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Chapter 14 Notes
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Capital Budgeting
This chapter examines various tools used to evaluate potential
projects or investments. Accountants advocate the use of the
Simple Rate of Return, which is based upon the accounting
concept of Net Income for this purpose. This return is also
referred to as the Accounting Rate of Return. Financiers do not
like to use Net Income as a basis for evaluating investments
because of the discretion that accountants have in determining
Net Income (e.g., estimates of various allowances, useful lives,
and the choice of depreciation methods). Financiers prefer to use
After-Tax Cash Flow as the basis for their analysis. Financiers
Capitol Investment?
advocate the use of the Payback Period, Net Present Value, and
Internal Rate of Return for purposes of evaluating investments.
Simple Rate of Return and Payback period are referred to as non-discounting models
because they do not utilize the time value of money. Net Present Value and Internal
Rate of Return are referred to as discounting models because the time value of money
is part of their analysis.
All of these tools are used to evaluate the merits of a particular project or investment.
The way that the project or investment will be financed is not included in the evaluation
(e.g., do not include interest costs). It is assumed that the project or investment is
funded with available capital, and the source of that capital is not in issue.
Present Value
"I'll gladly pay you Tuesday for a hamburger today." Instinctively, you
know that a dollar that Wimpy is willing to pay sometime in the future is
not as valuable as a dollar in your hand today. This inequality arises
because you can put the dollar that you currently have in the bank,
earn interest on that deposit, and have more than one dollar (the dollar
deposit plus the interest) in the future.
The Present Value tells you what you have to put in the bank today in order to have a
dollar in the future. While you can use your calculator, Excel or a Present Value table,
you should be able to figure out the Present Value of one dollar to be receive at the end
of a given period by using the following formula:
PVIF =
__1___
(1+ d)n
where “d” is the discount rate (the interest rate) and “n” is the number of periods until
you receive the one dollar. PVIF is the Present Value Interest Factor (PVIF) or discount
factor that is reported on a Present Value table. If you treat “n” as the number of years
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Chapter 14 Notes
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and “d” as the annual interest rate, then your Present Value is based on simple interest.
If you change the “n” to reflect the number of six-month periods, and the “d” to reflect
the amount of interest that is paid in each six-month period, then you have Present
Values that reflect semi-annual compounding of interest. By adjusting the “d” and “n” to
reflect various time periods and interest rates, you can vary the extent of the
compounding of interest.
For example, if one dollar is to be received at the end of one year, and you can receive
interest from your bank at the rate of 10% compounded annually, then you need to
deposit the following to have one dollar at the end of the year:
PVIF = (1/(1.10)1)
PVIF = 90.909¢
You can test this:
One Year's Interest Is .1 x 90.909¢
Original Principal
=
=
9.0909¢
90.9090¢
99.9999¢
We are a little off because of rounding.
As another example, assume that one dollar is to be received at the end of a two-year
period, and you can receive interest from your bank at the rate of 10% compounded
annually, then you need to deposit the following to have one dollar at the end of two
years:
PVIF = (1/(1.10)2)
PVIF = 82.6446¢
You can test this:
One Year's Interest Is .1 x 82.6446¢
Original Principal
Amount on Deposit After 1 year:
=
=
8.26446¢
82.64460¢
90.90906¢
Notice how this is similar to the Present Value of a dollar at the end of one year. The
only difference is due to rounding.
The Second Year's Interest Is .1 x 90.90906¢
Balance At Start of Year
Amount on Deposit After 2 years:
=
=
9.090906¢
90.909060¢
99.999966¢
If you are to receive a dollar at the end of each year for a given period of time, this is
called an annuity. You could figure out the Present Value of that annuity by calculating
the Present Value of each dollar you are going to receive using the above formula. This
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Chapter 14 Notes
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could get cumbersome if the annuity period is long. Alternatively, you could use the
formula for calculating the Present Value of an Annuity.
Conceptually, the Present Value of an Annuity is that amount that must be invested
today at a given interest rate in order to produce sufficient funds to enable annual
withdrawals of the annuity amount over the annuity period.
