Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 13
Capital Budgeting Under Uncertainty
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
1
Outline










13.1
13.2
13.3
13.4
Introduction
Risk-adjustment discount-rate method
Certainty equivalent method
The relationship of the risk-adjustment discount rate
method to the certainty equivalent method
13.5 Three other related stochastic approaches to capital
budgeting
13.6 Inflationary effects in the capital-budgeting procedure
13.7 Multi-period capital budgeting
13.8 Summary and concluding remarks
Appendix 13A. The use of the time-state preference and the
option-pricing models in capital budgeting
under uncertainty
Appendix 13B. Real Option approach for capital budgeting
decision.
2
13.2 Risk-adjustment discount-rate method
N
NPV = 
t=1
Xt
t I 0
(1 + r t )
(13.1)
where I0 = Initial outlay of the capital budgeting project; X t = A location
measure such as the median (or the mean) of the expected risky cashflow distribution Xt in period t; rt= Risk-adjusted discount rate appropriate
to the riskiness of the uncertain cash flow Xt; N = Life of the project.
13.3 Certainty equivalent method
N
Ct
NPV = 
t I0
t=1 (1 + i )
(13.2)
Where Ct = Certainty-equivalent cash flow at period t, I = Riskless interest
rate, N = Life of the project.
Ct  t X t ,
N
NPV = 
t=1
tX t
t
(1 + i )
- I0
(13.3) and (13.2’)
3
13.3 Certainty equivalent method
CAPM
E( Ri ) = R f + (E( Rm ) - R f ) (
Xi
 im )
 2m
= R f + (E( R m ) - R f ) Cov ( X i , R m ) ;
Vi
Vi
Vi : market
value of firm i
Multiply V and rearrange the equation above, we can have Equation (13.4)
V i=
0
i
V =
2
[E(
)]
[Cov
(
,
)]/

R
R
X
R
m
f
i
m
m
Xi
Rf
0
2
0
[E(
)
]
[Cov
(
,
)]/

R
R
X
R
m
f
i
m
m
Xi
Rf
(13.4)
(13.4’)
(13.2’’)
NPV  Vi 0  I 0
If Rf = 10%, E(Rm) = 15%, X = $300, I0 =$2,000, Bj = 0.80,  m=0.02,
0
then Cov ( X i , Rm) = (080)(0.02)(2,000) = 32, and from Eq. (13.4′) we have
2
V 
0
i
300 - [(0.15 - 0.10)(32)] / 0.02 300 - (2.5)(32) 220
=

 $2, 200.
0.10
0.10
0.10
4
13.4
The relationship of the risk-adjustment discount rate
method to the certainty equivalent method
N
NPV = 
t=1
tX t
Xt
Xt - ,
=
=
PV
I
t
0
t
t
t
(1 + i ) (1 + r t )
(1 + r t )
(13.5) and (13.6)
(1 + i )t
(1 + i )t+1
,  t+1 =
t=
t
(1 + r t )
(1 + r t+1 )t+1
rt =
1+ i

1/ t
t
- 1, r t+1 =
1+ i

1/(t+1)
t+1
-1
(13.7) and (13.8)
(13.9) and (13.10)
Under the Arrow (1971) and Pratt (1964) risk-aversion framework, α can be
derived as
E(z) - 
U ( X  E ( z ) )  EU [ X   ( Ez )], E ( z )     E ( z ),  =
E(z)
Where E( z ) = Actuarial value of risk; X is asset; and U′ (X) and U′(X) are second and first
derivatives with respect to utility function U(x) and  (risk _ premium)  1  z2 [U ''( X ) ]
2
U '( X )
5
13.4
The relationship of the risk-adjustment discount rate
method to the certainty equivalent method
Example 13.1
We assume that investors retain the
same attitude toward risk over
time, that is, α1 = α2 = α3 = 0.8.
Then the risk-adjusted discount rate
for the three periods is:

r1 =
(1 + 0.06)
- 1 = 0.325,
0.8
r2 =
(1 + 0.06)
- 1 = 0.185,
1/2
(0.8 )
r3 =
(1 + 0.06)
- 1 = 0.142.
1/3
(0.8 )
Therefore the risk-adjusted rate
decreases over time.
Example 13.2
In the capital-budgeting process, we
usually apply a constant riskadjusted discount rate to each
period’s cash flows, r1 = r2 = r3.
Assuming that this constant value is
0.185, we have:

