The paper compares numerically the results from two real option valuation methods, the Datar-Mathews method and the fuzzy pay-off method. Datar-Mathews method is based on using Monte Carlo simulation within a probabilistic valuation framework, while the fuzzy pay-off method relies on modeling the real option valuation by using fuzzy numbers in a possibilistic space. The results show that real option valuation results from the two methods seem to be consistent with each other. The fuzzy pay-off method is more robust and is also usable when not enough information is available for a construction of a simulation model.

Real option analysis (ROA) is slowly becoming a part of the investment analysis process in companies [

This paper concentrates on comparatively numerically analyzing two ROA methods, the Datar-Mathews method (DMM) that exploits Monte Carlo simulation in real option valuation [

This research, of which initial ideas are reported in [

This paper continues by presenting the constructs of the Datar-Mathews method and of the fuzzy pay-off method, followed by two case-based numerical illustrations that are used to showcase the practical use of the methods. Then, the results from the numerical illustrations are compared and finally the paper is closed with a discussion and conclusions.

In this part, we describe the structures of the two compared ROA methods, the Datar-Mathews method and the fuzzy pay-off method.

Datar-Mathews real option valuation method [

Managers are asked to define the type and details of the distribution of the possible values for each model input variable, from which the simulation procedure randomly draws values.

Simulation is run to generate a sufficient number of (typically thousands) pseudorandom profitability (NPV) outcomes with the model. From the outcomes a histogram is compiled, which is treated as a probability distribution of the project NPV pay-off.

The DMM treats the project as an option and in order to “move” from the NPV pay-off distribution to an option pay-off distribution the subzero outcomes from the project are mapped to zero, while they keep their original probability weight. This means that all the negative outcomes’ weights are truncated to zero.

The real option value (ROV) is calculated as the mean of the resulting option pay-off distribution.

The Datar-Mathews method is a relatively simple method for the user. In addition to the needed discounted cash-flow the user must have the ability to use a standard Monte Carlo simulation. Typically the analysis is conducted on spreadsheet software. The method has previously been used, for example, in the valuation of aircraft development projects [

Fuzzy pay-off method [

Three or four scenarios of the future project cash-flow streams are estimated. Typically the managers are asked to provide estimates for a “minimum possible” and a “maximum possible” and one or two “best estimate” scenarios. The estimated cash-flows are used in the calculation of the NPV for each scenario. Revenues and costs may be estimated separately and separate discount rates may be used for revenues and costs. The link between the operational costs and the revenues must be properly scrutinized; see [

A fuzzy pay-off distribution for the project is constructed from the scenario NPVs. As three or four scenarios are typically used, the fuzzy pay-off distribution is either triangular or trapezoidal. The minimum possible and maximum possible scenario NPVs are considered to establish the lower and the upper limits of the distribution and they are assigned a limit to zero degree of membership in the set of possible NPV outcomes. The best estimate value(s) is assigned full membership.

The real option value is directly calculated from the fuzzy pay-off distribution. The formula used adheres to the typical real option valuation logic and is simply the possibilistic mean of the positive side of the distribution weighted by the project “success ratio.” The project success ratio is the area of the pay-off distribution over the positive side divided by the total area of the pay-off distribution; see the following equation:

Details of the fuzzy pay-off method can be found in [

Two investment cases are used to compare results derived with the two methods. The investment cases are both industrial-scale investment projects into a solar photovoltaic (PV) power plant but under different renewable energy supporting schemes (subsidy schemes).

The first case analyzes an investment that falls within the scope of the Russian renewable energy (RE) support mechanism, based on long-term capacity contracts with a rather complex incentive system. A guaranteed capacity price within these contracts is calculated by the regulating authority as a variable rate annuity that is designed to provide a certain level of return on investment. The scheme takes into account changing market conditions and the project-specific performance [

The second case represents an investment into the same project under a simpler generic feed-in premium RE incentive scheme, such as that which is used more generally in Europe. The used incentive scheme guarantees a fixed premium over the spot electricity price over the long term (for the calculations twenty years are assumed). For the purposes of this illustration, the premium level is set in a way that it provides approximately the same level of profitability as the first case. This has no importance for the comparison of the results from the point of view of the comparative analysis of the two real option analysis methods but allows the comparison of the two RE incentive mechanisms for those interested. A more detailed description of the two cases and the set of assumptions made can be found in [

A typical “classical” NPV investment profitability analysis calculation model is used in both cases. The software used is Microsoft Excel® for the analyses with the fuzzy pay-off method and Matlab Simulink® for the analyses with the Datar-Mathews method. Even if the software used to run the analysis is different, the models used are identical. The information used in creating the three needed scenarios for the fuzzy pay-off method is presented in Table

Uncertain factors: see [

Factor | Range of values | ||
---|---|---|---|

Pessimistic | Best estimate | Optimistic | |

Electricity price, rub./MWh | 1000 | 2000 | 3000 |

Consumer price index (inflation) | 1.70 | 1.35 | 1.00 |

CapEx level | 150% | 100% | 80% |

Capacity factor (percent of target) | 30% | 75% | 120% |

Localization requirement | Failed | Fulfilled | Fulfilled |

The same values (Table

Results for the first case with the two methods are presented in Figure

NPV distributions for the first case. (a) Simulated NPV distribution, (b) triangular fuzzy NPV, and (c) stylized plot of both distributions on the same graph. Dashed lines: red, expected NPV; green, ROV.

