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FUZZY REAL OPTION VALUATION - A Better Way to Handle Giga-Investments. Christer Carlsson IAMSR / Åbo Akademi University [email protected]. GIGA-INVESTMENTS. Facts and observations - PowerPoint PPT Presentation
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FORS 8.05.2003 Christer Carlsson 2
GIGA-INVESTMENTS
Facts and observationsGiga-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 8.05.2003 Christer Carlsson 3
GIGA-INVESTMENTS
Facts and observationsGlobal 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 8.05.2003 Christer Carlsson 4
GIGA-INVESTMENTS
Facts and observationsThere 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 8.05.2003 Christer Carlsson 5
GIGA-INVESTMENTS
Facts and observationsThe 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 8.05.2003 Christer Carlsson 6
GIGA-INVESTMENTS
Facts and observationsGiga-investments are large enough to have an impact on
the market for which they are positioned:A 300 000 ton paper mill will change the relative competitive
positions; smaller units are no longer cost effectiveA new teechnology will redefine the CSF:s for the marketCustomer 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 8.05.2003 Christer Carlsson 7
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 8.05.2003 Christer Carlsson 8
EUR/USD, close daily 1.1.2001 - 16.8.2002, rates 19.6. 2001 ja 19.6.2002
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FORS 8.05.2003 Christer Carlsson 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 8.05.2003 Christer Carlsson 10
REAL OPTIONS
Table of Equivalences:
INVESTMENT OPPORTUNITY VARIABLE CALL 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 rfRisk-free interest rate
Risk of the project σ Standard deviation of returns on stock
FORS 8.05.2003 Christer Carlsson 11
REAL OPTION VALUATION (ROV)
The value of a real option is computed by
ROV =Se −δT N (d1) − Xe −rT N (d2)
whered1 = [ln (S0 /X )+(r −δ +σ2 /2)T] / σ√T
d2 =d1 − σ√T,
FORS 8.05.2003 Christer Carlsson 12
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 8.05.2003 Christer Carlsson 13
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
Ŝ0 =(s1, s2, α, β)
• In the same way we model the costs
FORS 8.05.2003 Christer Carlsson 14
FUZZY REAL OPTION VALUATION
We suggest the use of the following formula for computing fuzzy real option values
Ĉ0 = Ŝe −δT N (d1) − Xe −rT N (d2)
where
d1 = [ln (E(Ŝ0)/ E(X))+(r −δ +σ2 /2)T] / σ√T
d2 = d1 − σ√T,
FORS 8.05.2003 Christer Carlsson 15
FUZZY REAL OPTION VALUATION
• E(Ŝ0) 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(Ŝ0) is the possibilistic variance of the present value of expected cash flows.
FORS 8.05.2003 Christer Carlsson 16
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 8.05.2003 Christer Carlsson 17
FUZZY REAL OPTION VALUATION
FORS 8.05.2003 Christer Carlsson 18
SCREENSHOTS FROM MODELS
FORS 8.05.2003 Christer Carlsson 19
NUMERICAL AND GRAPHICAL SENSITIVITY ANALYSES
FORS 8.05.2003 Christer Carlsson 20
FORS 8.05.2003 Christer Carlsson 21
FUZZY OPTIMAL TIME OF INVESTMENT
Ĉt* = max Ĉt = Ŵt e-δt N(d1) – X e-rt N (d2 ) t =0 , 1 ,...,T where
Ŵt = PV(ĉf0, ..., ĉfT, βP) - PV(ĉf0, ..., ĉft, βP) = PV(ĉft +1, ..., ĉfT, βP)
Invest when FROV is at maximum:
FORS 8.05.2003 Christer Carlsson 22
OPTIMAL TIME OF INVESTMENT
Ct* = max Ct = Vt e-δt N(d1) – X e-rt N (d2 ) t =0 , 1 ,...,T
How long should we postpone an investment?Benaroch and Kauffman (2000) suggest:Optimal investment time = when the option value Ct* is atmaximum (ROV = Ct*)
Where
Vt = PV(cf0, ..., cfT, βP) - PV(cf0, ..., cft, βP) = PV(cft +1, ...,cfT, βP),
FORS 8.05.2003 Christer Carlsson 23
FUZZY OPTIMAL TIME OF INVESTMENT
We must find the maximising element from the set {Ĉ0, Ĉ1, …, ĈT}, this means that we need to rank the trapezoidal fuzzy numbers
In our computerized implementation we have employed thefollowing value function to order fuzzy real option values, Ĉt = (ct
L ,ctR ,αt, βt), of the trapezoidal form:
v (Ĉt) = (ctL + ct
R) / 2 + rA · (βt + αt) / 6
where rA > 0 denotes the degree of the investor’s risk aversion
FORS 8.05.2003 Christer Carlsson 24
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]