Transcript Slide 1

Tanja Magoč, François Modave, Xiaojing Wang, and Martine Ceberio
Computer Science Department
The University of Texas at El Paso
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Problems in the area of finance
 Portfolio selection
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Current techniques for portfolio selection
Utility based decision making
 Use of fuzzy measures
 Use of constraints
 Use of intervals
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Numerous problems in the area of finance
use computation techniques
 Data mining
 Option pricing
 Selection of optimal portfolio
Portfolio: a distribution of wealth among several
investment assets
 Problem: select optimal portfolio
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 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.
m
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maximize (minimize)  wi xi subject to
i 1
wi  0
m
w
i 1
i
1
m
w r
i 1
 risk
i i
m
w R
i 1

i
i
 return
Goal: find the vector w  (w1,...,wm )
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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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Learn from examples:
 Overfitting data
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Ignore relationships among characteristics of
an asset:
 Higher risk usually implies higher return
 Longer time to maturity usually leads to higher
risk
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Assume precise data:
 Return expected by an investor
 Return and risk associated with an asset
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Multi-criteria decision making
Fuzzy measure and fuzzy integration:
 Take care of dependence among characteristics
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Intervals:
 Take care of imprecise data
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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}
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 a preference relation for each criterion
Challenge: Combine partial preferences into a
global preference
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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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Maximax approach
 Optimistic situation
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Maximin approach
 Pessimistic situation
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Weighted sum approach
 Advantage: simple to calculate, O(n)
 Disadvantage: ignores the dependence among
criteria
▪ Longer time to maturity  higher return
  : P ( I )  [0,1]
is called a non-additive
measure if
(i)  ( )  0
(ii)  ( I )  1
(iii)  ( B )   (C ) if B  C  P(I )
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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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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:
(C )  fd 
I
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n

i 1


1















f
i

f
j
I

f
i

f
j
|
I
|

f
i
I



  i  | I ij | 
ij
ij
I ij  0
I ij  0
2 i j
Decision-maker inputs the importance of
each attribute and the importance of each
pair of attributes

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Advantages:
 Considers the interactions among attributes
 Quadratic complexity
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Drawback:
 Imprecise values of importance and interaction
indices
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 
x
 x,


A real interval
is a closed and connected
set of real numbers
Calculate the Choquet integral w.r.t. a 2additive measure over intervals
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Present a feasible computation approach in
portfolio selection, combining interval values
with 2-additive fuzzy measures and integrals.