Important Decisions and the Benefits of Fuzzy Models

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Important Decisions and the Benefits of Fuzzy
Models
FUZZY REAL OPTIONS

• Professor Christer Carlsson
• IAMSR/Abo Akademi University
• May 25, 2004
• christer.carlsson_at_abo.fi

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Combinatorial Explosion Example 1

• Shortest path through 10 points
• Assume a super-powered" computer analyzes,
  quantifies, and compares a million alternatives
  every second
• Answer found in less than one second
• Now find the shortest path through 20 points
• Same computer would take over 39,000 years to
  solve!

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Combinatorial Explosion Example 2

• Construct a five-stock portfolio out of 100
  possible stocks
• Evaluate for risk and return
• Same computer as example one
• Answer found in 1.25 minutes
• Now diversify to nine stocks - 220 Days
• Add one more stock - 5.5 years!

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HANDLING COMPLEX DECISIONS

• Brute force computing
• Real world problems are more complex than
  combinatorics
• Practical decision making
• Experience, guesswork, intuition because
  overwhelming amounts of data, facts handled by
  modern ICT
• Asking for advice and expert help
• Smarter decision making
• Optimisation models to find and make use of the
  core drivers of the decision problem
• Unnecessary precision precision and relevance in
  conflict

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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 CSFs for the
  market
• Customer needs are adjusting to the new
  possibilities of the giga-investment
• The estimation of future cash flows becomes much
  more complex and cumbersome as the
  giga-investment defines its own markets
  stochastic processes do not apply

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A TYPICAL GIGA-INVESTMENT

• A new paper mill
• Annual capacity of 300 000 ton fine paper width
  9 meters speed 2000 m/min a 70-80 g/m2 WFC
  paper with good opacity and superior running
  performance on printing machines investment
  about 470 M, expected to last for 15-25 years
• New technology to get a very light WFC paper with
  good opacity cannot be copied easily gives
  15-20 more printing area per ton, which is a
  significant cost reduction
• A huge paper machine normally gives scale
  benefits cost effectiveness when demand and
  prices are decreasing effective capacity when
  demand and prices are increasing
• Will redefine market segments and build new
  markets for WFC

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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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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

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REAL OPTION VALUATION (ROV)
The value of a real option is computed by ROV
Se -dT N (d1) - Xe -rT N (d2) where d1 ln
(S0 /X )(r -d s2 /2)T / svT d2 d1 - svT,
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FUZZY REAL OPTION VALUATION

• 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
• 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
• S0 (s1, s2, a, ß) ?
• 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 C0 Se -dT
N (d1) - Xe -rT N (d2)
where d1 ln (E(S0)/ E(X))(r -d s2 /2)T /
svT d2 d1 - svT,
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FUZZY REAL OPTION VALUATION

• E(S0) denotes the possibilistic mean value of
  the present value of expected cash flows
• E(X) is the possibilistic mean value of expected
  costs
• s s(S0) 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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OPTIMAL TIME OF INVESTMENT
How long should we postpone an investment? Benaroc
h and Kauffman (2000) suggest Optimal investment
time when the option value Ct is at maximum
(ROV Ct)
Ct max Ct Vt e-dt N(d1) X e-rt N (d2
) t 0 , 1 ,...,T
where Vt PV(cf0, ..., cfT, ßP) - PV(cf0,
..., cft, ßP) PV(cft 1, ...,cfT, ßP),
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FUZZY OPTIMAL TIME OF INVESTMENT
We must find the maximising element from the set
C0, C1, , CT, 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, Ct (ctL ,ctR ,at,
ßt), of the trapezoidal form v (Ct) (ctL
ctR) / 2 rA (ßt at) / 6
where rA gt 0 denotes the degree of the investors
risk aversion
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EXTENSIONS

• Fuzzy Real Options support system, which was
  built on Excel routines and implemented in four
  multinational corporations as a tool for handling
  giga-investments.
• Possibility vs. Probability Falling Shadows vs.
  Falling Integrals FSS accepted
• On Zadehs Extension Principle

