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Capital Budgeting
Wadia Haddaji
Duke University
January 14, 2008
• Topics:
1. Capital budgeting under certainty
• Readings:
1. Brealey, Myers and Allen, chapter 2;
2. Brealey, Myers and Allen, sections 7.1 and 7.2;
3. Brealey, Myers and Allen, sections 6.1, 6.2 and 6.3
1
Capital Budgeting under Certainty
• At the most general level, an investment is a claim to a stream of cash
flows.
• This definition encompasses both real and financial investments.
• In order to choose between alternative investments, we must therefore find
a way to compare cash flows differing in size (more is better than less),
timing (the sooner the better), and risk (the less the better).
• It is easy to compare two single cash flows along any of these dimensions;
it is hard to compare two streams of cash flows.
• The techniques of compounding and discounting allow us to compare cash
flow streams differing in size and timing.
• We will at first ignore risk and compare certain cash flows.
2
The NPV Formula
• We have seen that financial managers act in the best interest of the share-
holders by undertaking investments with positive NPV.
• Also, we have seen that for a one-period investment the NPV formula is
C1
N P V = C0 + 1+r1
where C0 is the initial cash flow (which is generally negative) and C1 is the
end-of-period cash flow (which is usually positive).
• The general formula is
T Ct T Ct
N P V = C0 + t=1 (1+rt )t = t=0 (1+r)t
• In this course, we will discuss how to find the cash flows and the appropriate
rate at which to discount them.
3
Estimating Cash Flows
• Only cash flows are relevant.
• Cash flows are simply the difference between dollars received and dollars paid out. Do not
confuse cash flows with accounting profits or losses.
• Estimate cash flows on an after-tax basis.
• Make sure that cash flows are recorded at the time they actually occur (example: credit
sales, tax liabilities).
• Forget sunk costs.
• Include opportunity costs.
• Treat inflation consistently.
1. Discount nominal cash flows at the nominal rate.
2. Discount real cash flows at the real rate.
4
Example: IM&C’s Fertilizer Project
• International Mulch and Company (IM&C) is considering a project requiring
an initial investment of $10 million in plant and machinery.
• The project is expected to generate the following sales and to require the fol-
lowing costs. The project will also require an initial and ongoing investment
in working capital.
C0 C1 C2 C3 C4 C5 C6
Sales 523 12,887 32,610 48,901 35,834 19,717
Costs of goods sold 837 7,729 19,552 29,345 21,492 11,830
Other costs 4,000 2,200 1,210 1,331 1,464 1,611 1,772
Working capital 550 1,289 3,261 4,890 3,583 2,002
• Even though the machinery is expected to be resold for $1.949 million in
7 years, it has been decided that it will be depreciated on a straight-line
basis over 6 years down to a $500, 000 book value.
9.5
• This means that the yearly depreciation will be 6
= 1.583M.
5
• The firm’s tax rate is 35% and its opportunity cost of capital (discount
rate)is 20%.
Example: IM&C’s Fertilizer Project (cont’d)
• First, let us calculate the after-tax operating profits of the firm.
C0 C1 C2 C3 C4 C5 C6
(1) Sales 523 12,887 32,610 48,901 35,834 19,717
(2) Costs of goods sold 837 7,729 19,552 29,345 21,492 11,830
(3) Other costs 4,000 2,200 1,210 1,331 1,464 1,611 1,772
(4) Depreciation 1,583 1,583 1,583 1,583 1,583 1,583
(5) Pre-tax profits -4,000 -4,097 2,365 10,144 16,509 11,148 4,532
(6) Tax -1,400 -1,434 828 3,550 5,778 3,902 1,586
(7) After-tax profits -2,600 -2,663 1,537 6,594 10,731 7,246 2,946
• The profits of the firm are not the project’s cash flows. We need to make
some adjustments to get the cash flows.
• Depreciation is not a cash flow; we need to add it back.
• Capital expenditures (and sales) have not been taken into account yet.
6
• Changes in working capital are cash flows.
