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Tanja Magoč, François Modave, Xiaojing Wang, and Martine Ceberio
Computer Science Department
The University of Texas at El Paso

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

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

 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

maximize (minimize)  wi xi subject to
i 1
wi  0
m
w
i 1
i
1
m
wr
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

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


Multi-criteria decision making
Fuzzy measure and fuzzy integration:
 Take care of dependence among characteristics

Intervals:
 Take care of imprecise data


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 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?

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

 : 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


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

The Choquet integral w.r.t. a 2-additive
measure:


1

(C )  fd    f i   f  j I ij    f i   f  j  | I ij |   f i  I i   | I ij | 
2 i j
I ij  0
I ij 0
i 1
I


n
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Decision-maker inputs the importance of
each attribute and the importance of each
pair of attributes

Advantages:
 Considers the interactions among attributes
 Quadratic complexity

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

Present a feasible computation approach in
portfolio selection, combining interval values
with 2-additive fuzzy measures and integrals.