5%

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Alternative measures of value

• Real-life firms commonly use other methods to evaluate projects
– Some alternatives have benefit of simple computations
– Other alternatives have intuitive appeal of return-on-revenue
(ROR)

• When results of alternative methods differ from NPV:
– This may mean that project go/no-go result is different, or that
– Priority ranking of projects is different.

• Any method producing results that differ from NPV is WRONG
(does not maximize wealth). Use such methods with great care

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Internal rate of return

•   IRR is the most popular alternative method, since it is intuitive and
usually gives the right result.
•   A project’s IRR is defined as the interest rate that sets the NPV of
expected cash flows equal to zero. The r that solves

C1        C2       C3
0  C0                             ...
1  r  1  r  1  r 
2        3

•   IRR rule: Go ahead with project if IRR > opportunity cost of capital for
the project

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Example 1 again

• Calculate IRR (cut and paste from Excel)

0             1             2    3
-100            50             55   10

Internal rate of return (IRR) function
8.923%(=irr(a2:d2))

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Example 1 again

• IRR rule: Take on project if opportunity cost of capital < 8.92%

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NPV vs. IRR

Calculation                      Investment
Decision
Taking discount rate as              Accept project if
NPV      given (r), calculate NPV             NPV > hurdle (zero)
Taking NPV as given (zero), Accept project if
IRR      calculate IRR.              IRR > hurdle (r)

   Most of the time, these rules are equivalent
   Always when negative expected cash flows occur prior to any
positive expected cash flows
   But …
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IRR pitfall 1: lending or borrowing

• Change sign of all expected cash flows in example 1.
– What is new NPV?
– What is new IRR?

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Example 2

• Example 2. Expected cash flows C0 = 100, C1 = 360, C2 = 431,
C3 = 171.6
• For 5% discount rate, what is NPV?
– NVP=0.162
• For 25% discount rate, what is NPV?
– NPV = 0.0192
• Calculate IRR

-100     360         -431       171.6
10.000%(=irr(a8:d8,0%))
30.000%(=irr(a8:d8,40%))

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IRR pitfall 2: multiple IRR values

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IRR pitfall 3: time-varying discount rates

• Example 3. Expected cash flows of Example 1 (-100, 50, 55, 10),
but discount rates for expected cash flows in years 1-3 are 8%,
8.5%,11% respectively. What is NPV?

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IRR pitfall 3: time-varying discount rates

• Example 3. Expected cash flows of Example 1, but discount rates
for expected cash flows in years 1-3 are 8%, 8.5%,11% respectively.
What is NPV?
– 0.3283
• What is IRR?
– Same as in Example 1, 8.92
• What does the IRR decision rule say?
– Go ahead with project if IRR > weighted average cost of capital, but
weights are complex to compute (depend on cash flows)
• Easier to just compute NPV

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IRR pitfall 4: no IRR

• Example 4. Expected
cashflows are 4, -8, 15.
• Try to calculate IRR

4        -8      15
#NUM! (=irr(a14:c14))

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Another alternative: payback period

• Some firms require investment “payback period” (e.g., adopt all
projects that pay back within two years)
• Example 5

Date Project A Project B Project C
0     -2000     -2000     -2000
1       500        500     1800
2       500       1800      500
3      5000          0         0
Payback 3 years 2 years 2 years
NPV (10%) \$2,624          (\$58)     \$50

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Mutually exclusive projects

• NPV rule: Take on highest NPV project

• Note that IRR rule is useless here; want to rank by generated
wealth, not highest % return on investment

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Resource constraints

• Definition: projects use a resource (say, managerial oversight) that
cannot be adjusted by the firm before the resource is needed

• Implication: firms must consider best use of resource

• Rule: choose set of projects that maximize sum of NPV subject to
resource constraint

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Example 6

• Project A requires all of a home builder’s available land; Projects
B and C each require half of the available land

Date        Cash Flows \$ Millions
Project A Project B Project C
0           -10         -5         -5
1            30          5          5
2             5        20          15
NPV (10%)          \$21       \$16         \$12
• Aside from very simple cases, programming techniques are
required to solve for max(NPV) subject to constraints

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VII. Fundamentals of capital
budgeting

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Project (e)valuation – the cash flow side

• We take a closer look at project analysis, taking the
choice of discount rate(s) as given
• Remaining question: What are the cash flows to
discount?
• The standard accounting framework is used to

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Step 1. Incremental effect of project on
accounting earnings
Sales
Cost of goods sold (COGS)
----------------
Gross profit

Selling, general, and administrative expenses (S,G&A)
Research and development (R&D)
Depreciation
Other
-----------------
Earnings before interest and taxes (EBIT)

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Step 1, continued

EBIT

Taxes
Interest expense
----------------------
Net unlevered income

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Why unlevered income?