IM&C’s Fertilizer Project: Depreciation
• Depreciation is an accounting number that does not directly affect cash
flows.
• Capital expenditures do not flow directly through the income statement;
instead the assets are depreciated over time to match their cost with their
use.
• For cash flow purposes we want to account for capital expenditures when
the cash is paid, i.e. when the assets are purchased.
• So if we did not add back depreciation, we would double-count the costs.
• However, depreciation still plays an important role because it is tax de-
ductible, so we cannot completely ignore it.
• This is why we calculate book income first, then adjust it to get cash flows.
7
• The adjustment here is simple – just add back the depreciation number each
year.
IM&C’s Fertilizer Project: Capital Expenditures
• Clearly, the initial capital investment of $10 million in plant and machinery
is a negative cash flow, but it has not been accounted for on the income
statement.
• In year 7, the machine sale will generate a positive cash flow of $1.949 million,
which will be taxed.
• Only the excess over the book value of the machine ($0.5 million) is taxed;
this is considered a taxable gain. The tax is 35% × (1.949 − 0.5) = 0.507.
• Therefore, the net capital inflow in year 7 is 1.949 − 0.507 = 1.442.
• There are no other interim capital expenditures for this project.
8
IM&C’s Fertilizer Project: Working Capital
• Working capital essentially represents a firm’s net investment in short-term
assets:
working capital = inventory + accounts receivable − accounts payable.
• For example:
1. An increase of $1 in accounts receivable means that part of the sales
figure in line (1) has not yet been received (i.e., it is not a cash inflow
flow yet, but it was counted as such on the income statement).
2. An increase of $1 in inventory means that the firm has spent cash to
buy products that have not yet been sold (ie, it is a cash outflow that
has not been accounted for on the income statement).
3. An increase of $1 in accounts payable means that some of the costs
have not yet been paid (so this is a cash outflow that is in the income
statement but is not yet a physical outflow).
9
• More generally, working capital is often defined as (non-cash) current assets
minus (non-debt) current liabilities. It is not always clear how to treat cash;
we will ignore this for most of the course.
IM&C’s Fertilizer Project: Working Capital (cont’d)
• The changes is working capital from year to year are as follows.
C0 C1 C2 C3 C4 C5 C6 C7
(10) Working capital 550 1,289 3,261 4,890 3,583 2,002
(11) Change in WC 550 739 1,972 1,629 -1,307 -1,581 -2,002
• We now have all of the ingredients to calculate the project’s cash flows.
10
Example: IM&C’s Fertilizer Project (cont’d)
• The project’s cash flows are as follows.
C0 C1 C2 C3 C4 C5 C6 C7
(7) After-tax profits -2,600 -2,663 1,537 6,594 10,731 7,246 2,946
(4) Depreciation 1,583 1,583 1,583 1,583 1,583 1,583
(11) Change in WC 550 739 1,972 1,629 -1,307 -1,581 -2,002
(12) Capital expenditure 10,000 -1,442
(13) Project cash-flows -12,600 -1,630 2,381 6,205 10,685 10,136 6,110 3,444
• We can now calculate the net present value (NPV) of the project as follows:
2,381 6,205 10,685 10,136 6,110 3,444
N P V = −12, 600 + −1,630 + (1.20)2 + (1.20)3 + (1.20)4 + (1.20)5 + (1.20)6 + (1.20)7 =
1.20
3, 519.
• Since the NPV is greater than zero, IM&C should undertake the project.
11
Alternatives to NPV: The Internal Rate of Return Rule
• The internal rate of return (IRR) of a project is defined as the constant
discount rate y which makes N P V = 0. In other words, y solves
T Ct
NP V = t=0 (1+y)t =0
• The IRR rule says that a project should be accepted if and only if y exceeds
the yield on financial securities (bonds) with comparable maturity, cash flows
and risk (the opportunity cost of capital or hurdle rate).
• Notice that with a flat term structure (discount rate is r for all maturities t),
the IRR rule implies that we should accept a project if and only if y > r.
• Given that we have assumed a flat term structure and considered only riskless
projects so far, the IRR rule implies that we should accept a project if y
exceeds the riskfree rate.
