1. Financial Analysis, Planning and Forecasting Theory and Application
Chapter 14
Capital Budgeting Under Uncertainty By
Cheng F. Lee
Rutgers University, USA
John Lee
Center for PBBEF Research, USA

2. Outline 14.1 Introduction 14.2 Risk-adjustment discount-rate method
14.3 Certainty equivalent method
14.4 The relationship of the risk-adjustment discount rate method to the certainty equivalent method
14.5 Three other related stochastic approaches to capital budgeting
14.6 Inflationary effects in the capital-budgeting procedure
14.7 Multi-period capital budgeting
14.8 Summary and concluding remarks
Appendix 14A. Time-state preference and the real option approaches in capital budgeting under uncertainty

3. 14.2 Risk-adjustment discount-rate method
(14.6)
where I0 = Initial outlay of the capital budgeting project; = A location measure such as the median (or the mean) of the expected risky cash-flow 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.
(14.7)
Where Ct = Certainty-equivalent cash flow at period t, I = Riskless interest rate, N = Life of the project.
(14.8) and (14.7’)
14.3 Certainty equivalent method

4. 14.3 Certainty equivalent method
(14.9)
(14.9’)
(14.7’’)
CAPM
Vi : market value of firm i
Multiply V and rearrange the equation above, we can have Equation (14.9)
If Rf = 10%, E(Rm) = 15%, = $300, I0 =$2,000, Bj = 0.80, =0.02,
then Cov ( , Rm) = (080)(0.02)(2,000) = 32, and from Eq. (14.9′) we have

5. 14.4 The relationship of the risk-adjustment discount rate method to the certainty equivalent method
(14.10) and (14.11)
(14.12) and (14.13)
(14.14) and (14.15)
Under the Arrow (1971) and Pratt (1964) risk-aversion framework, α can be derived as
Where E( ) = 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

6. 14.4 The relationship of the risk-adjustment discount rate method to the certainty equivalent method
Example 14.1
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:
Therefore the risk-adjusted rate decreases over time.
Example 14.2
In the capital-budgeting process, we usually apply a constant risk-adjusted discount rate to each period’s cash flows, r1 = r2 = r3.
Assuming that this constant value is 0.185, we have:
and we see that the value of the certainty equivalents will decrease over time.

7. 14.5.1 The Statistical Distribution Method
14.5 Three other related stochastic approaches to capital budgeting
The Statistical Distribution Method
(14.16) and (14.17a)
(14.17b)
(14.17b’) and (14.17b’’)
(14.19)
Cov(Ct, Cτ) = ρτt στ σt,
If the cash flows are mutually independent, then Eq. (14.17b’) reduces to (14.18)
Hillier (1969) combines the assumption of mutual independence and perfect correlation in developing a model to deal with mixed situations.

8. Hillier (1969) combines the assumption of mutual independence (Yt) and perfect correlation in developing a model to deal with mixed situations.
(14.19)
The derivation of Equation (14.19) is as follows
Assume the net cash flow at time t, Xt, is related to the sources as follows

10. 14.5.1 The Statistical Distribution Method
Table Expected cash flow for new product
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.
Year
Source
Expected Value of
Net Cash Flow
(in thousands)
Standard Deviation
Initial Investment
-$600
$50 1
Production Cash Outflow
-250 20 2
-200 10 3 4 5
Production Outflow – salvage Value
-100 15
Marketing
300 50
600
100
500
400

11. 14.5.1 The Statistical Distribution Method
Table 14.8a Illustration of conditional-probability distribution approach
Initial
Outlay
Period 0
(1)
Probability
P(1)
(2)
Net Cash
Flow
(3)
Conditional
P(3|2,1)
(4)
(5)
P(5|4,3,2,1)
(6)
Cash
(7)
Joint
(8)
0.2
1000
0.015
0.25
2000
0.5
0.0375
0.3
3000
0.0225
0.03
10,000
0.075
4000
0.045
5000
0.4
0.06
6000

12. 14.5.1 The Statistical Distribution Method
Table 14.8a Illustration of conditional-probability distribution approach (Cont’d)
Initial
Outlay
Period 0
(1)
Probability
P(1)
(2)
Net Cash
Flow
(3)
Conditional
P(3|2,1)
(4)
(5)
(6)
Cash
(7)
Joint
(8)
0.5
4000
0.4
5000
0.08
0.3
6000
0.06
0.045
7000
9000
0.25
0.0125
0.025
8000
0.2
0.05
11000

