A fuzzy real option valuation approach to capital budgeting under
uncertainty
Shin-Yun Wang
Department of Finance, National Dong Hwa University, 1, Sec.2, Da-Hsueh Rd., Shou-Feng, Hualien 974,
Taiwan
Cheng-Few Lee
Department of Finance and Economics, Rutgers University, New Jersey, USA
Abstract
The information needed for capital budgeting is generally not known with certainty. Therefore,
capital budgeting procedures under conditions of uncertainty should be developed to improve
the precision of assessment of the value of risky investment projects. The sources of
uncertainty may be either the net cash inflow, the life of the project, or the discount rate. We
propose a capital budgeting model under uncertainty in which cash flow information can be
specified as a special type of fuzzy numbers. Then, we can estimate the present worth of each
fuzzy project cash flow. At the same time, to select fuzzy projects under limited capital budget,
we give an example to compare and analyze the results of the capital budgeting problem using
a fuzzy real option model. Hence, the fuzzy numbers and the real option model can be jointly
used in solving capital budgeting under uncertainty.
Keywords: Capital budgeting; Real option; Fuzzy numbers; Uncertainty.
1. Introduction
Large investments are capital projects of strategic importance which have a long economic life cycle.
They often have many unknown that hard to estimate risks and potentials, difficult to be foreseen at their
initial planning stage. Hence these investments may change during their long economic life and the changes
can be fundamental. Such uncertainty and possibility of change in the fundamentals of large investments
which is essential, because of the larger the investments are, the more strategic importance they usually
have. However, the information needed for capital budgeting is generally not known with certainty.
Therefore, capital-budgeting procedures under conditions of uncertainty should be developed to improve
the precision of assessment of the value of risky investment projects. The sources of uncertainty may be
either the net cash inflow, the life of the project, or the discount rate.
In practice precise information concerning future investment projects is rarely obtained. Extensive
study has been undertaken to consider the capital budgeting problems under risk. Hence decision maker
does not have exact knowledge concerning future investment opportunities. Traditional approaches to
capital budgeting are based on the premise that probability theory is necessary and sufficient to deal with
the uncertainty that underlie the estimates of required parameters. It is argued that, in many circumstances,
this premise is invalid since the principal sources of uncertainty are often non-random in nature and relate
to the fuzziness rather than the frequency of data. In order to capture and quantify correctly the underlying
uncertainty present in non-statistical situations, the theory of possibility, an extension of the theory of fuzzy
sets introduced by Zadeh (1965). A possibility distribution can be viewed as the membership function of a
parameter.
In the real world, the data sometimes cannot be recorded or collected precisely. For instance, if
1
considering to ask a group of economists for predicting the rate of economical growth in the next year.
Their statements may be like “approximately 3%,” “should be 3–5%” or “may be below 6%,” etc. All of
those statements can be characterized as fuzzy sets. Suppose that we introduce a selection procedure which
will determine the probability that each economist will be selected. Then for each possible selection, their
statements are fuzzy sets. Therefore, to deal with a capital budgeting model under uncertainty, the cash
flow information can be specified as a special type of fuzzy number-triangular fuzzy numbers. Zadeh
(1965), Dubois and Prade (1988), and Carlsson and Fuller (2002) have investigated the usefulness of the
fuzzy set theory in decision making under uncertainty. The fuzzy capital budgeting approaches allow cash
flow estimates as fuzzy numbers and offer the means to integrate trend data into cash flows. Fuzzy cash
flows can better reflect the uncertainty in the project.
In this paper, fuzzy capital budgeting is to use fuzzy versions of the neo-classical capital budgeting
methods and real option valuation. In Section 2, we will discuss methods for capital budgeting and
investment decision making. Next section derives the fuzzy real option method. Section 4 gives a numerical
example to compare and analyze the results of the capital budgeting problem using a fuzzy real option
model and conclusion in section 5.
2. Methods for capital budgeting and investment decision making
Capital budgeting methods based on the discounted cash flow (DCF) have been the ruling instruments
for investment decision making. The most commonly used DCF based method is the net present value
(NPV). Under static circumstances and in truly now or never situations, DCF based methods provide
reliable results, but the real world situations are seldom static. Especially in cases of large investments with
long economic lives the static discounted cash flow based methods fail to present a highly reliable picture
of the profitability and possibilities offered by the investment project at hand. As DCF based methods have
been the best thing available, and it is better to use them than not to use any kind of decision tool for capital
budgeting, they have rooted to management practices during years of use.