• Essentially, the IRR measures the return of the project, i.e. the return on
the initial investment.
12
• If the project’s return (i.e.IRR) beats what can be obtained in capital markets
(i.e. r), then the project should be undertaken
The Internal Rate of Return Rule (cont’d)
• For example, consider the following project:
C0 C1 C2 C3
-5,000 2,000 2,000 2,000
• The internal rate of return (IRR) for this project is calculated as follows.
2,000 2,000
We want to find y that solves −5, 000 + 2,000 + (1+y)2 + (1+y)3 = 0. We find
1+y
y = 9.7%; this is the IRR.
• Notice that in this case the IRR rule corresponds exactly to the NPV rule.
• For any discount rate lower than 9.7%, the NPV is positive; this is precisely
when the IRR rule says that we should undertake the project.
• For any discount rate higher than 9.7%, the NPV is negative; this is precisely
13
when the IRR rule says that we should drop the project.
Alternatives to NPV: Multiples
• Multiples are used extensively by research analysts and investment bankers
for valuations of whole companies, but are less common in project valuation.
• The acquisition of a whole company is really just a type of project, though,
so the topic is important for this course.
• The multiple approach uses the market price multiples for comparable com-
panies to provide an appropriate valuation range.
• The use of multiples is predicated on the Efficient Markets Hypothesis
(EMH) and the Arbitrage Pricing Theory (APT). Simply stated, the multiple
approach says that comparable companies should sell at comparable prices.
14
How to use Multiples
• To implement the multiples method, first find a set of comparable compa-
nies.
• You are looking for firms with similar cash flow characteristics, i.e., similar
risk, timing, and expected growth rates (technically need proportionality).
• Next, divide the comparable firms’ value by some operating statistic to get
a pricing multiple for each. Depending on the operating statistic, use either
equity market value or total firm value (equity plus debt). Commonly used
multiples include:
1. Equity Value-to-Net Income (or P/E)
2. Equity Value-to-Book Value of Equity
3. Firm Value-to-EBITDA
4. Firm Value-to-EBIT (Operating Income)
5. Firm Value-to-Sales
15
6. Firm Value-to-Book Value of Assets
How to use Multiples (Cont’d)
• Next, just take an average of the comparable firms’ multiples (or use the
entire range), and multiply that by the analogous operating statistic for the
company you are trying to value.
• If you used an Equity Value multiple, this gives you an estimate of your
company’s (or project’s) Equity Value; if you used a Firm Value multiple, it
gives you an estimate of Firm Value (or just Total Value if it’s a project).
• To go from Equity Value to Firm Value, just add the value of existing debt.
• To go from Firm Value to Equity Value, just subtract existing debt.
16
Multiples: An Example
• Assume you are trying to value Project X, and you have three comparable
firms, A, B, and C. You have the following information on the three firms:
Shares Outs. Mkt Price Debt Outs. Revenues Op. Inc.
Company A 100 $5.00 $100 $100 $68
Company B 200 2.00 150 95 65
Company C 50 7.50 200 150 63
• From this information you calculate their multiples as follows:
Firm Value Firm Value
Equity Value Firm Value Revenues Op. Inc.
Company A $500 $600 6.0 8.8
Company B 400 550 5.8 8.5
Company C 375 575 3.8 9.1
17
Average 5.2 8.8
Multiples: An Example (cont’d)
• Now, just apply these multiples to get an estimate of the value of Project
X (note that we are looking for the total value here, which is analogous to
Firm Value for a stand-alone firm):
Valuation Based On
Revenues Op. Inc.
Average multiple 5.2 8.8
Project X operating statistic $195 $105
Implied value of Project X $1, 015 $924
• NOTE: Multiples are not really an alternative to NPV. They are just a
different way to estimate NPV.
• Our normal approach is to discount future cash flows to find PV, then
subtract the initial investment to get NPV.
18
• Multiples just give us an alternative way to estimate PV, so we can still
subtract the initial investment to get an estimate of NPV.