One can observe that the lowest and the highest values of the distributions match, which is expected, but in this case the shapes of the two distributions are very different. The rather complex construct of the underlying Russian RE incentive mechanism causes the shape of the simulated NPV pay-off histogram of the project to be atypical with multiple summits. This indicates that there are local maxima which the Monte Carlo simulation used in the Datar-Mathews method can capture. At the same time, it is quite clear that the fuzzy pay-off method may be too robust for the complex problem. What is interesting is that the real option valuation results are nevertheless similar in absolute numbers; see also Table

Comparison of result statistics (in rub. bln.).

First case | Second case | |||||
---|---|---|---|---|---|---|

DMM | FPOM | Difference | DMM | FPOM | Difference | |

ROV | 0.004 | 0.002 | 0.002 | 0.022 | 0.021 | 0.001 |

| −0.525 | −0.261 | 0.264 | −0.322 | −0.217 | 0.105 |

Standard deviation (×100%) | 0.441 | 0.358 | 0.083 | 0.384 | 0.406 | 0.022 |

“Success ratio” | 3% | 9% | 6% | 11% | 28% | 17% |

Results for the second case, an investment into a renewable energy project with a generic feed-in premium incentive system, are presented in Figure

NPV distributions for the second case. (a) Simulated NPV distribution, (b) triangular fuzzy NPV, and (c) stylized plot of both distributions on the same graph. Dashed lines: red, expected NPV; green, ROV.

As in the first case, the “limits” or the extreme high and low values of the distributions are almost equal; again this was expected; however, the shapes of the two distributions are much more uniform compared to the first case. This can be interpreted in the way that the problem complexity is at a level that is suitable also for the more robust fuzzy pay-off method. The results from the two methods are more similar to each other in the second case than in the first case; this indicates that when the type of problem analyzed is relatively simple the precision of the fuzzy pay-off method can be considered to be satisficing. The difference between the ROV in the second case is negligible. Lilliefors test shows that the simulated distributions in both cases are not “normal.” Descriptive statistics for the two cases are collected in Table

Table

Despite the comparable numeric indicators, the DMM and the FPOM are substantially different in terms of their implementation and computational performance (Table

Comparison of DMM and FPOM performances.

DMM | FPOM | |
---|---|---|

Computational time (first case), s. | 6.19 | 0.11 |

Computational time (second case), s. | 1.01 | 0.04 |

Ease of implementation | Requires simulation software and the skills to use it | Simple spreadsheet software is enough |

Information content of results | Histogram of the outcome, ability to capture irregularities of complex problems, for example, step-causal or nonlinear interdependency of variables | A triangular pay-off distribution has a fixed form regardless of the problem complexity, a simplification of results |

The computational time to run both analyses is short, but the simulation used in the DMM is time-consuming; it takes roughly 25–50 times more time than running three scenario calculations for the FPOM. The simulation time depends on the complexity of the problem. In addition, implementing the DMM requires building a model in a computational environment with Monte Carlo simulation capability and requires the user of the model to have the required skills to build simulation models and to run them. In contrast, the FPOM can be easily implemented with spreadsheet software, without any special skills.

The findings presented here support the previous findings [

The Datar-Mathews and the fuzzy pay-off method are both relatively new real option analysis methods that have been constructed, while keeping in mind managerial users. Both exploit the well-known real option valuation logic but are based on different theoretical foundations in terms of their computational procedure. The Datar-Mathews method is a simulation-based method that treats uncertainty in terms of probability theory, while the fuzzy pay-off method is a more robust method based on using fuzzy number representations of cash-flow information.

This paper has demonstrated with numerical illustrations the application and usability of these two methods in the analysis of two investment cases with different levels of complexity. The comparative analysis of the results of these analyses reveals that while the pay-off method simplifies the analysis, it still seems to offer sufficient precision for the analysis of problems with low complexity. On the other hand, the simulation-based Datar-Mathews method is able to treat problems that have more complex structures but requires more computational time and specialized software. The overall results obtained in terms of real option valuation show that the two methods return similar results. One has to observe that the two presented cases are not enough to draw definitive conclusions on the matter but illustrate well the difference in how robust these methods are.

The work presented in this paper can be used in understanding better the kinds of problems these methods are good for. The results are of use for practitioners navigating selection of proper valuation technique and support earlier findings on the usability of these methods. The comparison of these methods with other methods merits further study and specifically the amount of complexity and type of uncertainty that different methods can handle in terms of credible and usable results. In more general terms, the study of the usability of different analysis real option analysis methods is a topic that has been “under studied” in the past.

The authors declare that there are no competing interests regarding the publication of this paper.

The authors would like to acknowledge the support received by Mariia Kozlova from Fortum Foundation.