Capital budgeting techniques using discounted fuzzy versus probabilistic cash flows
Introduction
When evaluating a risky project using decision and risk analysis techniques, analysts typically construct a model that calculates project cash flows and the net present value of these cash flows given different settings of input variables. To examine the risks of the projects, probability distributions are assigned to the input variables and a probability distribution is calculated, called a risk profile, showing the likelihood of the different possible net present values.
By invoking the central limit theorem, which credits a normal distribution to the sum of independently distributed random variables as the number of terms in the summation increases, it can be considered that present worth (PW) normally distributed with a mean of E[PW] and a variance of V(PW). Thus, a probability of having a loss or a gain from the investment can be easily calculated.
There are some other methods to consider risky cash flows such as certainty equivalent method and risk-adjusted discount rate method. These methods do not use a probability distribution.
In the following, first the main equations for probabilistic cash flows will be given and then capital budgeting techniques using fuzzy cash flows are explained and developed.
Since the expected value of a sum of random variables equals the sum of the expected values of the random variables, then the expected present value (PV) is given bywhere Aj's are statistically independent net cash flows and N is the life of project. Then the variance of PV is given by
The central limit theorem establishes that the sum of independently distributed random variables tends to be normally distributed as the number of terms in the summation increases. Hence, as N increases, PV tends to be normally distributed with a mean value of E[PV] and a variance of V(PV).
In the case of a set of correlated cash flows (Aj's are not statistically independent) the variance calculation is modified as follows:where Cov[Aj,Ak] is the covariance between Aj and Ak. Cov[Aj,Ak] equals ρjkσ[Aj]σ[Ak], where ρjk is the correlation coefficient between Aj and Ak. If all Aj and Ak are perfectly correlated such that ρjk=+1, thenIn performing risk analyses involving correlated cash flows, it is suggested that the net cash flow in a year be separated into those components of cash flow one can reasonably expect to be independent from year to year and those that are correlated over time.
To deal with vagueness of human thought, Zadeh [1] first introduced the fuzzy set theory, which was based on the rationality of uncertainty due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague knowledge. The theory also allows mathematical operators and programming to apply to the fuzzy domain.
A fuzzy number is a normal and convex fuzzy set with membership function μA(x) which both satisfies normality: μA(x)=1, for at least one x∈R and convexity: μA(x′)⩾μA(x1)∧μA(x2), where μA(x)∈[0,1] and ∀x′∈[x1,x2]. `∧' stands for the minimization operator.
Quite often in finance future cash amounts and interest rates are estimated. One usually employs educated guesses, based on expected values or other statistical techniques, to obtain future cash flows and interest rates. Statements like approximately between $12,000 and $16,000 or approximately between 10% and 15% must be translated into an exact amount, such as $14,000 or 12.5%, respectively. Appropriate fuzzy numbers can be used to capture the vagueness of those statements.
A tilde will be placed above a symbol if the symbol represents a fuzzy set. Therefore, P̃, F̃, G̃, Ã, ĩ, r̃ are all fuzzy sets. The membership functions for these fuzzy sets will be denoted by , , , etc. A fuzzy number is a special fuzzy subset of the real numbers. The extended operations of fuzzy numbers are given in Appendix A. A triangular fuzzy number (TFN) is shown in Fig. 1. The membership function of a TFN defined bywhere m1≺m2≺m3, is a continuous monotone increasing function of y for 0⩽y⩽1 with and and is a continuous monotone decreasing function of y for 0⩽y⩽1 with and . is denoted simply as (m1/m2, m2/m3).
A flat fuzzy number (FFN) is shown in Fig. 2. The membership function of an FFN, Ṽ, is defined bywhere m1≺m2≺m3≺m4, is a continuous monotone increasing function of y for 0⩽y⩽1 with and and is a continuous monotone decreasing function of y for 0⩽y⩽1 with and . is denoted simply as (m1/m2, m3/m4).
The fuzzy sets P̃, F̃, G̃, Ã, ĩ, r̃ are usually fuzzy numbers but n will be discrete positive fuzzy subset of the real numbers [2]. The membership function is defined by a collection of positive integers ni, 1⩽i⩽K, where
Section snippets
Fuzzy present-value method
The present-value method of alternative evaluation is very popular because future expenditures or receipts are transformed into equivalent dollars now. That is, all of the future cash flows associated with an alternative are converted into present dollars. If the alternatives have different lives, the alternatives must be compared over the same number of years.