NPV’s Competitors
• In spite of the fact that the NPV rule always leads to investment decisions
which are in the shareholders’ best interests, alternative investment rules
have been—and to some extent are still—used by businesses.
• Four common alternatives to the NPV rule are:
1. Payback period.
2. Average return on book value (or average accounting return).
3. Internal rate of return.
4. Profitability index.
• The internal rate of return and the profitability index, when properly used,
lead to the same decisions as the NPV rule. We will concentrate on these
two rules.
19
NPV’s Competitors (cont’d)
• As the following survey shows, many of the above investment rules were still
broadly used in the 1980’s.
U.S. U.S. Japan
Capital Budgeting Method (1950’s) (1980’s) (1980’s)
Payback period 34% 12% 40%
Average accounting return 34% 8% 19%
Internal rate of return (IRR) 19% 49% 15%
Net present value (NPV) 19% 9%
Other 6% 10% 2%
None 6% 2% 15%
20
NPV’s Competitors (cont’d)
• In a 1984 survey of large U.S. multinational firms, Stanley and Block find
that over 80% of the responding firms used NPV or IRR as their primary
decision rule.
Primary Secondary
Capital Budgeting Method Technique Technique
Internal rate of return (IRR) 65.3% 14.6%
Net present value (NPV) 16.5% 30.0%
Average accounting return 10.7% 14.6%
Payback period 5.0% 37.6%
Other 2.5% 3.2%
21
NPV’s Competitors (cont’d)
• More recently, Graham and Harvey (2002) surveyed 392 CFOs about their
capital budgeting methods. The following table shows the percentage of
CFOs who “always or almost always” use a given method.
Internal rate of return (IRR) 75.6%
Net present value (NPV) 74.9%
Payback period 56.7%
Average accounting return 30.3%
Discounted payback period 29.5%
Profitability index 11.9%
22
Features of the NPV Rule
• When looking at NPV’s competitors, it is important to keep in mind the
main features of the NPV rule:
• Time value of money: a dollar today is worth more than a dollar tomorrow.
• NPV depends only on all the forecasted cash flows from the project and the
opportunity cost of capital.
• Any rule ignoring some of the project’s cash flows will lead to suboptimal
decisions.
• Any rule affected by the manager’s tastes such as the accounting methods,
the profitability of the company’s existing business, or the profitability of
other independent projects will lead to inferior decisions.
• Because present values are all measured in today’s dollars, you can add them
up and you can compare them.
• As a result, the NPV rule can easily identify whether joint projects are better
than single projects, and identify which mutually exclusive project is better.
23
• As we shall see, the alternatives to the NPV rule often fail to satisfy one or
more of these critical features.
The Payback Period Rule
• The payback period of a project is the number of years it takes to recover the initial
investment. The payback period rule says that a project should be accepted if the payback
period is less than some given cutoff.
• Here are some examples:
Cash flows Payback NPV
Project C0 C1 C2 C3 Period at 10%
A -2,000 2,000 1 -181.82
B -2,000 1,000 1,000 1,000 2 486.85
C -2,000 1,000 1,000 10,000 2 7,248.69
• The basic weaknesses of the payback rule are:
• It ignores the time value of money (as well as the risk of the project).
• It ignores the cash flows beyond the cutoff period.
• It gives no indications on what the cutoff rule should be.
24
• Some companies discount the cash flows before computing the payback rule. This modifi-
cation surmounts the first weakness, but is still subject to the others.
The Average Return on the Book Rule
• The average return on the book value is computed by dividing the average
yearly profit from a project (after depreciation and taxes) by the average
book value of the investment. This ratio is then compared with the book rate
of return for the firm as a whole (or some other equally absurd yardstick).
• This criterion suffers from several defects:
1. It ignores the relevant cash flow from investment and instead considers
the accounting profits (in particular, it depends critically on the accoun-
tants’ choice of a depreciation method).