PVIFannuity =
1 – 1/d[
__1___
]
(1+d)n
For example, if one dollar is to be received at the end of each year for a two-year
period, and you can receive interest from your bank at the rate of 10% compounded
annually, then you need to deposit the following to be able to withdraw one dollar at the
end of each year for two years:
The Present Value of one dollar received a year from now is:
The Present Value of one dollar received two years from now is:
90.9090¢
82.6446¢
$1.735536
Using the Present Value of an Annuity formula:
PVIFannuity = [1-(1/(1.10)2]/.1 PVIFannuity = $ 1.73553719
Again, the difference is due to rounding. If you deposit $1.73553719 with a bank at
10% interest, you can withdraw one dollar at the end of each year for two years:
Initial Deposit:
One Year’s Interest:
Amount On Deposit After One Year:
Withdrawal of Annuity:
Amount On Deposit After Withdrawal of Annuity:
Second Year’s Interest:
Amount On Deposit After Two Years:
Withdrawal of Annuity:
Amount On Deposit After Withdrawal of Annuity:
$1.735537190
17.355372¢
$1.909091
-$1.000000
90.9091¢
9.0909¢
$1.000000
-$1.000000
0
Net Present Value
The problem when evaluating a project or investment is that you invest money today,
and you get your payoff at some time in the future. As discussed above, we know that
comparing a dollar today with a dollar to be received some time in the future is like
comparing apples and oranges. Because the investment and the payoff occur at
different times you merely cannot say, "I've invested $1,000 and received a payoff of
more than $1,000 on that investment". It could be that when you factor in the time value
of money (Present Values), you have received a payoff that is less than your original
investment.
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Chapter 14 Notes
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With Net Present Value, you are attempting to compare apples and apples. You
convert all of the future dollars of the payoff into present dollars and then compare them
to the investment, which is already in present dollars. With Net Present Value you have
to take the Present Value of all of the cash to be received from an investment, and
subtract out the initial investment. If you have a positive number, then you know that
you have received your initial investment along with your minimum required return.
The discount (interest rate) is the minimum return that your firm requires on
investments. Traditionally, the discount rate is the weighted average cost of capital of
your firm. The thought is that you should not invest in a project or investment that will
not provide a return at least equal to the cost of the funds that you are investing in that
project or investment. A firm, however, is free to set any minimum return that it wishes.
The weights used in this calculation are the percentages of the total capital of the firm
coming from a particular source of capital (e.g., 20% of our capital comes from equity
and 80% of our capital comes from debt). The cost of that capital (debt or equity) is the
after-tax cost of that capital (e.g., interest is deductible while dividends are not).
Remember that you are considering only After-Tax Cash Flows (not Net Income). Net
Present Value is a Finance concept. Typically, the difference between Net Income and
After-Tax Cash Flow is the deduction for depreciation and other non-cash expenses
from Net Income. Think back to when you learned about the indirect method of
preparing a Cash Flow Statement. Under that method, you would start with a firm’s Net
Income and then add back the depreciation expense (because depreciation expense
does not require a current cash outlay in order to get the Cash Flow From Operations.
The traditional assumption is that the infusion of funds into an investment or project is
made on the first day of the investment or project, and that the payoffs (returns) are
received at the end of each year (e.g., the first payoff is received on the last day of the
first year). We will be making this assumption in the following discussion. This
assumption is not always the case. For example, cash inflows into an investment or
project may be made over a number of years, and payoffs can be received during a
year or in the year following the year in which it was earned.
Depreciation
Depreciation is not a cash expense, and therefore is not directly included in the
calculation of After-Tax Cash Flow. Depreciation is, however, an income tax deduction.
Thus, depreciation is still included in the calculation of the firm’s tax expense. The way
that depreciation is calculated for financial statement purposes is different than the way
that it is calculated for income tax purposes. While firms have many alternatives from
which to choose when calculating its depreciation expense, they must follow the rules of
the Modified Accelerated Cost Recovery System (MACRS) when preparing their income
tax returns. For purposes of this class, we will assume depreciation for both income tax
and financial reporting is the straight-line depreciation method, unless otherwise
provided.
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Chapter 14 Notes
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When calculating After-Tax Cash Flow, most books first subtract cash expenses
(without considering depreciation) from revenues to get the Before-Tax Cash Flow.