However, there are many enhancements to the original formulae, but the underlying unsatisfactory
assumptions still exist. Hence the adjustment models include the risk-adjusted discount-rate methods, and
the certainty-equivalent method. The risk-adjusted discount-rate method simply extends the cash-flow
valuation model under certainty to the uncertainty case. The advantage of this method is that the valuation
model gives us a formula that explicitly considers the uncertainty associated with future cash flows.
Although the model does not specify exactly what constitutes the risk of the cash flows, it can be used to
develop and explore the relationships among the variables of asset-valuation models. On the other hand, the
disadvantages of the risk-adjusted discount-rate model are clear. The value of interest rate is only a
subjective estimate, which could well differ from person to person. Therefore, an objective determination
of the value of a risky investment project will be almost impossible by simply applying this method. While
the risk-adjusted discount-rate method provides a means for adjusting the basic riskless discount rate, an
alternative method, the certainty-equivalent method, adjusts the estimated value of the uncertain cash flows.
The underlying rationale is that, given a risky cash flow, the decision maker will evaluate this risky cash
flow by attaching an expected utility to that cash flow, that utility estimate being hypothesized to be equal
to the utility derived from some certain amount. If the decision maker performs this process for each cash
flow, a series of certainty equivalents for the risky flows can be obtained. Above both methods are used to
evaluate future uncertain cash flows, the two models should yield the same value for a given stream of cash
flows. Moreover, the present value of each period's cash flows should be the same under these two
valuation models. Robichek and Myers (1966) showed that the risk-adjusted rate method tends to lump
together the pure rate of interest, a risk premium, and time (through the compounding process), while the
certainty-equivalent approach keeps risk and the pure rate of interest separate. This separation gives an
advantage to the certainty-equivalent method.
To remedy the problems of the DCF based methods new methods have been introduced. The real
option approach is a methodology that calculates the value of an investment with techniques originally
developed for valuation of financial options. This gives the possibility to take into consideration the
managerial flexibility to take action during the lifetime of an investment. The term real option was coined
in an article about corporate borrowing by Myers (1997). Since then there has been a growing literature
2
describing the different theoretical aspects of real options (Kulatilaka and Marcus, 1998; Dixit and Pindyck,
1994; Trigeorgis, 1995), as well as the managerial and strategic implications and application of real options
(Bowman and Hurry, 1993; Luehrman, 1998; Amram and Kulatilaka, 1999). A number of case based
articles are also available to give further insight into real world application (Kulatilaka, 1993; Nichols,
1994; Micalizzi, 1999). The value of a real option is computed by using the Black and Scholes (1973)
formula extended by Merton (1973).
Some Economic factors, such as competition, consumer preferences, technological development, and
labor market conditions are a few of the factors that make it virtually impossible to foretell the future.
Consequently, the economic life, revenues, and costs of investment projects are less than certain. With the
discuss of risk, a firm is no longer indifferent between two investment proposals having rates of return
equal or net present values. Both net present value and its standard deviation should be estimated in
performing capital-budgeting analyses under uncertainty. There are three related stochastic methods useful
in the making of capital-budgeting decisions which are the probability-distribution, decision-tree, and
simulation methods. The statistical distribution method which Chen and Moore (1982) have generalized this
model by introducing the estimation risk. Hillier (1963) combines the assumption of mutual independence
and perfect correlation in developing a model to deal with mixed situations. This model can be used to
analyze investment proposals in which some of the expected cash flows are closely related, and others are
fairly independent. A decision-tree approach to capital-budgeting decisions can be used to analyze some
investment opportunities involving a sequence of investment decisions over time. It is an analytical
technique used in sequential decisions, where various decision points are studied in relation to subsequent
chance events. This technique enables one to choose among alternatives in an objective and consistent
manner. Simulation is another approach to confronting the problems of capital budgeting under uncertainty.
Because uncertainty associated with capital budgeting is not restricted to one or two variables, every
variable relevant in the capital-budgeting decision can be viewed as a random variable. Facing so many
random variables, it may be impossible to obtain tractable results from an economic model. Simulation is a
useful tool designed to deal with this problem, and is the closest we can get to modeling in cases of
uncertainty.
Generally speaking, the probability-distribution, decision-tree, and simulation methods are three
alternative approaches that are available to deal with the problem of capital budgeting under uncertainty.
