Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 12
Capital Budgeting Under Certainty
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
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Outline
12.1 Introduction
12.2 Cash-flow evaluation of alternative investment
projects
12.3 Alternative capital-budgeting methods
12.4 Comparison of the NPV and IRR method
12.5 Equivalent Annual NPV and Equivalent Annual Cost
12.6 Capital rationing decision
12.7 Summary
Appendix 12A. NPV and break-even analysis
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Appendix 12C. Crossover rate
12.2 Cash-flow evaluation of alternative investment
projects
D
S

(11.18)
)  ke (
)
WACC  (1-  c )k d (
DS
DS
Rt + NtPt = Ntdt + WSMSt + It,
(12.1)
where Rt = Revenue in period t, NtPt= New equity in period t,
Ntdt= Total dividend payment in period t,
WSMSt= Wages, salaries, materials, and service payment in period t,
It= Investment in period t.
Annual After-Tax Cash Flow = ICFBT - (ICFBT - Δdep)τ
= ICFBT (1 - τ) + (dep)τ,
(12.2)
Where ICFBT = Annual incremental operating cash flows,
τ = Corporate tax rate, Δdep = Incremental annual depreciation charge, or
the annual depreciation charges on the new machine less the
annual depreciation on the old.
12.2 Cash-flow evaluation of alternative
investment projects
ICFBT(the annual incremental operating cash flows) = ΔRt - ΔWSMSt.
(12.3)
N
dep t
NPV (tax benefit)   
t
(1

k
)
t 1
(12.4)
12.3 Alternative capital-budgeting methods
TABLE 12.1
Year
A
B
C
D
0
-100
-100
-100
-100
1
20
0
30
25
2
80
20
50
40
3
10
60
60
50
4
-20
160
80
115
12.3 Alternative capital-budgeting methods
12.3.1 Accounting Rate of Return
N
APt

(12.5)
N
ARR  t 0
I
Where APt = After-tax profit in period t, I=
Initial investment, and N = Life of the
project, the depreciation is $25.
12.3 Alternative capital-budgeting methods
12.3.2 Internal Rate of Return
N
CFt
I

t
t 1 (1  r )
(12.6)
where
CFt = Cash flow (positive or negative) in
period t,
I = Initial investment,
N = Life of the project.
12.3 Alternative capital-budgeting methods
12.3.3 Payback Method
Assume R=Cf1=Cf2=…=Cfn, Equation 12.6 can be written as the
equation (12.6’)
R
1
1
1
I
[1 

 ... 
]
2
N -1
1 r
(1  r ) (1  r )
(1  r )
(12.6’)
By the geometric series, Equation (12.6’) can be written as
R R
1
r   ( )[
]
N
I
I (1  r )
(12.7)
Equation (12.7) define the special relationship of payback method
and IRR
12.3 Alternative capital-budgeting methods
12.3.4 Net Present Value (NPV) Method
 The
payback method calculates the time
period required for a firm to recover the cost
of its investment. It is that point in time at
which the cumulative new cash flow from
the project equals the initial investment.
12.3 Alternative capital-budgeting methods
12.3.4 Net Present Value (NPV) Method
N
NPV method
CFt
NPV  
I
t
t 1 (1  k )
NPV and break-even quantity
N
1
CF [
] I
t
t 1 (1  k )
[ I - (dep) ]/(1-  )
*
1
}(
).
Q { N
t
p-v
t 1 1/[(1  k ) ]
(12.8)
(12.8′)
12.3 Alternative capital-budgeting methods
12.3.5 Profitability Index
 [CF t /((1  k ) )]
PI 
.
I
t
N
t 1
(12.9)
Project
Initial Outlay
Present Value
of Cash Inflows
NPV
PI
A
100
200
100
2
B
1000
1300
300
1.3
12.4 Comparison of the NPV and IRR method
12.4.1 Theoretical criteria
Fig. 12.1 NPVs of Projects A and B at different discount rates.
12.4 Comparison of the NPV and IRR method
12.4.2 Multiple Rates-of-Return
Year
0
Cash Flow
－50
1
2
750
－800
750
800
NPV  0   50 

(1  IRR) (1  IRR) 2
50(1  IRR)  750(1  IRR)  800  0
2
50 IRR 2  650 IRR  100  0
(650 )2 - 4(50)(100)
IRR = 650 
,
100
IRR = 0.1557 or 12.84.
12.4 Comparison of the NPV and IRR method
12.4.2 Multiple Rates-of-Return
0
1
－100
Cash Flow
Year
0  100 
250
2
－160
250
160