They then multiply the Before-Tax Cash Flow by the tax rate to get a preliminary tax
expense, which is subtracted from the Before-Tax Cash Flow in order to get a
preliminary After-Tax Cash Flow. Because depreciation reduces a firm’s tax bill, they
then add back the reduction in taxes that the depreciation tax deduction produces (the
"Depreciation Tax Shield"). The Depreciation Tax Shield is the tax rate multiplied by the
amount of the depreciation tax deduction:
A
B
C
D
E
F
G
H
I
J
Revenue:
Less: Cash Expenses:
Before-Tax Cash Flow (A-B):
Multiplied by Tax Rate:
Preliminary Taxes (CxD):
Preliminary After-Tax Cash Flow (B-E):
Depreciation Deduction:
Multiplied by Tax Rate:
Depreciation Tax Shield (GxH):
After-Tax Cash Flow (F+I):
$100,000
-40,000
$60,000
X .4
$60,000
-$24,000
$36,000
$10,000
X ,4
+$4,000
$40,000
An alternate approach to calculating After-Tax Cash Flow, is to use Before-Tax Cash
Flow, and then subtract the amount of taxes due (which includes the depreciation
deduction in its calculation).
A
B
C
D
E
F
G
H
Revenue:
Less: Cash Expenses:
Before-Tax Cash Flow (A-B):
Depreciation Deduction:
Taxable Income (C-D):
Multiplied by Tax Rate:
Taxes (ExF):
After-Tax Cash Flow (C-G):
$100,000
-40,000
$60,000
-10,000
$50,000
X .4
$60,000
-$20,000
$40,000
While most books use the Depreciation Tax Shield approach, the calculation of the
income tax approach is easier to understand conceptually (and thus involves less
memorization).
Constant After-Tax Cash Flows
When you are dealing with the same savings in each year, remember that you can use
the Present Value of an Annuity formula.
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Chapter 14 Notes
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Salvage Value
Remember that if you have a salvage value, then you have to treat it as a cash flow in
the last year of the investment or project. If you have an income tax book value (e.g.,
“basis”) that is different than the salvage value, then you will have to take into account
the taxes due on the gain from the salvage sale of the asset in your analysis.
NPV Example
Troy, Inc. operates a theme park centered
around the classical ancient world. Troy is
considering opening a new attraction
simulating a ride in the Trojan Horse. The
new attraction would require an investment of
$420,000, and would produce the following
Before-Tax Cash Flow. Assume no salvage
value, and assume that Troy pays taxes at a
tax rate of 40%. Also assume that Troy
requires a minimum return of 10% from its
investments. Assume that depreciation is
calculated using the straight-line method with
no salvage value:
Cash Flow
$ 100,000
200,000
250,000
150,000
100,000
100,000
$ 900,000
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Most books would calculate the After-Tax Cash Flow using the Depreciation Tax Shield
We will assume that the depreciation deduction is $70,000 a year ($420,000/6):
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Cash Flow
-Taxes (CF x .4)
$ 100,000
200,000
250,000
150,000
100,000
100,000
$ 900,000
-
$40,000
$80,000
100,000
60,000
40,000
40,000
+
+
+
+
+
+
+Tax Shield
[.4(Tx])
$28,000
28,000
28,000
28,000
28,000
28,000
After-Tax Cash Flow
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88,000
148,000
178,000
118,000
88,000
88,000
$708,000
Chapter 14 Notes
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The alternative approach would be to calculate the income tax bill, because taxes are a
cash expense. Income for tax purposes include the cash received reduced by the cash
expenses, but there is also a deduction for the depreciation.
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
(A)
Bef-Tx Cash
$ 100,000
200,000
250,000
150,000
100,000
100,000
$ 900,000
-
(B)
Depreciation
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$420,000
=
=
=
=
=
=
(C=A-B)
Taxable Inc
$30,000
130,000
180,000
80,000
30,000
30,000
$480,000
Tax Rate
x.4=
x.4=
x.4=
x.4=
x.4=
x.4=
(D=.4C)
Taxes
12,000
52,000
72,000
32,000
12,000
12,000
$192,000
Next, subtract the tax bill from the Before-Tax Cash Flow in order to get the After-Tax
Cash Flow:
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
(A)
Before-Tax Cash Flow
$ 100,000
200,000
250,000
150,000
100,000
100,000
$ 900,000
(D)
Less Taxes
12,000
52,000
72,000
32,000
12,000
12,000
(A-D)
After-Tax Cash
88,000
148,000
178,000
118,000
88,000
88,000
$708,000
As you can see, you get the same After-Tax Cash Flow with either approach.