These methods have explicitly utilized the concepts of probability distributions and statistical distributions
to carry out the analysis. If there is only a single accept-reject decision at the outset of the project, then the
decision maker can use either statistical-distribution methods or simulation methods. If investment
opportunities involve a sequence of decisions over time, then a decision-tree method can be used to
perform the analysis. In addition, if want to reduce risk of the investment, the real option is a better choice.
The real option approach also supports the determination of the timing for the investment, and offers a
comprehensive way of presenting value of possibilities opened by the project, which is further enhanced
with frequently updated fuzzy cash flows.
3. The fuzzy real option valuation (FROV) approach
3.1. Fuzzy numbers
From Zadeh (1965), A is a convex fuzzy set if and only if its  -level set A  {x :  A ( x)   } is a
convex set for all any  . Therefore, if  is a fuzzy number, then the  -level set   is a compact (closed and
bounded in R) and convex set; that is,  is a closed interval. The  -level set of  is then denoted
by   [ L ,  U ] . A fuzzy number  is said to be nonnegative if  ( x)  0 for x<0. It is easy to see
that if  is a nonnegative fuzzy numbers then  L and  U are all nonnegative real numbers for
all   [0,1] .
3.2 Triangular Fuzzy Number, TFN
The concept of triangular fuzzy number attempts to deal with real problems by possibility. The
membership of a triangular fuzzy number is defined as follows:
3
( x  L) /( M  L)  M  x  U
 A ( x)  (U  x) /(U  M )  L  x  M
 0  otherwise
 A ( x)
1
0
L
M
R
x
Figure 1 The membership function of the triangular fuzzy number
According to the characteristics of triangular fuzzy numbers and the extension principle put forward by
Zadeh (1965), the operational laws of triangular fuzzy numbers, A  (l1 , m1 , r1 ) and B  (l2 , m2 , r2 ) are as
follows:
(1) Addition of two fuzzy numbers 
(l1 , m1 , r1 )  (l2 , m2 , r2 )  (l1  l2 , m1  m2 , r1  r2 )
(2) Subtraction of two fuzzy numbers 
(l1 , m1 , r1) (l2 ,m2 ,r
l(1 r2 ,m1 m2, r1 l2 )
2 ) 
(3) Multiplication of two fuzzy numbers 
(l1 , m1 , r1 
) (l2 ,m2 ,
r2 ) l(1 l2 ,m1 m2 , r1 r2 )
(4) Multiplication of any real number k and a fuzzy number 
k  (l1 , m1 , r1 )  (kl1 , km1 , kr1 )
(5) Division of two fuzzy numbers 
(l1 , m1 , r1 )(l2 , m2 , r2 )  (l1 / r2 , m1 / m2 , r1 / l2 )
3.2. A real option valuation
A real option means that the possibility for a certain period to either choose for or against making an
investment decision, without binding oneself up front. The real option rule is that one should invest today
only if the net present value is high enough to compensate for giving up the value of the option to wait.
Because the option to invest loses its value when the investment is irreversibly made, this loss is an
opportunity cost of investing. While ongoing evaluation effort is structured around key decision points, or
triggered by changes in the business environment, it is a process in which the valuation, even the
computing process, is not intended to provide a definite answer, but rather to provide decision makers an
ongoing dialogue about the project (Dahiberg and Porter, 2000). So a process view of the real option
approach is a relevant task in analyzing the decision support needs.
Real options in option thinking are based on the Black-Scholes model as financial options. The option
pricing theory is that you should invest today only if the net present value is high enough to compensate for
giving up the value of the option to wait. Because the option to invest loses its value when the investment is
irreversibly made, this loss is an opportunity cost of investing. Following Leslie and Michaels (1997), we
will compute the value of a real option as
ROV  S0 e  T N (d1 )  Xe  rT N (d 2 ),
where
ln( S0 / X )  (r     2 / 2)T
d1 
, d 2  d1   T
 T
and where ROV denotes the current real option value, S0 is the present value of expected cash flows, X is
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the (nominal) value of fixed costs, σ quantifies the uncertainty of expected cash flows, and  denotes the
value lost over the duration of the option.