1 =
(1 + 0.06)
= 0.8945;
(1 + 0.185)
2
1 + 0.06
] = 0.80;
2= [
1 + 0.185
3
1 + 0.06
=
[
] = 0.716,
3
1 + 0.185
and we see that the value of the
certainty equivalents will decrease
over time.
6
13.5 Three other related stochastic approaches
to capital budgeting
13.5.1 The Statistical Distribution Method
~
N

Ct + S n Ct
Sn
NPV
=
I0
 NPV = 
+
- I 0,
t
N
t
N
(1 + k )
(1 + k )
t=1 (1 + k )
t=1 (1 + k )
~
N
(13.11) and (13.12a)

1/2
+
Cov
(
,
)
(   t) (13.12b)
]
w
w
C
C
t


t

2t
N
 NPV  [ 
t=1
N
2
t
(1 + k )
N
 =1 t=1
Cov(Ct, Cτ) = ρτt στ σt,
If the cash flows are mutually independent, then Eq. (13.12b’) reduces to (13.13)
t
N
 NPV = 
t-1
t
(1 + k )
,  NPV =
N
 t2
 (1 + k )
(13.12’) and (13.13)
2t
t=1
Hillier (1969) combines the assumption of mutual independence and perfect
correlation in developing a model to deal with mixed situations.
=
N

2
yt
m
N

(h)
zt
2
)
+(

t
2t
t=1 (1 + k )
h 1 t=1 (1 + k )
(13.14)
7
Derivation of Equation (13.14)

Hillier (1969) combines the assumption of mutual
independence (Yt) and perfect correlation Z t ( h ) in
developing a model to deal with mixed situations.
 NPV =
N

2
yt
m
N

(h)
zt
2
)
+(

t
2t
t=1 (1 + k )
h 1 t=1 (1 + k )
(13.14)
The derivation of Equation (13.14) is as follows
 Assume the net cash flow at time t, Xt, is related to the
sources as follows
X t  Yt  Zt (1)  Zt (2)  ...  Zt ( m )
where Yt ,Zt (1) ,...Zt ( m ) follow normal distribution.
The random variables for Yt are mutually independent
while the random variables for Zt (1) ,...Zt ( m ) are perfectly correlated

8

 2Xt
N
2
NPV =[ 
t=1
(1 + k )
N
N
Cov ( X  , X t )
]

t
(1 + k ) (1 + k )
+2  
2t
 =1 t=1
where  2Xt   Yt2   2Z (1)   2Z ( 2 )  ...   2Z ( m )
t
t
t
Cov ( X  , X t )  Cov (Y  Z (1)  Z (2)  ...  Z ( m ) ,Yt  Z t (1)  Z t (2)  ...  Z t ( m ) )
 Cov ( Y  ,Y t )  Cov (Z (1) ,Z t (1) )  ...  Cov (Z ( m ) ,Z ( m ) )
Since Yt are mutually independent, Cov ( Y  ,Y t )  0
SinceZ t ( h ) are perfectly positive correlated ( t  1),
then Cov (Z ( h ) ,Z t ( h ) )   Z ( h )  Z ( h ) , h  1, 2,..., m


 Yt2   2Z   2Z  ...   2Z
N
2
NPV
N
=[
t=1
N
=[
t=1
N

t=1
=[
(1)
(2)
t
t
(1 + k )
t=1
2
Yt

m
(1 + k )
(1 + k )
N
+ ( 
2t
h=1 t=1
m
N
 Yt2
2t
 2yt
(1 + k )
t
2t
+ [
h=1 t=1
m
N
+(
h 1
t=1

