Christer Carlsson IAMSR / Åbo Akademi University

FORS Christer Carlsson2 GIGA-INVESTMENTS  Facts and observations  Giga-investments made in the paper- and pulp industry, in the heavy metal industry and in other base industries, today face scenarios of slow growth (2-3 % p.a.) in their key markets and a growing over-capacity in Europe  The energy sector faces growing competition with lower prices and cyclic variations of demand  Productivity improvements in these industries have slowed down to 1-2 % p.a

FORS Christer Carlsson3 GIGA-INVESTMENTS  Facts and observations  Global financial markets make sure that capital cannot be used non-productively, as its owners are offered other opportunities and the capital will move (often quite fast) to capture these opportunities.  The capital markets have learned “the American way”, i.e. there is a shareholder dominance among the actors, which has brought (often quite short-term) shareholder return to the forefront as a key indicator of success, profitability and productivity.

FORS Christer Carlsson4 GIGA-INVESTMENTS  Facts and observations  There are lessons learned from the Japanese industry, which point to the importance of immaterial investments. These lessons show that investments in buildings, production technology and supporting technology will be enhanced with immaterial investments, and that these are even more important for re-investments and for gradually growing maintenance investments.

FORS Christer Carlsson5 GIGA-INVESTMENTS  Facts and observations  The core products and services produced by giga- investments are enhanced with lifetime service, with gradually more advanced maintenance and financial add-on services.  New technology and enhanced technological innovations will change the life cycle of a giga-investment  Technology providers are involved throughout the life cycle of a giga-investment

FORS Christer Carlsson6 GIGA-INVESTMENTS  Facts and observations  Giga-investments are large enough to have an impact on the market for which they are positioned:  A ton paper mill will change the relative competitive positions; smaller units are no longer cost effective  A new teechnology will redefine the CSF:s for the market  Customer needs are adjusting to the new possibilities of the giga- investment  The proposition that we can describe future cash flows as stochastic processes is no longer valid; neither can the impact be expected to be covered through the stock market

FORS Christer Carlsson7 GIGA-INVESTMENTS  The WAENO Lessons: Fuzzy ROV  Geometric Brownian motion does not apply  Future uncertainty [15-25 years] cannot be estimated from historical time series  Probability theory replaced by possibility theory  Requires the use of fuzzy numbers in the Black-Scholes formula; needed some mathematics  The dynamic decision trees work also with fuzzy numbers and the fuzzy ROV approach  All models could be done in Excel

FORS Christer Carlsson8

9 REAL OPTIONS  Types of options  Option to Defer  Time-to-Build Option  Option to Expand  Growth Options  Option to Contract  Option to Shut Down/Produce  Option to Abandon  Option to Alter Input/Output Mix

FORS Christer Carlsson10 REAL OPTIONS  Table of Equivalences: INVESTMENT OPPORTUNITYVARIABLECALL OPTION Present value of a project’s operating cash flows S Stock price Investment costs X Exercise price Length of time the decision may be deferred t Time to expiry Time value of money rfrf Risk-free interest rate Risk of the project σ Standard deviation of returns on stock

FORS Christer Carlsson11 REAL OPTION VALUATION (ROV) The value of a real option is computed by ROV =Se −δT N (d 1 ) − Xe −rT N (d 2 ) where d 1 = [ln (S 0 /X )+(r −δ +σ 2 /2)T] / σ√T d 2 =d 1 − σ√T,

FORS Christer Carlsson12 FUZZY REAL OPTION VALUATION Fuzzy numbers (fuzzy sets) are a way to express the cash flow estimates in a more realistic way This means that a solution to both problems (accuracy and flexibility) is a real option model using fuzzy sets

FORS Christer Carlsson13 FUZZY CASH FLOW ESTIMATES Usually, the present value of expected cash flows cannot be characterized with a single number. We can, however, estimate the present value of expected cash flows by using a trapezoidal possibility distribution of the form Ŝ 0 =(s 1, s 2, α, β)  In the same way we model the costs

FORS Christer Carlsson14 FUZZY REAL OPTION VALUATION We suggest the use of the following formula for computing fuzzy real option values Ĉ 0 = Ŝe −δT N (d 1 ) − Xe −rT N (d 2 ) where d 1 = [ln (E(Ŝ 0 )/ E(X))+(r −δ +σ 2 /2)T] / σ√T d 2 = d 1 − σ√T,

FORS Christer Carlsson15 FUZZY REAL OPTION VALUATION E(Ŝ 0 ) denotes the possibilistic mean value of the present value of expected cash flows E(X) stands for the possibilistic mean value of expected costs σ: = σ(Ŝ 0 ) is the possibilistic variance of the present value of expected cash flows.

FORS Christer Carlsson16 FUZZY REAL OPTION VALUATION  No need for precise forecasts, cash flows are fuzzy and converted to triangular or trapezoidal fuzzy numbers  The Fuzzy Real Option Value contains the value of flexibility

FORS Christer Carlsson17 FUZZY REAL OPTION VALUATION

FORS Christer Carlsson18 SCREENSHOTS FROM MODELS

FORS Christer Carlsson19 NUMERICAL AND GRAPHICAL SENSITIVITY ANALYSES

FORS Christer Carlsson20

FORS Christer Carlsson21 FUZZY OPTIMAL TIME OF INVESTMENT Ĉ t* = max Ĉ t = Ŵ t e -δt N(d 1 ) – X e -rt N (d 2 ) t =0, 1,...,T where Ŵ t = PV(ĉf 0,..., ĉf T, β P ) - PV(ĉf 0,..., ĉf t, β P ) = PV(ĉf t +1,..., ĉf T, β P ) Invest when FROV is at maximum:

FORS Christer Carlsson22 OPTIMAL TIME OF INVESTMENT C t* = max C t = V t e -δt N(d 1 ) – X e -rt N (d 2 ) t =0, 1,...,T How long should we postpone an investment? Benaroch and Kauffman (2000) suggest: Optimal investment time = when the option value C t* is at maximum (ROV = C t* ) Where V t = PV(cf 0,..., cf T, β P ) - PV(cf 0,..., cf t, β P ) = PV(cf t +1,...,cf T, β P ),

FORS Christer Carlsson23 FUZZY OPTIMAL TIME OF INVESTMENT We must find the maximising element from the set {Ĉ 0, Ĉ 1, …, Ĉ T }, this means that we need to rank the trapezoidal fuzzy numbers In our computerized implementation we have employed the following value function to order fuzzy real option values, Ĉ t = (c t L,c t R,α t, β t ), of the trapezoidal form: v (Ĉ t ) = (c t L + c t R ) / 2 + r A · (β t + α t ) / 6 where r A > 0 denotes the degree of the investor’s risk aversion

FORS Christer Carlsson24 EXTENSIONS  Fuzzy Real Options support system, which was built on Excel routines and implemented in four mutlinational corporations as a tool for handling giga-investments.  Possibility vs Probability: Falling Shadows vs Falling Integrals [FSS accepted]  On Zadeh’s Extension Principle [FSS submitted]