,
(1  IRR) (1  IRR) 2
0  100(1  IRR) 2  250(1  IRR)  160,
0  10  20 IRR  10 IRR 2  25  25 IRR  16,
10 IRR 2  5 IRR  1  0,
IRR 
5  25  40
1 1
, IRR  
15i.
20
4 20
12.4 Comparison of the NPV and IRR method
TABLE 12.2
12.4.3 Reinvestment Rate Problem
Year
Project
0
1
2
3
NPV
A
-100
50
100
500
380.2537
B
-200
600
100
50
451.02269
C
-300
100
700
100
418.49945
A+B
-300
650
200
550
831.27505
B+C
-500
700
800
150
869.52214
A+C
-400
150
800
600
798.75182
TABLE 12.2a
>IRR
A
1.10438
B
2.18184
C
0.76360
A
1.67275
B+C
1.19227
A+C
0.87172
12.5 Equivalent Annual NPV and Equivalent Annual Cost

12.5.1 Mutually Exclusive Investment Projects with
Different lives
Year
Project A
Project B
0
－100
－100
1
70
50
2
70
50
3
NPV ( N , )  NPV ( N ) 
50
NPV ( N ) NPV ( N )

2 N  ...
N
(1  K )
(1  K )
12.5.1 Mutually Exclusive Investment Projects with
Different lives
NPV(N,t) = NPV(N)(1 + H + H2 + ... Ht).
(12.10)
H[NPV(N,t)] = NPV(N)(H + H2 + ... + Ht + Ht+1).(12.11)
NPV(N,t) - (H)NPV(N,t) = NPV(N)(1 - Ht+1),
NPV(N)(1- H t+1 )
NPV(N,t) =
.
1- H
1
limNPV ( N , t )  NPV ( N , )  NPV [1  [1/(1  K )
t 
 NPV ( N )[
N
]
],
(1  K ) N
(1  K )  1
N
].
(12.12)
12.5.1 Mutually Exclusive Investment Projects with
Different lives
For Project A:
For Project B:
3
 (1  0.12) 2 
(1

0.12
)
NPV (2, )  NPV (2) 
]
 NPV (3, )  NPV (3) [
2
3
(1  0.12) -1
 (1  0.12)  1 
1.4049
 1.2544 
 (18.30) 
 20.09 [
]

0.4049
 0.2544 
 69.71.
 90.23.
NPV ( N )
KNPV ( N , ) 
Annuity factor
where the annuity factor is
[1 - (1 + K)
-N
]/K.
(12.13)
Equivalent Annual Cost
NPV(N)= K × NPV(N,∞) × Annuity Factor
NPV(N) = C × Annuity Factor
C = K × NPV(N,∞)
(12.14)
(12.15)
(12.16)
250
250
250
250
NPV(A) 1000