• Method evaluates project as if owners of firm put new cash into firm
for incremental investment, take out cash flows as they arrive
– Initial cash flow (payment to owners) negative
• Alternate method assumes borrowing, then paying interest
– By law of one price, NPV is unchanged (conclusion modified later in
semester)

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Corporate Federal Income Tax Rates—2009

Taxable income                    Rate
\$0        to     \$ 50K            15%
\$50K      to     \$ 75K            25%
\$75K      to     \$100K            34%
\$100K     to     \$335K            39%     Selected state corporate income tax rates
\$335K     to     \$10MM            34%
\$10MM to         \$15MM            35%     California           flat   8.84%
\$15MM to         \$18.3MM          38%     DC                   flat   9.975%
\$18.3M and up                     35%     Hawaii     Below \$25K       4.40%
\$25K -- \$100K    5.40%
\$100K +          6.40%
Maryland             flat   8.30%
Pennsylvania         flat   9.99%
Wyoming              flat   0%

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Refining incremental earnings

• Opportunity costs or sunk costs?
– If a project at Cisco uses a large amount of a firm’s unused bandwidth,
is that an incremental expense for the project?

– If a project at Honeywell employs 20 mid-career Honeywell engineers
that were previously used on another project that the firm has decided
to close, are their salaries and benefits an incremental expense?

– If a project at Midwest Federal (a commercial bank) uses floor space in
retail bank branches, is that an incremental expense?

– If a research lab at 3M produces many new ideas for potential projects
each year, should the projects’ incremental earnings include the
expense of running the lab?
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• Principles of evaluating opportunity/sunk costs
– Are there competing projects that could use the resources, either
now or during the project’s life?
– Could the resources be sold or leased out if not used in the
project?
– Does the use of the resources lower the profitability of other
revenue streams?

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Refining incremental earnings

• Synergies and cannibalism
– Dr Pepper (1885)
– Diet Dr Pepper, Cherry Vanilla Dr Pepper, Diet Cherry Vanilla Dr
Pepper, Caffeine Free Dr Pepper, Caffeine Free Diet Dr Pepper, Dr
Pepper Berries & Cream, Diet Dr Pepper Berries & Cream
– Are new products cannibalizing old ones and/or helping to establish a
brand?
– Synergies/cannibalism refers not only to existing products, but potential
future products!

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From incremental earnings to incremental
cash flows
• A reminder
– Calculation of incremental earnings is difficult, and finance has
nothing helpful to say about the calculation – it is an input
• Incremental cash flows

Subtract any capital expenditures
Subtract any change in net working capital (e.g., inventories)
-----------------------------------------
Result is incremental free cash flow

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• Methods
– Straight-line
• Depreciation each year is cost/(tax life)
– Accelerated
• Greatest acceleration is Modified Accelerated Cost Recovery
System (MACRS)
• Depreciation each year is cost x rate from schedule (Ch 7
appendix)

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Depreciation and taxes

• Free cash flow formula
free cash flow  EBIT  1  tax rate   CapEx - NWC
 depreciati on

• Isolate depreciation component of EBIT
free cash flow  revenues - costs excluding depreciation - depreciation   1  tax rate 
 CapEx  NWC  depreciation

• Combine terms involving depreciation (next slide)

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Depreciation and taxes

• Free cash flow formula
free cash flow  revenues - costs excluding depreciation  1  tax rate 
 CapEx  NWC  depreciation  tax rate 

• Final term is “depreciation tax shield”

– The faster a firm is allowed to depreciate its assets, the more
after-tax cash flow it generates

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• Formula implicitly says that gross losses reduce taxes

• Does this make sense for all projects?
– Yes, for projects within profitable companies
– Yes, for projects within companies that made profits during the previous
two years
• Tax loss “carrybacks” – IRS lets you offset past profits with current losses – they send the
company a check for past taxes collected
– Yes, for projects within companies that will make profits during the next
20 years
• Tax loss “carryforwards” – IRS lets you offset future profits with current losses – but tax
loss should also be carried forward to future year (and discounted)