In real options, the options involve real assets. To have a real option means to have the possibility for
a certain period to either choose for or against making an investment decision. Real options can be valued
using the analogue option theories that have been developed for financial options, but do not mean that they
are the same. Real options are concerned about strategic decisions of a company, where degrees of freedom
are limited to the capabilities of the company. In these strategic decisions different stakeholders play a role,
especially if the resources needed for an investment are significant and thereby the continuity of the
company is at stake. In addition, it is quite different from traditional discounted cash flow investment
approaches. The traditional methods are very hard to make a decision when there is uncertainty about the
exact outcome of the investment. And since these methods ignore the value of flexibility and discount
heavily for external uncertainty involved, many interesting and innovative activities and projects are
cancelled because of the uncertainties.
If the result of NPV is negative, the company would obviously pass up the opportunity. But if we use
the real option valuation the result may be different of the same case. Such a valuation would recognize the
importance of uncertainty, which the NPV analysis effectively assumes away. The main question that a
company must answer for a deferrable investment opportunity is how long we postpone the investment up
to T time periods. To answer this question, Benaroch and Kauffman (2000) suggested the following
decision rule for optimal investment strategy: Where the maximum deferral time is T, make the investment
(exercise the option) at time t is positive and attends its maximum value. Of course, this decision rule has to
be reapplied every time new information arrives during the deferral period to see how the optimal
investment strategy might change in light of the new information.
3.3. A fuzzy real option valuation
We shall introduce a real option rule in a fuzzy setting, where the present values of expected cash
flows and expected costs are estimated by triangular fuzzy numbers, and determine the optimal exercise
time by using the fuzzy real option model. Usually, the present value of expected cash flows can not be
characterized by a single number. But managers are able to estimate the present value of expected cash
flows by using triangular possibility distribution of the form
F  [ f l , f m , f r ],
i.e. the most possible values of the present value of expected cash flows lie in the f m (which is the core of
the triangular fuzzy number F ), f r is the greatest value and f l is the smallest value for the present value of
expected cash flows.
In a similar manner one can estimate the expected costs by using a triangular possibility distribution of the
form
X  [ xl , xm , xr ],
i.e. the most possible values of expected costs lie in the interval xm (which is the core of the triangular fuzzy
number X ), xr is the greatest value and xl is the smallest value for expected costs.
In addition, the values also influence by the financial market, such as riskless interest rate (discount
rate) and the volatility of cash inflow. We use historic data to estimate them by using a triangular
possibility distribution of the form
R  [rl , rm , rr ],   [ l ,  m ,  r ],   [ l ,  m ,  r ].
i.e. the most possible values of discount rate, risk-adjusted discount-rate and the volatility of cash inflow lie
in the interval rm ,  m and  m (which is the core of the triangular fuzzy number R,  and  ), rr is the
greatest value and rl is the smallest value for discount rate,  r is the greatest value and  l is the smallest
value for the risk-adjusted discount-rate,  r is the greatest value and  l is the smallest value for volatility of
5
cash inflow.
The information needed for capital budgeting is generally not known with certainty. The sources of
uncertainty may be the net cash inflow, the discount rate, or the life of the project, etc. Hence in addition to
consider the fuzzy volatility  and investing cost X , we also consider the fuzzy cash inflow F , fuzzy
interest rate  and R . In these circumstances we suggest the use of the following formula for computing
fuzzy real option values
ROV ( F , X ,  , R,  , T )  F  e T  N (d1 )X  e RT  N (d2 ).
where
ln( F X )  ( R   2 / 2)  T
d1 
, d 2  d1  T
 T
and where, F denotes the possibilistic mean value of the present value of expected cash flows, X stands
for the possibilistic mean value of expected costs and  ( F ) is the variance of the present value expected
cash flows,  stands for the risk-adjusted discount-rate, R stands for the discount rate, T stands for the
interval, Using above formula for arithmetic operations on triangular fuzzy numbers we find
FROV  ( fl , f m , f r )  e T  N (d1 )( xl , xm , xr )  e RT  N (d 2 )
 ( fl N (d1 )  e T  xr N (d 2 )  e R T , f m N (d1 )  e  T  xm N (d 2 )  e  R T , f r N (d1 )  e  T  xl N (d 2 )  e  RT ],
Where
d1  [(ln fl / xr  rT
  r T   l  l T / 2) /  r T , (ln f m / xm  rmT   mT   m mT / 2) /  m T ,
l
(ln f r / xl  rr T   l T   r  r T / 2) /  l T ],
d 2  d1  T .
