A Fuzzy Real Option Valuation Approach To Capital Budgeting Under Uncertainty Environment

Abstract

The information needed for capital budgeting is generally not known with certainty. The sources of uncertainty may be the net cash inflows, the life of the project, or the discount rate. We propose a capital budgeting model under uncertainty environment in which the concept of probability is employed in describing fuzzy events and cash flow information can be specified as a special type of fuzzy numbers. The present worth of each fuzzy project cash flow can be subsequently estimated. At the same time, to select fuzzy projects and determine the optimal decision time under limited capital budget, we offer an example to analyze the results of the capital budgeting problem under uncertainty using a fuzzy real option valuation.

Appendix Supplementary Explanation for Fuzzy Set

The Definition of Fuzzy Set

Fuzzy sets have been introduced byZadeh (1965). What Zadeh proposed is very much a paradigm shift that first gained acceptance in the Far East, and its successful application has ensured its adoption around the world. Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. In classical set theory, the membership of elements in relation to a set is assessed in binary terms according to a crisp condition – an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets are an extension of classical set theory since, for a certain universe, a membership function may act as an indicator function, mapping all elements to either 1 or 0, as in the classical notion. Specifically, a fuzzy set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1]. A fuzzy set on a classical set Χ is defined as follows:

$$ \tilde{A}=\left\{\Big(x,{\mu}_A(x)\Big)|x\in X\right\}. $$

Membership Function

The membership function μ_A(x) quantifies the grade of membership of the elements x to the fundamental set Χ. An element mapping to the value 0 means that the member is not included in the given set; 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.

Fuzzy Logic

Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth values between “completely true” and “completely false.” As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.

The Operations of Fuzzy Set

After knowing about the characteristic of fuzzy set, we will introduce the operations of fuzzy set. A fuzzy number is a convex, normalized fuzzy set whose membership function is at least segmental continuous and has the functional value μ_A(x) = 1 at precisely one element. This can be likened to the funfair game “guess your weight,” where someone guesses the contestants weight, with closer guesses being more correct, and where the guesser “wins” if they guess near enough to the contestant’s weight, with the actual weight being completely correct (mapping to 1 by the membership function). A fuzzy interval is an uncertain set with a mean interval whose elements possess the membership function value μ_A(x) = 1. As in fuzzy numbers, the membership function must be convex, normalized, and at least segmental continuous.

Defuzzify

When we are through the operations of fuzzy set to get the fuzzy interval, we will convert the fuzzy value into the crisp value. Below are some methods that convert a fuzzy set back into a single crisp (nonfuzzy) value. This is something that is normally done after a fuzzy decision has been made, and the fuzzy result must be used in the real world. For example, if the final fuzzy decision were to adjust the temperature setting on the thermostat a “little higher,” then it would be necessary to convert this “little higher” fuzzy value to the “best” crisp value to actually move the thermostat setting by some real amount.

Fuzzy Decision

After the process of defuzzifying, the next step is to make a fuzzy decision. Fuzzy decision is a model for decision-making in a fuzzy environment, and the object function and constraints are characterized as their membership functions, the intersection of fuzzy constraints, and fuzzy objection function.