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• If a firm has substantial tax loss carryforwards from previous years,
should new projects be evaluated using a zero tax rate?
– Key issue: Would the carryforwards be used up if the project were not
implemented?
• When would they be used?
• Think of this as opportunity cost of using carryforwards on this project

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Net working capital

• Projects may require increased inventory
• Revenues will be accounts receivable before cash
flows
• Expenses will be accounts payable before negative
cash flows
• Net effect is change in net working capital, which must
be financed

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Example

• Question 1 of distributed problems

2010   2011    2012      2013          2014   2015   2016   2017

Equipment     -1      -3
Sales                      9.000 11.700 12.051 12.413 12.785            0.000

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Initial investment               \$ 3.000
Initial sales                      9.000
Sales growth rate from 1 to 2      30.00%
Discount rate                      11.00%
Sales growth rate after year 2      3.00%
COGS percentage                    62.00%
SG&A percentage                    22.00%
Depreciation rate                  20.00%

Year                                    0           1           2           3           4           5

Expected sales                               9.000      11.700      12.051      12.413      12.785
COGS                                        (5.580)     (7.254)     (7.472)     (7.696)     (7.927)
SG&A                                        (1.980)     (2.574)     (2.651)     (2.731)     (2.813)
Other expenses                    (0.200)      -           -           -           -           -
Depreciation                                (0.600)     (0.600)     (0.600)     (0.600)     (0.600)
Pretax profit                     (0.200)   0.840       1.272       1.328       1.386       1.446
After-tax profit                  (0.200)   0.840       1.272       1.328       1.386       1.446

Cash flow analysis
Investment                        (3.000)
Depreciation                                0.600       0.600       0.600       0.600       0.600

Net cash flow                     (3.200)   1.440       1.872       1.928       1.986       2.046
PV                                (3.200)   1.297       1.519       1.410       1.308       1.214
NPV                                3.549
IRR                                46.55%

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Financing

– “If the firm uses \$3MM of existing cash to make the initial
investment, the true cost of the equipment purchase is \$3MM
plus the lost interest that would have been earned on \$3MM.”

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Revised example

• Question 1 of distributed problems

– Marginal tax rate of 40%
– Implementation requires an increase in working capital of 25% of
sales. Working capital is needed at the beginning of the year
(i.e., end of previous year)

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Initial investment                        \$ 3.000
Initial sales                               9.000
Sales growth rate from 1 to 2               30.00%
Discount rate                               11.00%
Sales growth rate after year 2               3.00%
COGS percentage                             62.00%
SG&A percentage                             22.00%
Depreciation rate                           20.00%
Marginal tax rate                           40.00%
Working capital percentage                  25.00%

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Year                               0             1             2             3             4             5

Expected sales                          9.000        11.700        12.051        12.413        12.785
COGS                                   (5.580)       (7.254)       (7.472)       (7.696)       (7.927)
SG&A                                   (1.980)       (2.574)       (2.651)       (2.731)       (2.813)
Other expenses           (0.200)           -             -             -             -             -
Depreciation                           (0.600)       (0.600)       (0.600)       (0.600)       (0.600)
Pretax profit            (0.200)        0.840         1.272         1.328         1.386         1.446

Taxes paid                0.080        (0.336)       (0.509)       (0.531)       (0.554)       (0.578)
After-tax profit         (0.120)        0.504         0.763         0.797         0.832         0.867

Cash flow analysis
Investment               (3.000)
Depreciation                            0.600         0.600         0.600         0.600        0.600
Working capital          (2.250)       (0.675)       (0.088)       (0.090)       (0.093)       3.196

Net cash flow            (5.370)       0.429         1.275         1.307         1.339         4.664
PV                       (5.370)       0.386         1.035         0.955         0.882         2.768
NPV                       0.656
IRR                        14.48%

180.366, Spring 2011                                                          45
Break-even/sensitivity/scenario analyses

• All alter inputs to evaluate their effect on project NPV

– Break-even: Find values of particular inputs that set NPV to zero

• Break-even sales growth after 2012 (perhaps not 30%)?

• Break-even COGS percentage?

– Sensitivity: how does NPV change when inputs are set to
reasonable upper and lower bounds?

– Scenario: vary multiple inputs at once to see how they interact

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Interpreting the analyses

• What do we do with the results?

• Important principle: cash flow uncertainty owing to future events is
addressed in the choice of discount rates, not in direct adjustment of
cash flows

• Analyses should be used to determine which inputs are especially
important to NPV, so that effort in calculating NPV can be
concentrated on key inputs

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