tanja magoč, françois modave, xiaojing wang, and martine ceberio computer science department the...
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Application of Constraints and Fuzzy Measures in FinanceTanja Magoč, François Modave, Xiaojing Wang, and Martine CeberioComputer Science DepartmentThe University of Texas at El Paso
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Outline
Problems in the area of finance Portfolio selection
Current techniques for portfolio selection
Utility based decision making Use of fuzzy measures Use of constraints Use of intervals
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Problems in the area of finance
Numerous problems in the area of finance use computation techniques Data mining Option pricing Selection of optimal portfolio
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Portfolio selection
Portfolio: a distribution of wealth among several investment assets
Problem: select optimal portfolio Goal:▪ Maximize return for a given acceptable level of risk▪ Minimize risk to obtain required level of return
Constraints:▪ Minimal required return▪ Maximal acceptable risk▪ Time horizon▪ Transaction cost▪ Preferred portfolio structure▪ etc.
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Simple formalization of the problem
maximize (minimize) subject to
Goal: find the vector
m
iiixw
1
),...,( 1 mwww
m
iii
m
iii
m
ii
i
returnRw
riskrw
w
w
1
1
1
1
0
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Existing techniques
Return-based strategies Methods involving stochastic
processes Intelligent systems techniques
Genetic algorithms Rule-based expert systems Neural networks Support vector machines
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Drawbacks of the existing techniques
Learn from examples: Overfitting data
Ignore relationships among characteristics of an asset: Higher risk usually implies higher return Longer time to maturity usually leads to
higher risk Assume precise data:
Return expected by an investor Return and risk associated with an asset
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Tools for a novel approach
Multi-criteria decision making Fuzzy measure and fuzzy integration:
Take care of dependence among characteristics
Intervals: Take care of imprecise data
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Multi-criteria decision making Comparison of multidimensional alternatives
to select the optimal one Pick the best stock for investment
Elements of a MCDM setting: a set of alternatives
stocks (finitely many) a set of criteria
return, time to maturity, reputation of company a set of values of the criterion
return=[0,50], time={1,2,3}, reputation={bad, average, good}
a preference relation for each criterion Challenge: Combine partial preferences into
a global preference
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Utility function
Utility function is a transformation from an ordered set to a set of real numbers
Construct a utility function for each criterion that maps values of all criteria to a common scale
Combine monodimensional utilities into a global utility function using an aggregation operator. How?
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Combining monodimensional utilities
Maximax approach Optimistic situation
Maximin approach Pessimistic situation
Weighted sum approach Advantage: simple to calculate, O(n) Disadvantage: ignores the dependence
among criteria▪ Longer time to maturity higher return
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Non-additive (fuzzy) measure in a finite case
is called a non-additive measure if
(i) (ii) (iii) if
: ( ) [0,1]P I
( ) 0 ( ) 1I ( ) ( )B C )(IPCB
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Fuzzy integration: the Choquet integral
Decision maker inputs value of importance of each subset of the set of criteria
The Choquet integral is an aggregation operator evaluated w.r.t. a non-additive measure, which is defined by the values of importance of (subsets of) criteria
Drawback: exponential complexity
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2-additive measure
A non-additive measure where all interaction indices of order 3 and higher are null, and at least one interaction index of order 2 is not null.
Advantages of using 2-additive measure: Lower complexity than non-additive
measure Takes into consideration dependence
among criteria
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The Choquet integral w.r.t. a 2-additive measure
The Choquet integral w.r.t. a 2-additive measure:
Decision-maker inputs the importance of each attribute and the importance of each pair of attributes
I
n
i jiiji
Iij
Iij IIifIjfifIjfiffdC
ijij 100
||2
1||)(
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Advantages and drawbacks of 2-additive approach
Advantages: Considers the interactions among
attributes Quadratic complexity
Drawback: Imprecise values of importance and
interaction indices
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Intervals
A real interval is a closed and connected set of real numbers
Calculate the Choquet integral w.r.t. a 2-additive measure over intervals
xx,
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Conclusion
Present a feasible computation approach in portfolio selection, combining interval values with 2-additive fuzzy measures and integrals.