 1792.47
2
3
4
1.1 (1.1) (1.1) (1.1)
4
1792.47 C  A 0.1
 C  3.1699
C=565.47
4
4
NPV (N, ∞) = 1749.47×(1+0.1) / [(1+0.1) -1] = 5654.71
C = K* NPV (N, ∞) = 0.1× 5654.71 = 565.47
(12.17)
12.6 Capital rationing decision
12.6.1 Basic Concepts of Linear Programming
Maximize (or minimize) Z = c1X1 + c2X2 + ... + cnXn,
Subject to:
a11X1 + a12X2 + ... + a1nXn () b1,
a21X1 + a22X2 + ... + a2nXn () b2,
.
.
.
.
.
.
am1X1 + am2X2 + ... + amnXn () bm,
Xj  0, (j = 1, 2, ..., n).
12.6 Capital rationing decision
12.6.2 Capital Rationing
Year
Project
0
1
2
3
4
5
X
－100
30
30
60
60
60
Y
－200
70
70
70
70
70
Z
－100
－240
－200
400
300
300
Investment
NPV
X
65.585
Y
52.334
Z
171.871
Year 0
Year 1
Year 2
$300
$70
$50
12.6.2 Capital Rationing
Maximize V = 65.585X + 52.334Y + 171.871Z
+ 0C + 0D + 0E
100X + 200Y + 100Z + C + 0D + 0E = 300.
-30X - 70Y + 240Z - C + D + 0E = 70,
-30X - 70Y + 200Z + 0C - D + E = 50.
X  1,Y  1, Z  1.
If V=$208.24, then X = 1.0, Y = 0.6586, Z = 0.6305.
Funds Constraint
Shadow Price
1st period
0.4517
2nd period
0.4517
3rd period
0.0914
12.7 Summary
Important concepts and methods related to
capital-budgeting decisions under certainty were
explored in this chapter. Cash-flow estimation
methods were discussed before alternative
capital-budgeting methods were explored. A
comparison of the NPV and IRR methods was
investigated in accordance with both theoretical
and practical viewpoints. Issues relating different
project lives were explored in some detail. Finally,
capital-rationing decisions in terms of linear
programming were discussed. In the next chapter,
issues relating to capital budgeting under
uncertainty will be explored. In Chapter 14, the
lease-vs.-buy decision will be investigated.
Appendix 12A. NPV and break-even analysis
T
NPV(k) =  R(t) - C(t)e dt,
- t
(12.A.1)
0
where
NPV(k) = Net present value of the project
discounted at cost-of-capital rate k;
R(t) = Stream of cash revenues at time t;
C(t) = Stream of cash outlays at time t;
T = Investment time horizon;
ρ = Continuously compounded discount rate
which is equal to loge(1 + k)
Appendix 12A. NPV and break-even analysis
RI
Ct 
A
for 0  t  A,
(12.A.2)
Where R = RDTE costs, I = Total initial outlay on production
facilities, A = Time up to the onset of production.
YQ = Y1Q-b
(12.A.3)
where Q = Number of aircraft produced; YQ = Cumulative
average production cost for Q aircraft produced;
b = –log (γ)/log(2); γ = “Learning coefficient,” which remains
constant over all Q;Y1 = First unit cost of production.
Appendix 12A. NPV and break-even analysis
(1-b)
The cumulative total production cost TC(t) = Y1[Q(t)]
(12.A.4)
The rate of production cost
Q t (12.A.5)
C (t )  (1  b)Y 1Q(t ) (
).
t
-b
Q(t) = (t - A)N.
(12.A.6)
C(t) = (1 - b)Y1(t - A)-bN(1-b)
(12.A.7)
for A = 42, B = 0.369188, Y1 = $100 million, t > A.
j n
k   (W j k j )
(12.A.8)
where kj = Effective annual after-tax cost per dollar of the jth
source of funds; Wj = Proportion of the jth source of funds
in the long-run capital structure.
j 1
Appendix 12A. NPV and break-even analysis
k = 0.3kd + 0.7ke
(12.A.9)
where kd = after-tax cost of debt, and
ke = after-tax cost of equity.
D
k e = + g,
P
(12.A.10)
Where D = dividend per share, P = Net
proceeds per share after flotation costs,
g = Average annual compound growth rate
Appendix 12A. NPV and break-even analysis
R(t) = PN for t > A,
T
R+ I
- t
NPV (k ) = [1 - tx][ NP  e dt A
A
(12.A.11)
A
e
- t
0
dt
(12.A.12)
T
 (1 - b)Y 1 N (1-b)  (t - A )-b e - t dt ] ,
A
where
tx = Effective tax rate on corporate profits
for Lockheed, and
ρ = Discount rate.
Appendix 12A. NPV and break-even analysis
1  e   (T  A )
R  I 1  e  A
NPV (k )  [1  tx][ NP (
)(
)(
)
 e A
A

 (1  b)Y 1 (
N

)
(1b )   A
Fig. 12.A.1 (From Reinhardt, H. E., “Breakeven analysis for Lockheed’s Tri Star: An
Application,” Journal of Finance 28
(September 1973): 830. Reprinted by
permission.)
e
( 1) j[  (T  A)] j 1b
[
]]