In the following we shall generalize the probabilistic decision rule for optimal investment strategy to a
fuzzy setting: Where the maximum deferral time is T, make the investment (exercise the option) at time t  ;
0  t   T , for which the option, Ct  , attends its maximum value,
Ct  max Ct  F  e T  N (d1 )X  e R T  N (d 2 )
t  0,1,...,T
Where
Ft  PV (cf0 , cf1 ,..., cfT ,  )  PV (cf0 , cf1 ,..., cft ,  )  PV (cft 1 , cf1 ,..., cfT ,  ),
that is,
T
t
T
cf j
cf j
cf j
Ft  cf 0  

cf


,


0
j
j
j
j 1 (1   )
j 1 (1   )
j  t 1 (1   )
where cf j denotes the expected (fuzzy) cash flow at time j,  is the risk-adjusted discount rate (or
required rate of return on the project). However, to find a maximizing element from the set
{C0 , C1 ,..., CT } is not an easy task because it involves ranking of triangular fuzzy numbers.
In our computerized implementation we have employed the following function to determine the expected
fuzzy real option values {Cl , Cm , Cr }, of triangular form:
E (Ct )  [(Cr  Cl )  (Cm  Cl )]/ 3  Cl
where, Cr, Cm and Cl are the largest, middle, and smallest values of the fuzzy real option values.
4. Computational method and example
We give an example to compare and analyze the results of the capital budgeting problem using a fuzzy
real option model, where the present values of expected cash flows, expected costs, the discount rate, the
risk-adjusted discount rate and volatility of cash inflows are estimated by triangular fuzzy numbers.
6
Table 1 Illustration of conditional-probability distribution approach
Initial
Initial
Net
Conditional
Net Cash
Conditional
Outlay
Probability Cash
Probability
Flow
Probability
Period 0 P(1)
Flow
P(2|1)
P(3|2,1)
10,000
0.3
0.5
0.2
2000
4000
6000
0.25
2000
0.5
3000
0.25
4000
0.3
3000
0.4
5000
0.3
7000
0.25
5000
0.5
7000
0.25
8000
0.2
0.5
0.3
0.2
0.5
0.3
0.2
0.5
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.3
0.4
0.3
0.25
0.5
0.25
0.25
0.5
0.25
0.25
0.5
0.25
Cash
Flow
Joint
Probability
1000
2000
3000
2000
3000
4000
3000
4000
5000
2000
4000
6000
4000
5000
6000
5000
7000
9000
4000
6000
8000
5000
7000
9000
7000
9000
11000
0.015
0.0375
0.0225
0.03
0.075
0.045
0.015
0.0375
0.0225
0.045
0.06
0.045
0.06
0.08
0.06
0.045
0.06
0.045
0.0125
0.025
0.0125
0.025
0.05
0.025
0.0125
0.025
0.0125
Firstly, let we refer to Table 1 by Bonini (1975), where we consider a project requiring an initial outlay
of $10,000 (Column 1). In each of the following time periods (Columns 3, 5, and 7), the cash flow to be
received is not known with perfect certainty, but the probabilities associated with each cash flow in each
period are assumed to be known (Columns 2, 4, and 6), so that we are dealing with a case of risk, and not
strict uncertainty. We note that Columns (4) and (6) are conditional-probability figures, where the later
periods' expected cash flows depend highly on what occurs in earlier time periods. Given our cash flow and
simple probability estimates for the periods 1, 2, and 3, we find 27 possible joint probabilities (Column 8),
each of which corresponds to a cash-flow series. Thus in the uppermost path we find the joint probability
0.015 being associated with a cash flow of $2000 in Period 1, followed by cash flows of $2000 and $1000
in Periods 2 and 3, respectively. This particular path is the worst possible result; in non-discounted dollar
terms there is a 50 percent loss on the investment. At the other end of the spectrum, we find the best
possible outcome offers a return $12,945 with a joint probability of 0.0125. The returns and joint
probabilities can be calculated similarly for the other 25 possible patterns.