Chiu and Park [3] propose a present-value formulation of a fuzzy cash flow. The result of the present value is also a fuzzy number with
Fuzzy capitalized value method
A specialized type of cash flow series is a perpetuity, a uniform series of cash flows which continues indefinitely. An infinite cash flow series may be appropriate for such very long-term investment projects as bridges, highways, forest harvesting, or the establishment of endowment funds where the estimated life is 50 years or more.
In the nonfuzzy case, if a present value P is deposited into a fund at interest rate r per period so that a payment of size A may be withdrawn each and every period
Fuzzy future value method
The future value (FV) of an investment alternative can be determined by using the relationshipwhere FV(r) is defined as the future value of the investment using a minimum attractive rate of return (MARR) of r%. The future-value method is equivalent to the present-value method and the annual-value method.
Chiu and Park's [3] formulation for the fuzzy future value has the same logic of fuzzy present-value formulation:
Fuzzy benefit/cost ratio method
The benefit/cost ratio (BCR) is often used to assess the value of a municipal project in relation to its cost; it is defined aswhere B represents the equivalent value of the benefits associated with the project, D represents the equivalent value of the disbenefits, and C represents the project's net cost. A BCR greater than 1.0 indicates that the project evaluated is economically advantageous. In BCR analyses, costs are not preceded by a minus sign.
When only one alternative must be
Fuzzy equivalent uniform annual value method
The equivalent uniform annual value (EUAV) means that all incomes and disbursements (irregular and uniform) must be converted into an equivalent uniform annual amount, which is the same each period. The major advantage of this method over all the other methods is that it does not require making the comparison over the least common multiple of years when the alternatives have different lives [5]. The general equation for this method iswhere NPV is the net
Fuzzy payback period method
The payback period method involves the determination of the length of time required to recover the initial cost of investment based on a zero interest rate ignoring the time value of money or a certain interest rate recognizing the time value of money. Let Cj0 denote the initial cost of investment alternative j, and Rjt denote the net revenue received from investment j during period t. Assuming no other negative net cash flows occur, the smallest value of mj ignoring the time value of money
Fuzzy internal rate of return method
The internal rate of return (IRR) method is referred to in the economic analysis literature as the discounted cash flow rate of return, internal rate of return, and the true rate of return. The internal rate of return on an investment is defined as the rate of interest earned on the unrecovered balance of an investment. Letting r* denote the rate of return, the equation for obtaining r* iswhere Pt is the net cash flow at the end of period t.
Assume the cash flow
An expansion to geometric and trigonometric cash flows
When the value of a given cash flow differs from the value of the previous cash flow by a constant percentage, j%, then the series is referred to as a geometric series. If the value of a given cash flow differs from the value of the previous cash flow by a sinusoidal wave or a cosinusoidal wave, then the series is referred to as a trigonometric series.
Conclusions
In this paper, probabilistic cash flows and capital budgeting techniques in the case of fuzziness and discrete compounding have been studied. The cash flow profile of some investments projects may be geometric or trigonometric. For these kind of projects, the fuzzy present, future, and annual value formulas have been also developed under discrete and continuous compounding in this paper. Fuzzy set theory is a powerful tool in the area of management when sufficient objective data have not been
References (14)
Fuzzy sets
Inf. Control
(1965)The fuzzy mathematics of finance
Fuzzy Sets and Systems
(1987)- et al.
Justification of manufacturing technologies using fuzzy benefit/cost ratio analysis
Int. J. Product. Econom.
(2000) - et al.
Ranking fuzzy numbers in the setting of possibility theory
Inf. Sci.
(1983) - et al.
Fuzzy cash flow analysis using present worth criterion
Eng. Econom.
(1994) Discounted fuzzy cash flow analysis
- et al.
Engineering Economy
(1987)
Cited by (190)
Decision making for energy investments by using neutrosophic present worth analysis with interval-valued parameters
2020, Engineering Applications of Artificial IntelligenceCitation Excerpt :Nachtmann and Needy (2001) proposed PWA technique by using Type-1 fuzzy sets, and they evaluated knowledge information investment alternatives. Kahraman et al. (2002) developed fuzzy PWA formulas, and they expanded the examined cash flows to geometric and trigonometric cash flows. Tercenño et al. (2003) presented fuzzy PWA formula to select portfolios of tangible investments.
Maximizing returns through investment analysis: An overview of analytical tools
2023, Advancement in Business Analytics Tools for Higher Financial PerformanceA New Scenario-Based Simulation Model for Cost Management of Healthcare Services through Improving the Efficiency of the Health Centers
2023, Journal of Healthcare Engineering