j
![
j

1

b
]
j 0

(12.A.13)
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
In an attempt to determine exactly what tools were needed by
practitioners and what methods they were currently using in capital
budgeting, Mao (1970), Hastie (1974), Fremgen (1973), Brigham and
Pettway (1973), Shall, Sundem, Geijsbeek (1978), and Oblak and
Helm (1980) conducted surveys and field studies of companies. The
papers that emerged from these studies provide great insight into the
gulf that exists between theory and practice, and attempt to explain the
reasons for this gulf.
Mao (1970) in “Survey of Capital Budgeting: Theory and Practice,”
specifically examines three areas of capital budgeting and the disparity
between theory and practice in each area. He first considers the
objective of financial management, which, according to theory, is to
maximize the market values of the firm’s common shares. Price per
share is, according to theory, a function of its expected earnings, the
pure rate of interest, the price of risk, and the amount of risk as
measured by covariance between its return and other returns. Of
course, current theory does not provide any all-encompassing criteria
by which to choose between alternative time patterns of share prices
within the planning horizon, so the businessman has no way to
accurately implement plans to increase share price.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Nevertheless, most executives interviewed implied that maximization of
the value of the firm was their goal, although they phrased the idea in
more operationally meaningful terms. However, in a break from theory,
most executives didn’t consider diversification by investors as having
much impact on the value of the firm. According to theory, in a portfolio
context only the nondiversifiable risk is relevant. While the major
institutional investors, with large staffs of investment analysts, may fit
into this portfolio context, many other investors will not. The executives
saw consistent growth as a more important factor determining share
value.
Mao next considered the theory and practice of risk analysis. The
theoreticians measure risk by the variance of returns. Mao suggests,
and I agree, that semi variance is a better measure of risk because it
measures only downside risk. Management will not see the possibility
of excess returns as a risk, but will focus on the risk of failing to earn an
adequate return. The executives interviewed also emphasized
downside risk and one called the chance of excess returns “a negative
risk (a sweetener).” Those interviewed also expressed risk as a
danger of insolvency when a large amount of capital was to be
invested.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Theory recommends either of two methods for incorporating risk into
investment analysis: the certainty-equivalent approach and the riskadjusted discount-rate approach. These two approaches will be
explored in Chapter 13. When more than one investment may be
made, the theory advocates the use of the portfolio approach.
The practitioners depended in general on a risk-adjusted discount rate
approach to incorporate risk, although their actual methods may be
more rudimentary than the purely theoretical approach dictates.
Consideration is given to the human factors of enthusiasm and
dedication to the project, qualities that are nonquantifiable. Interviews
also disclosed a definitional difference between theorists and
practitioners about the word diversification. In theory, every project
should be evaluated in terms of its covariance with other projects in the
portfolio. In practice, diversification is a much more subjective, longrange process where only major activities and their impact on
diversification are considered.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Mao next revives a topic considered earlier: how to measure returns on
projects. Theory immediately discounts payback period and accounting profit
in favor of internal rate-of-return and net present value. Interview results show
that only two of the eight companies use Internal Rate-of-Return alone,
whereas six use payback and accounting profit alone or in conjunction with
internal rate-of-return. Theorists have advanced two explanations for this
incongruence. First, internal rate-of-return and net present value do not
consider the effect of an investment on reported earnings. Stability of
estimated EPS is important to management and investors alike, and these two
criteria do not give management an indication of expected stability of earnings.
Many companies neglect the net-present-value method because of the
extreme difficulty of determining the appropriate discount rate. Individual
company characteristics also determine, to a large extent, which measurement
criteria are most appropriate. Lastly, Mao recommends types of research that
can make theory more useful and meaningful to practitioners.
K. Larry Hastie (1974), himself a practitioner, also tries to give the academic
world some advice on how to better aid the businessman. According to Hastie,
in “One Businessman’s View of Capital Budgeting,” what is needed is not
refinement or multiplication of measurement techniques but a re-evaluation of
the assumptions inherent in the capital-budgeting process.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Hastie outlines the major problems practitioners face in capital
budgeting. First, most companies are limited by capital rationing, so
the problem becomes not one of finding adequate projects, but of
choosing from among the acceptable projects. Theory offers no means
of ranking projects with different risks, strategic purposes, and quality
of analytical support. Ranking per se is not an adequate selection
method unless the more qualitative criteria can somehow be
incorporated into the process.
Judgments enter into any process in which uncertain profits must be
estimated. Hastie highlights two types of errors in judgment that can
lead to failure to achieve expected returns on projects. The first is
caused by excessive pessimism or optimism, with only the second
posing a serious problem. Overpessimism is akin to “upside” risk in
that the company will not fail to meet its goal. Overoptimism is caused
by poor judgment concerning future uncertainties, which in many cases
could be cured only by hiring accurate fortune tellers.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Hastie also recognizes that, in many cases, it is not the measurement
method but the financial analyst who fails. The financial analyst must
have a good grasp of the quantitative and qualitative impacts of each
project and must be able to communicate this information to the
decision makers. Those preparing expenditure requests should be
objective and realistic.
Hastie recommends several methods to improve capital-budgeting
techniques. First, corporate strategy must be clarified and
communicated so that projects incompatible with this strategy will not
be needlessly analyzed. Second, analytical techniques must be
evaluated. They should be understood by all who work with them and
should generate the type of information used by the company in
decision making. Hastie recommends the use of sensitivity analysis to
isolate critical variables and give an expanded, more realistic range of
estimated profits. What is essential is that those involved in the capitalbudgeting process understand corporate strategy and policy and
generate realistic data, which can effectively communicate to top-level
management.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
James Fremgen (1973) in “Capital Budgeting Practices: A Survey,” continues
the analysis of practitioner use of capital-budgeting techniques, and offers
some support for Hastie’s position that measurement techniques are not the
only important factor in capital budgeting. His survey again finds that payback
period and accounting profit are widely used as selection criteria, contrary to