2. It ignores the time value of money (as well as the risk of the project).
3. The choice of a yardstick is totally arbitrary.
25
Limitations of the IRR Rule: Non-Flat Term Structure
• One of the main limitations of the IRR rule is that it is very difficult to
apply with a non-flat term-structure, since in this case the opportunity cost
of capital depends on the maturity and cash-flow profile of each investment
and is a complicated average of the interest rates r1 , r2 , . . ., rT .
• As an example, assume the following term structure,
t
1 2 3 4 5
rt 4.00% 4.50% 5.00% 5.50% 6.00%
• and consider the two projects:
Project C0 C1 C2 C3 C4 C5 IRR NPV
A -1,000 20 20 20 20 1,200 5.24% -32.32
B -1,000 50 50 1,050 5.00% 0.89
26
Why does project A have higher IRR but lower NPV?
Limitations of the IRR Rule:Non-Flat Term Structure (cont’d)
• The answer is that the IRR of project A should be compared to a cutoff
different from that of project B. In particular, the IRR for project A (B)
should be compared to the yield on a 5-year (3-year) bond with the same
cash flows.
• The prices P5 and P3 of such 5-year and 3-year bonds are
20 20 20 20 1,200
P5 = 1.04
+ (1.045)2
+ (1.045)3
+ (1.045)4
+ (1.06)5
= 967.68
50 50 1,050
P3 = 1.04
+ (1.045)2
+ (1.05)3
= 1, 000.89.
The yields on these two bonds can be calculated as follows:
20 20 20 20 1,200
967.68 = 1+yA
+ (1+yA )2
+ + (1+yA )3 + (1+yA )4 (1+y )5
A
⇒ yA = 5.96%.
50 50 1,050
1, 000.89 = + + ⇒ yB = 4.97%.
27
1+yB (1+yB )2 (1+yB )3
Limitations of the IRR Rule: Non-Flat Term Structure (cont’d)
• Since IRRA < yA , we should reject project A. However, Since IRRB > yB ,
we should accept project B.
• Note that, since N P VA = −1, 000 + 967.68 and N P VB = −1, 000 + 1, 000.89,
we have essentially gone back to the NPV rule!
• Typically (in practice), do we perform this type of computation for the hurdle
rate? No, because the step before that essentially involves calculating the
project’s NPV (i.e., we rediscovered the NPV formula).
• Do we often take the riskfree term structure into account with IRR? No
but, as we will see later, a different risk implies a different discount rate.
Many projects will have different phases in which their risk differs, and so
the appropriate hurdle rate may not be obvious with the IRR. This slide
shows how to get the hurdle rate in those situations too.
28
Limitations of the IRR Rule: Multiple IRRs
• In the example presented in the previous lecture, we noted that the IRR rule
corresponded exactly to the NPV rule. This is not always the case.
• For example, a project can have more than one IRR (in general, there can
be as many different IRRs as there are changes in the sign of cash flows).
In fact, it is also possible that the IRR does not exist for some projects.
• Consider the following two projects:
• and consider the two projects:
Project C0 C1 C2 IRR
A -1,000 2,300 -1,320 10% and 20%
B -1,000 3,000 -2,300 none
• If the appropriate hurdle rate is 15%, it is difficult to tell whether these
projects should be undertaken based solely on their IRR.
29
• With a discount rate of 15%, the NPV of project A is 1.89 > 0 (undertake),
and that of project B is −130.44 < 0 (reject).
Limitations of the IRR Rule: Mutually Exclusive Projects
• The IRR rule can be misleading when choosing between mutually exclusive
projects, as the following example shows (we assume that the hurdle rate is
15%):
Project C0 C1 C2 IRR NPV at 15%
A -2,000 1,500 1,500 31.9% 439
B -5,000 1,000 6,500 24.5% 784
• Essentially, it is better to realize a yield of 24.5% on a larger project than
31.9% on a smaller project.
• The IRR rule can be salvaged in the case of mutually exclusive projects by
computing the IRR for the incremental cash flows.
Project C0 C1 C2 IRR
B-A -3,000 -500 5,000 21.0%
30
• Since the IRR of the incremental cash flows is greater than 15%, we should
choose project B over project A.