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References

• C. Carlsson and R. Fullér, On the Fuzzy Capital
  Budgeting Problem, Proceedings of the
  International ICSC Congress on Computational
  Intelligence Methods and Applications, ICSC
  Academic Press, Rochester 1999, 634-638
• C. Carlsson and R. Fullér, Capital Budgeting
  Problems with Fuzzy Cash Flows, Mathware Soft
  Computing, Vol VI, n.1, 1999, pp 81-89
• C. Carlsson and R. Fullér, Benchmarking in
  Linguistic Importance Weighted Aggregations,
  Fuzzy Sets and Systems, Vol 114/1, 2000, pp 35-41
• C. Carlsson and R. Fullér, Optimization Under
  Fuzzy If-Then Rules, Fuzzy Sets and Systems, Vol
  119, No.1, 2001, pp 111-120
• C. Carlsson and R. Fullér, Multi-Objective
  Linguistic Optimization, Fuzzy Sets and Systems,
  Vol 115, Nr 1, October 2000, 5-10
• C. Carlsson and R. Fullér, On Possibilistic Mean
  Value and Variance of Fuzzy Numbers, Fuzzy Sets
  and Systems, Vol. 122, No.2, 2001, 315-326
• C. Carlsson and R. Fullér, Decision Problems with
  Interdependent Objectives, International Journal
  of Fuzzy Systems, Vol. 2, No. 2, June 2000, pp
  98-107
• C. Carlsson, R. Fullér and P. Majlender, Project
  Selection with Fuzzy Real Options, Proceedings
  of the 2nd International Symposium of Hungarian
  researchers on Computational Intelligence,
  Budapest 2001, 81-88
• C. Carlsson and R. Fullér, On Optimal Investment
  Timing with Fuzzy Real Options, EUROFUSE 2001
  Workshop on Preference Modeling and Applications,
  Granada 2001, 235-239
• C. Carlsson and R. Fullér, Project Scheduling
  with Fuzzy Real Options, in Trappl (ed)
  Cybernetics and Systems2002, Vienna 2002

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References

• C. Carlsson, R. Fullér and P. Majlender, A
  possibilistic approach to selecting portfolios
  with highest utility score, Fuzzy Sets and
  Systems, Vol. 131/1, 2002, pp 13-21
• C. Carlsson, R. Fullér and P. Majlender,
  Possibility Distributions A Normative View, SAMI
  2003 Proceedings, Herlany 2003, pp 1-9
• C. Carlsson, R. Fullér and P. Majlender,
  Possibility versus Probability Falling Shadows
  versus Falling Integrals, IFSA 2003 Proceedings,
  Istanbul 2003, pp 5-8
• C. Carlsson, M. Collan and P. Majlender, Fuzzy
  Black and Scholes Real Option Pricing, Journal of
  Decision Systems, Vol. 12, No. 3-4, November 2003
• C. Carlsson and R. Fullér, A Fuzzy Approach to
  Real Option Valuation, Fuzzy Sets and Systems,
  Vol 139, 2003, pp 297-312
• C. Carlsson, R. Fullér and P. Majlender, A
  Normative View on Possibility Distributions, in
  Nikravesh, Masoud Zadeh, Lotfi A. Korotkikh,
  Victor (Eds.) Fuzzy Partial Differential
  Equations and Relational Equations Reservoir
  Characterization and Modeling, Springer Verlag,
  Heidelberg 2004, pp 186-205
• C. Carlsson and R. Fullér, Fuzzy Reasoning in
  Decision Making and Optimization, Studies in
  Fuzziness and Soft Computing Series,
  Springer-Verlag, Berlin/Heidelberg, 2002, 340 p
• C. Carlsson, M. Fedrizzi and R. Fullér Fuzzy
  Logic in Management, Kluwer, Dordrecht 2003, 296 p