theoretical approval of these methods, but also finds strong support for the use
of internal rate-of-return. His results, however, do highlight a problem
encountered when using the internal rate-of-return method -- the multiple
internal rate-of-return. His results also give some support to Doenges’
recommendation that firms try to predict reinvestment rates for the funds to be
received form projects being currently evaluated. Although of the 29 percent
which projected reinvestment rates, the majority used current rates-of-return or
costs of capital, some tried to estimate future reinvestment rates based on
predicted future rate-of-returns or cost of capital.
A majority of those questioned used some technique to measure risk and
uncertainty when analyzing investment projects. Again, however, a problem
arises when deciding how to quantify this risk into the analysis. Most firms
appear to require an unspecified amount of additional profit for additional risk.
Of course, much of the analysis of projects is based on nonfinancial or
nonquantitative judgments, and companies may feel that risk is best handled in
this manner.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Fremgen confirmed the previously mentioned conclusion that capital
rationing is a major influence on the capital-budgeting process. This
rationing, commonly caused by a limitation on borrowing, was dealt
with by most of the surveyed companies through ranking of projects.
Although Hastie says this is not an adequate method of project
selection, Fremgen makes little mention of non-financial, subjective
methods of selection. Since project selection must be based on both
financial and nonfinancial data, the results received must be due to
wording of the question, which disallowed nonfinancial answers.
Providing impressive support for Hastie’s position, Fremgen next
described three stages of capital budgeting, only one of which dealt
with financial analysis of the project. The results clearly reveal that
financial analysis is considered neither the most critical nor the most
difficult stage of the capital-budgeting process. More academic
attention should be focused on the stage of project definition and
estimation of cash flows, and the implementation and review stage of
the process. Although these two stages are more difficult to adapt to
quantitative methods, they would be more useful for the practitioner.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
One final analysis of capital-budgeting theory and practice
deals with a specific, fairly unique industry. Eugene
Brigham and Richard Pettway (1973), in “Capital Budgeting
by Utilities,” studied the practices in this heavily regulated
industry. Regulation has a profound effect on capitalbudgeting practice, and the theory behind this regulation
has become antiquated with the advent of double-digit
inflation.
The regulators specify a target rate-of-return for utility
companies, which then determine the rates they can
charge consumers. However, inflation has caused the
actual rate-of-return to fall below the “reasonable” rate-ofreturn, and, due to the lags in the regulatory process, new
targeted rates-of-returns, when implemented, already have
fallen behind inflation.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Another unique feature of the utility industry is that, due to legal requirements,
they must make “mandatory” investments when needed to provide service
upon demand. These mandatory investments, the major component of the
capital budget, frequently offer rates-of-return below the utility’s cost of capital.
Although discretionary investments may provide higher returns, rarely can
these excess returns counterbalance the effects of inflated operating costs,
rising cost of capital, mandatory investments, and regulatory lags. Thus, the
cost of capital exceeds the actual rate-of-return in the capital-investment
budget.
Because of this unique situation, utilities must be very cautious when deciding
which discretionary projects to accept. Projects with high rates of return are
needed to help compensate for other losses. For mandatory investments,
revenues are disregarded and alternatives evaluated solely on the basis of
costs on a discounted cash-flow basis. Due to the urgency of keeping costs as
low as possible for mandatory investments, and profits as high as possible for
discretionary investments, 94 percent of the companies use the discounted
cash-flow method to project future financial results more accurately. Risk is
also formally analyzed by over 50 percent of the utilities questioned.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
Surprisingly, only 49 percent of these companies indicated that they have
experienced capital rationing in the past five years, and most of these indicated
that their response would be to apply for a rate increase to alleviate the
problem. The most serious problem they face is securing permission to build
new generating plants, a problem not shared with other industries. Since it is
so crucial for rate determination, most utilities have ready cost-of-capital figures
to use in capital-budgeting analysis.
Obviously, many of the problems facing the utility industry are unique to the
industry, and the managers have developed different perspectives and policies
on capital budgeting to cope with these problems. There is a message in this
for all those involved with financial management. Regardless of the
academician’s recommendations, the competition’s practices, and the market’s
signals, capital-budgeting policy and practice must be adapted to suit the
individual firm’s characteristics and needs. Theory and practice are helpful
only to the extent that they can be successfully integrated into the individual
company’s financial structure. Theorists must try to recognize the needs of
financial practitioners, but the practitioners must also realize that no mere
formula will guarantee success, and realistic theories will help their financial
analysis and planning decisions.
Appendix 12B. Managers’ views on Alternative
capital-budgeting methods
The reader should be aware that, in practice, most firms use a
combination of capital budgeting techniques to arrive at investment
decisions. For instance, in a survey of large firms, Schall, Sundem,
and Geijsbeek (1978) found that 17 percent of those firms responding
used four of the capital-budgeting techniques outlined above, and 34
percent used three of the four in making decisions. More surprisingly,
although 86 percent of the firms used at least one of the discounting
methods, the most popular technique was found to be the payback
method, despite its disregard of several important factors. Perhaps the
continued use of simpler methods combined with the more accurate
NPV or IRR, points to the importance of ease of calculation for
practitioners. In addition, despite the frequently noted ambiguities
accompanying use of the IRR, the method enjoys a substantial and
continuing popularity in practice. In their paper “Survey and Analysis of
Capital-Budgeting Methods used by Multinationals,” Oblak and Helm
(1980) found that the IRR method and the payback method are two
most popular capital-budgeting methods used by multinational firms.
The continued use of IRR may be due to the fact that the rate-of-return
of a project has a more intuitive appeal and is therefore easier to
explain and justify within the firm than the more esoteric NPV criterion.
Appendix 12C. Derivation of Crossover Rate
Period
0
1
2
3
Project A
-10,000
10,000
1,000
1,000
Project B
-10,000
1,000
1,000
12,000
Cash flows of B-A
0
-9,000
0
11,000
Appendix 12C. Derivation of Crossover Rate
Figure 12.C.1 Net Present Value and IRR for Mutually
Exclusive Projects
NP V (A)
$5,000
NP V (B)
$4,000
$3,000
$2,000
$1,000
$0
-$1,000
-$2,000
0 .0 0 %
5 .0 0 %
1 0 .0 0 %
1 5 .0 0 %
2 0 .0 0 %
Appendix 12C. Derivation of Crossover Rate