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Table 2 NPV and joint probability
t
PVA
NPV
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
4,661.2
5,550.2
6,439.2
6,474.76
7,363.76
8,252.76
8,288.32
9,177.32
10,066.32
8,297.84
10,175.84
11,953.84
12,024.96
12,913.96
13,802.96
14,763.08
16,541.08
18,319.08
13,948.04
15,726.04
17,504.04
16,686.16
18,464.16
20242.16
19,388.72
21,166.72
22,944.72
-5,338.8
-4,449.8
-3,560.8
-3,525.24
-2,636.24
-1,747.24
-1,711.68
-822.68
66.32
-1,602.16
175.84
1,953.84
2,024.96
2,913.96
3,802.96
4,763.08
6,541.08
8,319.08
3,948.04
5,726.04
7,504.04
6,686.16
8,464.16
10,242.16
9,388.72
11,166.72
12,944.72
Probability
0.015
0.0375
0.0225
0.03
0.075
0.045
0.015
0.0375
0.0225
0.045
0.06
0.045
0.06
0.08
0.06
0.045
0.06
0.045
0.0125
0.025
0.0125
0.025
0.05
0.025
0.0125
0.025
0.0125
Standard
Deviation
1021.913
1443.632
984.8824
1131.085
1544.94
1008.12
577.6831
741.2431
440.8144
998.557
693.0202
223.0023
240.0801
25.77347
195.4392
372.9275
866.1388
1127.269
105.4258
430.2211
502.9987
582.0294
1220.687
1144.282
713.7123
1290.468
1111.285
While the primary hurdle in this process is in estimating the cash flows and the probabilities, we
recognize that it is the NPV figures we are ultimately interested in. Table 2 gives the present value of the
cash flows and the net present value of the project for each cash-flow series (PVA), where a constant 4
percent discount rate was employed. When we multiply each joint probability by the expected net present
value associated with that probability, we obtain the expected net present value for the project as a whole,
there shown to be $2,517.18.
If the cash flows probabilities do not know, then the cash flows become an uncertainty case. In
X 0  [9000,10000,11000], R [0.03,0.04,0.05],  [0.0 2,0.03,0.04], we can use fuzzy real option to
approach the capital budgeting under uncertainty. We use above example and fuzzy real option model to
compute the expected fuzzy real option values. From above data, we transform them into the triangular
fuzzy number, and then we substitute into following function.
FROV  ( fl , f m , f r )  e T  N (d1 )( xl , xm , xr )  e RT  N (d 2 )
 ( fl N (d1 )  e T  xr N (d 2 )  e R T , f m N (d1 )  e  T  xm N (d 2 )  e  R T , f r N (d1 )  e  T  xl N (d 2 )  e  RT ],
Where
d1  [(ln f l / xr  rT
  r T   l  l T / 2) /  r T ,
l
(ln f m / xm  rmT   mT   m mT / 2) /  m T , (ln f r / xl  rr T   l T   r  r T / 2) /  l T ],
d 2  d1  T .
8
Table 3 Expected fuzzy real option value
t
F
 (F )
FROV
E (Ct )
1
F1  [2617,4661,6705]
1  [960,1022,1084]
C1  [-3765,-2532,-1097]
-2465
2
F2  [2663,5550,8437]
 2  [1182,1444,1705]
C2  [-5856,-1899,2721]
-1678
3
F3  [4469,6439,8409]
 3  [920,985,1050]
C3  [-3387,-1480,809]
-1353
4
F4  [4213,6475,8737]
 4  [1005,1131,1257]
C4  [-4545,-1352,2483]
-1138
5
F5  [4274,7364,10454]
 5  [1119,1545,1970]
C5  [-6851,-744,7922]
109
6
F6  [6237,8253,10269]
 6  [906,1008,1110]
C6  [-3896,-473,3729]
-213
7
F7  [7133,8288,9444]
 7  [531,578,624]
C7  [-3198,-408,2833]
-258
8
F8  [7695,9177,10660]
 8  [731,741,752]
C8  [-1163,-25,1368]
60
9
F9  [9185,10066,10948]
 9  [343,441,539]
C9  [-5432,295,7906]
923
10
F10  [6301,8298,10295]
10  [901,999,1096]
C10  [-3972,-266,4316]
26
11
F11  [8790,10176,11562]
11  [656,693,730]
C11  [-2140,433,3541]
611
12
F12  [11508,11954,12400]
12  [-8,223,454]
C12  [842,2152,3717]
2237
13
F13  [11545,12025,12505]
13  [-19,240,499]
C13  [770,2196,3899]
2289
14
F14  [12862,12914,12966]
14  [-394,26,446]
C14  [6765,2773,9496]
6345
15
F15  [13412,13803,14194]
15  [-85,195,476]
C15  [1619,3313,6015]
3649
16
F16  [14017,14763,15509]
16  [205,373,541]
C16  [-5411,337,11259]
2062
17
F17  [14809,16541,18273]
17  [818,866,914]
C17  [-1135,2400,7003]
2756
18
F18  [16065,18319,20574]
18  [975,1127,1280]
C18  [-3213,2656,11864]
3769
19
F19  [13737,13948,14159]
19  [-43,105,254]
C19  [1334,3211,5723]
3423
20
F20  [14866,15726,16586]
 20  [323,430,537]
C20  [-4289,1435,11019]
2722
21
F21  [16498,17504,18510]
 21  [444,503,562]
C21  [-2794,2333,9709]
3083
22
F22  [15522,16686,17850]
 22  [523,582,641]
C22  [-2547,2122,8664]
2747
23
F23  [16023,18464,20906]
 23  [1018,1221,1423]
C23  [-3453,2263,12298]
3702
24
F24  [17954,20242,22531]
