Triangular fuzzy number

The use of triangular fuzzy numbers is applicable for a standard valuation as the concept of a triangular fuzzy number harmonizes with a standard cash flow set-up with three different scenarios.

A fuzzy set A is called a triangular fuzzy number if it is noted by a peak (centre) a, a left width α

> 0 and a right width defined by β > 0 and its membership function has the following form

1

1

0,

4.3

, with the use of the notation A = (a, α, β) and the support of A as (a-α, a+β).

The triangular fuzzy number is visualized in figure 4.1 below. The peak, or centre of the triangle is represented by a, which can be seen as a fuzzy quantity “x is equal to a”. The left width is represented by alpha, α, and represents the distance from a to the lower extreme, hence a – α.

The right width is represented by beta, β, and represents the distance from a to the higher

extreme, a + β. It is important to remember that it is the distance that is represented by α and β as

“real positive numbers”, hence α > 0 and β > 0.

We now end up with a fuzzy distribution where (a – α) and (a +β) represent the smallest and the largest possible value respectively. The shape of the fuzzy distribution is defined from its

membership function which was defined above. This means that the centre, a, has a membership degree of one, while the extremes have a membership degree of zero.

1

Degree of  membership

0

(a‐α)  0   a E(A+) (a+β)

positive NPV outcomes,

valued at weighted mean, weight * E(A+) negative NPV outcomes,

all valued at 0

Figure 4.1: Own construction. Source: Collan, Fullér & Mezei (1), 2009 

The support of A (membership function) is a crisp subset of real numbers ranging from (a – α) to (a + β). In figure 4.1 above the triangular fuzzy number has negative values (a – α) to zero and positive values from zero to (a + β) which means that the negative numbers represent the negative NPV of payoffs from a given project (which is valued at 0) and vice versa for the positive side.

4.1.1 The fuzzy payoff method for real option valuation

The fuzzy payoff method was introduced by Mezei, Collan and Fullér in 2009. They presented a new method to calculate the real option value (referred to as ROV) with the use of triangular fuzzy numbers (Mezei, Collan & Fullér (1), 2009, p.6). Their method is inspired by a real option valuation approach applied to a project in the Boeing Corporation (Datar & Mathews, 2007).

In order to calculate the ROV a process is needed to create an expected payoff distribution. To achieve this models like Black-Scholes (Black & Scholes, 1973) utilise stochastic processes while the binomial approach (Cox, Ross & Rubinstein, 1979) uses binomial processes. However, some experts argue that it is not in line with the reality of real investments to use stochastic processes, because managerial actions can affect the value, hence the value is not just random (Kinnunen, 2010, p.12).

The Datar-Mathews method uses a three-way cash flow scenario forecast, i.e. with a bad case, a base case and a good case, as the basis for the real option valuation. The method uses an

expected probability distribution found by a Monte Carlo simulation. Mezei, Collan and Fullér

instead use the fuzzy pay-off method to create a possibility distribution by rearranging the forecasted cash flows. The cash flows are rearranged to depict every possible outcome of the project and hence become a fuzzy cash flow. To portrait the most extreme positive outcome (the good case for the fuzzy cash flow) the lowest possible costs are deducted from the highest possible revenue. And vice versa for the most extreme negative outcome (the bad case for the fuzzy cash flow). The base case from the original cash flow forecast represents the base case of the fuzzy cash flow. By performing this rearrangement of the cash flows the possibility

distribution is created and becomes a triangular fuzzy number (also referred to as the fuzzy NPV), which the real option value can be calculated from.

In the calculation of the ROV the value can be found by weighing the positive values (NPV > 0) by their expected probability. For the negative outcomes (NPV < 0) NPV is set to zero (NPV < 0

 NPV = 0), as the managerial flexibility allows the management to terminate projects with a projected negative NPV and thus avoid further investment in a non-profitable project.

Mezei, Collan and Fullér present the following formula to calculate the ROV,

∞

∞

∞

4.4

The ROV is found by calculating the area of the positive side divided by the entire area of the triangle and then multiplied by the possibilistic (fuzzy) mean value of the positive side of the fuzzy distribution.

The relationship between the areas can easily be calculated by simple integral calculus but the calculation of the fuzzy mean value of the positive side needs further derivation which is presented next.

Carlsson and Fullér derived the general formula for calculation of the possibilistic mean value (4.1). Collan, Mezei and Fullér were able to derive four additional formulas for the calculation of the E(A+) from (4.1), due to different cut-off points (Collan, Mezei and Fullér (1), 2009, p. 7).

The cut-off levels determine if a, α, β is above or below zero, respectively. The four different cases are presented below (Collan, Mezei and Fullér (2), 2009, p. 7).

First case is where the whole fuzzy distribution is above zero, when 0 < (a- α). The mean value of the positive area can then be calculated as shown in equation 4.5.

6 4.5

Second case is where the fuzzy distribution is partly above zero, which means that a is above zero but (a – α) is below zero; (a-α < 0 < a), which resembles the situation that is shown in figure 4.1. The mean value of the positive area can then be calculated as shown in equation 4.6.

6 6 4.6

Third case is where fuzzy distribution is partly above zero, but with the centre, a, below zero but a + β still above zero; (a < 0 < a + β). The mean value of the positive area can then be calculated as shown in equation 4.7.

6 4.7

Fourth case is when the whole fuzzy distribution is below zero. The mean value of the positive area can then be calculated as shown in equation 4.8.

0 4.8

In order to ensure a better understanding of triangular fuzzy numbers a small illustrative example with the fuzzy pay-off method have been created in order to aid the understanding of how we create the triangular fuzzy number and how we calculate the ROV which is shown in appendix 6.

The shape of the triangular fuzzy number made it ideal dealing with the standard three-way cash flow scenario analysis but in other cases a different cash flow scenario set-up could be

experienced. For instance we could image a situation where it would be difficult to estimate the base case expressed as a single value. Instead we could consider expressing the estimates in an interval. Such a situation is just what the use of fuzzy trapezoidal numbers in valuations purposes can help with (Carlsson & Fullér, 2000, p. 70) as demonstrated by Ucal and Kahraman (Ucal &

Kahraman, 2009). Although based on many of the same assumptions as the triangular fuzzy number it lacks in aiding to the mental understanding of the possible future strategic and operational decision-making. Also it is based on a cash flow forecast type which none of the participants in our survey used, why we will focus on the application of triangular fuzzy

numbers. Hence a thorough examination of fuzzy trapezoidal number’s applicability to perform a real options valuation is presented along with a numerical example in appendix 7.