Now, you need to calculate the Present Value of the After-Tax Cash Flow:
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Artist’s Rendering of Project
After Tx Cash
x PVIF PV of Cash Flow
88,000 x .90909
$ 80,000
148,000 x .82645
122,315
178,000 x .75131
133,733
118,000 x .68301
80,595
88,000 x .62092
54,641
88,000 x .56447
49,673
$708,000
$520,957
Less Original Investment:
-420,000
NPV:
$100,957
If the NPV is greater than zero, then you know that you are getting at least your
minimum required return. Unfortunately, you do not know the actual return that you are
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Chapter 14 Notes
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getting. It turns out that you are receiving an 18.1% return, but you have no way of
knowing that from the NPV calculation.
Internal Rate of Return
Along with the Net Present Value, financiers also examine whether to make an
investment by examining the Cash Payback Period and Internal Rate of Return. Internal
Rate of Return represents a modification of the calculation of the Net Present Value that
we have discussed above. With these modifications you assume that the Net Present
Value is zero, and you solve for the discount (interest) rate. In other words, you are
trying to find out the return that the investment produces. You need a computer or
calculator to calculate the Internal Rate of Return efficiently.
The Internal Rate of Return is very popular in the business world. Among academics,
however, its use is discouraged because it has theoretical problems. One complaint is
that IRR assumes that cash payoffs that are received are reinvested at the same rate as
the IRR, which may not be reasonable for high IRRs. Another objection comes with
cash flows from the investment that alternate from positive to negative. In your Finance
classes you will be taught to use a modified IRR to counter these objections.
On the other hand, NPV is not really used much in the business world. The problem
with NPV is that it does not tell you the amount of the return that you are receiving. The
NPV approach is, “If you tell me what return you want to make, I will tell you if you made
it.”
Using Excel to Calculate IRR and NPV
There is a major problem using Excel in calculating NPV. Excel makes the following
assumption about the investments and payoffs, which is described in the Excel Help
notes:
“The initial cost [investment] … occurs at the end of the first period.”
Excel also assumes that the first payoff is received at the end of the second year of the
investment. These are not traditional assumptions. Excel explains that if you want to
assume that the investment occurred on the first day of the investment, you have to add
it separately. As the Excel Help notes state:
“If your first cash flow occurs at the beginning of the first period, the first
value must be added to the NPV result, not included in the values
arguments.”
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Chapter 14 Notes
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NPV
Using Excel’s assumptions regarding the date of the investments and payoffs, you
would calculate the NPV using Excel by setting up the Investment (a negative number)
and the After-Tax Cash Flows as shown below:
The Formula is “=NPV(Required Rate of Return, Cells Containing Investment and AfterTax Cash Flow”. A 10% Required Rate of Return would be written as “.1”:
If you cannot remember the NPV notation, then select “Insert” from the Menu Bar, and
then select “Function”. NPV is the name of the function, and it is classified as a
Financial function.
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Chapter 14 Notes
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As noted above, Excel assumes that the investment is made on the last day of the first
year, and the first payoff is received on the last day of the second year. Because of this,
the answer that Excel returns is not the same as the one that we just calculated. We
assumed that the investment was made on the first day of the investment and the first
payoff occurs at the end of the first year.
Excel will tell you that the NPV is $91,780. It has moved the investment and the each
cash flow payoff back one year. If you wish to assume that the investment is made on
the first day of the investment, and that the first payoff is received on the last day of the
first year, then you should use NPV function to value the payoff (not including the initial
investment) and then subtract the initial investment. This is the approach suggested by
Excel Help Notes:
Rather than using this approach, you could counter the assumption that Excel is
making. Excel is pushing everything back one year. If you add one year’s interest to
each number, then when Excel pushes it back one year, the end result will be the
proper value. For example, Excel assumes that the investment is made one year from
now, which gives it a present value of $381,818 ($420,000 x .90909). If you increase
the $420,000 by one year’s interest of 10%, you give increase the investment to
$462,000 ($420,000 x 1.1). Now, when Excel pushes the investment back one year, it
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Chapter 14 Notes
Page 11
will give the investment a Present Value of $420,000 ($462,000 x .90909), which is the
correct value. Using this approach, gives you the correct NPV:
IRR
You can calculate the IRR using Excel by setting up the problem the same as with NPV.
This time you use the IRR function: “=IRR(cells containing investment and cash flow,
guess of the IRR)”. A guess of 10% would be written as “.1”. If an error appears, it
means that your guess was not close, and you should try again.
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Chapter 14 Notes
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If you cannot remember the formula for IRR, you can select “Insert” on the Menu Bar.