(m)
t
2t
m
N
+ 2 
 Z  Z
N


(h)
h=1  =1 t= 1
2
Zt ( h )
m
N
+ 2
N
N
Z
Z

(h)
t
t
(1 + k ) (1 + k )
]

t
) + 2 
]
2t

t
(1 + k )
h=1  =1 t= 1 (1 + k ) (1 + k )
 2Z
N
(h)
t
(1 + k )
2t
 (zth ) )
t
(1 + k )
2


 =1 t= 1
(h)
 Z  Z
(h)

(h)
(h)
t
t
(1 + k ) (1 + k )
]]
9
13.5.1 The Statistical Distribution Method
Table 13.1 Expected cash flow for new product
Year
Source
Expected Value of
Net Cash Flow
(in thousands)
Standard Deviation
(in thousands)
0
Initial Investment
-$600
$50
1
Production Cash Outflow
-250
20
2
Production Cash Outflow
-200
10
3
Production Cash Outflow
-200
10
4
Production Cash Outflow
-200
10
5
Production Outflow – salvage Value
-100
15
1
Marketing
300
50
2
Marketing
600
100
3
Marketing
500
100
4
Marketing
400
100
5
Marketing
300
100
5
Ct
t
t=0 (1.04 )
NPV = 
= - 600 +
300 - 250
(1.04)