 24  [1025,1144,1263]
C24  [-2345,2840,10969]
3821
25
F25  [17961,19389,20816]
 25  [702,714,726]
C25  [493,2736,5643]
2957
26
F26  [18586,21167,23748]
 26  [1125,1290,1456]
C26  [-2703,2802,12303]
4134
27
F27  [20722,22945,25167]
 27  [1035,1111,1188]
C27  [29493,74345,165301]
4070
where, t: time(in years), F :cash flow,  ( F ) : variance of the present value expected cash flows, FROV
:
fuzzy real option value, E (Ct ) : expected fuzzy real option value.
When the environment is uncertainty, with uncertain cash flows, the investor will face a distribution of
future cash flows rather than a single-point estimate of future cash flow in each period. The method utilized
to deal with these uncertain cash flows is that of the fuzzy real option model. What would option valuation
make of the same case? To begin with, such a valuation would recognize the importance of uncertainty,
which the NPV analysis effectively assumes away. We use the fuzzy real option model to evaluate the
project in the same case, which holding the option also obliges one to incur the annual fixed costs of
keeping the reserve active—let us say $300, This represents a dividend-like payout of three percent (i.e.
300/1000) of the value of the assets, in the discount rate is around 4%, cost is around $10,000 and dividend
rate is around 3%, then we obtain the expected fuzzy real option values for the project as a whole,
9
Ct  max Ct , where the maximum deferral time is T=27, make the investment (exercise the option) at
t  0,1,...,T
time t ; 0  t   T , for which the option, Ct  , there shown to be $6,345 at t=14, which is positive and

attends its maximum value.
5. Conclusions
The FROV will increase with the current value of increasing expected cash flows estimates. A
proactive management can Influence this by developing market strategies to enhance the sale and save the
cost. If volatility of cash flow will increase, then the FROV will increase. The corporate management can
be proactive and find the ways to expand to new markets, product innovations and innovation product
combinations as end results of their strategic decisions. The longer the time to maturity, the greater will be
the FROV. A proactive management can make sure of this development by maintaining protective barriers,
communicating implementation possibilities and maintaining a technological lead. On the other hand, if the
expected value of fixed costs goes up, the FROV will decrease as opportunities of operating with less cost
are lost. This can be countered by using the postponement period to explore and implement production
scalability benefits. An increase in risk-less returns will increase the FROV, and this can be further
enhanced by closely monitoring changes in the interest rates. The FROV will decrease if value is lost
during the postponement of the investment, but this can be countered by either creating business barriers
for competitors or by better managing key resources.
The fuzzy capital budgeting approaches allow cash flows estimate as fuzzy numbers and offer the
means to integrate trend data into cash flows. Fuzzy cash flows can better reflect the uncertainty in the
project. The integration of qualitative information into the process gives better support. The real option
approach also supports the determination of the timing for the investment, and offers a comprehensive way
of presenting value of possibilities opened by the project, which is further enhanced with frequently
updated fuzzy cash flows. Finally, we would like to stress that advanced decision methods such as real
options and fuzzy capital budgeting open the chance to explore the value of flexibility inside and outside a
project, and give further insight into the real uncertainty of large investments. The uncertainty is genuine,
i.e. we simply do not know the exact levels of future cash flows. Without introducing fuzzy real option
model it would not be possible to formulate this genuine uncertainty. The proposed model that incorporates
subjective judgments and statistical uncertainties may give investors a better understanding of the problem
when making investment decisions.
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