Then select “Function”. IRR is the name of the function, and it is classified as a
financial function:
Excel’s assumption about the timing of the investment and payoffs does not affect the
calculation of the IRR. Whether the investment and payoffs begin now or at end of this
year, the return that the payoffs provide on the investment is the same. In other words
the return on the investment over the life of the investment (IRR) is the same, whether
you make the investment this year or anytime in the future:
As you can see, failure to correct for Excel’s assumption will provide you with the wrong
NPV ($91,780.41), but still provides the correct IRR (18.1075%). The investment and
payoffs that are increased by the discount factor (10%) provide you with the correct
NPV ($100,958) and IRR (18.1075%)
Payback Period
A simple tool used to evaluate potential investments is the Payback Period. This
method is also referred to as the Cash Payback Period. The Payback Period does not
incorporate present value concepts. The Payback Period merely reflects the feeling that
if you get your investment back quickly then there is little risk. For example, if you can
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Chapter 14 Notes
Page 13
refinance a loan at a lower interest rate by paying a loan fee of $1,000, and if your
interest savings are $100 a month, then you will get your money back in 10 months.
After that you will save $100 a month for a number of years. What is the downside to
making the investment when you recoup your investment so quickly? So, with Payback
Period, you merely state how long it takes to recoup your investment.
Besides a rough approximation of risk, the Payback Period is also important to many
managers because they are not interested in making investments that provides a longterm return. Such an investment will not help them get their bonus this year and it may
be received after they have moved to their next job.
If the After-Tax Cash Flow is the same every year, the calculation of the Payback Period
is a simple matter:
Payback Period =
____Original Investment____
Annual After-Tax Cash Flow
For example, If the investment is $420,000 and it generates an After-Tax Cash Flow of
$100,000 a year, then the Payback Period is 4.2 years:
Payback Period = $420,000/$100,000 = 4.2 years
If the After-Tax Cash Flow is uneven, then you have to figure it out by examining the
cumulative cash received over the life of the investment.
Payback Period Example
We will calculate the Payback Period for the Trojan Horse
attraction. You can see that after three years, Troy has
not yet recouped its investment, but after the fourth year,
Troy has received more than its initial investment. At the
beginning of the fourth year, Troy still needs $6,000 to
recoup its investment in the Trojan Horse attraction:
After-Tax
Cumulative
Cash Flow
Cash
After-Tax
Needed to
Flow
Cash Flow
Recoup Inv.
Year 1
88,000
$ 88,000
$332,000
Year 2
148,000
236,000
184,000
Year 3
178,000
414,000
6,000
Year 4
118,000
532,000
Year 5
88,000
Year 6
88,000
$708,000
New Attraction
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Chapter 14 Notes
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Troy will receive $118,000 over the entire fourth year, but it only needs part of that
year’s cash flow in order to recoup its investment, which is calculated:
Portion of Year Needed To
Recoup Troy’s Investment
=
___Cash Needed___
Total 4th Year Cash
=
_$6,000_
$118,000
= .0508474
Troy needs approximately .05 of the fourth year to recoup its investment. The Payback
Period is 3.05 years.
Simple Rate of Return
The Simple Rate of Return is an accounting concept, so it uses Net Income (not AfterTax Cash Flow). This return is also known as the Accounting Rate of Return. There is
a different Simple Rate of Return for each year of the investment, but this formula
averages them all together, which is why it is also known as the Average Accounting
Rate of Return. The definition is:
Simple Rate of Return =
Average Annual Net Income
Original Investment
Simple Rate of Return Example
We will now calculate the Simple Rate of Return of the investment in the Trojan Horse
attraction. We must first calculate the Average Net Income:
Yr.
1
2
3
4
5
6
(A)
Cash Flow
$ 100,000
200,000
250,000
150,000
100,000
100,000
$ 900,000
(B)
-
Deprec.
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$420,000
=
=
=
=
=
=
(C=A-B)
BeforeTax
Income
$30,000
130,000
180,000
80,000
30,000
30,000
$480,000
-
(D=.4C)
Taxes
(40%)
12,000
52,000
72,000
32,000
12,000
12,000
$192,000
=
=
=
=
=
=
(E=C-D)
Net
Income
18,000
78,000
108,000
48,000
18,000
18,000
$288,000
Average Annual Net Income = $288,000 / 6
Average Annual Net Income = $48,000
Accounting Rate of Return = Average Net Income / Original Investment
Accounting Rate of Return = $48,000/$420,000
Accounting Rate of Return = 11.4%
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Chapter 14 Notes
Page 15
Theme Park Visitors Enjoying New Attraction
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