600 - 200 500 - 200
+
2
(1.04 )
(1.04 )3

400 - 200 300 - 100
+
4
(1.04 )
(1.04 )5
 $419.95
2
2
50
100 2 1/2
20
15
=
[
+
+
...
+
+
(
+
...
+
) ]  $398. by (13.14)
 NPV 50
2
10
5
1.04
(1.04 )
(1.04 )
(1.04 )
2
Reprinted by permission of Hillier, F., “The derivation of probabilistic information for the evaluation of risky
investments,” Management Science 9 (April 1963). Copyright 1963 by The Institute of Management Sciences.
10
13.5.1 The Statistical Distribution Method
Table 13.2a Illustration of conditional-probability distribution approach
Initial
Outlay
Period 0
(1)
Initial
Probability
P(1)
(2)
Net Cash
Flow
(3)
Conditional
Probability
P(3|2,1)
(4)
0.25
10,000
0.3
2000
0.5
0.25
0.3
Net Cash
Flow
(5)
2000
3000
4000
3000
Conditional
Probability
P(5|4,3,2,1)
(6)
Cash
Flow
(7)
Joint
Probability
(8)
0.2
1000
0.015
0.5
2000
0.0375
0.3
3000
0.0225
0.2
2000
0.03
0.5
3000
0.075
0.3
4000
0.045
0.2
3000
0.015
0.5
4000
0.0375
0.3
5000
0.0225
0.3
2000
0.045
0.4
4000
0.06
0.3
6000
0.045
0.3
4000
0.06
11
13.5.1 The Statistical Distribution Method
Table 13.2a Illustration of conditional-probability distribution approach (Cont’d)
Initial
Outlay
Period 0
(1)
Initial
Probability
P(1)
(2)
Net Cash
Flow
(3)
Conditional
Probability
P(3|2,1)
(4)
Net Cash
Flow
(5)
Conditional
Probability
P(3|2,1)
(6)
Cash
Flow
(7)
Joint
Probability
(8)
0.5
4000
0.4
5000
0.4
5000
0.08
0.3
6000
0.06
0.3
5000
0.045
0.4
7000
0.06
0.3
9000
0.045
0.25
4000
0.0125
0.5
6000
0.025
0.25
8000
0.0125
0.25
5000
0.025
0.5
7000
0.05
0.25
9000
0.025
0.25
7000
0.0125
0.5
9000
0.025
0.25
11000
0.0125
0.3
0.25
0.2
6000
0.5
0.25
7000
5000
7000
8000
12
13.5.1 The Statistical Distribution Method
Table 13.2b NPV and joint probability
PVA
NPV
Probability
PVA
NPV
Probability
12,024.96
2,024.96
0.06
4,661.2
-5,338.8
0.015
12,913.96
2,913.96
0.08
5,550.2
-4,449.8
0.0375
13,802.96
3,802.96
0.06
6,439.2
-3,560.8
0.0225
14,763.08
4,763.08
0.045
6,474.76
-3,525.24
0.03
16,541.08
6,541.08
0.06
7,363.76
-2,636.24
0.075
18,319.08
8,319.08
0.045
8,252.76
-1,747.24
0.045
13,948.04
3,948.04
0.0125
8,288.32
-1,711.68
0.015
15,726.04
5,726.04
0.025
9,177.32
-822.68
0.0375
17,504.04
7,504.04
0.0125
10,066.32
66.32
0.0225
16,686.16
6,686.16
0.025
8,297.84
-1,602.16
0.045
18,464.16
8,464.16
0.05
10,175.84
175.84
0.06
20242.16
10,242.16
0.025
11,953.84
1,953.84
0.045
19,388.72
9,388.72
0.0125
21,166.72
11,166.72
0.025
22,944.72
12,944.72
0.0125
Discount Rate
NPV
Variance
Standard Deviation
= 4 percent
= $2,517.182
= $20,359,090.9093
= $4,512.105
13
13.5 Three other related stochastic approaches
Fig. 13.1 Decision tree. Numbers in parentheses are
to capital budgeting
probabilities; numbers without parentheses are NPV
13.5.2 The Decision-Tree Method
Decision
Variables
For the
First Stage
Net
Standard
Present Deviation
Value*
*
Coefficient
Of
Variation
Regional
$610.5
0
$666.98
1.093
National
767.27
1067.20
1.3909
International
629.71
1533.80
2.4357
Expressed in thousands of dollars.
NPV = 0.7(1,248.84) + 0.2(814.73)
+ 0.1(-14.54) = $1,035.68.
14
13.5.2 The Decision-Tree Method
Table 13.3 Expected cash flows for various branches of the decision tree
0
1
2
3
4
5
6
NPV
Regional Distribution throughout:
High
-2000
250
500
600
900
600
500
893.70
Medium
-2000
0
400
550
800
550
400
310.80
Low
-2000
-250
200
450
600
450
200
-614.61
National Distribution throughout:
High
-3000
350
700
1100
1400
1000
600
1,454.47
Medium
-3000
0
500
900
1200
800
700
498.90
Low
-3000
-350
100
700
1000
600
300
-1,036.73
International Distribution Throughout:
High
-4000
450
900
1600
2200
1400
800
2,350.71
Medium
-4000
0
600
1400
2000
1200
600
969.44
Low
-4000
-450
100
700
1500
700
400
-1,544.26
15
13.5.2 The Decision-Tree Method
Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d)
0
1
2
3
4
5
6
NPV
High National – High Inter.
-3000
350
-300
1600
2200
1400
800
2,145.08
High National – Medium Inter.
-3000
350
-300
1400
2200
1200
600
1,473.88
High National – Low Inter.
-3000
350
-300
700
1500
700
400
-144.85
Medium National – High Inter.
-3000
0
-500
1600