13. 14.5.1 The Statistical Distribution Method
Table 14.8b NPV and joint probability
Discount Rate = 4 percent
NPV = $2,
Variance = $20,359,
Standard Deviation = $4,
PVA
NPV
Probability
12,024.96
2,024.96
0.06
12,913.96
2,913.96
0.08
13,802.96
3,802.96
14,763.08
4,763.08
0.045
16,541.08
6,541.08
18,319.08
8,319.08
13,948.04
3,948.04
0.0125
15,726.04
5,726.04
0.025
17,504.04
7,504.04
16,686.16
6,686.16
18,464.16
8,464.16
0.05
10,242.16
19,388.72
9,388.72
21,166.72
11,166.72
22,944.72
12,944.72
PVA
NPV
Probability
4,661.2
-5,338.8
0.015
5,550.2
-4,449.8
0.0375
6,439.2
-3,560.8
0.0225
6,474.76
-3,525.24
0.03
7,363.76
-2,636.24
0.075
8,252.76
-1,747.24
0.045
8,288.32
-1,711.68
9,177.32
10,066.32
66.32
8,297.84
-1,602.16
10,175.84
175.84
0.06
11,953.84
1,953.84

14. 14.5.2 The Decision-Tree Method
14.5 Three other related stochastic approaches to capital budgeting
Fig Decision tree. Numbers in parentheses are probabilities; numbers without parentheses are NPV
The Decision-Tree Method
Decision
Variables
For the
First Stage
Net
Present
Value*
Standard
Deviation*
Coefficient Of
Variation
Regional
$610.50
$666.98
1.093
National
767.27
1.3909
International
629.71
2.4357
Expressed in thousands of dollars.
NPV = 0.7(1,248.84) + 0.2(814.73)
+ 0.1(-14.54) = $1,

15. 14.5.2 The Decision-Tree Method
Table Expected cash flows for various branches of the decision tree 1 2 3 4 5 6
NPV
Regional Distribution throughout:
High
-2000
250
500
600
900
893.70
Medium
400
550
800
310.80
Low
-250
200
450
National Distribution throughout:
-3000
350
700
1100
1400
1000
1,454.47
1200
498.90
-350
100
300
-1,036.73
International Distribution Throughout:
-4000
1600
2200
2,350.71
2000
969.44
-450
1500
-1,544.26

16. 14.5.2 The Decision-Tree Method
Table Expected cash flows for various branches of the decision tree (Cont’d) 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.
1200
600
1,473.88
High National – Low Inter.
700
1500
400
Medium National – High Inter.
-500
1,623.63
Medium National – Medium Inter
2000
952.43
Medium National – Low Inter
Low National – High Inter.
-350
-900
917.27
Low National – Medium Inter
246.07
Low National – Low Inter
-1,372.66

17. 14.5.2 The Decision-Tree Method
Table Expected cash flows for various branches of the decision tree (Cont’d) 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
900
1200
800
700
814.73
High Region – Low National
300
-14.54
Medium Region – High National
-600
916.00
Medium Region – Medium National
481.89
Medium Region – Low National
Low Region – High National
-250
-800
490.71
Low Region – Medium National
56.59
Low Region – Low National

18. 14.5 Three other related stochastic approaches to capital budgeting
Simulation Analysis
Table 14.10a 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 (1990) are used to determine the value of the variable for simulation.
b. Useful life of facilities is an integer.
Random number Year

19. 14.5.3 Simulation Analysis Table 14.10b 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
(73)$54.6
(93)$58.6
(59)51.8
(78)55.6
(61)$52.2
GROW
(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
(37)8,740,000
(97)9,940,000
(87)9,740,000
(61)9,220,000
(65)9,300,000
KUSE
(02)5 years
(68)8 years
(47)7 years
(23)6 years
(71)8 years
RES
(87)1,870,000
(41)1,410,000
(56)1,560,000
(15)1,150,000
(20)1,200,000
VAR
(98)$44.7
(91)$43.65
(22)33.3
(58)38.7
(17)$32.55
FIX
(10)$410,000
(80)$480,000
(19)419,000
(93)493000
(48)448,000
TAX
40%
0.4
0.40
DIS
12%
0.12
NPV
$197,
$7,929,
$12,146,
$1,169,846.55
$15,306,
Number in parentheses is the generated random number.

