Transcript Slide 1
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
w r
i 1
risk
i i
m
w R
i 1
i
i
return
Goal: find the vector w (w1,...,wm )
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
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 )
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:
(C ) fd
I
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
Advantages:
Considers the interactions among attributes
Quadratic complexity
Drawback:
Imprecise values of importance and interaction
indices
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.