Limitations of the IRR Rule: Mutually Exclusive Projects (cont’d)
• This can be seen more easily in the following figure:
• Notice that the incremental project’s IRR of 21.0% also represents the break-
even discount rate between the two projects.
NPV
1500
Project B
1000
Project A
500
35% 40% Discount
0
10% 15% 20% 25% 30% rate
-500
-1000
31
Limitations of the IRR Rule: Mutually Exclusive Projects (cont’d)
• With project scaling, the IRR rule can also be adjusted by thinking about
what the firm can do with the excess cash when it invests in the smaller
project.
• In the previous example, there are no other projects, so the excess $3,000
(assuming that the firm has $5,000 in cash) can be invested in capital
markets at an annual rate of 15%.
• Over two years, such a project will generate the cash flows of a coupon
bond with a 15% annual coupon (check that the NPV is zero):
Project C0 C1 C2 NPV at 15%
Capital markets -3,000 450 3,450 0
• So, if project A is undertaken and the rest of the money is invested in capital
markets, the project’s cash flows will be as follows:
Project C0 C1 C2 IRR NPV at 15%
A with cap. mkt. -5,000 1,950 4,950 20.9% 439
32
• Notice that the IRR is now lower than that of project B (24.5%).
Limitations of the IRR Rule: Mutually Exclusive Projects (cont’d)
• Because IRR assumes that the proceeds of a project are reinvested at a
rate equal to the IRR, cash flow timing may also create problems when
comparing mutually exclusive projects. In what follows, we assume a hurdle
rate of 10%.
Project C0 C1 C2 C3 IRR NPV at 10%
A -1,200 1,000 500 100 22.8% 197
B -1,200 100 600 1,100 17.4% 213
• As before, the IRR rule can be salvaged by computing the IRR for the
incremental cash flows.
Project C0 C1 C2 C3 IRR
B-A 0 -900 100 1,000 11.1%
• Since the IRR of the incremental cash flows is greater than 10%, we should
choose project B over project A.
33
Limitations of the IRR Rule: Mutually Exclusive Projects (cont’d)
• Again, this can be seen graphically:
NPV
600 Project B
400
Project A
200
Discount
5% 10% 15% 20% 25% 30% rate
-200
34
Potential Advantages of the IRR Rule
• In short, the IRR rule yields the same answers as the NPV rule when it is
used properly.
• The IRR rule is sometimes preferred to the NPV rule for a number of reasons.
1. People understand and can relate to rates better than to present values.
2. The IRR is the yield on each dollar I invest.
3. This can be useful when different investors invest a different amount in
a given project/firm/fund (e.g., venture capital, hedge funds).
4. Also, the size of a project’s net present value can be misleading when
capital is not easy to raise. For example, a net present value of $1 million
is good if the initial investment is $1 million, but not so good (and maybe
not worth it) if the initial investment is $100 million.
5. Because the IRR rule splits the capital budgeting problem into two dis-
tinct parts (cash flow estimation, hurdle rate estimation), the source of
35
value may be more apparent than with the NPV rule.
The Profitability Index Rule
• The profitability index is defined as the present value of future cash flows
divided by the initial investment:
T
PV Ct /(1+rt )t
PI = −C0
= t=1
−C0
.
• The PI rule consists in accepting a project if and only if P I > 1.
PV N P V − C0 NP V
• Note that since P I = − =− =1+ , the PI rule is equiv-
C0 C0 −C0
alent to the NPV rule (provided that C0 < 0).
36
The Profitability Index Rule (cont’d)
• As with the IRR rule, the PI rule can be misleading when applied to mutually
exclusive projects, unless we look at the incremental cash flows.
• This is shown in the following example (where we assume that the hurdle is
10%):
Project C0 C1 PI NPV at 10%
A -1,000 2,000 1.82 818
B -10,000 15,000 1.36 3,636
B-A -9,000 13,000 1.31 2,818
• Even though project A has a larger profitability index than project B, project B
should be undertaken because the incremental project from A to B has a PI
over one.
37
