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1. Real Options Valuation for South African Nuclear Waste Management Using a Fuzzy Mathematical Approach by Obakeng Montsho, Rhodes University Energy Postgraduate Conference 2013

2. Contents • Aim of the Study • Fuzzy Set Theory • Real Options Theory • Fuzzy Set Theoretic Real Option Valuation Models • Empirical Analysis - Data and Methods - Results • Conclusion

3. Aim of the Study To estimate the costs of deferring the decommissioning of an electricity-generating nuclear power plant which operates in South Africa. Fuzzy Set Theory What does “fuzzy” mean in a mathematical context? • Assist in finding the precise solutions from the approximated data. • It is a logic that go beyond true and false. • Everything is a matter of degree of truthfulness and/or falsehood. Zadeh (1965) introduced the word fuzzy as a formalization of uncertainty or vagueness in complex systems. Fuzzy Numbers • Fuzzy set theory employs fuzzy numbers to quantify subjective fuzzy estimates. • A fuzzy set A is called a fuzzy (real) number, if and only if A is convex and if there exists exactly one real number c with A(c) = 1. • Most common types of fuzzy numbers are triangular and trapezoidal. • Triangular and trapezoidal fuzzy numbers are defined by three and four parameters, respectively.

4. Fuzzy Set Theory • A triangular fuzzy number A with membership function µA(x) is defined on ℝ by (1.1) Where [a₁, a₂ ] is the supporting interval. • A trapezoidal fuzzy number A is defined on ℝ by (1.2) The supporting interval is and the flat segment on level α = 1 has projection on the x-axis. Real Options Theory • In many project evaluation settings, the firm has more than one option to make a strategic changes to the project during its life. • For example, a nuclear waste management authority may decide to defer investments due to the uncertainty of future costs. • These strategic options are called Real Options.

5. Real Options Theory • Real options are applicable to many of the situations where management can decide a course of action. Tabl Table 1.1: Types of Real Options

6. Fuzzy Set Theoretic Real Option Valuation Models • To evaluate investment projects that are embedded, the fuzzy binomial approach has been introduced by Liao and Ho (Liao and Ho, 2010). • Most of the cash flow models used for financial decision making involve some degree of uncertainty. • Using fuzzy set theory to rationalize the uncertainty, triangular fuzzy numbers are often used to test knowledge of profitability indices. Fuzzy Binomial Model • If the jumping factors can be written as = [] and = [, ], the risk-neutral probabilities can be expressed in terms of fuzzy numbers as follows: (1.3) where= [, , and = [, , . Therefore, the above equation can be written as: (1.4) for i = 1, 2, 3. Its solution is: (1.5) and (1.6)

7. Fuzzy Set Theoretic Real Option Valuation Models • The fuzzy option values and are expressed in terms of the fuzzified jumping factors and which yields the fuzzy call option price: = x + x ] (1.7) Fuzzy Black-Scholes Formula The formula for calculating fuzzy real option values is: (1.8) where , (1.9) and - the expected present value of cash flow; - the value of expected costs; - the possibilistic mean value of the expected cash flows; ) - the possibilistic mean value of expected costs; The above equation for computing fuzzy real option values can be expressed as ,, α, β) (2.0)

8. Empirical Analysis • Important historic and current qualitative information about Koeberg’s operations are concealed by Eskom’s management. Data and Methods The fuzzy binomial valuation approach and FROV using the Black-Scholes formula, will be used to assess a proposal to defer Koeberg’s Life of Plant Plan (LOPP) from 40 years to 50 years. The following assumptions are being considered in this project: • The total number of Koeberg spent fuel assemblies (SFAs) generated over 50 years LOPP and to be finally disposed of, will be 3903. • The mean of the SFAs is 73.41 per year and the standard deviation is 19.34 SFAs. • The method of disposal to be used will be directly in a suitable deep geological repository (Eskom, 2005). • The NPV of the direct disposal option is estimated to be R1.336 billion • The future cost is R7.680 billion for the period of 50 years since the commissioning. • Based on what has been done in (Liao and Ho, 2010) we estimate the coefficient of variation to be ±30% per year. The volatility of SFA production will be represented by the triangular fuzzy number: = [(1 – 0.3) x 0.26, 0.26, (1 + 0.3) x 0.26] = [0.182, 0.26, 0.338].

9. Empirical Analysis • Python code has been written to create the binomial tree of project value and the decision tree with the option to defer. • One should note that in computing the decision tree, we follow Liao and Ho  in estimating: (2.0) where and are computed using Equations (1.5) and (1.6) respectively, and are the top and bottom decision nodes respectively, I represents future costs of direct disposal and r is the annual discount rate. Referring to the extension of Koeberg’s LOPP, the related parameters are estimated as follows: • Koeberg’s expected cash flow ( would be worth between R891 billions and R1603.8 billions. • The expected costs () are estimated to be worth between R6,498 and R8,447.4 billions. • Sensitivity analysis conducted for time are 30, 40 and 50 years; and for risk-free interest rate are 10%, 15% and 20% values. • The value lost over the duration of the option (δ) is set at 0.03 following Carlsson and Fuller (Carlsson and Fuller, 2003).

10. Empirical Analysis Results – Fuzzy Binomial Model Table 1.2: Fuzzy binomial approach using discount-rate (r) = 20% per year The values of E(FENPV) and option premium with 10% and 15% cannot be accepted. Results – Fuzzy Black-Scholes Formula The variables that change simultaneously with time and annual discount rates when computing FROV using Equation 1.8 are the normal cumulative distribution function. Table 1.3: FROV using discount-rate (r) = 10% per year

11. Empirical Analysis Results - Fuzzy Black-Scholes Formula Table 1.4: FROV using discount-rate (r) = 15% per year Table1.5: FROV using discount-rate (r) = 20% per year • The maximal value is obtained when the interest rate is set to be 10% and the year of exercising the option is 30 years. • The expected value of FROV is minimal when the discount-rate is 15% during the fortieth year. • We deduce from Table 1.3, Table 1.4, Table 1.5 that the expected value of FROV decreases as the number of years and discount-rate increases.

12. Empirical Analysis • Thus, deferral of decommissioning is valuable because FROV decreases as the number of years increases. Conclusion • The two valuation methods used do not yield the same results and also behaves differently to the sensitivity analysis. • However, we take note that FROV was computed using the trapezoidal fuzzy numbers whereas the triangular fuzzy numbers were used to compute FENPV. • We only consider the annual SFA production as the source of uncertainty although multiple uncertainties may occur in a practical case. • Results were based on estimates given lack of disclosure of information about relevant variables. • It will be useful to obtain more accurate information about Koeberg’s operation and then estimate values using the binomial, trinomial and Black-Scholes formula considering both triangular and trapezoidal fuzzy numbers.