Christer Carlsson
IAMSR / Åbo Akademi University
[email protected]
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
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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.
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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.
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GIGA-INVESTMENTS
 Facts and observations
 The core products and services produced by gigainvestments 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
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GIGA-INVESTMENTS
 Facts and observations
 Giga-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 effective
 A new teechnology will redefine the CSF:s for the market
 Customer needs are adjusting to the new possibilities of the gigainvestment
 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
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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
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1.8.2002
0,95
1.7.2002
1.6.2002
1.5.2002
1.4.2002
1.3.2002
1.2.2002
1.1.2002
1.12.2001
1.11.2001
1.10.2001
1.9.2001
1.8.2001
0,85
1.7.2001
1.6.2001
1.5.2001
1.4.2001
1.3.2001
1.2.2001
1.1.2001
kurssi
EUR/USD, close daily 1.1.2001 - 16.8.2002,
rates 19.6. 2001 ja 19.6.2002
1,1
1,05
1
0,9566
0,9
0,8579
0,8
pvm
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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
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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
rf
Risk-free interest rate
Risk of the project
σ
Standard deviation of returns
on stock
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REAL OPTION VALUATION (ROV)
The value of a real option is computed by
ROV =Se −δT N (d1) − Xe −rT N (d2)
where
d1 = [ln (S0 /X )+(r −δ +σ2 /2)T] / σ√T
d2 =d1 − σ√T,
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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
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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
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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,
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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.
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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
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FUZZY REAL OPTION
VALUATION
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SCREENSHOTS FROM MODELS
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NUMERICAL AND GRAPHICAL
SENSITIVITY ANALYSES
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FUZZY OPTIMAL TIME OF
INVESTMENT
Invest when FROV is at maximum:
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)
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OPTIMAL TIME OF INVESTMENT
How long should we postpone an investment?
Benaroch and Kauffman (2000) suggest:
Optimal investment time = when the option value Ct* is at
maximum (ROV = Ct*)
Ct* = max Ct = Vt e-δt N(d1) – X e-rt N (d2 )
Where
t =0 , 1 ,...,T
Vt = PV(cf0, ..., cfT, βP) - PV(cf0, ..., cft, βP) = PV(cft +1, ...,cfT, βP),
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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 the
following value function to order fuzzy real option values,
Ĉt = (ctL ,ctR ,αt, βt), of the trapezoidal form:
v (Ĉt) = (ctL + ctR) / 2 + rA · (βt + αt) / 6
where rA > 0 denotes the degree of the investor’s risk aversion
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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]

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