2200
1400
800
1,623.63
Medium National – Medium Inter
-3000
0
-500
1400
2000
1200
600
952.43
Medium National – Low Inter
-3000
0
-500
700
1500
700
400
-666.30
Low National – High Inter.
-3000
-350
-900
1600
2200
1400
800
917.27
Low National – Medium Inter
-3000
-350
-900
1400
2000
1200
600
246.07
Low National – Low Inter
-3000
-350
-900
700
1500
700
400
-1,372.66
16
13.5.2 The Decision-Tree Method
Table 13.3 Expected cash flows for various branches of the decision tree (Cont’d)
0
1
2
3
4
5
6
NPV
High Region – High National
-2000
250
-500
1100
1400
1000
600
1,248.84
High Region – Medium National
-2000
250
-500
900
1200
800
700
814.73
High Region – Low National
-2000
250
-500
700
1000
600
300
-14.54
Medium Region – High National
-2000
0
-600
1100
1400
1000
600
916.00
Medium Region – Medium National
-2000
0
-600
900
1200
800
700
481.89
Medium Region – Low National
-2000
0
-600
700
1000
600
300
-347.38
Low Region – High National
-2000
-250
-800
1100
1400
1000
600
490.71
Low Region – Medium National
-2000
-250
-800
900
1200
800
700
56.59
Low Region – Low National
-2000
-250
-800
700
1000
600
300
-772.68
17
13.5 Three other related stochastic approaches
to capital budgeting
13.5.3 Simulation Analysis
Table 13.4a Variables for simulation
Variables
Range
1. Market size (units)
2,500,000 – 3,000,000
2. Selling price ($/unit)
40 – 60
3. Market growth
0 – 5%
4. Market share
10 – 15%
5. Total investment required ($)
8,000,000 – 10,000,000
6. Useful life of facilities (years)
5-9
7. Residue value of investment ($)
1,000,000 – 2,000,000
8. Operating cost ($)
30 – 45
9. Fixed cost ($)
400,000 – 500,000
10. Tax rate
40%
11. Discount rate
12%
Notes:
a. Random numbers from Wonnacott and
Wonnacott (1977) are used to determine
the value of the variable for simulation.
b. Useful life of facilities is an integer.
Random number
01-19
20-39
40-59
60-79
80-99
00
Year
5
6
7
8
9
10
18
13.5.3 Simulation Analysis
Table 13.4b Simulation
Variables
1
2
3
4
5
VMARK
1
(39)2,695,000
(47)2,735,000
(67)2,835,000
(12)2,580,000
(78)2,890,000
PRICE
2
(73)$54.6
(93)$58.6
(59)51.8
(78)55.6
(61)$52.2
GROW
3
(72)3.6%
(21)1.05%
(63)0.0315
(03)0.0015
(42)0.021
SMARK 4
(75)13.75%
(95)14.75%
(78)0.139
(04)0.102
(77)0.1385
TOINV
5
(37)8,740,000
(97)9,940,000
(87)9,740,000
(61)9,220,000
(65)9,300,000
KUSE
6
(02)5 years
(68)8 years
(47)7 years
(23)6 years
(71)8 years
RES
7
(87)1,870,000
(41)1,410,000
(56)1,560,000
(15)1,150,000
(20)1,200,000
VAR
8
(98)$44.7
(91)$43.65
(22)33.3
(58)38.7
(17)$32.55
FIX
9
(10)$410,000
(80)$480,000
(19)419,000
(93)493000
(48)448,000
TAX
10
40%
0.4
0.4
0.4
0.40
DIS
11
12%
0.12
0.12
0.12
0.12
$197,847.561
$7,929,874.28
7
$12,146,989.5
79
$1,169,846.55
$15,306,245.2
93
NPV
Number in parentheses is the generated random number.
19
13.5.3 Simulation Analysis
Table 13.4b Simulation (Cont’d)
VARIABLES
6
7
8
9
10
VMARK
1
(89)2,945,000
(26)2,630,000
(60)2,800,000
(68)2,840,000
(23)2,615,000
PRICE
2
(18)43.6
(47)49.4
(88)$57.6
(39)$47.8
(47)$49.4
GROW
3
(83)0.0415
(94)0.047
(17)0.0085
(71)0.0355
(25)0.0125
SMARK 4
(08)0.104
(06)0.103
(36)0.118
(22)0.111
(79)0.1395
TOINV
5
(90)9,800,000
(72)9,440,000
(77)9,540,000
(76)9,520,000
(08)8,160,000
KUSE
6
(05)5 years
(40)7 years
(43)7 years
(81)9 years
(15)5 years
RES
7
(89)1,890,000
(62)1,620,000
(28)1,280,000
(88)1,880,000
(71)1,710,000
VAR
8
(18)$32.7
(47)37.05
(31)$34.65
(94)44.1
(58)$38.7
FIX
9
(08)408,000
(68)468,000
(06)406,000
(76)476,000
(56)456,000
TAX
10
0.4
0.4
0.4
0.4
0.4
DIS
11
0.12
0.12
0.12
0.12
0.12
$-1,513,820.475
$11,327,171.67
$839,650.211
$-6,021,018.052
$563,687.461
NPV
NPV=4,194,647.207
Random number
Useful life
Notes. 1. Definitions variables can be found in Table 13.4a.
2. NPV calculator procedure can be found in Table 13.4c.
01-19 20-39 40-59 60-79 80-99 00
5
6
7
8
9
10
20
13.5.3 Simulation Analysis
[Sales Volume]t = [(Market Size)  (1 + Market Growth Rate)t]
 (Share of Market),
EBITt = [Sales Volume]t  [Selling Price - Operating Cost]
- [Fixed Cost];
[Cash Flow]t = [EBIT]t  [1 - Tax Rate];
Useful life
NPV =