NPV(A) = -10,000 + 10,000 / (1+Rc) + 1,000 /
(1+Rc)2 + 1,000 / (1+Rc)3
(12.C.1)

NPV(B) = -10,000 + 1,000 / (1+Rc) + 1,000 /
(1+Rc)2 + 12,000 / (1+Rc)3
(12.C.2)

NPV(A)=NPV(B)
Appendix 12C. Crossover rate
10,000 1,000
1,000
10,000 


2
1  Rc (1  Rc ) (1  Rc )3
1,000 1,000
12,000
 10,000 


2
1  Rc (1  Rc ) (1  Rc )3
1,000 10,000
0  [10,000  (10,000)]  [

]
1  Rc 1  Rc
1,000
1,000
12,000 1,000
[

] [

].....
2
2
3
3
(1  Rc ) (1  Rc )
(1  Rc ) (1  Rc )
(12.C.3)
Appendix 12C. Crossover rate
CF1 ( B  A)
0  CF0 ( B  A) 
1  Rc
where
CF2 ( B  A) CF3 ( B  A)


2
(1  Rc )
(1  Rc )3
CF0(B-A) =The different of net cash inflow between project A and project
B at time 0.
CF1(B-A) =The different of net cash inflow between project A and project
B at time 1.
CF2(B-A) =The different of net cash inflow between project A and project
B at time 2.
CF3(B-A) =The different of net cash inflow between project A and project
B at time 3.
In other word, Rc is the IRR of the project (B – A).