(This article will discuss a new real options technique using fuzzy payoff method developed by Collan, Fuller and Mezei in 2009. The method is simpler than other available real option valuation methods. No simulation required and easy to construct in the excel spreadsheet)

**Intuition behind Real Options**

Before we go through the theory background on the fuzzy pay off method, we refresh the concept of Real Options.

Real options method focuses on the potential upside rather than downside. As seen in the below figure, it shows if we produce 100 million ton today, we’ll get the revenue around $ 6 billion using the current price of $60/ton. The cost requirement to produce 100 million ton is $6,050 billion, so the net pay off (NPV) today is $ – 50 million

Supposed next year, there will be 2 price scenarios, i.e: price will increase to $62/ton dan decrease to $58/ton with probability of 50% for each scenario.

Assuming the cost to produce is not changed in the next year, there will be two pay off scenarios, i.e:

1. If price is going up to $62/ton, NPV = (100 x 62) – 6.05 = $ 150 million

2. If price is going down to $58/bbl, NPV = (100 x 58) – 6.05 = $ – 250 million

If we look at the scenario 2 when the price is down to $58 next year, management will decide not to operate when the payoff is negative. In fact, scenario 2 is potentially not happened or value of the scenario 2 is nil.

As a result, if we delay the operation to the next year, we can expect the pay off as follows:

NPVT=1 = (50% x 150) + (50% x 0) = $ 75 million ? this is Real Options Value (ROV)

In the probabilistic approach, we can described ROV distribution using below figure

The white line described the traditional NPV distribution from negative to positive value.

The yellow line described ROV distribution that has less negative value than Traditional NPV has. The ROV distribution represents that management will always take an action in the future to minimize downside losses and maximize upside gain of the project.

**Issues on Real Options (RO)**

The issues why real options is still not applied widely for project evaluation is some peoples thought that most of RO techniques are not practical, and it’s only for academic research. The RO techniques concentrated on design on advance models rather than simple model to explain what the RO is. The “black box” in the real options techniques made lack of interest of many companies to adopt real options into their daily routines of investment profitability analysis. It seems it requires an alternative real option technique that is clearly presentable and intuitively understandable. And the most important thing, this alternative technique must be able to adapt to the way profitability analysis is presently done in companies.

**Fuzzy Pay-Off Technique**

This technique was developed by Collan, Fuller and Mezei in 2009. The method is based on:

- Real option value can be calculated from the pay-off distribution of a project by calculating the probability weighted average of the positive values of the distribution
- The pay-off distribution is the fuzzy net present value (FNPV) of the project
- The weighted average of the fuzzy pay-off distribution is the fuzzy mean of the positive values of the fuzzy pay-off distribution
- The fuzzy mean of the positive values of the fuzzy NPV is the real option value

When fuzzy pay-off is described with triangle as seen below

Assumed that 85% of the area NPV is positive outcome (area from point 0 to point ?) and 15% of the area NPV is negative income (area from point ? to point 0), the mean value of the positive area is M+ by considering the all negative values at 0.

The ROV Formula using fuzzy pay-off technique is seen in the below formula:

There will be two parts in the above formula.

The first part of the formula is the success ratio (yellow circle) that would be a percentage between the positive area of the pay-off distribution and the whole area of the pay-off distribution.

In a triangular distribution, we can calculate success ratio using area of a triangle between positive area and the whole area of the triangle as seen in the figure below

The second part of the formula is the possibilistic mean of the positive area of the pay-off distribution [E(A+)].

There will be 4 conditions for the E(A+), i.e:

1. The whole of pay-off distribution is above zero

2. When the pay-off distribution is partly above zero (zero is between the minimum possible NPV and the base case NPV)

3. When the pay-off distribution is partly above zero (zero is equal to the base case NPV or between the base case NPV and the maximum NPV)

4. Whole pay-off distribution is below zero

* E(A+) = 0*

Note:

a = the best guest NPV

alpha = different betweeen minimum and best guess NPV

beta = different between best guess and maximum NPV

**Fuzzy Pay Off Mechanism**

There are three (3) steps to calculate RO with fuzzy pay off technique, i.e:

1. Build 3 cash flow scenarios and perform present value & NPV calculations as seen in the example below.

2. Create the fuzzy pay off distribution chart, as seen below

3. Calculate the Possibilistic Mean of the positive area of the Pay off distribution, as described with the red line in the below chart

**Case Study**

An oil field would be developed with 3 reserve scenarios using each of development cost. The data is summarized in the below table.

**First step**: Build 3 Scenarios and perform PV.

In performing Present Value (PV) for each scenario, we use two different costs of capital for operating and investment cash flows.

The operating cash flows are subject to market risk and these cash flows should therefore be adjusted by an appropriate discount rate that incorporates the same risks as the WACC

Investment cash flows of these costs have a relatively low risk as they are actively managed by the management.

Cummulative NPV is derived from cummulative PV of operating cash flow minus PV of invesment CF. The result is seen in the below table

**Second Step**: Create Fuzzy NPV Distribution

Fuzzy NPV (FNPV) is constructed in order to model the most extreme Cash Flow (CF) possibly. It is calculated by :

- Operational CF from High case minus investment CF from Low case -> Optimistic
- Operational CF from Base case minus investment CF from Base case -> Base
- Operational CF from Low case minus investment CF from High case -> Pessimistic

The above figures show the cummulative net cash flow for each scenario. The result of fuzzy NPV calculation is seen in the below table.

**Third Step**: Calculate Possibilistic Mean of the Fuzzy NPV Distribution

The above figure shows two results, i.e:

1. The possibilistic mean of the fuzzy NPV distribution is $ 14.3 million (green line)

2. The Real Options value which is derived from two parts from the above formula:

a. Success ratio: the positive area is 63% of the whole area of triangle.

b. Possibilistic mean of the positive side of the fuzzy NPV distribustion is $ 26.6 million.

The Real Option Value is 63% x 26.6 = $ 16.7 million (red line)

**Wrap Up**

The Fuzzy pay off technique is simpler than other available real option valuation techniques. This technique can utilize directly scenarios ”already” available in the company for profitability analysis. There is no extra work in the company such as doing a simulation. Based on the technique, it is easy to construct usable applications with the most commonly used spread sheet software.