t=1
[CashFlow ] t
- I0 ,
t
(1 + DiscountRa te )
21
13.5.3 Simulation Analysis
Table 13.4c Cash Flow Simulations
Period
1
2
3
4
5
1
$2,034,382.33
5
$3,368,605.53
1
$4,260,506.32
7
$2,376,645.06
4
$4,549,425.96
1
2
2,116,476.099
3,406,999.889
4,402,631.377
2,380,653.731
4,650,608,707
3
2,201,525.239
3,445,797.388
4,549,233.365
2,384,668.412
4,753,916.289
4
2,289,636.147
3,485,002.261
4,700,453.316
2,388,689.114
4,859,393.331
5
2,380,919.049
3,524,618.785
4,856,436.695
2,392,715.848
4,967,085.391
6
3,564,651.282
5,017,333.551
2,396,748.622
5,077,038.985
7
3,605,104.120
5,183,298.658
8
3,645,981.714
5,189,301.603
5,303,921.737
22
13.5.3 Simulation Analysis
Table 13.4c Cash Flow Simulations (Cont’d)
Notes. 1. NPVs are listed in Table 13.4b
Period
6
7
8
9
10
1
$1,841,398.6
55
$1,820,837,7
60
$4,344,679.6
68
$439,076.86
4
$2,097,642.4
48
2
1,927,975.89
9
1,919,614.73
5
4,383,680.04
5
464,802.893
2,127,282.97
9
3
2,018,146.09
9
2,023,034.22
8
4,423,011.92
6
491,442.196
2,157,294,01
6
4
2,112,058.36
2
2,131,314,43
6
4,462,678.12
7
519,027.194
2,187,680.19
1
5
2,209,867.98
4
2,244,683.81
5
4,502,681.49
1
547,591.459
2,218,446.19
4
6
2,363,381.55
4
4,543,024.88
4
577,169.756
7
2,487,658.08
7
4,583,711.19
5
607,798.082
8
639,513.714
23
13.6 Inflationary effects in the capital-budgeting
procedure
N
C
NPV t = - I 0 + 
t=1
t
(13.15)
(1 + k )t
Where k = A real rate of return in the absence of inflation (i) plus an inflation
premium (η) plus a risk adjustment to a riskless rate of return (ρ).
Ct
Ct
V t-1
,
V 0=
=
,
=
V t-1
V0
t
1 + k t 
(1 + k t )t-1(13.16a), (13.16b), and (13.17)
1+ k t
N
Ct
Ct
NPV
=
=
,
V0