20. 14.5.3 Simulation Analysis Table 14.10b Simulation (Cont’d) VARIABLES6 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
(18)43.6
(47)49.4
(88)$57.6
(39)$47.8
(47)$49.4
GROW
(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
(90)9,800,000
(72)9,440,000
(77)9,540,000
(76)9,520,000
(08)8,160,000
KUSE
(05)5 years
(40)7 years
(43)7 years
(81)9 years
(15)5 years
RES
(89)1,890,000
(62)1,620,000
(28)1,280,000
(88)1,880,000
(71)1,710,000
VAR
(18)$32.7
(47)37.05
(31)$34.65
(94)44.1
(58)$38.7
FIX
(08)408,000
(68)468,000
(06)406,000
(76)476,000
(56)456,000
TAX
0.4
DIS
0.12
NPV
$-1,513,
$11,327,171.67
$839,
$-6,021,
$563,
NPV=4,194, Notes. 1. Definitions variables can be found in Table 14.10a.
2. NPV calculator procedure can be found in Table 14.10c.
Random number
Useful life

21. [Sales Volume]t = [(Market Size)  (1 + Market Growth Rate)t]
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];

22. 14.5.3 Simulation Analysis Table 14.10c Cash Flow Simulations Period 12 3 4 5
$2,034,
$3,368,
$4,260,
$2,376,
$4,549,
2,116,
3,406,
4,402,
2,380,
4,650,608,707
2,201,
3,445,
4,549,
2,384,
4,753,
2,289,
3,485,
4,700,
2,388,
4,859,
2,380,
3,524,
4,856,
2,392,
4,967, 6
3,564,
5,017,
2,396,
5,077, 7
3,605,
5,183,
5,189, 8
3,645,
5,303,

23. 14.5.3 Simulation Analysis Table 14.10c Cash Flow Simulations (Cont’d)
Notes. 1. NPVs are listed in Table 13.4b
Period 6 7 8 9 10 1
$1,841,
$1,820,837,760
$4,344,
$439,
$2,097, 2
1,927,
1,919,
4,383,
464,
2,127, 3
2,018,
2,023,
4,423,
491,
2,157,294,016 4
2,112,
2,131,314,436
4,462,
519,
2,187, 5
2,209,
2,244,
4,502,
547,
2,218,
2,363,
4,543,
577,
2,487,
4,583,
607,
639,

24. 14.6 Inflationary effects in the capital-budgeting procedure
(14.20)
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).
(14.21a), (14.21b), and (14.22)
(14.23), and (14.24)
(14.25)
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 nominal cash flow.

25. 14.6 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.
(14.26), and (14.27)
(14.28)
(14.29)

26. 14.6 Inflationary effects in the capital-budgeting procedure
(14.29’)
(14.30)
(14.31), and (14.32)
TABLE 14.11
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.
Condition L
X/TA
1 + r
D/X
L > L’
L = L’
L < L’ + -

27. 14.6 Inflationary effects in the capital-budgeting procedure
TABLE Regression results of (Figures in parentheses are t values)
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.
Period a b c d e
.0884
.0882
.1044
.0103
(14.74)
.0131
(9.93)
.0097
(5.90)
.0957
(4.88)
-.0079
(0.37)
.1046
(4.05)
(2.35)
.0670
(0.15)
-.8642
(1.87)
-.0545
(4.42)
-.0297
(1.46)
-.0870
(5.63)

28. 14.7 Multi-period capital budgeting
(14.34)
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.
(14.35)
(14.36)
E[Vp1] - L0E[Cov (Vp1, VM1)] - Vp0 > 0.

29. 14.7 Multi-period capital budgeting
(14.37)
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;
= Variance of the market asset’s rate-of-return; σIm = ; 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))] (14.38)
where Ri = Rate-of-return on the risky asset I; rf = Riskless rate-of-return 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.
(14.39), and (14.40)

30. 14.8 Summary
The preceding discussion outlined three alternative capital-budgeting 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.

31. 14.8 Summary and concluding remarks
Lastly we investigated some of the more generalized mean-variance 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 earlier 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.

32. Appendix 14A. Time-state preference and the real option approaches for capital budgeting under uncertainty
(14.A.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 14.A.1 Expected cash flows for Project,
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.
State of Economy
Year 1
Year 2
Year 3
Boom
Normal
Recession
$1000
$800
$500
$400
$200
$300
$100

33. ΔVj = Vj - Vj+1, Vj = e-1(N[d2 (Mj)] - N [d2 Mj+1])
Appendix 14A. Time-state preference and the real option approaches for capital budgeting under uncertainty
Vj = e-i N [d2 (Mj)],
(14.A.2), and(14.A.3)
where i= Interest rate, = Instantaneous variance of the rate-of-return on the market portfolio, N(•) = Probability of d2(Mj) obtained from a normal distribution.
(14.A.4)
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])
(14.A.5), and (14.A.6)
vj = e-i (N[D2 (rMj)] - N[d2(rM,(j+1))], Vj = vj + Vj+1.
(14.A.7), and(14.A.8)

34. Appendix 14A. Time-state preference and the real option approaches for capital budgeting under uncertainty