t-1 I 0
t-1
(1 + k t ) (1 + i i )
t=1 (1 + k t ) (1 + i )
NPVt   I 0
(13.18), and (13.19)
[ R t  tj=1 (1 +  j ) - O t  tj=1 (1 +  j ) - F t ](1 -  ) ( dep t )(  )
+
+
, (13.20)
t-1
t
t
t=1
(1 + i +  t )(1 + i )  j=1 (1 + j )
(1 + i )
Where Rt= Expected growth in cash flow, Ot= Outflow for variable
operating expense, θj= The percentage change in Ot induced by
inflation in period j. Ft = Expected fixed cash charge, dept= Fixed
noncash charge, τ= Marginal corporate tax rate, i= Real risk-free rate,
η= Inflation rate, ρ= Risk premium associated with uncertainty of
24
nominal cash flow.
N
13.6 Inflationary effects in the capital-budgeting
procedure
Define PV= present value of a one-period project, X= Net cash flow
received at the end of the period, I= net investment outlay at time 0,
r=risk-adjusted noninflation-adjusted required rate of return, τ=tax rate
applicable to the firm, and p’= change in the price level expected to
occur over the coming period.
PV = - I +
X -  (X - I)
(1 + p) X -  [(1 + p) X - I]
, PV = - I +
1+ r
(1 + r)(1 + p)
dPV
(1 -  ) dX

= - 1+
+
= 0.
dI
(1 + r) dI (1 + r)(1 + p)
dPV
(1 -  ) dX  -  p
= - 1+
+
dI
(1 + r) dI (1 + r)
N
X
t(1 -  ) + D
-I
NPV 0 = 
t
(1 + r )
t=1
(13.21), and (13.22)
(13.23)
(13.24)
25
13.6 Inflationary effects in the capital-budgeting
procedure
(1 -  ) + D
(1 -  ) + D
NPV = X 1
+X t
-I
t
1+ r
(1 + r )
t=2
N
(13.24’)
n
X L(1 -  )(1 + p) + D
X
t(1 -  ) + D
+
NPV P =
t
(1 + p)(1 + r)
(13.25)
(1
+
r
)
t=2
D p
D p
X
1(1 -  )
[L - 1 ] , L = 1 +
NPV p - NPV 0 =
(1 + r)
X 1(1 -  )(1 + p)
X 1(1 -  )(1 + p)
(13.26), and (13.27)
TABLE 13.5
Condition
L
X/TA
1+r
D/X
L > L’
L = L’
L < L’
+
+
+
+
0
-
0
+
-
From Kim, M. K. L., “Inflationary effects in the capital-investment process: An empirical examination,”
Journal of Finance 34 (September 1979): Table 1. Reprinted by permission.
26
13.6 Inflationary effects in the capital-budgeting
procedure
TABLE 13.6 Regression results of Eq. (13.28) (Figures in
parentheses are t values)
Gj = a + bLj + cXτAjZj - drjZj - eDXj + uj,
(13.28)
Period
a
b
c
d
e
1965-76
1965-70
1971-76
.0884
.0882
.1044
.0103
(14.74)
.0131
(9.93)
.0097
(5.90)
.0957
(4.88)
-.0079
(0.37)
.1046
(4.05)
-1.0007
(2.35)
.0670
(0.15)
-.8642
(1.87)
-.0545
(4.42)
-.0297
(1.46)
-.0870
(5.63)
From Kim, M. K. L., “Inflationary effects in the capital-investment process: An empirical examination.” Journal of
Finance 34 (September 1979): Table 4. Reprinted by permission.
27
13.7 Multi-period capital budgeting
E ( V 1 ) - L0E[Cov ( V 1,V M1 )]
V0=
1+ r f
(13.29)
Where V1 = Random value of the firm at the end of the time period;
VM1 = Random value of the market value of all firms at the
end of the time period; L0 = Market-determined price of risk;
rf = Riskless rate-of-return available to all investors.
V 0 + V p1 =
E[ V 1 + V p1 ] - L0E[Cov ( V 1 + V p1,V M 1 )]
1+ r f
(13.30)
E[Vp1] - L0E[Cov (Vp1, VM1)] - Vp0 > 0.
E[ C t ] - Lt-1E[Cov ( C t ,V mt )]
V P(t-1) =
1 + r f(n-1)
(13.31)
28
13.7 Multi-period capital budgeting
1 + Q1 b Im
=
Q0
 2m
(13.32)
where η = Elasticity of expectations of future earnings stream;
b = Firm-specific constant measuring sensitivity of the disturbance term to
unanticipated changes in the economic index;
2
 m = Variance of the market asset’s rate-of-return; σIm = Cov( IT , RmT ) ;
IT represents the unanticipated changes in some general economic index;
RmT= Market return; Qt= Cash-flow multiplier for period t.
E[Ri] = rf + L[cov (Cov (Ri, Rm))]
(13.33)
where Ri = Rate-of-return on the risky asset I; rf = Riskless rate-ofreturn available to all market participants;L = Price of risk in the
market, (E(Rm) - rf)/ [variance of the returns on the market];
RM = Return on a market portfolio of risky securities.
E( D j ) + E( P j1 ) dE( X i )
I d cov ( X i , R m )
>r f +
,
P 0j =
dI
dI
1 + E( R j )
(13.34), and (13.35)
29
13.8 Summary and concluding remarks
The preceding discussion outlined three alternative capitalbudgeting procedures, each useful when cash flows are not known
exactly but only within certain specifications. Depending on the
correlation of the cash flows and the number of possible outcomes,
they will all yield meaningful results. Simulation was introduced as
a tool to deal with those situations in which the most uncertainty
existed.
Also discussed were various means of forecasting cash flows, most
notably the product life-cycle approach. The emphasis in the later
portion of the chapter was on the effects inflation has on the ability
to forecast cash flows and on the appropriate discount-rate
selection. We stress the importance of this factor as its
nonrecognition can lead to disastrous results.
In addition, we touched upon more theoretical issues by attempting
to apply the CAPM, which created problems, but the basic
approach is still applicable.
30
13.8 Summary and concluding remarks
Lastly we investigated some of the more generalized meanvariance pricing frameworks and found that they were essentially
equivalent. The application of these approaches is much the same
as that of the CAPM, and further supports the increasing use of this
technique in dealing with a very large, if not the largest problem
area in applied finance theory, capital budgeting under uncertainty.
The concepts, theory, and methods discussed in Chapters 12 and
13 are essentially based upon a myopic view of capital budgeting
and decision making. This weakness can be improved by using
Pinches’ (1982) recommendations. Concern should be given to
how capital budgeting actually interfaces with the firm’s strategic
positioning decision: to deal with risk effectively; to improve the
control phase; and to take advantage of related findings from other
disciplines. A broader examination of the capital-budgeting process,
along with many effective business/academic interchanges, can go
a long way toward improving the capital-budgeting process.
31
Appendix 13A. The use of the time-state preference and the
option-pricing models in capital budgeting under uncertainty
n
t
(13.A.1)
PV    (Vst ) ( Z st )
s 1 t 1
where Vst= current value (price) of a dollar for state s and time
t, Zst= present value of cash flow for state s and time t, PV=
present value of project.
TABLE 13.A.1 Expected cash flows for Project,
State of Economy
Year 1
Year 2
Year 3
Boom
Normal
Recession
$1000
$800
$500
$500
$400
$200
$300
$200
$100
PV = $1000(0.1672) + $800(0.2912) + $500(0.5398)+ $500(0.1693)
+ $400(0.2915) + $200(0.5333) + $300(0.1686) + $200(0.2903)
+ $100(0.5313) = $1,140.
32
Appendix 13A. The use of the time-state preference and the
option-pricing models in capital budgeting under uncertainty
2
ln
(
/
)
+
[(i
S
M
M
0
j
M ) / 2]
-i
Vj = e N [d2 (Mj)], d 2 =
SM
(13.A.2), and(13.A.3)
2
where i= Interest rate,S M = Instantaneous variance of the rateof-return on the market portfolio, N(•) = Probability of d2(Mj)
obtained from a normal distribution.
M0 = 1 ,
(13.A.4)
M j 1+ r Mj
where rMj = Rate-of-return on the market portfolio for the
period if M0  Mj.
ΔVj = Vj - Vj+1, Vj = e-1(N[d2 (Mj)] - N [d2 Mj+1])
(13.A.5), and (13.A.6)
vj = e-i (N[D2 (rMj)] - N[d2(rM,(j+1))], Vj = vj + Vj+1.
33
(13.A.7), and(13.A.8)
Appendix 13A. The use of the time-state preference and the
option-pricing models in capital budgeting under uncertainty
V2  V
HH

 LH
V
HL 
LL 
HH
L
 H
 H H  H L LH 
 H H
  LH H H  LL LH 
HL 
LL 
 H H H L  H L LL 
.
 LH H L  LL LL  
34